Abstract
This is the first part of a work on generalized variational inequalities and their applications in optimization. It proposes a general theoretical framework for the solvability of variational inequalities with possibly nonconvex constraints and objectives. The framework consists of a generic constrained nonlinear inequality (\(\exists\hat{u}\in\Psi(\hat {u})\), \(\exists \hat{y}\in\Phi(\hat{u})\), with \(\varphi(\hat{u},\hat{y},\hat{u})\leq \varphi(\hat{u},\hat{y},v)\), \(\forall v\in\Psi(\hat{u})\)) derived from new topological fixed point theorems for setvalued maps in the absence of convexity. Simple homotopical and approximation methods are used to extend the Kakutani fixed point theorem to upper semicontinuous compact approachable setvalued maps defined on a large class of nonconvex spaces having nontrivial EulerPoincaré characteristic and modeled on locally finite polyhedra. The constrained nonlinear inequality provides an umbrella unifying and extending a number of known results and approaches in the theory of generalized variational inequalities. Various applications to optimization problems will be presented in the second part to this work to be published ulteriorly.
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1 Introduction
The theory of variational inequalities was initiated to study equilibrium problems in contact mechanics. Fichera’s treatment of the existence and uniqueness for the Signorini problem in 19621963 is viewed as the birth of the theory (see [1] for a first hand historical account). In that problem, at equilibrium, contact points between an elastic body and a rigid surface must satisfy the equilibrium equations in addition to a set of boundary conditions expressed as equalities (on the free boundary of the elastic body) together with inequalities involving displacement and tension along tangent and normal directions to the contact boundary of the body. The analyses of the problem by both Signorini and Fichera were based on a crucial variational argument, namely that the solution of the equilibrium problem ought to be the displacement configuration \(\hat{u}\) minimizing the total elastic potential energy functional \(I(u)\) amongst admissible displacements u. Naturally, such a minimizer must solve the variational inequality \(\frac{d}{dt}I(u+tv)_{t=0}\geq0\) for all admissible directions v.^{Footnote 1} The directional variation \(\frac{d}{dt}I(u+tv)_{t=0}\) takes on the form of a superposition \(a(u,v)F(v)\) of a bilinear form and a linear functional defined on admissible displacements in an appropriate Hilbert space (namely a Sobolev space \(H^{1}\)). The functional analytic framework for the use of variational inequalities as a tool for solving boundary value problems owes much to the pioneering work of Stampacchia. The celebrated existence and uniqueness theorem of Stampacchia (1964) remains a corner stone of the theory in normed spaces of any dimension (see [2]). It intimately links variational inequalities to the minimization of energy functionals and states as follows: given a closed convex subset X of a reflexive (real, for simplicity) Banach space E and a continuous coercive bilinear form \(a(\cdot,\cdot)\) on \(E\times E\), then
If in addition \(a(\cdot,\cdot)\) is symmetric, then \(\hat{u}\) is characterized by \(I(\hat{u})=\min_{v\in X}I(v)\), where \(I(v)=\frac{1}{2}a(v,v)p(v)\).
For the Signorini problem, the variational inequality (1) corresponds to the EulerLagrange necessary condition expressing stationarity in the Hamilton principle for the minimization of the energy \(I(v)\). Fixed point arguments are at the heart of (1) in more than one respect. On one hand, it can be derived from the Banach contraction principle (see, e.g., [3]). Indeed, the bilinear continuous and coercive^{Footnote 2} bilinear form \(a(\cdot,\cdot)\) defines an inner product whose norm \(\Vert u\Vert _{a}=a(u,u)^{1/2}\) is equivalent to the original norm on E. By the RieszFréchet representation theorem, we may write \(p(v\hat{u})=a(p,v\hat{u})\) with \(p\in E\) and view (1) as \(\forall p\in E\), \(\exists!\hat{u}\in X\), \(a(\lambda p\lambda\hat{u}+\hat{u}\hat{u},v\hat{u})\leq0\), \(\forall v\in X\) for any given scalar \(\lambda>0\). This formulation is equivalent to a fixed point problem \(\hat{u}=P_{X}(\lambda p+(1\lambda)Id_{X})(\hat{u})\) for the orthogonal projection \(P_{X}\) onto X. The operator \(T(v)=P_{X}(\lambda p+(1\lambda)v)\) is a contraction whenever the scalar λ is chosen so that \(0<\lambda<2\alpha/C\). The Banach contraction principle applies to yield the solution’s existence and uniqueness. This point of view highlights the intimate relationship between variational inequalities and minimization problems. For applications of contraction principles to variational relations, the reader is referred to [4].
On the other hand, one may adopt an alternate fixed point approach via setvalued analysis  ultimately calling upon the Brouwer theorem or some of its topological generalizations.^{Footnote 3} This is the approach adopted in this work in order to study variational inequalities in the presence of nonconvexity.
To set the tone, let us note that (1) could alternately and quite easily be obtained as a consequence of two distinct fundamental topological fixed point principles for setvalued maps. The first approach uses the BrowderKy Fan fixed point theorem (which is equivalent to the KnasterKuratowskiMazurkiewiczKy Fan principle) much as in [5] and relies heavily on convexity. Here, the pointtoset map \(\Phi :X\rightrightarrows X\), \(\Phi(u):=\{v:a(u,vu)p(vu)<0\}\) turns out to be a socalled Ky Fan map without fixed points on a bounded closed convex subset of X.^{Footnote 4} It must have a ‘maximal element’ \(\hat{u}\) with \(\Phi(\hat{u})=\emptyset\), i.e., \(\hat{u}\) solves (1) (the uniqueness follows at once from the additivity and the coercivity of the form a). The reader is referred to the early work by Minty [6], to DugundjiGranas [7] for pioneering the KKM maps approach,^{Footnote 5} to Allen [8] for an early concise account and to Lassonde [9] for a comprehensive treatment based on KKM theory.
The second approach is based on a generalization of the Kakutani fixed point theorem much as in BenElMechaiekhIsac [10]. This is the point of view we shall focus on here.
In geometric terms of convex analysis, (1) can be written as an orthogonality property \(p\hat{u}\in N_{X}(\hat{u})\), where \(N_{X}(\hat{u})\) is the normal cone to X at \(\hat{u}\) in the sense of convex analysis. Indeed, viewing E as a Hilbert space with inner product \(a(\cdot,\cdot)\), (1) amounts to \(p\hat{u}\in(X\hat{u})^{}\) the negative polar cone of \(X\hat{u}\). But \((X\hat{u})^{}=(\bigcup_{t>0}\frac{1}{t}(X\hat{u}))^{}=T_{X}(\hat{u})^{}=N_{X}(\hat{u})\), where \(T_{X}(\hat{u})=\operatorname{cl}(\bigcup_{t>0}\frac{1}{t}(X\hat{u}))\) is the tangent cone to X at \(\hat{u}\). Thus, (1) can be seen as a setvalued fixed point problem \(\hat{u}\in\Psi_{p}(\hat{u}):=pN_{X}(\hat{u})\) or, equivalently, as an equilibrium (or a zero) problem
for the setvalued map \(\Phi_{p}:X\rightrightarrows E\) defined as \(\Phi _{p}(u):=p(u+N_{X}(u))\). Observe first that if \(p\notin X\), then \(\hat {u}=P_{X}(p)\) verifies \(a(p\hat{u},v\hat{u})\leq0\), \(\forall v\in X\), i.e., \(p\hat{u}\in(X\hat{u})^{}=N_{X}(\hat{u})\) which amounts to \(0\in p\hat {u}N_{X}(\hat{u})=\Phi_{p}(\hat{u})\) and (1) is solved. If \(p\in X\), then \(p\in u+T_{X}(u)\), \(\forall u\in X\), thus \(pu=pu+0_{E}\in (puN_{X}(u))\cap T_{X}(u)\). Generalizations to infinite dimensions of the BolzanoPoincaré intermediate value theorem (see, e.g., BenElMechaiekh [11]) can be used to solve (2) as the map \(\Phi_{p}\) has closed convex values, is upper semicontinuous, and satisfies the tangency boundary condition \(\Phi (u)\cap T_{X}(u)\neq\emptyset\), \(\forall u\in X\). This approach lends itself to the treatment of nonconvex problems through the consideration of natural and appropriate topological substitutes to convexity as well as corresponding notions of tangency from nonsmooth analysis.
We have briefly described above the intimate relationships between the Stampacchia variational inequality (1), minimization problems, general nonlinear inclusions, and fixed point principles. The theory of variational inequalities is playing an increasingly central role in the study of problems not only in mechanics, physics, and engineering but also in optimization, game theory, finance, economics, population dynamics, etc. The theory has vastly expanded in the past five decades with the intensive production of literature on numerous functional analytic, qualitative, and computational aspects. The interested reader is referred to the books by Baiocchi and Capelo [12], Kinderlehrer and Stampacchia [13], Nagurney [14], Granas [15], Cottle et al. [16], Isac [17], Murty [18], Facchinei and Pang [19], Konnov [20], Ansari et al. [21], as well as the papers by Gwinner [22], Blum and Oettli [23], Agarwal and O’Regan [24] and recently the survey paper by Ansari [25].
This paper is the first part of a work devoted to the study of generalized variational inequalities on nonconvex sets. It describes the constrained inequalities umbrella framework for variational and quasivariational inequalities. The main existence results on general systems of constrained inequalities (Theorems 17, 20 below) are derived from new topological generalizations of the fixed point theorem of Kakutani without convexity (Theorems 12 and 15). The domains considered are spaces modeled on locally finite polyhedra having nontrivial EulerPoincaré characteristic which are not necessarily compact. Rather, compactness is imposed on the maps. Solvability of generalized variational inequalities expressed as coequilibria problems for nonself nonconvex setvalued maps defined on Lipschitzian retracts is established in the last section (Theorem 32 and Corollary 33). The paper also illustrates how the general results apply to particular situations in the theories of variational inequalities, complementarity, and optimal control.
2 A general constrained nonlinear inequality
We assume that vector spaces are over the real number field and topological vector spaces are Hausdorff. Setvalued maps (simply called maps) are denoted by capital letters and double arrows ⇉.
Given two nonempty sets X, Y, two maps \(\Psi:X\rightrightarrows X\), \(\Phi :X\rightrightarrows Y\), and a proper real function \(\varphi:X\times Y\times X\rightarrow(\infty,+\infty]\), consider the constrained nonlinear inequality
The solvability results for CNI will be discussed in Section 4. For now, let us make clear that the CNI framework includes several types of variational inequalities.

Generalized quasivariational inequalities
Given a vector space E and a dual pair \(\langle F^{\ast},F\rangle\) of vector spaces,^{Footnote 6} given two nonempty subsets \(X\subseteq E\), \(Y\subseteq F\) and two maps \(\Psi:X\rightrightarrows X\), \(\Phi :X\rightrightarrows Y\), given two mappings \(\theta:X\times Y\rightarrow F^{\ast}\), \(\eta:X\times X\rightarrow F\) and a functional \(\phi :X\rightarrow \mathbb{R} \), the generalized quasivariational inequality problem QVI associated to the data \((X,Y,\Psi,\Phi,\theta,\eta,\phi)\) is
Obviously, the existence part in the variational inequality (1) is a very particular case of QVI  hence of CNI  whereby \(E=F\) is a reflexive Banach space that is Hilbertisable by the bilinear continuous and coercive form \(a(\cdot,\cdot)\) [thus E identifies with its topological dual \(E^{\prime}\), the dual pairing being obviously \(\langle p,u\rangle=a(p,u)\)], and \(X=Y\subseteq E\), \(\Phi(u)=\Psi(u)=X\) for all u are constant maps, \(\theta (u,y)=u\) for all y, \(\eta(v,u)=vu\), and \(\phi=p\).
In fact, quite clearly, QVI contains the socalled variationallike inequalities of the Stampacchia type: given a dual pair of vector spaces \(\langle E,F\rangle\), a nonempty closed subset \(X\subseteq E\), a setvalued map \(\Phi:X\rightrightarrows F\), and a mapping \(\eta:X\times X\rightarrow E\),
This problem is clearly equivalent to the equilibrium problem
where \(\eta(X,u)^{}=\{y\in F:\langle\hat{y},\eta(v,u)\rangle\leq 0 \mbox{ for all }v\in X\}\) is a negative polar cone.
In case \(\eta(v,u)=vu\), then \(\eta(X,u)^{}=(Xu)^{} =(\bigcup_{t>0}\frac{1}{t}(X\hat{u}))^{}\). If in addition E is equipped with the weak topology \(\sigma(E,F)\)  or with any topology for which the linear forms \(x\mapsto\langle y,x\rangle\), \(y\in F\), are continuous on E  then \(\eta(X,u)^{}=N_{X}^{R}(\hat{u})=T_{X}^{R}(\hat{u})^{}\), where \(T_{X}^{R}(u)=\operatorname{cl}(\bigcup_{t>0}\frac{1}{t}(X\hat{u}))\) is the radial cone (which is simply the tangent cone of convex analysis \(T_{X}(\hat{u})\) whenever X is locally convex at \(\hat{u}\)). Thus, (3) writes
The latter inclusion, referred to as a generalized variational inequality, covers (2) whenever \(E=F\) and \(\Phi(u)=p+u\) is singlevalued. This case strongly relates to the minimization of functionals as described next.

Quasiconvex programming
It is well established that for a proper Gâteauxdifferentiable (on its effective domain, assumed to be open and convex) function \(f:E\rightarrow(\infty,+\infty]\) of a topological vector space E, quasiconvexity^{Footnote 7} is equivalent to the proposition [given \(u,v\in X=\operatorname{dom}(f)\), \(\langle\nabla f(u),vu\rangle>0\Longrightarrow f(u)\leq f(v)\)]  see, e.g., Proposition 4.12 in [26]. Thus, the strict ^{Footnote 8} variational inequality \(\exists\hat{u}\in X\), \(\langle\nabla f(\hat{u}),v\hat{u}\rangle>0\) implies that \(f(\hat{u})=\min_{X}f(v)\).
This characterization of quasiconvexity extends to a nonsmooth function. Indeed, if f is only lower semicontinuous (not necessarily differentiable) on its effective domain X, and if the space E is a Banach space with a Gâteaux differentiable (equivalent) norm, then quasiconvexity is equivalent to
where \(\partial f(u)\) is the lower Dini subdifferential of f at u. Obviously, if the special instance of (VIS)
holds for a quasiconvex and lower semicontinuous function f, then \(f(\hat{u})=\min_{X}f(v)\).
As pointed out by Aussel [27], a better adapted approach to quasiconvex programming relies on the socalled adjusted normal cone to sublevel sets. Consider the sublevel set operators \(S_{f<},S_{f\leq }:X\rightrightarrows X\) defined by \(S_{f\leq}(u):=\{y\in X:f(y)\leq f(u)\}\) and \(S_{f<}:=\{y\in X:f(y)<f(u)\}\), where X is the effective domain of \(f:E\rightarrow(\infty,+\infty]\), E being a metrizable topological vector space. Clearly, \(S_{f<}\) has nonempty values at points \(u\in X\backslash\arg\min f\) and is a selector of \(S_{f\leq}\). Let \(\rho _{u}=d(u,S_{f<}(u))\) and consider, for each \(u\in X\), the adjusted level set
Always, \(S_{f<}(u)\subseteq S_{f}^{a}(u)\subseteq S_{f\leq}(u)\) for all \(u\in X\); if f is l.s.c., then \(S_{f<}(u)\subseteq \operatorname{cl}(S_{f<}(u))\subseteq S_{f}^{a}(u)\subseteq S_{f\leq}(u)\) and at points u with \(\rho_{u}=0\), we have \(\operatorname{cl}(S_{f<}(u))=S_{f}^{a}(u)\). All four sublevel set operators \(S_{f<}(\cdot)\), \(\operatorname{cl}(S_{f<}(\cdot))\), \(S_{f}^{a}(\cdot)\), and \(S_{f\leq}(\cdot)\) have convex values if and only if f is quasiconvex. The adjusted normal cone operator \(N_{f}^{a}:X\rightrightarrows E^{\prime}\) is defined as the polar cone:
Denoting the negative polar cones of \(S_{f<}(u)u\) and \(S_{f\leq }(u)u\) by \(N_{f<}(u)\) and \(N_{f\leq}(u)\), respectively, we have \(N_{f\leq }(u)\subseteq N_{f}^{a}(u)\subseteq N_{f<}(u)\) for all \(u\in X\), with all sets being closed convex cones. If f is quasiconvex, then for all \(u\in X\setminus\arg\min_{X}f\), the set \(N_{f}^{a}(u)\) has nontrivial elements. Proposition 5.16 in [27] provides a sufficient optimality condition for quasiconvex programming: for a quasiconvex function \(f:E\rightarrow (\infty,+\infty]\) radially continuous on its effective domain X, given a (not necessarily convex) subset \(C\subseteq \operatorname{int}(X)\), if a special instance of (VIS) is solvable over C with \(\Phi=N_{f}^{a}\setminus\{0\}\) (or \(N_{f<}\setminus\{0\}\), obviously) and \(\eta(v,u)=vu\), i.e., if \(\exists \hat{u}\in C\), \(\exists\hat{y}\in N_{f}^{a}(\hat{u})\setminus\{0\}\) such that \(\langle\hat{y},v\hat{u}\rangle\geq0\) for all \(v\in C\), then \(f(\hat {u})=\inf_{C}f\). Taking (4) into account, we have
Conversely, if C is a closed convex subset of the effective domain of a semistrictly quasiconvex^{Footnote 9} continuous function \(f:E\rightarrow(\infty ,+\infty]\) such that \(f(\hat{u})=\inf_{C}f\) and \(\operatorname{int}(S_{f\leq}(\hat{u} ))\neq\emptyset\), then \(0\in(N_{f}^{a}(\hat{u})\setminus\{0\} )+N_{C}^{R}(\hat{u})\) (here, \(N_{C}^{R}\) is the normal cone of convex analysis).

Multivalued complementarity problems
Multivalued complementarity problems are also very particular cases of QVI. Recall that given a dual pair of vector spaces \(\langle F,E\rangle\) and a cone \(X\subset E\) with dual cone \(X^{\ast}=\{y\in F;\langle y,x\rangle\geq0,\forall x\in X\}\), and given a setvalued map \(\Phi :X\rightrightarrows F\), a mapping \(f:X\times F\rightarrow F\) and a functional \(\phi:X\rightarrow \mathbb{R} \), the multivalued complementarity problem (associated to \((X,\Phi ,f,\phi ) \)) is
The classical generalized multivalued complementarity problem corresponds to \(\phi(u)\) being identically zero and \(f(u,y)=y\) (see, e.g., [17]).

A general optimal control problem
Let I = interval in \(\mathbb{R} \), X closed ⊂E separable Banach space, \(F:I\times X\rightrightarrows E\), \(S_{F}(u)=\) solutions viable in X for the Cauchy problem
(assuming such solutions exist). Starting at a point \(u\in X\), consider the journey along a trajectory \(y(t)\) of (6) followed by a path to a point v in a return set \(\Psi(u)\). Assume that a cost \(\varphi(u,y,v)\) is associated to this journey (e.g., \(\varphi(u,y,v)=\varphi _{1}(u,y)+\varphi_{2}(y,v)\)). We will discuss in Section 5 the case of the particular instance of CNI, namely the general optimal control problem
3 Fixed points without convexity
The main general existence results for constrained nonlinear inequalities of this paper (Theorems 17, 20 below) derive from new fixed point theorems for approachable setvalued maps in the sense on BenElMechaiekhDeguire ([28]; see also [29]) defined on spaces modeled over locally finite polyhedra, in particular ANRs (Theorems 12, 15 and Corollaries 16, 33). Before getting to the fixed point and equilibrium results, we briefly recall fundamental topological concepts used as a substitute for convexity together with the definition, examples, and properties of approachable maps.
3.1 Approachable maps on ANRs
Definition 1
([5])
Let \((X,{\mathcal{U}})\) and \((Y,{\mathcal{V}})\) be two topological spaces with compatible uniformity structures \({\mathcal{U}}\) and \({\mathcal{V}}\). A map \(\Phi :X\rightrightarrows Y\) is said to be approachable if and only if, for each entourage W of the product uniformity \({\mathcal{U}}\times{ \mathcal{V}}\) on \(X\times Y\), there exists a continuous singlevalued mapping \(s:X\rightarrow Y\) satisfying the inclusion \(\operatorname{graph}(s)\subset W[\operatorname{graph}(\Phi)]\).
Thus, approachable maps are those maps admitting arbitrarily close singlevalued continuous graph approximations, also known as continuous approximative selections. Indeed, it is easy to see that Φ is approachable if and only if \(\forall U\in{ \mathcal{U}}\), \(\forall V\in {\mathcal{V}}\), Φ admits a continuous \((U,V)\)approximative selection, i.e., a continuous singlevalued function \(s:X\rightarrow Y\) verifying
This continuous graph approximation property turns out to be, in presence of some compactness, a byproduct of the upper semicontinuity of the map Φ together with a qualitative topological/geometric property of its values. The classical convex example (which can be traced back to von Neumann’s proof of its famous minimax theorem) is a case in point.
Example 2
(Convex case, [30])
Let X be a paracompact topological space equipped with a compatible uniformity \({\mathcal{U}}\), and let Y be a convex subset of a locally convex topological vector space F. Let \(\Phi :X\rightrightarrows Y\) be an upper semicontinuous^{Footnote 10} (u.s.c. for short) map with nonempty convex values. Then Φ is approachable.
This landmark result has been extended to natural topological notions extending convexity which we consider in this work. Recall that a topological space X is said to be contractible (in itself) if there exist a fixed element \(x_{0}\in X\) and a continuous homotopy \(h:X\times [0,1]\rightarrow X\) such that \(h(x,0)=x\) and \(h(x,1)=x_{0}\), \(\forall x\in X\). Clearly, every convex and, more generally, every starshaped subset of a topological vector space is contractible.
Absolute retracts are important examples of contractible spaces and occupy a central place in topological fixed point theory as initiated by Karol Borsuk. We recall here basic facts on retracts that are crucial for the sequel. For a detailed exposition on absolute retracts and absolute neighborhood retracts (ARs and ANRs for short), we refer to the book of Jan Van Mill [31].
Definition 3

(i)
A subspace A of a topological space X is a neighborhood retract of X if some open neighborhood of A in X can be continuously retracted into A, i.e., there exist an open neighborhood V of A in X and a continuous mapping \(r:V\rightarrow A\) such that \(r(a)=a\) for all \(a\in A\). If \(V=X\), A is simply said to be a retract of X.^{Footnote 11}

(ii)
A metric space A is an absolute (neighborhood) retract  written \(A\in AR\) (\(A\in ANR\), resp.)  if and only if A is an absolute (neighborhood) retract of every metric space in which it is imbedded.

(iii)
A metric space A is an approximative absolute neighborhood retract (\(A\in AANR\) for short) if and only if A is an approximative neighborhood retract of any metric space \((X,d)\) in which it is imbedded as a closed subspace; i.e., for any \(\epsilon>0\), there exists an open neighborhood V of A in X and a continuous mapping \(r:V\rightarrow A\) such that \(d(r(a),a)<\epsilon\) for all \(a\in A\).
Note that AR ⊂ ANR ⊂ AANR.^{Footnote 12} Observe also that if A is a retract of a topological space X with retraction \(r:X\rightarrow A\), then any continuous mapping \(f_{0}:A\rightarrow Y\) into any topological space Y extends to the continuous mapping \(f=r\circ f_{0}:X\rightarrow A\rightarrow Y\). Thus, retracts and neighborhood retracts are characterized by extension properties. In effect, every AR is an absolute extensor for metric spaces. This implies that each AR is contractible in itself.^{Footnote 13} Also, every AR is a retract of some convex subspace of a normed linear space. Conversely, the Dugundji extension theorem (see, e.g., [7] or [5]) asserts that convex sets in locally convex spaces are absolute extensors for metric spaces. Hence, any metrizable retract of a convex subset of a locally convex topological linear space is an AR. Every infinite polyhedron endowed with a metrizable topology is an AR. Similarly, every ANR is an absolute neighborhood extensor of metric spaces. Even more precisely, ANRs are characterized as retracts of open subsets of convex subspaces of normed linear spaces. The class ANR include all compact polyhedra. Every Fréchet manifold is an ANR. The union of a finite collection of overlapping closed convex subsets in a locally convex space is an ANR provided it is metrizable (see [31]). AANRs as characterized as metrizable spaces that are homeomorphic to approximative neighborhood retracts of normed spaces.
We now state some extensions of Example 2 to maps with nonconvex values. We start with the contractible case.
Example 4
(Contractible case, [28, 32, 33])
Given two ANRs X and Y with X compact, every u.s.c. map \(\Phi:X\rightrightarrows Y\) with compact contractible values is approachable.
Contractibility is not sufficient to describe qualitative properties of solution sets to some differential or integral equations and inclusions. A seminal result of Aronszajn [34] establishes that such solution sets satisfy a more general proximal contractibility property tantamount to being contractible in each of their open neighborhoods (sets with trivial shape and \(R_{\delta}\) sets^{Footnote 14}). To be more precise, let us consider the following notion.
Definition 5
(Dugundji [35])
A subspace Z of a topological space Y is said to be ∞proximally connected in Y if for each open neighborhood U of Z in Y, there exists an open neighborhood V of Z in Y contained and contractible in U.
Example 6

(i)
The set \(\{(t,\sin(\frac{1}{t}));0< t\leq1\}\cup(\{0\}\times [ 1,1])\) is not contractible in itself, but it is contractible in each of its open neighborhoods in \(\mathbb{R}^{2}\).

(ii)
If a subspace Z of an ANR Y has trivial shape in Y (that is, Z is contractible in each of its neighborhoods in Y), then Z is ∞proximally connected in Y (see [31]).

(iii)
Let \(\{Z_{i}\}_{i=1}^{\infty}\) be a decreasing sequence of compact spaces having trivial shape in an ANR Y. Then \(Z=\bigcap_{i=1}^{\infty }Z_{i}\) is ∞proximally connected in Y (see [35]). In particular, every \(R_{\delta}\) set in an ANR Y is ∞proximally connected in Y.
We now state extensions of Example 4 to maps with noncontractible values.
Example 7
(Noncontractible cases)

(i)
(Compact domains, [28, 29]) Let X be a compact AANR and let Y be a uniform space. Then every u.s.c. map \(\Phi :X\rightrightarrows Y \) with nonempty compact ∞proximally connected values in Y is approachable.

(ii)
(Noncompact domains, [36]) Let X be an ANR and let Y be a metric space. Then every u.s.c. map \(\Phi:X\rightrightarrows Y\) with nonempty compact ∞proximally connected values in Y is approachable.
Case (i) is a particular version of a result in [29] (see Corollary 2.17 there or Corollary 3.4 in [5]), where the nonmetrizable case  X is an approximative absolute neighborhood extension space (AANES) for compact topological spaces  is considered (compact AANRs are AANES for compact spaces). In the special case where X and Y are ANRs with X compact, this result first appeared in [37].
Examples 2, 4, and 7 indicate that some compactness of the domain plays a key role in the approachability of a map (ANRs are paracompact spaces). Compactness can be weakened by simply requesting approachability on finite polyhedra. More precisely:
Proposition 8
Let X be an ANR, \((Y,{\mathcal{V}})\) be a uniform topological space, and let \(\Phi:X\rightrightarrows Y\) be a u.s.c. map with nonempty values. If the restriction of Φ to any finite polyhedron \(P\subset X\) is approachable on P, then the restriction of Φ to any compact subset K of X is also approachable on K.
Proof
We only sketch the proof. Recall that given an open subset U of a normed space and a compact subset K of U, there exists a compact ANR C such that \(K\subset C\subset U\) (Girolo [38]). Since X can be seen as a retract of an open set in a normed space (namely, the space of bounded continuous real functions on X), one concludes that if K is any compact subset of X, then there exists a compact ANR C such that \(K\subset C\subset X\). Since compact ANRs are dominated by finite polyhedra and since the enlargement of classes of topological spaces by domination of domain preserves approachability (see Proposition 3.17 in [5]), it follows that the restriction \(\Phi_{C}\) of Φ to C is also approachable. Invoking the fact that restrictions of approachable maps to compact subsets are also approachable (Proposition 3.10 in [5] or Proposition 2.3 in [29]), it follows that the restriction \(\Phi_{K}\) of Φ to K is also approachable. □
We now formulate (without proofs) two stability properties for approachable maps essential to the proofs of the main results in Section 3.2 below.
Proposition 9
(See [5])
Given three topological spaces equipped with compatible uniformity structures \((X,{\mathcal{U}})\), \((Y,{\mathcal{V}})\) and \((Z,{\mathcal{W}})\), let \(\Phi:X\rightrightarrows Y\) be a u.s.c. approachable map with nonempty compact values, and let \(\Psi:Y\rightrightarrows Z\) be a u.s.c. map with nonempty values such that the restriction of Ψ to the set \(\Phi(X)\) is approachable. Then the composition product \(\Psi\Phi :X\rightrightarrows Z\) is u.s.c. and approachable provided the space X is compact.
This implies the following.
Example 10
Let \(\Phi:X_{0}\rightrightarrows X_{n}\) be a map that admits a decomposition \(\Phi(x)=(\Phi_{n}\circ\cdots\circ\Phi_{1})(x)\), where each map \(\Phi_{i}:X_{i1}\rightrightarrows X_{i}\) is u.s.c. with ∞proximally connected in an ANR \(X_{i}\) for all \(i=1,\ldots,n\). Then the restriction of Φ to each compact subspace of \(X_{0}\) is approachable.
3.2 Fixed point theorems
The general nonlinear inequality presented in Section 4 below is based on Theorem 15 which is a generalization of the Borsuk and the EilenbergMontgomery fixed point theorems to approachable compact setvalued maps defined on spaces dominated by locally finite polyhedra and having nonzero EulerPoincaré characteristic.
The following observation by the second author [29] provides the essence of the passage from ‘almost fixed point’ to fixed point for u.s.c. maps.
Lemma 11
(Lemma 3.1 in [29])
Let X be a regular topological space and \(\Gamma :X\rightrightarrows X\) be a u.s.c. map with nonempty closed values. Assume that there exists a cofinal family \(\{\omega\}\) of open (in X) covers of \(K=\operatorname{cl}(\Gamma(X))\) such that Γ has an ωfixed point ^{Footnote 15} for each open cover ω. Then Γ has a fixed point.
The case of convex domains is much simpler as the next generalization of the fixed point theorems of Ky Fan [39] and Himmelberg [40] to approachable maps shows.
Theorem 12
Let X be a nonempty convex subset of a Hausdorff locally convex space E and \(\Gamma:X\rightrightarrows X\) be a u.s.c. map with nonempty closed values such that:

(i)
Γ is a compact map, i.e., \(K=\operatorname{cl}(\Gamma(X))\) is compact in X;

(ii)
for each finite subset N of K, the restriction of the map Γ to the convex hull \(\operatorname{conv}\{N\}\) of N is approachable.
Then Γ has a fixed point.
Given any open convex symmetric neighborhood U of the origin in E, the compact set K can be covered by \(\bigcup_{i=1}^{n}(y_{i}+U)\cap X\), \(N=\{y_{i}\in K:i=1,\ldots,n\}\). The Schauder projection associated to U, \(\pi_{U}:\bigcup_{i=1}^{n}(y_{i}+U)\cap X\rightarrow \operatorname{conv}\{N\}\), verifies
(note that \(\pi_{U}(y)=\frac{1}{\sum_{i=1}^{n}\mu_{i}(y)}\sum_{i=1}^{n}\mu _{i}(y)y_{i}\), where \(\mu_{i}(y):=\max\{0,1p_{U}(yy_{i})\}\), \(p_{U}\) being the Minkowski functional associated to U, is a convex combination). By hypothesis (ii) and Proposition 9, the composition product \(\Gamma _{U}=\pi_{U}\circ\Gamma_{\operatorname{conv}\{N\}}: \operatorname{conv}\{N\}\rightrightarrows \bigcup_{i=1}^{n}(y_{i}+U)\cap X\rightarrow \operatorname{conv}\{N\}\) is approachable. (Note that since \(\Gamma(X)\subset K\), any suitable continuous \((V,V)\)approximative f selection of \(\Gamma_{\operatorname{conv}\{N\} }\) (\(V\subset U\)) has values in the uniform open neighborhood \(K+U\) of K; thus \(\pi_{U}f\) is a continuous approximative selection of \(\Gamma_{U}\).) By the extension of Kakutani fixed point theorem to approachable maps (see [28]), \(\Gamma_{U}\) has a fixed point \(\bar{x}_{U}\) which is a Ufixed point for Γ. As K is compact and Γ is u.s.c. with compact values, Lemma 11 ends the proof.
Note that if Γ is approachable, then its restriction to any compact subset of its domain is also approachable (Proposition 2.3 in [29] for the case of topological vector spaces and Proposition 3.10 in [5] for the general case); thus (ii) always holds true in case Γ is approachable.
We turn our attention to the case of the domain being an ANR. It is well established that for any given ANR X and any open cover ω of X, the geometric nerve \(N(\omega)\) of the cover ωhomotopy dominates X in the following sense: there exist continuous mappings \(s:X\rightarrowN(\omega)\) and \(r:N(\omega)\rightarrow X\) such that \(r\circ s\) and \(\mathrm{id}_{X}\) are ωhomotopic.^{Footnote 16} Since ANRs are paracompact, the polyhedron \(N(\omega)\) is locally finite. This motivates the following definition.
Definition 13

(i)
Given an open cover ω of a topological space X and a topological space P, we say that the space P ωdominates X (\(\omega_{H}\)dominates X, respectively) if there are continuous mappings \(s:X\rightarrow P\) and \(r:P\rightarrow X\) such that \(r\circ s\) and \(\mathrm{id}_{X}\) are ωnear (ωhomotopic, respectively).

(ii)
Given a class of topological spaces \({\mathcal{P}}\), the classes \(D({\mathcal{P}}) \) and \(D_{H}({\mathcal{P}})\) of topological spaces dominated and homotopy dominated by \({\mathcal{P}}\) are defined as: \(X\in D({\mathcal{P}})\) (\(X\in D_{H}( {\mathcal{P}})\), respectively) if and only if ∀ω open cover of \(X,\exists P\in{ \mathcal{P}}\) such that P ωdominates X (\(\omega _{H} \)dominates X, respectively).
Clearly \(D_{H}({\mathcal{P}})\subset D({\mathcal{P}})\). It is well established that ANRs \(\subset D_{H}({\mathcal{P}})\), where \({\mathcal{P}}\) is the class of polyhedra endowed with the CWtopology (see, e.g., Example 3, Section 1.3 in [41]). Compact ANRs as well as \({\mathcal{C}}\)convex subsets of locally \({\mathcal{C}}\)convex metrizable topological spaces (where \({\mathcal{C}}\) is a convexity structure (linear or topological) are dominated by finite polyhedra (see [5])).
We will require the following specialization of the property of domination by locally finite polyhedra for spaces with nontrivial EulerPoincaré characteristic.
Lemma 14
Let \({\mathcal{P}}\) be the class of polyhedra and let \(X\in D({\mathcal{P}})\) be a paracompact space with a welldefined nontrivial EulerPoincaré characteristic \({\mathcal{E}}(X)\).^{Footnote 17} Given any open cover ω of X, then there exists a locally finite polyhedron P such that:

(i)
P ωhomotopy dominates X, and

(ii)
\({\mathcal{E}}(P)\) is well defined and nontrivial.
Proof
Let ω be any given locally finite open cover of X and let \(N(\omega)\) be its geometric nerve. By Lemma 7.2 in [42], since \(N(\omega)\) is a locally finite polyhedron (thus an ANR), there exists an open cover α of \(N(\omega)\) such that any two continuous mappings \(f,g:Z\rightarrowN(\omega)\) of a space Z that are αnear are homotopic. By a theorem of Withehead (see, e.g., p.419 in [43]), there exists a triangulation τ of \(N(\omega)\) finer than the cover α. Let us choose a (possibly iterated) star refinement \(\omega^{\ast}\) of ω, the locally finite polyhedron \(P_{1}=N(\omega^{\ast})\) and continuous mappings \(s_{1}:X\rightarrow P_{1}\), \(r_{1}:P_{1}\rightarrow X\) such that \(r_{1}\circ s_{1}\) and \(\mathrm{id}_{X}\) are \(\omega^{\ast}\)homotopic, and let us consider the cover \(\alpha^{\ast}=r_{1}^{1}(\omega^{\ast})\) of \(P_{1}\). All of the above can be chosen in such a way that:

(1)
\(P_{2}=N(\alpha^{\ast})\) is a subpolyhedron (with the same set of vertices) of \(P_{1}\) which is in turn a subpolyhedron of \((N(\omega ),\tau )\), and

(2)
there are mappings \(s_{2}:P_{2}\rightarrow P_{1}\), \(r_{2}:P_{1}\rightarrow P_{2}\) with \(r_{2}\circ s_{2}\) and \(\mathrm{id}_{P_{1}}\) being \(\alpha^{\ast}\)homotopic, and finally,

(3)
the cover \(\alpha^{\prime}=r_{2}^{1}(\alpha^{\ast})\) of \(P_{2}\) refines the trace of the cover α on \(P_{2}\).
Let \(P=P_{2}\). It is clear that P ωhomotopy dominates X, and that the mappings \(\mathrm{id}_{P},s_{2}\circ s_{1}\circ r_{1}\circ r_{2}:P\rightarrow P\) being \(\alpha^{\prime}\)near are homotopic. Note that by construction, the mappings \(r_{1}\circ r_{2}\circ s_{2}\circ s_{1}\) and \(\mathrm{id}_{X}\) are homotopic. By the homotopy invariance of the Lefschetz number, \({\mathcal{E}}(X)=\lambda(\mathrm{id}_{X})=\lambda(r_{1}\circ r_{2}\circ s_{2}\circ s_{1})\) and \({\mathcal{E}}(P)=\lambda(\mathrm{id}_{P})=\lambda(s_{2}\circ s_{1}\circ r_{1}\circ r_{2})\). It is well known (see, e.g., [41]) that, when defined for a pair of mappings f and g, the Lefschetz numbers \(\lambda(f\circ g)\) and \(\lambda(g\circ f)\) are equal. Hence \({\mathcal{E}}(X)=\lambda(r_{1}\circ r_{2}\circ s_{2}\circ s_{1})=\lambda(s_{2}\circ s_{1}\circ r_{1}\circ r_{2})={\mathcal{E}}(P)\). □
Lemmas 11 and 14, together with the definition of approachability (Definition 1) imply the first purely topological fixed point property.
Theorem 15
Let \({\mathcal{P}}\) be the class of polyhedra, \(X\in D({\mathcal{P}})\) be a paracompact space with \({\mathcal{E}}(X)\neq0\) and \(\Gamma :X\rightrightarrows X\) be a u.s.c. approachable map with nonempty closed values. If Γ is compact, then it has a fixed point.
Proof
By Lemma 14, for an arbitrarily chosen open cover ω of X, there exists a locally finite polyhedron P with nontrivial EulerPoincaré characteristic and a pair of continuous mappings \(X\stackrel{s}{\rightarrow}P\stackrel{r}{\rightarrow}X\) such that, for each \(x\in X\), \((r\circ s)(x)\) and x are ωnear. Consider the following commutative diagram:
We show that the u.s.c. compactvalued map \(s\Gamma r:P\rightrightarrows P\) is approachable in a stronger sense, sufficient for the existence of a fixed point. Being a compact subset of P, the set \(K^{\prime}=\operatorname{cl}((s\Gamma r)(P)) \) admits a cofinal family of open covers \(\{\vartheta\}\). Any given arbitrary open cover ϑ of \(K^{\prime}\), has a uniform refinement of the form \(\{W[p_{i}]:p_{i}\in K^{\prime},i=1,\ldots,n\}\), where W is an entourage of a uniformity structure generating a topology equivalent to the initial CWtopology on P. In addition, the entourage W can be chosen small enough so as to satisfy the following: \(W[K^{\prime}]\subset O\), where O is an open neighborhood of a compact ANR C such that \(K^{\prime }\subset C\subset O\subset P\) provided by the main result (Theorem) in [38], with neighborhood retraction \(\rho:O\rightarrow C\).
The map \(s:\operatorname{cl}(\Gamma(X))\rightarrow P\) being continuous on the compact set \(\operatorname{cl}(\Gamma(X))\) is uniformly continuous. Thus, there exists a member V of a uniform structure \({\mathcal{V}}\) defining an equivalent topology on X such that \(s(V[x])\subset W[s(x)]\), \(\forall x\in \operatorname{cl}(\Gamma(X))\). By hypothesis, according to (7) there exists a continuous \((V,V)\)approximative selection \(f:X\rightarrow X\) of Γ, i.e.,
Thus, \(\forall p\in P\),
The continuous mapping \(s\circ f\circ r\) maps P into the tubular open neighborhood \(W[K^{\prime}]\) in P. Being a continuous compact singlevalued mapping of a locally finite polyhedron with nonzero EulerPoincaré characteristic, \(\rho\circ s\circ f\circ r:P\rightarrow C\hookrightarrow P\) has a fixed point \(p_{W}=(\rho \circ s\circ f\circ r)(p_{W})\in\rho(W[s\Gamma(V[r(p_{W})])])\). Hence, \(p_{W}\in W^{2}[s\Gamma(V[r(p_{W})])]\), i.e., \(p_{W}\in W^{2}[s(y_{W})]\) with \(y_{W}\in\Gamma(x_{W})\), \(x_{W}\in V[r(p_{W})]\). By compactness, there exist nets \(\{p_{W}\}\) converging to some \(\bar{p}\in P\), \(r(p_{W})\rightarrow r(\bar{p})=\bar{x}\) in X, \(x_{W}\rightarrow\bar{x}\) in X, \(y_{W}\rightarrow\bar{y}\) in \(\operatorname{cl}(\Gamma(X))\), \(s(y_{W})\rightarrow s(\bar{y})\) in P. Consequently, \(\bar{p}=s(\bar{y})\in s\Gamma(\bar {x})=s\Gamma r(\bar{p})\).
By commutativity of diagram (8), the map \(rs\Gamma:X\rightrightarrows X\) also has a fixed point \(x_{\omega}=rs(y_{\omega})\), \(y_{\omega}\in \Gamma (x_{\omega})\) satisfying \(\{x_{\omega},y_{\omega}\}\) are ωnear, i.e., \(x_{\omega}\) is an ωfixed point for Γ. Since \(\operatorname{cl}(\Gamma(X))\) is compact, Lemma 11 ends the proof. □
The novelty in Theorem 15 is that compactness is on the map rather than the domain and in the use of simple homotopy and approximation methods (see, e.g., Theorem 3.21 in [5] for the case where X is compact and P is the class of finite polyhedra). Surely, the theorem could be obtained using Lefschetz theory and homological methods, but the methods used here are notably simpler.
Corollary 16
Every compact u.s.c. approachable map with nonempty closed values \(\Gamma :X\rightrightarrows X\) of an ANR X with \({\mathcal{E}}(X)\neq0\) has a fixed point.
Corollary 16 is a significant improvement on the KakutaniHimmelberg fixed point theorem and on the Borsuk fixed point theorem for ARs. It holds true if the values of Γ are ∞proximally connected in the ANR X (or Γ admits a decomposition as in Example 10), thus extending the main theorem in [37] whereby X is a compact ANR.
4 Solvability results for CNI and applications
The main solvability results for the convex as well as the nonconvex CNI problem are presented in this section together with applications to QVI, MCP, and GOCP.
4.1 CNI with quasiconvex objectives
For simplicity, we start with CNI in the case of a convex domain and a convexvalued constraint Ψ by extending the main result in [10] to noncompact domains.
Theorem 17
Let X, Y be nonempty convex subsets in locally convex spaces E, F and let:

(i)
\(\Phi:X\rightrightarrows Y\) be a compact u.s.c. map with nonempty closed values such that the restriction \(\Phi_{\operatorname{conv}\{N\}}\) is approachable for each finite subset N of X;

(ii)
\(\Psi:X\rightrightarrows X\) be a compact l.s.c. map with nonempty closed (hence compact) convex values; and

(iii)
\(\varphi:X\times Y\times X\rightarrow (\infty,+\infty] \) be a continuous extended proper real function with \(\varphi(u,y,\cdot)\) quasiconvex on \(\Psi(u)\), \(\forall(u,y)\in X\times Y\).
Then CNI is solvable, i.e.,
Proof
Define the marginal map \(M:X\times Y\rightrightarrows X\) by putting, for \((u,y)\in X\times Y\),
The compactness and convexity of the values of Ψ, together with the lower semicontinuity of φ and its quasiconvexity in the third argument when restricted to \(\Psi(u)\), imply that M has nonempty convex compact values. We verify that M is u.s.c. To do this, in view of the fact that both Φ and Ψ are compact maps, it suffices to verify that the graph of M is closed (indeed, a compact map with closed graph is u.s.c.). To do this, let \((u_{\alpha},y_{\alpha},v_{\alpha })_{\alpha}\) be a net in \(\operatorname{graph}(M)\) converging to \((u,y,v)\in X\times Y\times X\). Then,
where the first inequality above follows from the lower semicontinuity of φ and the last inequality from the upper semicontinuity of the marginal functional \(\inf_{w\in\Psi(\cdot)}\varphi(\cdot,\cdot,w)\) (this upper semicontinuity follows at once from the facts that φ is an upper semicontinuous functional and Ψ is an l.s.c. setvalued map^{Footnote 18}). Hence, \((u,y,v)\in \operatorname{graph}(M)\).
Cellina’s approximation theorem (Example 2 above) asserts that the restriction of the map M to any compact subset of \(X\times Y\) (in particular to finite convex polytopes) is approachable.
Define a map \(\Gamma:X\times Y\rightrightarrows X\times Y\) by putting
Now as a product map of u.s.c. approachable maps, the map Γ is also u.s.c. and approachable on finite convex polytopes (and on compact subsets of its domain). It has nonempty compact values. Moreover, \(\Gamma (X\times Y)\subset\Psi(X)\times\Phi(X)\subset K\) a compact subset in \(X\times Y\). All conditions of Theorem 12 are thus satisfied. Therefore, Γ has a fixed point \((\hat{u},\hat{y})\in\Gamma(\hat{u},\hat{y})\), that is, \(\hat{u}\in\Psi(\hat{u})\), \(\hat{y}\in\Phi(\hat{u})\) and \(\varphi(\hat {u},\hat{y},\hat{u})\leq\varphi(\hat{u},\hat{y},v)\), \(\forall v\in\Psi(\hat {u})\). □
Remark 1

(1)
Theorem 3.1 in [10] corresponds to the case where, instead of the maps Φ, Ψ being compact, the space X is compact and Ψ is both u.s.c. and l.s.c.

(2)
If in addition, \(\forall u\in X\) with \(u\in\Psi(u)\), \(\forall y\in \Phi(u)\) one has \(\varphi(u,y,u)\geq0\), then \(\varphi(\hat{u},\hat {y},v)\geq0\), \(\forall v\in\Psi(\hat{u})\).

(3)
If \(\Psi(u)=X\), \(\forall u\in X\), the continuity assumptions on φ can be slightly relaxed to φ is l.s.c. and \(\varphi (\cdot,\cdot,u)\) is u.s.c.; in which case, Theorem 17 extends Theorem 1 in [44] to infinite dimensional spaces and to the case where Φ is a composition of convex as well as nonconvex maps Φ.
The map Ψ may be a noncompact nonself map. In such a case, a compactness coercivity condition of the Karamardian type can be used to solve CNI (see, e.g., [45] for an early use in the context of variational inequalities).
Given two subsets X and C in a topological space E, denote by \(\partial_{X}(C)=\operatorname{cl}(C)\cap \operatorname{cl}(X\setminus C)\) the boundary of C relative to X, and by \(\operatorname{int}_{X}(C)=C\cap(E\setminus \partial_{X}(C))\) the interior of C relative to X.
Theorem 18
Let X, Y be nonempty convex subsets in locally convex spaces E, F, and C be a nonempty compact convex subset of X. Let \(\Phi :C\rightrightarrows Y\) and \(\Psi:C\rightrightarrows X\) be maps satisfying:

(i)
Φ is u.s.c. with nonempty compact values and approachable on convex finite polytopes in C;

(ii)
the compression map \(\Psi_{C}:C\rightrightarrows C\) defined by \(\Psi _{C}(x):=\Psi(x)\cap C\) is l.s.c. with nonempty compact convex values;

(iii)
\(\varphi:C\times Y\times X\rightarrow (\infty,+\infty] \) is an extended proper function continuous on \(C\times Y\times C\) and with \(\varphi(u,y,\cdot)\) convex on \(\Psi(u),\forall(u,y)\in X\times Y\);

(iv)
\(\forall u\in C\) with \(u\in\partial_{\Psi(u)}(\Psi_{C}(u))\), \(\exists v\in \operatorname{int}_{\Psi(u)}(\Psi_{C}(u))\) with \(\varphi(u,y,v)\leq \varphi (u,y,u)\), \(\forall y\in\Phi(u)\).
Then CNI is solvable.
Proof
By Theorem 17, \(\exists\hat{u}\in\Psi_{C}(\hat{u})\), \(\exists\hat{y}\in \Phi(\hat{u})\) such that \(\varphi(\hat{u},\hat{y},v)\geq\varphi(\hat {u},\hat{y},\hat{u})\), \(\forall v\in\Psi_{C}(\hat{u})\). Given \(v\in\Psi (\hat{u})\setminus C\), two cases are possible.
Case 1: \(\hat{u}\in \operatorname{int}_{\Psi(\hat{u})}(\Psi_{C}(\hat{u}))\). One can choose \(0<\lambda<1\) small enough so that \(w=\lambda v+(1\lambda)\hat {u}\in\Psi_{C}(\hat{u})\). Hence \(\varphi(\hat{u},\hat{y},\hat{u})\leq \varphi(\hat{u},\hat{y},w)\), and by convexity of \(\varphi(\hat{u},\hat {y},\cdot)\) it follows that \(\varphi(\hat{u},\hat{y},\hat{u})\leq\lambda \varphi (\hat{u},\hat{y},v)+(1\lambda)\varphi(\hat{u},\hat{y},\hat{u})\). Thus, \(\varphi(\hat{u},\hat{y},\hat{u})\leq\varphi(\hat{u},\hat{y},v)\).
Case 2: \(\hat{u}\in\partial_{\Psi(\hat{u})}(\Psi_{C}(\hat{u}))\). By (iv), \(\exists\hat{v}\in \operatorname{int}_{\Psi(\hat{u})}(\Psi_{C}(\hat{u}))\) with \(\varphi(\hat{u},\hat{y},\hat{v})\leq\varphi(\hat{u},\hat{y},\hat{u})\). One can choose \(0<\lambda<1\) small enough so that \(w=\lambda v+(1\lambda)\hat{v}\in\Psi_{C}(\hat{u})\). Hence \(\varphi(\hat{u},\hat{y},\hat {u})\leq \lambda\varphi(\hat{u},\hat{y},v)+(1\lambda)\varphi(\hat{u},\hat {y},\hat{v})\leq\lambda\varphi(\hat{u},\hat{y},v)+(1\lambda)\varphi (\hat{u},\hat{y},\hat{u})\), where the last inequality follows from (iv). Thus, \(\varphi(\hat{u},\hat{y},\hat{u})\leq\varphi(\hat{u},\hat{y},v)\) thus completing the proof. □
Remark 2
Again, if \(\Psi(u)=X\), \(\forall u\in C\), the continuity assumptions on φ can be slightly relaxed to φ is l.s.c. and \(\varphi (\cdot,\cdot,w)\) is u.s.c., extending Theorem 1 in [44].
4.2 CNI with nonconvex objectives
Nonconvexity occurs naturally in optimization. For example it is well known that Paretooptimal sets in multiobjective programming are not necessarily convex. Rather, under suitable hypotheses on the objectives and constraints, they may be contractible retracts of the feasible set (see, e.g., [46]). Topological properties of solution sets of vector optimization have been extensively studied with central themes being compactness, (path) connectedness, contractibility, retractability, etc. (see, e.g., the works by Benoist [47] and Huy [48] and his group for a number of nonconvex vector optimization settings).
Our goal in this section is to establish, based on the topological fixed point Theorem 15, topological solvability result for CNI involving functions whose sublevel sets are absolute retracts. Such functions are not unusual in nonconvex optimization, as the example by Ricceri below suggests.
Example 19
([49])
Given a closed convex subset X in a Banach space E and two functions \(\phi,J:X\rightarrow \mathbb{R} \) such that:

(i)
ϕ is l.s.c. convex such that \(\exists v_{0}\in X\) with \(\phi (v_{0})=0\) and \(\alpha:=\inf_{v\in X,v\neq v_{0}}\frac{\phi(v)}{\Vert vv_{0}\Vert}>0\);

(ii)
J is Lipschitzian with constant \(L<\alpha\).
Then each nonempty sublevel set of \(\phi+J\) is an AR.
Indeed, given \(\lambda\in \mathbb{R} \) such that \(\{v\in X:(\phi+J)(v)\leq\lambda\}\) is nonempty, define \(F:X\stackrel{\lambdaJ}{\rightarrow}\mathbb{R} \stackrel{\Phi}{\rightarrow}X\) as \(F(v):=\Phi(\lambdaJ(v))\), where
and observe that \(\{v\in X:(\phi+J)(v)\leq\lambda\}=\operatorname{Fix}(F)\). It is easy to verify that F is a setvalued contraction with closed convex values. It is well known that the fixed point set of such maps is an absolute retract. Similar arguments based on the topological structure of fixed points sets of setvalued maps can be used to construct other examples of functions whose level sets are retracts of sorts.
We shall now substantially weaken the convexity assumptions in Theorem 17. Note first that if Ψ and φ are as in Theorem 17 (e.g., Ψ has convex values and φ is quasiconvex in its third argument), then for all \((u,y)\in X\times Y\), the subset \(\arg\min_{\Psi(u)}\varphi (u,y,\cdot) \) as well as the sublevel sets of \(\varphi(u,y,\cdot)\) are convex sets, thus retracts of every convex set containing them, in particular of \(\Psi (u)\).
Theorem 20
Let X, Y be ANRs with \({\mathcal{E}}(X),{\mathcal{E}}(Y)\neq0\) and let:

(i)
\(\Phi:X\rightrightarrows Y\) be an approachable compact u.s.c. map with nonempty closed values;

(ii)
\(\Psi:X\rightrightarrows X\) be a compact l.s.c. map whose values are ARs;

(iii)
\(\varphi:X\times Y\times X\rightarrow (\infty,+\infty] \) be a continuous proper real function.
Assume that for all \((u,y)\in X\times Y\) any one of the following conditions holds:
 (iv)_{1} :

\(\arg\min_{\Psi(u)}\varphi(u,y,\cdot)\) is a retract of \(\Psi(u)\); or
 (iv)_{2} :

for any \(n\in \mathbb{N} \) large, the sublevel set \(S_{\varphi(u,y,\cdot)\leq}^{(n)}:=\{v\in\Psi (u):\varphi(u,y,v)\leq\min_{\Psi(u)}\varphi(u,y,\cdot)+1/n\}\) is a retract of \(\Psi(u)\); or
 (iv)_{3} :

for any \(n\in \mathbb{N} \) large, for any \(\epsilon>0\), there exists an ϵdeformation \(h:S_{\varphi(u,y,\cdot)\leq}^{(n)}\times [0,1]\rightarrow S_{\varphi(u,y,\cdot)\leq}^{(n)}\) such that \(h(\cdot,1)\) can be extended to \(\Psi (u)\).
Then CNI has a solution.
Proof
The proof goes along the lines of that of Theorem 17. The constrained marginal map
\((u,y)\in X\times Y\) is u.s.c. with nonempty compact values for exactly the same reasons. Its values are precisely the sets \(\arg\min_{\Psi (u)}\varphi(u,y,\cdot)\).
In case (iv)_{1} holds, being a retract of the compact absolute retract \(\Psi(u)\), the set \(\arg\min_{\Psi(u)}\varphi(u,y,\cdot)\) is also a compact AR, thus contractible. Therefore, the map M is approachable by Example 4.
If (iv)_{2} holds, then the set \(\arg\min_{\Psi(u)}\varphi(u,y,\cdot)\) can be written as a decreasing sequence
of compact sublevel sets that are retracts of \(\Psi(u)\). Such level sets are therefore themselves compact ARs, and the values \(\arg\min_{\Psi (u)}\varphi(u,y,\cdot)\) of the map M are compact \(R_{\delta}\)sets in view of Example 6(iii) above. Thus, M is approachable on the AR \(X\times Y\) by Example 7(ii).
As of (iv)_{3}, by an important result in the theory of retracts, it is necessary and sufficient for the closed subset \(S_{\varphi(u,y,\cdot)\leq }^{(n)}\) of the absolute retract \(\Psi(u)\) to be an AR as well (see, e.g., Lemma 5.6.3 in [31]).
In all three cases, as in the proof of Theorem 17, the map \(\Gamma (u,y):=M(u,y)\times\Phi(u)\), \((u,y)\in X\times Y\), is u.s.c. and approachable. It has nonempty compact values. Moreover, \(\Gamma (X\times Y)\subset\Psi(X)\times\Phi(X)\subset K\) a compact subset in \(X\times Y\). All conditions of Theorem 15 are thus satisfied (note that \({\mathcal{E}}(X\times Y)={\mathcal{E}}(X)\times{ \mathcal{E}}(Y)\neq0\)). Therefore, Γ has a fixed point and the proof ends as in Theorem 17. □
Remark 3
Theorem 20 contains Theorem 3.1 of [50]. Indeed, recall that an ANR is contractible if and only if it is an AR. A compact AR is acyclic and has the fixed point property for singlevalued continuous functions (Borsuk’s theorem). In addition, condition (i) of Theorem 3.1 in [50] is relaxed. On the other hand, if the values of the map Φ are ∞proximally connected (in particular contractible), then by Example 7 hypothesis (i) holds true.
As a particular case of Theorem 20, we have the following.
Corollary 21
Let X, Y, Ψ, and φ be as in Theorem 20, and let \(\Phi :X\rightrightarrows Y\) be a map satisfying either one of the following conditions:

(a)
Φ is a compact u.s.c. map with closed ∞proximally connected values in Y.

(b)
Φ admits a decomposition \(\Phi(x)=(\Phi_{n}\circ\cdots \circ \Phi_{1})(x)\), where each map \(\Phi_{i}:X_{i1}\rightrightarrows X_{i}\) is u.s.c. with ∞proximally connected in an ANR \(X_{i}\) for all \(i=1,\ldots,n\), \(X_{0}=X\), \(X_{n}=Y\) and X is compact.
Then problem CNI has a solution.
Proof
It suffices to observe that since X is an ANR, then by Examples 7(ii) and 10, Φ is approachable. The conclusion follows immediately from Theorem 20. □
The solvability results for CNI above apply to the various problems described in Section 2: namely, generalized quasivariational inequalities QVI, variationallike inequalities of the Stampacchia type VIS, multivalued complementarity problems MCP, or general optimal control problem GOCP. The remainder of the paper is devoted to illustrating a few cases of applications. Space constraints impose limits on the discussion.
4.3 Solving quasivariational inequalities
Consider QVI associated to the data \((X,Y,\Phi,\Psi,\theta,\eta ,\phi)\) as defined in Section 2:
where \(X\subseteq E\), \(Y\subseteq F\), E being a vector space, \(\langle F^{\ast},F\rangle\) a dual pair of vector spaces, \(\Psi :X\rightrightarrows X\), \(\Phi:X\rightrightarrows Y\), \(\theta:X\times Y\rightarrow F^{\ast}\), \(\eta:X\times X\rightarrow F\) and \(\phi :X\rightarrow \mathbb{R} \).
Particular instances of QVI were studied in [44, 45, 50–54], and many others. We refer to [17] and [21] for comprehensive discussions on the various aspects as well as the many applications of variational inequalities.
We shall apply now Corollary 21(a) and Theorem 17 to the functional \(\varphi:X\times Y\times X\rightarrow \mathbb{R} \) given by \(\varphi(u,y,v)=\langle\theta(u,y),\eta(v,u)\rangle +\phi (v) \) to obtain the following results.
Theorem 22
Given X a convex subset of the normed space E and Y an ANR with \({\mathcal{E}}(Y)\neq0\) imbedded in the normed space F, let:

(i)
\(\Phi:X\rightrightarrows Y\) be a compact u.s.c. map with closed ∞proximally connected values in Y;

(ii)
\(\Psi:X\rightrightarrows X\) be a compact l.s.c. map with closed convex values;

(iii)
ϕ be continuous and convex and verify \(\forall u\in X\), \(\exists v_{u}\in\Psi(u)\) with \(\phi(v_{u})=0\) and \(\alpha_{u}=\min_{v\in \Psi (u),v\neq v_{u}}\frac{\phi(v)}{\Vert vv_{u}\Vert}>0\);

(iv)
η be continuous and \(\forall u\in X,\eta(\cdot,u)\) be Lipschitzian with constant \(L_{u}>0\);

(v)
θ be continuous and \(\forall(u,y)\in X\times Y\), \(\Vert\theta (u,y)\Vert<\frac{\alpha_{u}}{L_{u}}\).
Then QVI has a solution.
Proof
Observe that \(\varphi(u,y,v)=J(v)+\phi(v)\) with \(J(v)=\langle\theta (u,y),\eta(v,u)\rangle\) Lipschitzian with constant \(\Vert\theta (u,y)\Vert L_{u}\). By Example 19 applied to a convex compact (hence complete) set \(\Psi(u)\), the level sets of \(\varphi(u,y,\cdot)\) restricted to \(\Psi(u)\) are absolute retracts. Thus, all hypotheses of Theorem 20 including (iv)_{2} are satisfied (a convex set in a normed spaces is an AR by the Dugundji’s extension theorem). This ends the proof as QVI is a particular case of CNI. □
Suppose now that X is a subset in a normed space E, and let, for a given \(\rho>0\), \(X_{\rho}\) be the set \(X\cap D_{\rho}\), where \(D_{\rho}\) is the closed disk of radius ρ centered at the origin in E. Assuming that \(X_{\rho}\) is nonempty, denote by \(\Psi_{\rho}\) the compression of Ψ restricted to \(X_{\rho}\) given by \(\Psi_{\rho}(u):=\Psi(u)\cap D_{\rho }\), \(u\in X_{\rho}\). An immediate application of Theorem 18 is the following solvability result with a coercivity condition in lieu of the compactness of the set X.
Theorem 23
Consider QVI for the data \((X,Y,\Phi,\Psi,\theta,\eta,\phi)\) with X being a nonempty convex subset of a normed space E and Y a nonempty convex subset of a locally convex space F, and θ and η being continuous. Assume that:

(i)
\(\langle\theta(u,y),\eta(u,u)\rangle\geq0\), \(\forall(u,y)\in \operatorname{graph}(\Phi)\);

(ii)
\(\forall(u,p)\in X\times F^{\ast}\), \(\langle p,\eta(\cdot,u)\rangle \) is convex on \(\Psi(u)\).
Assume, furthermore, that \(\exists\rho_{0}>0\) such that \(\forall\rho \geq \rho_{0}\):

(iii)
\(X_{\rho}\) is compact nonempty and the map \(\Psi_{\rho}\) is l.s.c. and has nonempty compact convex values on \(X_{\rho}\);

(iv)
the restriction of the map Φ to \(X_{\rho}\) admits a decomposition as a finite composition of u.s.c. maps with nonempty compact ∞proximally connected values through a sequence of ANRs;

(v)
ϕ is convex and its restriction to \(X_{\rho}\) is continuous;

(vi)
\(\forall u\in\Psi(u)\), \(\Vert u\Vert=\rho\), \(\exists v\in\Psi (u)\), \(\Vert v\Vert<\rho\) with
$$\max_{y\in\Phi(u)}\bigl\langle \theta(u,y),\eta(v,u)\bigr\rangle \leq \phi(u)\phi(v). $$
Then problem QVI has a solution.
Proof
Take \(C=X_{\rho}=X\cap D_{\rho}\) and \(\varphi(u,y,v)=\langle\theta (u,y),\eta(v,u)\rangle+\phi(v)\) in Theorem 18. □
Remark 4

(1)
Let \(\phi\equiv0\). If there exists \(v_{0}\in\bigcap_{u\in X}\Psi(u)\) such that
$$\lim_{\Vert u\Vert\rightarrow\infty,u\in\Psi(u)}\max_{y\in\Phi (u)}\bigl\langle \theta(u,y), \eta(v_{0},u)\bigr\rangle < 0, $$then hypothesis (vi) is satisfied. We thus obtain a generalization of a result in [50].

(2)
It is easy to verify that an alternative coercivity condition to (vi) is:
 (iv)′:

there exists a nonempty compact convex subset C of X such that
$$\forall u\in X\setminus C,\exists v\in X\text{ with }\max_{y\in \Phi(u)} \bigl\langle \theta(u,y),\eta(v,u)\bigr\rangle < \phi(u)\phi(v). $$

(3)
If \(E=F\) and \(\eta(v,u)=vu\), then hypotheses (i)(ii) are obviously satisfied. If \(\eta(u,u)=0\), \(\forall u\in X\), then (i) is obviously satisfied. However, it may happen that η is not identically zero on the diagonal of \(X\times X\) and yet problem QVI has a solution (see, e.g., [52]).
Note that given any subset X of a normed space E, \(X_{\rho}=X\cap D_{\rho}\) is a retract of X (because \(D_{\rho}\) is a retract of E). In our next result, we shall assume more, namely that \(X_{\rho}\) is a deformation retract of X (thus has the same homotopy type as X).
Corollary 24
Assume that X and Y are ANRs in normed spaces with \({\mathcal{E}}(Y)\neq0\), and let \(\Phi:X\rightrightarrows Y\) be a map admitting a decomposition as a finite composition of u.s.c. maps with nonempty compact ∞proximally connected values through a sequence of ANRs. Assume also that θ and η are continuous and verify:

(i)
\(\langle\theta(x,y),\eta(x,x)\rangle\geq0\), \(\forall(x,y)\in \operatorname{graph}(\Phi)\).
Assume that \(\exists\rho_{0}>0\) such that \(\forall\rho\geq\rho_{0}\):

(ii)
\(X_{\rho}\) is a compact deformation retract of X and \({\mathcal{E}}(X)\neq0\) (or more generally \(X_{\rho}\) is compact and \({\mathcal{E}}(X_{\rho })\neq0\));

(iii)
\(\Psi_{\rho}\) is continuous with nonempty compact values;

(iv)
\(\forall(u,y)\in X_{\rho}\times Y\), the marginal set
$$M(u,y)=\Bigl\{ v\in\Psi_{\rho}(u);\bigl\langle \theta(u,y),\eta(v,u)\bigr\rangle =\inf_{w\in\Psi_{\rho}(u)}\bigl\langle \theta(u,y),\eta(w,u)\bigr\rangle \Bigr\} $$is ∞proximally connected in \(X_{\rho}\).
Then:

(1)
QVI associated to the data \((X_{\rho},Y,\Psi_{\rho},\Phi ,\theta,\eta,\phi)\) has a solution \(u_{\rho}\), \(\forall\rho\geq \rho _{0}\);

(2)
if the set \(\{u_{\rho}\}_{\rho\geq\rho_{0}}\) has a cluster point, then problem QVI has a solution.
Proof
For \(\rho\geq\rho_{0}\), since \(X_{\rho}\) is a deformation retract of the ANR X, it is a compact ANR with \({\mathcal{E}}(X_{\rho})\neq0\) (the EulerPoincaré characteristic being a homotopy invariant). Conclusion (1) readily follows from a general formulation of Theorem 20, whereby the marginal map \(M:X_{\rho}\times Y\rightrightarrows X_{\rho}\) is u.s.c. and approachable (Example 7). Assume now that the set \(\{u_{\rho}\}\) of solutions to the problems \(QVI(X_{\rho},Y,\Psi_{\rho},\Phi,\theta ,\eta ) \) has a subsequence \(\{u_{\rho_{n}}\}_{n}\) converging to \(\hat{u}\in X\) (an ANR is a closed set). For each n, \(u_{\rho_{n}}\in\Psi_{\rho _{n}}(u_{\rho_{n}})\) and for some \(y_{n}\in\Phi(u_{\rho_{n}})\), \(\langle \theta(u_{\rho_{n}},y_{n}),\eta(v,u_{\rho_{n}})\rangle\geq0\), \(\forall v\in\Psi_{\rho_{n}}(u_{\rho_{n}})\). Since for any large ρ, \(\Psi _{\rho}\) is u.s.c. with closed values, it follows that \(\hat{u}\in \Psi(\hat{u})\). Furthermore, the sequence \(\{y_{n}\}\) being contained in the compact set \(\Phi(\{u_{\rho_{n}}\}\cup\{\hat{u}\})\) has a cluster point \(\hat{y}\in\Phi(\hat{u})\). The continuity of θ and η implies that \(\langle\theta(\hat{u},\hat{y}),\eta(v,\hat{u})\rangle\geq 0\), \(\forall v\in\Psi(\hat{u})\). □
Corollary 24 generalizes Theorem 3.8 of [50] in several ways.
Theorems 22, 23 and Corollary 24 for the solvability of QVI can be applied to generalize results by Isac and the second author [10] for QVIs involving monotone maps in a generalized sense defined on neighborhood retracts including Riemannian manifolds. This is the object of a subsequent work.
4.4 Multivalued complementarity problem
Recall that given a dual pair of vector spaces \(\langle F,E\rangle\) and a cone \(X\subset E\) with dual cone \(X^{\ast}=\{y\in F;\langle y,x\rangle \geq 0,\forall x\in X\}\), and given a setvalued map \(\Phi :X\rightrightarrows F\), a mapping \(f:X\times F\rightarrow F\) and a functional \(\phi :X\rightarrow \mathbb{R} \), the multivalued complementarity problem MCP (associated to \((X,\Phi ,f,\phi ) \)) is
The classical generalized multivalued complementarity problem corresponds to \(\phi(u)\) being identically zero and \(f(u,y)=y\) (see, e.g., [17]).
We formulate a typical existence result for MCP that generalizes to nonconvex maps classical results in [45] and their generalizations. Their proofs are similar to those presented there for convexvalued Φ and are left to the reader.
Theorem 25
Assume that Φ is u.s.c. with nonempty compact ∞proximally connected values and that \(\phi:X\rightarrow(\infty,0]\) is an l.s.c. convex functional. Assume also that there exists a compact convex subset C of X with nonempty interior relative to X such that for each \(u\in \partial_{X}(C)\) there exists \(v\in \operatorname{int}_{X}(C)\) with \(\inf_{y\in\Phi (u)}\langle y,uv\rangle\geq\phi(v)\phi(u)\).
Then:

(1)
MCP has a solution provided \(\phi(0)=0\) and \(\phi(\lambda u)=\lambda \phi(u)\), \(\forall(\lambda,u)\in [1,+\infty)\times X\).

(2)
\(\exists\hat{u}\in C\), \(\exists\hat{y}\in\Phi(\hat{u})\cap X^{\ast}\) with \(0\leq\langle\hat{y},\hat{u}\rangle\leq\phi(\hat{u})\) provided \(\phi(0)=0\) and \(\phi(u+v)\leq\phi(u)\), \(\forall(u,v)\in X\times X\).
As an example of a byproduct of Theorem 25, we obtain a generalization of known results that could be applied to finding stationary points of the KuhnTucker type for nonsmooth programming problems with general objective functions. Assume that \(E=\mathbb{R}^{n}\), \(X=\mathbb{R}_{+}^{n}\), and \(\phi :X\rightarrow(\infty,0]\) is an l.s.c. convex functional with \(\phi (0)=0\) and \(\phi(\lambda u)=\lambda\phi(u)\), \(\forall(\lambda,u)\in [1,+\infty)\times X\). Let g be a locally Lipschitz real function on X, and let us assume that \(\Phi(u):=h(\partial f(u))\) is a homeomorphic image, lying in X, of the Clarke generalized gradient [51] of g at u (such a mapping Φ is of course u.s.c. and has nonempty compact contractible values). If there exist a constant \(\beta>0\) and a vector \(d\in X\) such that
then, with \(C:=\{u\in X;\langle d,u\rangle\leq\beta\}\) (a compact set), one immediately obtains the solvability of MCP. Note that our coercivity condition above is independent of the mapping f (which could be of the form \(f(u,y):=Mu+y+r\), \(M\in\mathbb{R}^{n\times n}\), \(r\in\mathbb{R}^{n}\), as in [54], or not).
Corollary 26
Let X be a closed convex cone in \(\mathbb{R}^{n}\), and let \(\Phi :X\rightrightarrows\mathbb{R}^{n}\) be such that for any compact convex subset C of X, the restriction \(\Phi_{C}\) is compactvalued u.s.c. and approachable. Assume that \(f(u,y)=y\), \(\phi=0\), and that \(\exists \alpha>0\) such that \(\langle yz,u\rangle\geq\alpha\Vert u\Vert^{2}\), \(\forall (u,y)\in \operatorname{graph}(\Phi)\), \(\forall z\in\Phi(0)\). Then MCP has a solution.
5 Generalized variational inequalities and coequilibria on Lipschitzian ANRs
The last section of this work establishes the existence of a solution for generalized variational inequalities as a coequilibrium for an upper hemicontinuous nonself map with convex values defined on a Lipschitzian ANR.
Recall that, given a closed subset X of a normed space E, an element \(x_{0}\in X\) is an equilibrium for a setvalued map \(\Phi :X\rightrightarrows E\) if \(0\in\Phi(x_{0})\) (i.e., \(x_{0}\) is a zero for Φ). Naturally, such solvability theorems are always subject to tangency boundary conditions. In the absence of convexity, concepts of tangent and normal cones of nonsmooth analysis are required. We briefly recall few facts about the contingent and circatangent cones (see, e.g., Mordukhovich [55], AubinFrankowska [56], AubinCellina [57]).
Definition 27

(i)
The BouligandSeveri contingent cone \(T_{X}(x)\) to X at x is the upper limit in the sense of PainlevéKuratowski when \(t\downarrow0\) of the family \(\{\frac{1}{t}(Xx)\}_{t>0}\).

(ii)
The Clarke circatangent tangent cone \(T_{X}^{C}(x)\) is the lower limit (i.e., the set of all limit points) when \(t\downarrow0\) and \(x^{\prime}\rightarrow_{X}x\) of the family \(\{\frac{1}{t}(Xx^{\prime })\}_{t>0,x^{\prime}\in X}\).
A useful characterization of the Clarke cone is
where \(d_{X}^{0}(x)(v)\) is the Clarke directional derivative (see [58]) of the locally Lipschitzian distance function \(x\mapsto \operatorname{dist}(x;X)\) at x in the direction v ^{Footnote 19}.
\(T_{X}^{C}(x)\) is a closed convex cone contained in the closed cone \(T_{X}(x)\). At interior points of X, \(T_{X}^{C}(x)=T_{X}(x)=E\), the whole space. If X is locally convex at \(x\in X\), then \(T_{X}^{C}(x)=T_{X}(x)=T_{X}^{R}(x)=\operatorname{cl}(\bigcup_{t>0}\frac{1}{t}(Xx))\) the tangent cone of convex analysis.
Definition 28
The set X is said to be sleek at a point \(x\in X\) if the setvalued map \(x\mapsto T_{X}(x)\) is l.s.c. at x. X is sleek if it so at each of its points.
If X is sleek at x, then \(T_{X}^{C}(x)=T_{X}(x)\) (hence, X is regular at x), both cones being convex and closed cones; moreover, the Clarke’s normal cone \(N_{X}^{C}(x)=T_{X}^{C}(x)^{}=(\partial ^{0}\operatorname{dist}(x;X)^{})^{}=\operatorname{cl}(\bigcup_{\lambda>0}\lambda\partial ^{0}\operatorname{dist}(x;X))\), where \(\partial^{0}\) is the Clarke’s generalized gradient. Most importantly:
Proposition 29
If X is sleek, then the map \(N_{X}^{C}:X\rightrightarrows E^{\prime }\) has a closed graph and closed convex values.
The existence of an equilibrium is subject to the boundary condition of the BolzanoPoincaréHalpern type being satisfied:
This tangency condition always implies a Ky Fan type condition expressed in terms of the normal cone \(N_{X}^{C}(x)=T_{X}^{C}(x)^{}\) (the negative polar cone to \(T_{X}^{C}(x)\)):
The reader is referred to [11] for a detailed discussion on equilibria for setvalued maps on nonsmooth domains.
In view of the characterizations (3) and (4) of generalized variational inequalities, one introduces the following concept.
Definition 30
An element \(x_{0}\in X\) is a coequilibrium for Φ if it solves the generalized variational inequality \(0\in\Phi(x_{0})N_{X}^{C}(x_{0})\).
Remark 5

(i)
Clearly, an interior coequilibrium is an equilibrium since, for such a point, \(N_{X}^{C}=\{0\}\).

(ii)
Observe that \(x_{0}\) is a coequilibrium for Φ if and only if the maps Φ and \(N_{X}^{C}\) coincide at \(x_{0}\), i.e., \(\Phi (x_{0})\cap N_{X}^{C}(x_{0})\neq\emptyset\). As \(N_{X}^{C}(x_{0})=T_{X}^{C}(x_{0})^{}\), this coincidence implies the infsup inequality \(\inf_{y\in\Phi (x_{0})}\sup_{v\in T_{X}^{C}(x)}\langle y,v\rangle\leq0\).

(iii)
Conversely, \(\inf_{y\in\Phi(x_{0})}\sup_{v\in T_{X}^{C}(x)}\langle y,v\rangle\leq0\) implies that \(x_{0}\) is a coequilibrium for Φ, provided \(\Phi(x_{0})\) is weakly compact. Indeed, the extended realvalued function \(y\mapsto\sup_{v\in T_{X}^{C}(x)}\langle y,v\rangle\) is l.s.c. and convex, hence weakly l.s.c. Thus it achieves its infimum on \(\Phi (x_{0})\) at some \(y_{0}\) verifying \(\langle y_{0},v\rangle\leq 0\), \(\forall v\in T_{X}^{C}(x)\), i.e., \(y_{0}\in N_{X}^{C}(x_{0})\).
For simplicity, assume in this section, that the underlying space is a real Hilbert space \((E,\langle\cdot,\cdot\rangle)\) identified with its dual.^{Footnote 20} By a Hilbert space pair we mean a pair \((X,E)\) with E a real Hilbert space and X a closed subset of E. Recall that a map Φ is upper hemicontinuous on X (u.h.c.) if for each \(p\in E^{\prime}\), the support functional \(x\mapsto \sigma_{\Phi(x)}(p)=\sup_{y\in\Phi(x)}\langle p,y\rangle\) is upper semicontinuous as an extended realvalued function on X. Always u.s.c. ⟹ u.h.c. The converse holds whenever Φ has convex weakly compact values. Define
Definition 31
Let us say that a Hilbert space pair \((X,E)\) has the equilibrium property for the class \(\mathbf{H}_{\partial}\) if and only if any map \(\Phi \in\mathbf{H}_{\partial}(X,E)\) has an equilibrium in X.
Theorem 32
Assume that a Hilbert pair \((X,E)\) has the equilibrium property for the class \(\mathbf{H}_{\partial}\). If X is sleek, then any compact map \(\Psi\in\mathbf{H}(X,E)\) has a coequilibrium, i.e., \(\exists x_{0}\in X\) solving the generalized variational inequality \(0\in\Psi (x_{0})N_{X}^{C}(x_{0})\).
Proof
The image \(\Psi(X)\) of Ψ is contained in a closed disk D centered at the origin with radius \(M>0\) in E. Consider the map \(\Phi :X\rightrightarrows E\) given by \(\Phi(x):=\Psi(x)(N_{X}^{C}(x)\cap D)\). By Proposition 29 and since X is sleek, the map \(N_{X}^{C}:X\rightrightarrows E\) has a closed graph. The values \(N_{X}^{C}(x)\cap D\) are closed, convex, and bounded, hence weakly compact. Thus, the map \(x\mapsto N_{X}^{C}(x)\cap D\) is u.h.c. with closed convex and bounded values. Being a linear combination of u.h.c. maps, Φ is also u.h.c. As the sum of a compact convex set and a closed bounded convex set, \(\Phi(x) \) is closed and convex for each \(x\in X\), i.e., \(\Phi\in \mathbf{H}(X,E)\). We verify that Φ verifies the boundary condition (10). For any given \(x\in \partial X\), since the cone \(T_{X}^{C}(x)\) is closed and convex, by the Moreau decomposition theorem, any \(y\in\Psi(x)\) has the form \(y=y_{T}+y_{N}\) with \(y_{T}=\operatorname{Proj}_{T_{X}^{C}(x)}(y)\) and \(y_{N}= \operatorname{Proj} _{N_{X}^{C}(x)}(y)\), \(\langle y_{N},y_{T}\rangle=0\). Therefore, \(0=\langle y_{N},y_{T}\rangle=\langle y_{N},yy_{N}\rangle =\langle y_{N},y\rangle \Vert y_{N}\Vert^{2}\). By the CauchySchwarzBunyakowsky inequality, \(\Vert y_{N}\Vert\leq\Vert y\Vert\leq M\), that is, \(y_{T}=yy_{N}\in \Psi (x)(N_{X}^{C}(x)\cap D)\), i.e., \(\Phi(x)\cap T_{X}^{C}(x)\neq\emptyset\). The fact that \((X,E)\) has the equilibrium property for \(\mathbf {H}_{\partial}\) ends the proof. □
Recall that a subset X of a metric space \((E,d)\) is an Lretract (of E) if there is a continuous neighborhood retraction \(r:U\rightarrow X\) (U an open neighborhood of X in E) and \(L>0\) such that \(d(r(x),x)\leq L\operatorname{dist}(x;X)\) for all \(x\in U\). An Lretract is clearly a neighborhood retract of E and, in particular, is closed in E. The class of Lretracts is quite large and contains many subclasses of nonconvex sets of interest in analysis and topology, e.g., closed subset of normed spaces that are biLipschitz homeomorphic to closed convex sets, epiLipschitz subsets of normed spaces, proxregular sets, etc. (see [59] and [11]). The following general variational inequality on Lretracts follows at once from Theorem 32 above and Theorem 5.3 in [59], which establishes that compact Lretracts belong to \(\mathbf{H}_{\partial}(X,E)\).
Corollary 33
If X is a compact Lretract in a Hilbert space E with \({\mathcal{E}}(X)\neq0\), and \(\Psi\in\mathbf{H}(X,E)\) is a compactvalued map, then Ψ has a coequilibrium.
Note that one can make use of a generalized Moreau decomposition theorem in Banach spaces to prove that Corollary 33 holds true in a Banach space E.
5.1 Solvability for GOCP on compact epiLipschitz domains
Given an interval I in \(\mathbb{R} \), a closed subset X in a separable Banach space E, a map \(F:I\times X\rightrightarrows E\), let \(S_{F}(t_{0},u)\) be the solutions viable in X for the Cauchy problem
(assuming such solutions exist). Starting at a point \(u\in X\), consider the journey along a trajectory \(y(t)\) followed by a path to a point v in a return set \(\Psi(u)\subset X\). Assume that a cost \(\varphi (u,y,v)\) is associated to this journey (e.g., \(\varphi (u,y,v)=\varphi _{1}(u,y)+\varphi_{2}(y,v)\)). We are interested in the particular control problem GOCP (see Section 2; we may assume with no loss in generality that \(t_{0}\) is fixed)
Consider for an illustration the particular case treated in [60] where \(F(t,y(t))=Ay(t)+R(t,y(t))\) with \(A=\lim_{t\downarrow0}\frac{1}{t}(U(t)Id_{E})\) being a closed densely defined linear operator which is the infinitesimal generator of a \(C_{0}\)semigroup \({\mathcal{U}}=\{U(t)\} _{t\geq0}\) of bounded linear operators on E such that \({\mathcal{U}}(X)\subseteq X\). Let \(R:I\times X\rightrightarrows E\) be a Carathéodory map^{Footnote 21} with linear growth (i.e., \(\sup_{z\in R(t,y)}\Vert y\Vert\leq\mu(t)(1+\Vert y\Vert)\) for some \(\mu\in L_{\mathrm{loc}}^{1}(I,E)\)). The set of mild solutions \(S_{F}(t_{0,}u)\) is the fixed point set of the composition
where \(N_{R}(y):=\{f\in L_{\mathrm{loc}}^{1}(I,E):f(t)\in R(t,y(t))\mbox{ a.a. } t\in I\}\) is the Nemetskij operator associated to R, and \(M(t_{0},u;f)(t):=U(tt_{0})u+\int_{t_{0}}^{t}U(ts)f(s)\,ds\) is the mild solution of the Cauchy problem \(y^{\prime}(t)\in Ay(t)+f(t)\), \(y(t_{0})=u\). The solvability of GOCP is based on two crucial observations on the qualitative properties of the solution set of by Bothe [61] and Kryszewski [60].
Theorem 34

(i)
([61]) Assume that the semigroup \({\mathcal{U}}\) is compact and R maps precompact subsets of \(I\times X\) into compact sets in E. If the tangency condition with the BouligandSeveri cone \(R(t,y)\cap T_{X}(y)\neq\emptyset\) a.e. \(t\in I\) for all \(y\in X\) holds, then the map \(S:I\times X\rightrightarrows C(I,X)\) given by \(S(t_{0},u)=S_{F}(t_{0},u)\) is u.s.c. and has nonempty compact values.

(ii)
([60]) If in addition X is epiLipschitz in E, and the more restrictive tangency condition with the Clarke’s cone \(R(t,y)\cap T_{X}^{C}(y)\neq\emptyset\) for a.a. \(t\in I\) and all \(y\in X\) holds, then the values of the map S are also \(R_{\delta}\) sets.
These properties are setvalued generalizations to differential inclusions in infinite dimensions of Aronszajn’s celebrated theorem on the \(R_{\delta}\)set structure of the solution set of the classical singlevalued Cauchy problem with continuous righthand side [34]. They extend results by Plaskacz [62] where X was a nonempty closed proximate retract^{Footnote 22} of \(\mathbb{R}^{n}\). We conclude the paper with an extension of Theorem 4.1 in [10] for the solvability of GOCP.
Theorem 35
Assume that X is a compact epiLipschitz set in a separable Banach space E with \({\mathcal{E}}(X)\neq0\) and that the above hypotheses on \(F(t,y(t))=Ay(t)+R(t,y(t))\) hold with the semigroup \({\mathcal{U}}\) being compact. If \(\Psi:X\rightrightarrows X\) is l.s.c. with AR values and φ is continuous on \(X\times C(I,E)\times X\), and quasiconvex with respect to the return variable v. Then GOCP has a solution provided Φ verifies the tangency condition
Proof
Apply Theorem 20 with \(\Phi=S_{F}\) which is a u.s.c. compact approachable map by Example 7. □
Notes
One could thus reasonably argue that variational inequalities go as far back as the establishment of optimality conditions for minimization problems, i.e., to Pierre de Fermat’s necessary optimality condition for an equilibrium.
Continuity: \(\exists C>0\), \(a(u,v)\leq C\Vert u\Vert\Vert v\Vert \), \(\forall u,v\in E\). Coercivity: \(a(u,u)\geq\alpha\Vert u\Vert^{2}\), \(\forall u\in E\).
Using the Banach contraction principle presents a clear computational advantage of approximation by Picard iterations.
A Ky Fan map has convex values and open preimages. Boundedness of domain (thus weak compactness) follows from the coercivity of a.
Based on Ky Fan’s extension of the KnasterKuratowskiMazurkiewicz principle to vector spaces of arbitrary dimension.
That is a pair of real vector spaces E, F together with a bilinear form \(\langle\cdot,\cdot\rangle:F\times E\rightarrow \mathbb{R} \) such that \(\forall x\in E\setminus\{0\}\), \(\exists y\in F\) with \(\langle y,x\rangle \neq0\) and \(\forall y\in F\setminus\{0\}\), \(\exists x\in E\) with \(\langle y,x\rangle \neq0\).
A real function f on a convex subset of a vector space is quasiconvex if \(f(z)\leq\max\{f(x),f(y)\}\) for all \(z\in [ x,y]\). It is semistrictly quasiconvex if \(f(x)>f(y)\Longrightarrow f(x)>f(z)\) for all \(z\in\,]x,y]\).
The strict inequality cannot be replaced by a large inequality as the quasiconvex differentiable function \(f(x)=x^{3}\), \(x\in [1,1]\) indicates. The strict inequality can be replaced by the large inequality for the smaller class of differentiable pseudoconvex functions (which includes convex functions). In such a case, \([\exists\hat{u}\in X,\forall v\in X , \langle\nabla f(\hat{u}),v\hat{u}\rangle\geq 0]\Longleftrightarrow f(\hat{u})=\min_{X}f(v)\).
Note that if f is semistrictly quasiconvex and l.s.c., then \(\operatorname{cl}(S_{f<}(u))=S_{f\leq}(u)\) for all \(u\in X\backslash\arg\min f\). Indeed, if \(y\in S_{f\leq}(u)\) and \(f(y)=f(u)\), consider \(y_{1}\in X\) with \(\inf f\leq f(y_{1})< f(y)=f(u)\). By semistrict quasiconvexity, \(f(y_{i})< f(y)=f(u)\) for any net \(\{y_{i}\}\) converging to y along the line segment \([y_{1},y)\subset X\).
A map \(\Phi:X\rightrightarrows Y\) of two topological spaces X and Y is said to be upper semicontinuous at a point \(x_{0}\in X\) if for any open neighborhood V of \(\Phi(x_{0})\) in Y, there exists an open neighborhood U of \(x_{0}\) in X such that \(\Phi(U)\subset V\). The map Φ is said to be upper semicontinuous (u.s.c. for short) on X if it is upper semicontinuous at every point of X. Note that Φ is u.s.c. on X if and only if the upper inverse image \(\{x\in X;\Phi(x)\subset V\}\) of any open subset V of Y is open in X.
Note that since topological spaces are assumed to be Hausdorff, a neighborhood retract A of X is closed in X.
The inclusions are strict. A Euclidean sphere is an ANR but not an AR. The set \(\Gamma:=\{(x,\sin(\frac{1}{x}))\in\mathbb {R}^{2} :0< x\leq1\}\cup\{(0,y) : 1\leq y\leq1\}\) is an AANR but not an ANR (it is not locally contractible!).
It is well known that all homology, cohomology, homotopy, and cohomotopy groups of an AR are trivial. Also, every retract of an AR is also an AR.
An \(R_{\delta}\) set is the intersection of a countable decreasing sequence of compact contractible metric spaces.
A point \(x_{\omega}\in X\) such that, for some \(y_{\omega}\in\Phi (x_{\omega})\), both \(x_{\omega}\) and \(y_{\omega}\) belong to a common member W of ω.
For the definition of the nerve of a covering, see Definition 5.3, p.172 in Dugundji [43]. Given a topological space Y and an open cover ω of Y, two mappings \(f,g:X\rightarrow Y\) are said to be ωnear if for each \(x\in X\), \(\{f(x),g(x)\}\subset W\) for some member W of ω. They are said to be ωhomotopic if there exists a deformation \(h:X\times[0,1]\rightarrow Y\) joining f and g satisfying \(\exists W\in\omega\) with \(h(\{x\}\times [ 0,1])\subset W\) \(\forall x\in X\). If Y is an ANR, every open cover ω of Y admits a refinement α such that any two continuous mappings \(f,g:X\rightarrow Y\) that are αnear are ωhomotopic (Lemma 7.2 in [42]).
The EulerPoincaré characteristic \({\mathcal{E}}(X) \) of a space X is assumed to be a homotopy invariant. This is the case when X is compact, with \({\mathcal{E}}(X)\) being the signed finite sum of Betti numbers \(\sum_{i\geq 0}(1)^{i}\beta_{i}\), \(\beta_{i}=\dim H^{i}(X;\mathbf{Q})\), where the cohomology graded linear space \(\{H^{i}(X;\mathbf{Q})\}\) is of finite type. It turns out that, in this case, \({\mathcal{E}}(X)=\lambda(\mathrm{id}_{X})\) the Lefschetz number of the identity mapping on X. A homotopically invariant Euler characteristic can be defined for large classes of noncompact spaces, e.g., finite unions of convex sets, noncompact complex algebraic varieties, ndimensional hyperbolic Riemannian manifolds with finite volume, etc. (see, e.g., Chen [63], Gromov [64], and Harder [65]).
If a setvalued map Ψ is l.s.c. and a real function \(f(u,w)\) is u.s.c., then the marginal function \(g(u)=\inf_{w\in\Psi(u)}f(u,w)\) is u.s.c. (see [56]).
The mapping \(v\mapsto d_{X}^{0}(x)(v)\) is finite, positively homogeneous, subadditive, and Lipschitz continuous on E. In addition, \((x,v)\mapsto d_{X}^{0}(x)(v)\) is u.s.c. on \(X\times E\). The generalized gradient \(\partial^{0}\operatorname{dist}(x;X)\) is the convex weak^{∗}convex set of linear forms \(\{p\in E^{\prime}:\langle p,v\rangle\leq d_{X}^{0}(x)(v)\}\).
The results below remain valid with a dual pair \((E,E^{\prime})\) of a normed space and its topological dual.
That is, R has convex values, is measurable in t for all y, and is u.s.c. in y for a.a. \(t\in I\).
That is, there exists a continuous neighborhood retraction \(r:U\rightarrow X\) with \(r(x)=x\), \(\forall x\in X\) and \(\Vert r(x)x\Vert =\operatorname{dist}(x,X)\), \(\forall x\in U\).
References
Fichera, G: La nascita della teoria delle disequazioni variazionali ricordata dopo trent’anni. Accademia Nazionale dei Lincei 114, 4753 (1995)
Hartmann, P, Stampacchia, G: On some nonlinear elliptic differentialfunctional equations. Acta Math. 115, 271310 (1966)
Brézis, H: Analyse fonctionelle, Théorie et applications. Masson, Paris (1983)
Latif, A, Luc, DT: Variational relation problems: existence of solutions and fixed points of setvalued contraction mappings. Fixed Point Theory Appl. 2013, 315 (2013)
BenElMechaiekh, H: Approximations and selections methods for setvalued maps. In: AlMezel, SAR, AlSolamy, FRM, Ansari, QH (eds.) Fixed Point Theory, Variational Analysis, and Optimization, pp. 77138. Chapman and Hall/CRC, Boca Raton (2014)
Minty, GJ: On variational inequalities for monotone operators I. Adv. Math. 30, 17 (1978)
Dugundji, J, Granas, A: KKMmaps and variational inequalities. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 5, 679682 (1978)
Allen, G: Variational inequalities, complementarity problems and duality theorems. J. Math. Anal. Appl. 58, 110 (1977)
Lassonde, M: On the use of KKM multifunctions in fixed point theory and related topics. J. Math. Anal. Appl. 97, 151201 (1983)
BenElMechaiekh, H, Isac, G: Generalized multivalued variational inequalities. In: AndreianCazacu, C, Letho, O, Rassias, TM (eds.) Topology and Analysis, pp. 115142. World Scientific, River Edge (1998)
BenElMechaiekh, H: On nonlinear inclusions in nonsmooth domains. Arab. J. Math. 1, 395416 (2012)
Baiocchi, C, Capelo, A: Variational and Quasivariational Inequalities. Applications to FreeBoundary Problems. Wiley, New York (1984)
Kinderlehrer, D, Stampacchia, G: An Introduction to Variational Inequalities and Their Applications. Classics in Applied Mathematics, vol. 31. SIAM, Philadelphia (2000)
Nagurney, A: Network Economics: A Variational Inequality Approach. Academic Publishers, Dordrecht (1983)
Granas, A: Méthodes topologiques en analyse convexe. Séminaire de Mathématiques Supérieures, vol. 110. Les Presses de l’Université de Montréal, Montreal (1990)
Cottle, RW, Pang, JS, Stone, RE: The Linear Complementarity Problem. Academic Press, Boston (1992)
Isac, G: Complementarity Problems. Lecture Notes in Math., vol. 1528. Springer, Berlin (1992)
Murty, KG: Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988)
Facchinei, F, Pang, JS: Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. I and II. Springer, New York (2003)
Konnov, IV: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)
Ansari, QH, Lalitha, CS, Mehta, M: Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization. CRC Press, Boca Raton (2014)
Gwinner, J: On fixed points and variational inequalities  a circular tour. Nonlinear Anal. 5, 565583 (1981)
Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123145 (1994)
Agarwal, RP, O’Regan, D: Nonlinear generalized quasivariational inequalities: a fixed point approach. Math. Inequal. Appl. 6, 133143 (2003)
Ansari, QH: An introduction to variationallike inequalities. In: AlMezel, SAR, AlSolamy, FRM, Ansari, QH (eds.) Fixed Point Theory, Variational Analysis, and Optimization, pp. 207245. Chapman and Hall/CRC, Boca Raton (2014)
Hadjisavvas, N: Convexity, generalized convexity and applications. In: AlMezel, SAR, AlSolamy, FRM, Ansari, QH (eds.) Fixed Point Theory, Variational Analysis, and Optimization, pp. 139171. Chapman and Hall/CRC, Boca Raton (2014)
Aussel, D: New developments in quasiconvex optimization. In: AlMezel, SAR, AlSolamy, FRM, Ansari, QH (eds.) Fixed Point Theory, Variational Analysis, and Optimization, pp. 173204. Chapman and Hall/CRC, Boca Raton (2014)
BenElMechaiekh, H, Deguire, P: Approximation of nonconvex setvalued maps. C. R. Acad. Sc. Paris 312, 379384 (1991)
BenElMechaiekh, H: Continuous approximation of setvalued maps, fixed points and coincidences. In: Florenzano, M, Guddat, J, Jimenez, M, Jongen, HT, Lagomasino, GL, Marcellan, F (eds.) Proceedings of the Second International Conference on Approximation and Optimization, pp. 6997. Peter Lang, Frankfurt (1995)
Cellina, A: A theorem on the approximation of compact multivalued mappings. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8) 47, 429433 (1969)
Van Mill, J: Infinite Dimensional Topology. North Holland, Amsterdam (1989)
Mas Colell, A: A note on a theorem of F. Browder. Math. Program. 6, 229233 (1974)
McLennan, A: Approximation of contractible valued correspondences by functions. J. Math. Econ. 20, 591598 (1991)
Aronszajn, N: Le correspondant topologique de l’unicité dans la théorie des équations différentielles. Ann. Math. 43, 730738 (1942)
Dugundji, J: Modified Vietoris theorems for homotopy. Fundam. Math. LXVI, 223235 (1970)
Kryszewski, W: Graphapproximation of setvalued maps on noncompact domains. Topol. Appl. 83, 121 (1998)
Górniewicz, L, Granas, A, Kryszewski, W: On the homotopy method in the fixed point index theory of multivalued mappings of compact absolute neighborhood retracts. J. Math. Anal. Appl. 161, 457473 (1991)
Girolo, J: Approximating compact sets in normed linear spaces. Pac. J. Math. 98, 8189 (1982)
Fan, K: Fixed point and minimax theorems in locally convex topological linear spaces. Proc. Natl. Acad. Sci. USA 38, 121126 (1952)
Himmelberg, CJ: Fixed points for compact multifunctions. J. Math. Anal. Appl. 38, 205207 (1972)
Granas, A: Points fixes pour les applications compactes: espaces de Lefschetz et la théorie de l’indice, vol. 68. Les Presses de l’Université de Montréal, Montreal (1980)
Dugundji, J: An extension of Tietze’s theorem. Pac. J. Math. 1, 353367 (1951)
Dugundji, J: Topology. Allyn and Bacon, Boston (1966)
Parida, J, Sen, A: A variationallike inequality for multifunctions with applications. J. Math. Anal. Appl. 124, 7381 (1987)
Itoh, S, Takahashi, W, Yanagi, K: Variational inequalities and complementarity problems. J. Math. Soc. Jpn. 30, 2328 (1978)
Slavov, ZD, Slavova Evans, C: Compactness, contractibility and fixed point properties of the Pareto sets in multiobjective programming. Appl. Math. 2, 556561 (2011)
Benoist, J: Contractibility of the efficient set in strictly quasiconcave vector maximization. J. Optim. Theory Appl. 110, 325336 (2001)
Huy, NQ: Arcwise connectedness of the solution sets of a semistrictly quasiconcave vector maximization problem. Acta Math. Vietnam. 27, 165174 (2002)
Ricceri, B: A class of functions whose sublevel sets are absolute retracts. Topol. Appl. 155, 871873 (2008)
Yao, JC: The generalized quasivariational inequality problem with applications. J. Math. Anal. Appl. 158, 139160 (1991)
Chan, D, Pang, JS: The generalized quasivariational inequality problem. Math. Oper. Res. 7, 211222 (1982)
Dien, NH: Some remarks on variationallike and quasivariationallike inequalities. Bull. Aust. Math. Soc. 46, 335342 (1992)
Gowda, MS, Pang, JS: Some existence results for multivalued complementarity problems. Math. Oper. Res. 17, 657670 (1992)
Parida, J, Sen, A, Kumar, A: A linear complementarity problem involving a subgradient. Bull. Aust. Math. Soc. 37, 345351 (1988)
Mordukhovich, B: Variational Analysis and Generalized Differentiation, Vol. I and II. Springer, Berlin (2006)
Aubin, JP, Frankowska, H: SetValued Analysis. Birkhäuser, Boston (1990)
Aubin, JP, Cellina, A: Differential Inclusions. Springer, Berlin (1984)
Clarke, F: Optimization and Nonsmooth Analysis. Les publications CRM, Montréal (1989)
BenElMechaiekh, H, Kryszewski, W: Equilibria of setvalued maps on nonconvex domains. Trans. Am. Math. Soc. 349, 41594179 (1997)
Kryszewski, W: Topological structure of solution sets of differential inclusions: the constrained case. Abstr. Appl. Anal. 6, 325351 (2003)
Bothe, D: Multivalued differential equations on graphs and applications. Ph.D. thesis, Universität Paderborn (1992)
Plaskacz, S: On the solution sets for differential inclusions. Boll. Un. Math. Ital. A (7) 6A, 387394 (1992)
Chen, B: On the Euler characteristic of finite unions of convex sets. Discrete Comput. Geom. 10, 7993 (1993)
Gromov, M: Volume and bounded cohomology. Publ. Math. IHÉS 56, 5100 (1982)
Harder, G: A GaussBonnet formula for discrete arithmetically defined groups. Ann. Sci. Éc. Norm. Super. 4, 409455 (1971)
Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130127D1435 ). The authors acknowledge with thanks DSR’s technical and financial support.
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To the memory of George Isac
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Latif, A., BenElMechaiekh, H. Topological fixed point theory and applications to variational inequalities. Fixed Point Theory Appl 2015, 85 (2015). https://doi.org/10.1186/s136630150329y
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DOI: https://doi.org/10.1186/s136630150329y
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Keywords
 constrained nonlinear inequality
 variational and quasivariational inequalities
 variationallike inequalities
 complementarity problems
 approximation methods in setvalued analysis
 fixed points
 equilibria and coequilibria for setvalued maps
 control problems
 differential inclusions
 quasiconvex programming
 normal and tangent cones