On the Stability of the Directional Regularity

Abstract

In this paper we select two tools of investigation of the classical metric regularity of set-valued mappings, namely the Ioffe criterion and the Ekeland Variational Principle, which we adapt to the study of the directional setting. In this way, we obtain in a unitary manner new necessary and/or sufficient conditions for directional metric regularity. As an application, we establish stability of this property at composition and sum of set-valued mappings. In this process, we introduce directional tangent cones and the associated generalized primal differentiation objects and concepts. Moreover, we underline several links between our main assertions by providing alternative proofs for several results.

Appendix

In this section, we illustrate some other connections between our results and several well-known tools in variational analysis. In this sense, we provide some different proofs for two of the key results in this work.

On one hand, we present a direct and constructive proof for the general criterion for the directional regularity of single-valued maps. This underlines again the fact that the use of the Ekeland Variational Principle is an alternative for explicit iterative procedures. On the other hand, we provide as well another proof for the result about the stability at composition of the directional regularity. For this, we employ now, instead of Proposition 13, a variant of the directional Ekeland Variational Principle, formulated on product spaces on the basis of Lemma 12.

1.1 A.1 Proof of the Criterion for the Directional Regularity by an Iterative Procedure

Let us present next the announced constructive proof of the criterion for the directional regularity of single-valued maps.

Proof (of Proposition 11 by Iterative Procedure)

Let $\lambda :=\text {dirsur}_{L\times M}g(\overline {x})$ and s be the supremum from the statement. We only prove that s ≤ λ, since the opposite inequality is straightforward, as shown in the proof given in Section 3.

Define a function φ : X × X → [0, +∞] by φ(u, v) = T_L(u, v), (u, v) ∈ X × X, and a function ψ : Y × Y → [0, +∞] by ψ(y, z) = T_M(y, z), (y, z) ∈ Y × Y. Observe that the convexity of coneL implies that

$$ \varphi(u,v)\leq\varphi(u,w)+\varphi(w,v),\quad\text{for all}\quad u,v,w\in X. $$

(A.1)

To show that s ≤ λ, fix an arbitrary c ∈ (0, s) (if there is any) for which there is r > 0 such that for all $(x,y)\in (B[\overline {x} ,r]\cap \text {Dom} g)\times B[g(\overline {x}),r]$, with 0 ≠ g(x) − y ∈coneM, there is a point x′ ∈ Dom g such that

$$ c \varphi(x,x^{\prime})<\psi(y,g(x))-\psi(y,g(x^{\prime})). $$

(A.2)

Make r > 0 smaller, if necessary, so that the set $B[\overline {x} ,r]\cap \text {Dom} g$ is complete and g is continuous on this set. By the continuity of g, there is ε ∈ (0, r) such that

$$ B[g(u),c\varepsilon]\subset B[g(\overline{x}),r]\ \text{and}\ B[u,\varepsilon ]\subset B[\overline{x},r]\ \text{whenever}\ u\in B[\overline{x} ,\varepsilon]\cap\text{Dom} g. $$

(A.3)

Fix any t ∈ (0, ε) and any $u\in B[\overline {x},\varepsilon ]\cap \text {Dom} g$. Let Λ := B[u, t] ∩ (u + coneL) ∩ Dom g. As L ⊂ S_X is closed, so is coneL. Consequently, Λ is complete. We have to show that

$$g({\Lambda} )\supset B[g(u),ct]\cap(g(u)-\text{cone} M). $$

Consider any fixed y ∈ B[g(u), ct] ∩ (g(u) −coneM); we will find x ∈Λ such that y = g(x). If y = g(u), take x := u and we are done. Assume further that y ≠ g(u). We will construct a sequence x₁, x₂, … in Λ satisfying

$$ c \varphi(u,x_{m})\leq\psi(y,g(u))-\psi(y,g(x_{m})),\quad m\in\mathbb{N}. $$

(A.4)

As φ(u, u) = 0 and g(u) − y ∈coneM (thus ψ(y, g(u)) is finite), the point x₁ := u satisfies (A.4) with m = 1. Let $n\in \mathbb {N}$ and assume that x_n ∈Λ satisfying (A.4) with m = n was already found. If g(x_n) = y, then take x := x_n, and stop the construction. Assume further that g(x_n) ≠ y. Then (A.4), with m := n, implies that ψ(y, g(x_n)) is finite, meaning that g(x_n) − y ∈coneM. Using (A.3) and (A.2), we find x_n+ 1 ∈ Dom g such that

$$ c \varphi(x_{n},x_{n + 1})<\psi(y,g(x_{n}))-\psi(y,g(x_{n + 1}))\quad\text{and that}\quad\varphi(x_{n},x_{n + 1})\geq\tfrac{1}{2}s_{n} $$

(A.5)

where

$$s_{n}:=\sup\left\{ \varphi(x_{n},x^{\prime}):\ x^{\prime}\in \text{Dom} g\ \ \text{and}\ \ c \varphi(x_{n},x^{\prime} )<\psi(y,g(x_{n}))-\psi(y,g(x^{\prime}))\right\} . $$

Note that $0\leq s_{n}\leq \frac {1}{c}\psi (y,g(x_{n}))<+\infty$. Using (A.1), the first inequality in (A.5), and (A.4) with m := n, we get

$$c \varphi(u,x_{n + 1})\leq c \varphi(u,x_{n})+c \varphi(x_{n},x_{n + 1} )<\psi(y,g(u))-\psi(y,g(x_{n + 1})), $$

which is (A.4) with m := n + 1. In particular, we have cφ(u, x_n+ 1) ≤ ψ(y, g(u)) = ∥y − g(u)∥≤ ct; thus x_n+ 1 ∈ u + coneL and φ(u, x_n+ 1) = ∥u − x_n+ 1∥. Consequently, x_n+ 1 ∈Λ. If the process stops at some $n\in \mathbb {N}$, we are done. Assume that this was not the case, that is, g(x_n) ≠ y for every $n\in \mathbb {N}$. From (A.5) and (A.1) we have, for all 1 ≤ n < m, that

$$\begin{array}{@{}rcl@{}} 0\leq c \varphi(x_{n},x_{m}) & \leq& c \varphi(x_{n},x_{n + 1})+\cdots +c \varphi(x_{m-1},x_{m})\\ & <&(\psi(y,g(x_{n})) - \psi(y,g(x_{n + 1})))\!+\cdots+\!(\psi(y,g(x_{m-1} )) - \psi(y,g(x_{m})))\\ & =&\psi(y,g(x_{n}))-\psi(y,g(x_{m})), \end{array} $$

(A.6)

and so, ψ(y, g(x_n)) > ψ(y, g(x_m)). Thus $\ell :=\lim _{n\rightarrow +\infty }\psi (y,g(x_{n}))$ exists and is finite. By (A.6), for all 1 ≤ n < m, we have φ(x_n, x_m) < +∞, and hence φ(x_n, x_m) = ∥x_n − x_m∥. Consequently, (x_n) is a Cauchy sequence in Λ (which is a complete metric space). Put $x:=\lim _{n\rightarrow +\infty }x_{n}$. Then x ∈Λ and ψ(y, g(x)) ≤ ℓ < +∞ because ψ(y,⋅) is lower semicontinuous and g is continuous. Moreover, for any $n\in \mathbb {N}$, using the lower semi-continuity of φ(x_n,⋅) and (A.6) we get that

$$c \varphi(x_{n},x)\leq c\liminf_{p\rightarrow+\infty}\varphi(x_{n} ,x_{n+p})\leq\psi(y,g(x_{n}))-\ell. $$

Consequently, $\lim _{n\rightarrow +\infty }\varphi (x_{n},x)= 0$. Suppose that y ≠ g(x). By (A.2), there is x′ ∈ Dom g such that

$$ c \varphi(x,x^{\prime})<\psi(y,g(x))-\psi(y,g(x^{\prime}))\leq\ell -\psi(y,g(x^{\prime})). $$

(A.7)

Then (A.1) implies that $\limsup _{n\rightarrow +\infty }\varphi (x_{n},x^{\prime })\leq \lim _{n\rightarrow +\infty }\varphi (x_{n},x)+\varphi (x,x^{\prime })=\varphi (x,x^{\prime })$. This and (A.7) imply that, for each $n\in \mathbb {N}$ sufficiently large, we have

$$c \varphi(x_{n},x^{\prime})<\psi(y,g(x_{n}))-\psi(y,g(x^{\prime})). $$

As x ≠ x′ by (A.7), we have φ(x, x′) > 0. The lower semicontinuity of φ(⋅, x′), the choice of s_n, and (A.5) yield that

$$0<\varphi(x,x^{\prime})\leq\liminf_{n\rightarrow+\infty}\varphi(x_{n} ,x^{\prime})\leq\limsup_{n\rightarrow+\infty}s_{n}\leq2\lim_{n\rightarrow +\infty}\varphi(x_{n},x_{n + 1})= 0, $$

a contradiction. Therefore y = g(x). We proved that c ≤ λ, and thus s ≤ λ.

1.2 A.2 Proof of Directional Openness Stability at Composition by Directional EVP

As mentioned before, in the second part of this appendix, we discuss the possibility to give an alternative proof of the main result of the paper, namely Theorem 16, by the use of the next variant of the directional Ekeland Variational Principle.

Theorem 32

Let (X₁, ∥⋅∥), … , (X_n,∥⋅∥) be Banach spaces andA ⊂ X₁ × ... × X_nbe a nonempty closed set.Consider nonempty closed sets$L_{i}\subset S_{X_{i}}$, i = 1, … , nsuchthat coneL_iare convex. Then, for every lower semicontinuous bounded from below function$f:A\rightarrow {\mathbb {R}\cup \{+\infty \}}$, everya₀ := (x₀₁, ... , x_0n) ∈ Asuch thatf(a₀) < +∞, and everyδ, α₁, ... , α_n > 0, thereexistsa_δ := (x_δ1, ... , x_δn) ∈ Asuch that

$$f(a_{\delta})\leq f(a_{0})-\delta\max\{\alpha_{1}T_{L_{1}}(x_{\delta1} ,x_{01}), ... ,\alpha_{n}T_{L_{n}}(x_{\delta n},x_{0n})\} $$

and, for every a := (x₁, ... , x_n) ∈ A ∖{a_δ},

$$f(a_{\delta})<f(a)+\delta\max\{\alpha_{1}T_{L_{1}}(x_{1},x_{\delta 1}), ... ,\alpha_{n}T_{L_{n}}(x_{n},x_{\delta n})\}. $$

Proof

Take $\widetilde {L}$ as in the Lemma 12 and observe that $\text {cone}\widetilde {L}=\text {cone} L_{1}\times ...\times \text {cone} L_{n}$ is convex. Apply Theorem 10 with X := X₁ ×⋯ × X_n and $M:=\widetilde {L}$ to get the statement. □

Now, we are ready to provide the announced proof of the main (and the essential) part of Theorem 16.

Proof (of Theorem 16 by Directional EVP)

Again, as in the proof of Theorem 16, we only have to consider the case where the right-hand side of the inequality (4.2) is positive. We find again positive constants α, β, β′, γ, and δ such that c := αγ − βδ > 0, and inequalities (4.4) and (4.5) hold. Moreover, keeping the notation of Theorem 16, there is ε > 0 such that (4.6), (4.7) and (4.9) hold. Also, taking into account Proposition 3, we may suppose that for any $z\in B(\overline {z},\varepsilon ),$ the mapping $G_{z}^{-1}$ is directionally Aubin continuous around $(\overline {w},\overline {y})$ with respect to P and − M with modulus γ^− 1, i.e.,

$$ e_{-M}\left( G_{z}^{-1}(w)\cap B(\overline{y},\varepsilon),G_{z}^{-1}(w^{\prime})\right) \leq\gamma^{-1}T_{P}(w^{\prime},w)=\gamma^{-1} T_{-P}(w,w^{\prime}), $$

(A.8)

for any $z\in B(\overline {z},\varepsilon ),$ and any $w,w^{\prime }\in B(\overline {w},\varepsilon )$.

Also, in view of the local closedness of the graphs of F₁, F₂ and G, we can consider that GrF₁ ∩ (B [x, ε] × B [y, αε]), GrF₂ ∩ (B [x, ε] × B [z, βε]) and GrG ∩ (B [y, αε] × B [z, βε] × B [w, (αγ + βδ)ε]) are closed, for any $(x,y,z,w)\in B(\overline {x},\varepsilon )\times B(\overline {y} ,\alpha \varepsilon )\times B(\overline {z},\beta \varepsilon )\times B\left (\overline {w},\left (\alpha \gamma +\beta \delta \right ) \varepsilon \right )$ with y ∈ F₁(x), z ∈ F₂(x) and w ∈ G(y, z).

Take

$$\rho:=\min\left\{ 3^{-1}\varepsilon,(3\alpha)^{-1}\varepsilon,(3\beta )^{-1}\varepsilon,(3\alpha\gamma+ 3\beta\delta)^{-1}\varepsilon\right\} . $$

Fix t ∈ (0, ρ) and $(x,y,z,w)\in B(\overline {x},\rho )\times B(\overline {y},\alpha \rho )\times B(\overline {z},\beta \rho )\times B\left (\overline {w},\left (\alpha \gamma +\beta \delta \right ) \rho \right )$ with y ∈ F₁(x), z ∈ F₂(x) and w ∈ G(y, z). We want to prove that

$$B(w,ct)\cap\left[ w-\text{cone} P\right] \subset\mathcal{E} _{G,(F_{1},F_{2})}\left( B_{X\times Y\times Z}((x,y,z),t)\cap ((x,y,z)+\text{cone} \widetilde{L})\right) , $$

where the norm on X × Y × Z and $\widetilde {L}\subset S_{X\times Y\times Z}$ are as in the proof of Theorem 16.

Denote

$$\begin{array}{@{}rcl@{}} A & :=&B(x,2\rho)\times B(y,2\alpha\rho)\times B(z,2\beta\rho)\times B(w,2\left( \alpha\gamma+\beta\delta\right) \rho),\\ \overline{A} & :=&B\left[ x,2\rho\right] \times B\left[ y,2\alpha \rho\right] \times B\left[ z,2\beta\rho\right] \times B\left[ w,2\left( \alpha\gamma+\beta\delta\right) \rho\right] ,\\ {\Omega} & :=&\left\{ (x^{\prime},y^{\prime},z^{\prime},w^{\prime})\in X\times Y\times Z\times W\mid(y^{\prime},z^{\prime})\in(F_{1},F_{2})(x^{\prime})\text{ and }w^{\prime}\in G(y^{\prime},z^{\prime})\right\} . \end{array} $$

Take an arbitrary v ∈ w − [0, ct) ⋅ P. We must prove that $v\in \mathcal {E}_{G,(F_{1},F_{2})}((x,y,z)+[0,t)\cdot \widetilde {L})$.

We can find τ ∈ (0, 1) such that ∥v − w∥ < τct. Remark that ${\Omega }\cap \overline {A}$ is closed (since 2ρ < ε). Define

$$h:{\Omega}\cap\overline{A}\rightarrow\lbrack0,+\infty],\quad h(p,q,r,s):=T_{P} (v,s)=T_{-P}(s,v), $$

and observe that it is lower semicontinuous and bounded from below. Thus, we can apply Theorem 32, for τc > 0 instead of δ, and a₀ = (x, y, z, w) and − L, M,−N, and S_W as sets in X, Y, Z and W, respectively, to find $(\widetilde {a},\widetilde {b},\widetilde {c},\widetilde {d})\in {\Omega }\cap \overline {A}$ satisfying

$$\begin{array}{@{}rcl@{}} T_{P}(v,\widetilde{d})&\leq& T_{P}(v,w)-\tau c\max\{T_{-L}(\widetilde {a},x),\alpha^{-1}T_{M}(\widetilde{b},y),\beta^{-1}T_{-N}(\widetilde{c},z),\\ &&(\alpha\gamma+\beta\delta)^{-1}\left\Vert \widetilde{d}-w\right\Vert \}\\ T_{P}(v,\widetilde{d})&\leq& T_{P}(v,s)+\tau c\max\{T_{-L}(p,\widetilde {a}),\alpha^{-1}T_{M}(q,\widetilde{b}),\beta^{-1}T_{-N}(r,\widetilde{c}),\\ &&(\alpha\gamma+\beta\delta)^{-1}\left\Vert s-\widetilde{d}\right\Vert \}, \end{array} $$

for every $(p,q,r,s)\in {\Omega }\cap \overline {A}$. As an immediate consequence, $\widetilde {b}\in F_{1}(\widetilde {a})$, $\widetilde {c}\in F_{2}(\widetilde {a})$, $\widetilde {d}\in G(\widetilde {b},\widetilde {c}),$ and

$$\max\{T_{-L}(\widetilde{a},x),\alpha^{-1}T_{M}(\widetilde{b},y),\beta^{-1}T_{-N}(\widetilde{c},z),(\alpha\gamma+\beta\delta)^{-1}\left\Vert \widetilde{d}-w\right\Vert \}<\infty, $$

which implies the following:

$$\begin{array}{@{}rcl@{}} \widetilde{a} & \in& x+\text{cone} L,\quad\widetilde{b}\in y-\text{cone} M,\quad\widetilde{c}\in z+\text{cone} N,\\ T_{-L}(\widetilde{a},x) & =&\left\Vert \widetilde{a}-x\right\Vert ,\quad T_{M}(\widetilde{b},y)=\left\Vert \widetilde{b}-y\right\Vert ,\quad T_{-N}(\widetilde{c},z)=\left\Vert \widetilde{c}-z\right\Vert . \end{array} $$

Moreover, since v ∈ w −coneP, we also have

$$\begin{array}{@{}rcl@{}} &&\tau c\max\{T_{-L}(\widetilde{a},x),\alpha^{-1}T_{M}(\widetilde{b} ,y),\beta^{-1}T_{-N}(\widetilde{c},z),(\alpha\gamma+\beta\delta)^{-1} \left\Vert \widetilde{d}-w\right\Vert \}\\ &\leq& T_{P}(v,w)=\left\Vert v-w\right\Vert <\tau ct, \end{array} $$

so

$$\begin{array}{@{}rcl@{}} \widetilde{a} & \in& B(x,t)\cap(x+\text{cone} L)=x+[0,t)\cdot L\subset B(x,t)\subset B(x,\rho),\\ \widetilde{b} & \in& B(y,\alpha t)\cap(y-\text{cone} M)=y-[0,\alpha t)\cdot M\subset B(y,\alpha t)\subset B(y,\alpha\rho),\\ \widetilde{c} & \in& B(z,\beta t)\cap(z+\text{cone} N)=z+[0,\beta t)\cdot N\subset B(z,\beta t)\subset B(z,\beta\rho),\\ \widetilde{d} & \in& B(w,(\alpha\gamma+\beta\delta)t)\subset B(w,(\alpha \gamma+\beta\delta)\rho). \end{array} $$

Hence, $(\widetilde {a},\widetilde {b},\widetilde {c},\widetilde {d})\in A$. Now, if $v=\widetilde {d},$ then

$$\begin{array}{@{}rcl@{}} v\in\mathcal{E}_{G,(F_{1},F_{2})}(\widetilde{a},\widetilde{b},\widetilde{c}) & \subset&\mathcal{E}_{G,(F_{1},F_{2})}(x+[0,t)\cdot L,y-[0,\alpha t)\cdot M,z+[0,\beta t)\cdot N)\\ & =&\mathcal{E}_{G,(F_{1},F_{2})}((x,y,z)+[0,t)\cdot\widetilde{L}), \end{array} $$

which is exactly what we need. We will prove that $v=\widetilde {d}$ is the only possibility.

Assume, on the contrary, that $v\neq \widetilde {d}$. Remark that $T_{P} (v,\widetilde {d})\leq T_{P}(v,w)<\infty ,$ which means that $v-\widetilde {d} \in -\text {cone} P$. Then

$$v^{\prime}:=\frac{v-\widetilde{d}}{\left\Vert v-\widetilde{d}\right\Vert } \in-P, $$

since its norm equals 1 and it belongs to −coneP. Fix σ ∈ (0, αγ) such that

$$ c-\sigma>\tau c, $$

(A.9)

and choose $\zeta \in \left (0,\min \left \{ 3^{-1}\rho ,(\alpha \gamma -\sigma )^{-1}\left \Vert v-\widetilde {d}\right \Vert \right \} \right )$.

We have that

$$\begin{array}{@{}rcl@{}} \left\Vert \widetilde{a}-\overline{x}\right\Vert & \leq&\left\Vert \widetilde{a}-x\right\Vert +\left\Vert x-\overline{x}\right\Vert <\rho +\rho<\varepsilon,\\ \left\Vert \widetilde{b}-\overline{y}\right\Vert & \leq&\left\Vert \widetilde{b}-y\right\Vert +\left\Vert y-\overline{y}\right\Vert <\alpha \rho+\alpha\rho\ <\ \varepsilon,\\ \left\Vert \widetilde{c}-\overline{z}\right\Vert & \leq&\left\Vert \widetilde{c}-z\right\Vert +\left\Vert z-\overline{z}\right\Vert <\beta \rho+\beta\rho\ <\ \varepsilon,\\ \left\Vert \widetilde{d}-\overline{w}\right\Vert & \leq&\left\Vert \widetilde{d}-w\right\Vert +\left\Vert w-\overline{w}\right\Vert <(\alpha\gamma+\beta\delta)\rho+(\alpha\gamma+\beta\delta)\rho\ <\ \varepsilon ,\\ \left\Vert \widetilde{d}+(\alpha\gamma-\sigma)\zeta v^{\prime}-\overline {w}\right\Vert & \leq&\left\Vert \widetilde{d}-w\right\Vert +\left\Vert w-\overline{w}\right\Vert +(\alpha\gamma-\sigma)\zeta<(\alpha\gamma +\beta\delta)\rho\\ &&+(\alpha\gamma+\beta\delta)\rho+ 3^{-1}\varepsilon \ <\ \varepsilon, \end{array} $$

hence by (A.8),

$$\begin{array}{@{}rcl@{}} T_{-M}(\widetilde{b},G_{\widetilde{c}}^{-1}(\widetilde{d}+(\alpha\gamma -\sigma)\zeta v^{\prime})) & \leq& e_{-M}\left( G_{\widetilde{c}}^{-1}(\widetilde{d})\cap B(\overline{y},\varepsilon),G_{\widetilde{c}} ^{-1}(\widetilde{d}+(\alpha\gamma-\sigma)\zeta v^{\prime})\right) \\ & \leq&\gamma^{-1}T_{-P}(\widetilde{d},\widetilde{d}+(\alpha\gamma -\sigma)\zeta v^{\prime})=\gamma^{-1}(\alpha\gamma-\sigma)\zeta\\ &&<\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta, \end{array} $$

hence there exists m ∈coneM with ∥m∥ < 1 such that $\widetilde {b}-\gamma ^{-1}(\alpha \gamma -2^{-1}\sigma )\zeta m\in G_{\widetilde {c}}^{-1}(\widetilde {d}+(\alpha \gamma -\sigma )\zeta v^{\prime })$ or, equivalently,

$$\widetilde{d}+(\alpha\gamma-\sigma)\zeta v^{\prime}\in G(\widetilde{b} -\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta m,\widetilde{c}). $$

Now, since ζ < ε and $\widetilde {b}-\gamma ^{-1}(\alpha \gamma -2^{-1}\sigma )\zeta m\in \widetilde {b}-[0,\alpha \zeta )\cdot M,$ it follows using (4.6) that

$$\widetilde{b}-\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta m\in\widetilde {b}-[0,\alpha\zeta)\cdot M\subset F_{1}(\widetilde{a}+[0,\zeta)\cdot L), $$

hence there exists ℓ ∈coneL with ∥ℓ∥ < 1 such that $\widetilde {b}-\gamma ^{-1}(\alpha \gamma -2^{-1}\sigma )\zeta m\in F_{1}(\widetilde {a}+\zeta \ell )$. But we have

$$\left\Vert \widetilde{a}+\zeta\ell-\overline{x}\right\Vert \ <\ \left\Vert \widetilde{a}-x\right\Vert +\left\Vert x-\overline{x}\right\Vert +\zeta <\rho+\rho+\rho\leq\varepsilon, $$

and since $\widetilde {c}\in B(\overline {z},\varepsilon ),$ we can apply the directional Aubin property of F₂ (4.8) to find that

$$T_{N}(\widetilde{c},F_{2}(\widetilde{a}+\zeta\ell))\leq e_{N}(F_{2} (\widetilde{a})\cap B(\overline{z},\varepsilon),F_{2}(\widetilde{a}+\zeta \ell))\leq\beta T_{L}(\widetilde{a},\widetilde{a}+\zeta\ell)=\beta \zeta\left\Vert \ell\right\Vert <\beta\zeta. $$

It follows that we can find n ∈coneN with ∥n∥ < 1 such that $\widetilde {c}+\beta \zeta n\in F_{2}(\widetilde {a}+\zeta \ell )$.

Finally, since

$$\begin{array}{@{}rcl@{}} \left\Vert \widetilde{b}-\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta m-\overline{y}\right\Vert & <& \left\Vert \widetilde{b}-y\right\Vert +\left\Vert y-\overline{y}\right\Vert +\alpha\zeta<\alpha\rho+\alpha \rho+\alpha\rho\leq\varepsilon,\\ \left\Vert \widetilde{c}+\beta\zeta n-\overline{z}\right\Vert & <& \left\Vert \widetilde{c}-z\right\Vert +\left\Vert z-\overline {z}\right\Vert +\beta\zeta<\beta\rho+\beta\rho+\beta\rho\leq\varepsilon, \end{array} $$

we can use the directional Aubin property of G with respect to z (4.9) to get that

$$\begin{array}{@{}rcl@{}} &&T_{P}(\widetilde{d}+(\alpha\gamma-\sigma)\zeta v^{\prime}, G_{\widetilde {b}-\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta m}(\widetilde{c}+\beta\zeta n))\\ && \leq e_{P}(G_{\widetilde{b}-\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta m}(\widetilde{c})\cap B(\overline{w},\varepsilon), G_{\widetilde{b}-\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta m}(\widetilde{c}+\beta\zeta n))\\ &&\leq\delta T_{N}(\widetilde{c},\widetilde{c}+\beta\zeta n)=\beta\delta \zeta\left\Vert n\right\Vert <\beta\delta\zeta, \end{array} $$

hence there exists p ∈ coneP with ∥p∥ < 1 such that

$$\widetilde{d}+(\alpha\gamma-\sigma)\zeta v^{\prime}+\beta\delta\zeta p\in G(\widetilde{b}-\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta m,\widetilde {c}+\beta\zeta n). $$

Observe that

$$\begin{array}{@{}rcl@{}} \left\Vert \widetilde{d}+(\alpha\gamma-\sigma)\zeta v^{\prime}+\beta \delta\zeta p-\overline{w}\right\Vert & <& \left\Vert \widetilde {d}-w\right\Vert +\left\Vert w-\overline{w}\right\Vert +(\alpha\gamma -\sigma+\beta\delta)\zeta\\ & <&(\alpha\gamma+\beta\delta)\rho+(\alpha\gamma+\beta\delta)\rho +(\alpha\gamma+\beta\delta)\rho\ <\varepsilon, \end{array} $$

hence

$$\left( \widetilde{a}+\zeta\ell,\widetilde{b}-\gamma^{-1}(\alpha\gamma -2^{-1}\sigma)\zeta m,\widetilde{c}+\beta\zeta n,\widetilde{d}+(\alpha \gamma-\sigma)\zeta v^{\prime}+\beta\delta\zeta p\right) \in{\Omega} \cap\overline{A}, $$

and we can use the second relation in the Ekeland variational principle to find that

$$\begin{array}{@{}rcl@{}} \left\Vert v-\widetilde{d}\right\Vert & \leq& T_{P}(v,\widetilde{d} +(\alpha\gamma-\sigma)\zeta v^{\prime}+\beta\delta\zeta p)\\ & +&\tau c\max\left\{ \begin{array} [c]{c} T_{-L}(\widetilde{a}+\zeta\ell,\widetilde{a}),\alpha^{-1}T_{M}(\widetilde {b}-\gamma^{-1}(\alpha\gamma-2^{-1}\sigma)\zeta m,\widetilde{b}),\\ \beta^{-1}T_{-N}(\widetilde{c}+\beta\zeta n,\widetilde{c}),(\alpha\gamma +\beta\delta)^{-1}\left\Vert(\alpha\gamma-\sigma)\zeta v^{\prime}+\beta \delta\zeta p\right\Vert \end{array} \right\} . \end{array} $$

Remark that

$$\begin{array}{@{}rcl@{}} \widetilde{d}+(\alpha\gamma-\sigma)\zeta v^{\prime}+\beta\delta\zeta p & =&v+(\widetilde{d}-v)-(\alpha\gamma-\sigma)\zeta\frac{\widetilde{d} -v}{\left\Vert v-\widetilde{d}\right\Vert }+\beta\delta\zeta p\\ & =&v+\left( 1-\zeta\frac{\alpha\gamma-\sigma}{\left\Vert v-\widetilde {d}\right\Vert }\right) (\widetilde{d}-v)+\beta\delta\zeta p\\ & \in& v+\text{cone} P+\text{cone} P=v+\text{cone} P. \end{array} $$

Then the previous relation becomes

$$\begin{array}{@{}rcl@{}} \left\Vert v-\widetilde{d}\right\Vert & \leq&\left\Vert v-\left[ v+\left( 1-\zeta\frac{\alpha\gamma-\sigma}{\left\Vert v-\widetilde{d}\right\Vert }\right) (\widetilde{d}-v)+\beta\delta\zeta p\right] \right\Vert \\ &&+\tau c\max\{\zeta\left\Vert \ell\right\Vert ,\alpha^{-1}\gamma^{-1} (\alpha\gamma-2^{-1}\sigma)\zeta\left\Vert m\right\Vert ,\beta^{-1}\beta \zeta\left\Vert n\right\Vert ,\\ &&(\alpha\gamma+\beta\delta)^{-1}\zeta\left\Vert (\alpha\gamma-\sigma)v^{\prime}+\beta\delta p\right\Vert \}\\ & \leq&\left\Vert \left( 1-\zeta\frac{\alpha\gamma-\sigma}{\left\Vert v-\widetilde{d}\right\Vert }\right) (\widetilde{d}-v)\right\Vert +\beta \delta\zeta+\tau c\zeta\\ & =&\left\Vert v-\widetilde{d}\right\Vert -\zeta\left( \alpha\gamma -\sigma\right) +\beta\delta\zeta+\tau c\zeta. \end{array} $$

Using this and (A.9), we get

$$\tau c\zeta\geq\zeta\left( \alpha\gamma-\beta\delta-\sigma\right) =\zeta(c-\sigma)>\zeta\tau c, $$

a contradiction. This finishes the proof. □

Taking into account that the proof of the directional EVP is based on an iterative procedure, we can summarize the implications between the assertions in this work as follows:

$$\begin{array} [c]{ccccccc} \text{{directional EVP}} & {\Rightarrow} & \begin{array} [c]{c} \text{{criterion for directional}}\\ \text{{regularity of }}\\ \text{{single-valued maps}} \end{array} & {\Rightarrow} & \begin{array} [c]{c} \text{{criterion for directional}}\\ \text{{regularity of }}\\ \text{{set-valued maps}} \end{array} & {\Rightarrow} & \begin{array} [c]{c} \text{{directional regularity}}\\ \text{{of compositions}} \end{array} \\ & & {\Uparrow} & & & & {\Uparrow}\\ & & \text{{iterative procedure}} & & {\Rightarrow} & & \text{{directional EVP.}} \end{array} $$