Fixed point–critical point hybrid theorems and application to systems with partial variational structure

In this paper, fixed point arguments and a critical point technique are combined leading to hybrid existence results for a system of two operator equations where only one of the equations has a variational structure. An application to periodic solutions of a semi-variational system is given to illustrate the theory.


Introduction
Many mathematical models are expressed as operator equations of fixed point type: N (u) = u associated with some operators N, or as critical point type: for Fréchet differentiable functionals E. Those equations which are equivalent -in the sense of having the same solutions-with some critical point equation are said to have a variational structure. Solving fixed point equations is the topic of fixed point theory, while critical point equations are the object of critical point theory. Both theories offer lots of principles and methods for the existence, uniqueness, stability, and approximation of solutions. In the last years, the idea of combining different existence principles has become useful for the treatment of many problems. We use the term hybrid to refer to such kind of combined principles and methods. Applied to both fixed point and critical point theory, the term hybrid introduces a certain splitting of the operator N and of the functional E, respectively. A typical result in this sense ≤ Ld(S (y) , S (y)) + d(N 1 (S (y) , y) , N 1 (S (y) , y)).
For a fixed y ∈ D 2 , since N 1 (S (y) , ·) is continuous, from (1.3), we immediately see that S is continuous at y, as claimed. Next, we look at the composed mapping N 2 (S (·) , ·) : D 2 → D 2 . Clearly, it is continuous as a composition of two continuous mappings. In addition, from assumption (b), its image is a relatively compact subset of Y. Thus, Schauder's fixed point theorem applies and guarantees the existence of a point y * ∈ D 2 with: N 2 (S (y * ) , y * ) = y * .
can be replaced by the Leray-Schauder boundary condition: Indeed, under this assumption, instead of Schauder's fixed point theorem, the Leray-Schauder fixed point theorem applies to the mapping N 2 (S (·) , ·) : U → Y. (c) Based on the key remark that in the proof of Theorem 1.2, it is essential that a fixed point result can be applied to the mapping N 2 (S (·) , ·) , in [13] it is given a meta version of Avramescu's theorem also involving a more general contraction property. The result is stated in terms of fixed point structures in [15].
The connection between Theorems 1.2 and 1.1 is given by the following remark. Namely, in case that D 1 and D 2 are two closed bounded convex subsets of two Banach spaces X and Y , respectively, and on N 1 we assume a Lipschitz condition of the form and to the mappings A, B : D 1 × D 2 → X × Y given by: Note that Avramescu's theorem is a particular case of a more heterogeneous fixed point theorem given in [6] that combines Banach-Perov contraction principle with Schauder fixed point theorem and its analogue for the weak topology.
In critical point theory, the idea of splitting the functional as a sum of two functions is particularly used for nonsmooth extensions of the classical results (see, e.g., [8,Ch. 3] and [14]). As concerns the idea of splitting the space and correspondingly the mapping E , we mention the recent papers [11,12].
The main idea of this paper is to combine fixed point arguments with a critical point technique in a hybrid existence result for a system of two operator equations where only one of the equations has a variational structure. The motivation for this problem comes from the class of boundary value problems related to systems of second-order equations where part of the nonlinearities do not depend on the gradient, but the others do. Thus, the first equation has a variational structure, while for the rest, such a structure does not exist.
The paper is organized as follows: first, in Section 2.1, we give a variational analogue of Avramescu's theorem in terms of a couple (N, E) of an operator N and a functional E. More exactly, we find sufficient conditions for the existence of a solution (x, y) to the problem: see Corollary 2.3. Here, D is a complete metric space, U is an open subset of a Banach space, N : D × U → D, E : D × U → R, and by E y (x, ·), we mean the Fréchet derivative of the functional E (x, ·) . The Corollary 2.3 is improved in Section 2.2 in case that U is a ball. Furthermore, in Section 2.3, we look for a solution (x, y) with y localized in a bounded convex subset of a wedge (cone, in particular) imposing to y a barrier from below. The result can be particularly useful in establishing the existence of multiple positive solutions for operator systems having only a partial variational structure. An application is given in Section 3 to suggest the way that our theoretical results apply to boundary value problems.
Our theoretic results are based on the weak form of the Ekeland variational principle (see, e.g., [10]) and the notion of a contraction with respect to a vector-valued metric in Perov's sense. Recall that a map d : X ×X → R n is said to be a vector-valued metric on the set X if, for every x, y, z ∈ X, one has that: (1) d(x, y) ≥ 0 n and d(x, y) = 0 n if and only if x = y where 0 n is the null vector in R n , where if x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ), then by x ≤ y, one means that A map f : D ⊂ X → X is said to be a Perov contraction on D with respect to the vector-valued metric d if there exists a matrix A ∈ M n×n (R + ) with spectral radius ρ (M ) less than one, such that: Details about these notions can be found in [10].

A variational analogue of Avramescu's theorem
The conclusion of Avramescu's theorem says that x is a fixed point of N 1 (·, y) and y is a fixed point of N 2 (x, ·) . We now give its analogue hybrid fixed pointcritical point result involving instead of the couple (N 1 , N 2 ) of two mappings, a couple (N, E) of a mapping and a functional. It will guarantee the existence of a couple (x, y), such that x is a fixed point of N (·, y) , while y is a critical point (a minimum point) of a functional E (x, ·) .
The theory will apply to the treatment of systems of two nonlinear equations in case that only one of them has a variational structure.
Assume that the following conditions are satisfied: Then, for every k ∈ N, there exists (x k , y k ) ∈ D × U , such that: Under a stronger condition on the couple (N, I Y − E y ), we can guarantee in the conclusion of Corollary 2.2 that y = y. More exactly, we have: Under the hypotheses of Corollary 2.2 with, instead of (iii), (iii * ) there are two constants R 0 , γ > 0, such that: for all x ∈ D and y ∈ U with y > R 0 , and the mapping N, In addition, (x, y) is the unique solution of the system N 1 (x, y) = x, E y (x, y) = 0.
Proof of Theorem 2.1. As in the proof of Theorem 1.2, we can consider the mapping S : U → D with: Like there, the mapping S is continuous based on the continuity of N (x, ·) for every x ∈ D.
We now proceed to the construction of a sequence (y k ) and its accompanying sequence (x k ) with x k = S (y k−1 ) which in view of (2.6) gives: (2.7) We shall define this sequence for indices k > k 0 , where k 0 ≥ 1/a. The first element y k0 is chosen arbitrarily in U. At step j (j ≥ 1) , having already introduced y i for i = k 0 , · · ·, k 0 + j − 1 =: k − 1, we apply the weak form of the Ekeland variational principle to the functional E (S (y k−1 ) , ·), and we get an element y k ∈ U satisfying: contrary to assumption (2.1). Hence, y k ∈ U which makes possible to choose y in (2.9) of the form: for t > 0 sufficiently small that y ∈ U. From (2.9), using the definition of the Fréchet derivative gives: and replacing y becomes: Dividing by t and letting t go to zero imply: Now, (2.7), (2.8), and (2.11) give the conclusion of the theorem.
Proof of Corollary 2.2. From (iii), the sequence y k − E y (x k , y k ) is relatively compact, so it has a convergent subsequence. Since, from (2.11), the sequence E y (x k , y k ) converges to zero, we have that the corresponding subsequence of (y k ) is convergent. Let this subsequence be y kj and its limit be y. Similarly, the sequence y kj −1 has a convergent subsequence to some y (not necessary equal to y). Thus, passing to a subsequence, we may assume that y kj → y and y kj −1 → y as j → +∞. Furthermore, from x kj = S y kj −1 , one has x kj → S (y) := x. Now, taking into account the continuity of N, E and E y , we can pass to limit in ( 2.7), (2.8), and (2.11), thus obtaining (2.3).

Proof of Corollary 2.3.
We first note that if, for the sequence (y k ) in (2.2), there would exist a convergent subsequence y kj , such that the translated sequence y kj −1 converges to the same limit as y kj , then (2.5) immediately follows from (2.2) after passing to the limit, with y = lim j→+∞ y kj = lim j→+∞ y kj −1 and x = S (y) . Thus, the problem is to guarantee the same limit for two out of phase subsequences y kj and y kj −1 . Obviously, the simplest way to do this is to ensure the convergence of the entire sequence (y k ) . We can do this using condition (iii * ), as first shown in [11]. Let z k := E y (x k , y k ) . Since the mapping N, I Y − E y is a Perov contraction on D×Y with respect to the vector-valued metric d, there is a square matrix: of nonnegative entries with the spectral radius less than one, such that for all k and p, one has: (2.13) Notice that the spectral radius of M is less than one if and only if: (2.14) 63 Obviously, we may assume that α = L. From (2.13), we deduce that: (2.15) while, from (2.12), since N (x k+p , y k+p−1 ) = x k+p and N (x k , y k−1 ) = x k , we have: Then: Here, by (2.14),  [11] applies and guarantees that the sequence (y k ) is Cauchy, as wished.
The uniqueness of the solution of the system is an immediate consequence of the fact that the mapping N, I Y − E y is a Perov contraction.

A Leray-Schauder-type variant in balls
Here, we discuss the possibility of replacing the boundary condition (2.1 ) expressed in terms of the functional E, by a Leray-Schauder-type boundary condition involving the derivative E y . We shall do it in case that U is a ball. In particular, in the same setting of Corollary 2.3, we can prove the following result. Proof. Coming back to the proof of Theorem 2.1 shows us that without condition (2.1), the element y k giving by Ekeland's principle may belong to the boundary of U. Let k > 0 be such that y k = R. Three cases are possible: (a) E y (x k , y k ) = 0. In this case, the conclusion (2.11) is true without any proof.

Vol. 23 (2021)
Fixed point-critical point hybrid theorems and application Page 9 of 19 63 (b) E y (x k , y k ) , y k > 0. If so, the choice of y prescribed by (2.10) is still admissible for all small enough t > 0. Indeed: The proof will then continue as in Theorem 2.1 getting (2.11).
(c) E y (x k , y k ) = 0 and E y (x k , y k ) , y k ≤ 0. In this case, we let y in (2.9) be of the form: where t > 0, μ k = − E y (x k , y k ) , y k /R 2 and ε > 0 is arbitrarily fixed. Then, since: as above y < R for all small enough t > 0. Replacing in (2.9), dividing by t, letting t go to zero, and finally letting ε → 0 yield: Note that in virtue of (2.17), one has 0 ≤ μ k ≤ C R 2 . Thus, concerning the sequence (y k ), two situations are possible: (I) There is a subsequence of (y k ) for which (2.11) holds. Then, we follow the proof of Corollary 2.3 with this subsequence instead of the whole sequence (y k ) .
(II) If case (I) does not hold, then after eliminating a finite number of terms, we may assume that for the sequence (y k ) , inequality (2.20) holds, y k ∈ ∂U andμ k > 0. Moreover, passing if necessary to a subsequence, we may assume that μ k → μ for some μ ≥ 0. Then, (2.20) implies E y (x k , y k ) + μy k → 0, and we continue with the proof of the convergence of (y k ) as for Corollary 2.3, where now we let z k = E y (x k , y k ) + μy k . Passing to the limit in (2.20) gives E y (x, y) + μy = 0, where y ∈ ∂U andμ ≥ 0. The case μ > 0 being excluded by (2.18), it remains that in any case one has E y (x, y) = 0. Analogously, by the continuity of N and E, we get N (x, y) = x and E (x, y) = inf U E (x, ·) .

A variant in convex conical sets
Next, we shall look for y in a wedge (a cone, in particular) and we shall introduce for y a barrier from below. The new result can be particularly useful in establishing the existence of multiple positive solutions for operator systems having only a partial variational structure. To this aim, we shall adapt the technique used in [12] for systems in which all equations have a variational structure.
Let K be a wedge (a closed convex set with R + K ⊂ K, K = {0}) of the Hilbert space Y and let l : K → R + be a concave, upper semicontinuous 63 Page 10 of 19 I. Benedetti et al. JFPTA function with l (0) = 0. For any two numbers r, R > 0 denote by K rR the conical set: K rR := {y ∈ K : r ≤ l(y)and y ≤ R}, which will replace the set U. The set K rR is convex, since l is concave, and closed since l is upper semicontinuous. We assume that K rR = ∅ and we denote: ∂K R = {y ∈ K : y = R}. We continue to consider the complete metric space D. E (x, y) ≥ m + εfor all x ∈ D and y ∈ K rR which simultaneously satisfy l (y) = r and y = R; (2.21) Then, there exists (x, y) ∈ D × K rR with: In addition, (x, y) is the unique solution of the system N (x, y) = x, E y (x, y) = 0.
Proof. First note that in view of (2.21), we may assume that the terms of the sequence (y k ) constructed as explained in the proof of Theorem 2.1 do not simultaneously satisfy l (y k ) = r and y k = R. Hence, for each k, we have either (a) l (y k ) ≥ r and y k < R, or (b) l (y k ) > r and y k = R.
In case (a), the choice (2.10) of y to replace in inequality (2.9) now given on K rR instead of U is admissible for all t > 0 small enough. Indeed, using the concavity of l and (h2) yields: for all t ∈ [0, 1] . In addition, the inequality y ≤ R follows from y k < R provided that t is sufficiently small. Thus, in case (a), we derive the estimate ( 2.11).
In case (b), if E y (x k , y k ) , y k > 0, then the same choice of y is possible based on the previous explanation and that from the proof of Theorem 2.4. It remains to consider the case when: As in the proof of Theorem 2.4, we now choose y of the form (2.19 ), and like there y ≤ R for all sufficiently small t. It remains to guarantee that l (y) ≥ r. From l (y k ) > r, we have σl (y k ) = r for some number σ ∈ (0, 1) . Then, for any ρ ∈ [σ, 1] , one has: l (ρy k ) = l (ρy k + (1 − ρ) 0) ≥ ρl (y k ) + (1 − ρ) l (0) = ρl (y k ) ≥ σl (y k ) = r.
For the rest, we follow the proof of Theorem 2.4 .

Remark 2.2. (Multiplicity)
Assume that there is a constant c > 0, such that: for all y ∈ K. Then, from the assumption K rR = ∅, one finds r ≤ cR. Indeed, if y ∈ K rR , then r ≤ l (y) ≤ c y ≤ cR. If now : r 1 ≤ cR 1 , r 2 ≤ cR 2 and cR 1 < r 2 , (2.25) then the sets K r1R1 and K r2R2 are disjoint. Indeed, if y ∈ K r1R1 , then : Hence, l (y) < r 2 which shows that y / ∈ K r2R2 . The same conclusion holds if :

Application
As an application, we consider the system: subject to the periodic conditions: