Topological fixed point theory and applications to variational inequalities
 Abdul Latif^{1} and
 Hichem BenElMechaiekh^{2}Email author
https://doi.org/10.1186/s136630150329y
© Latif and BenElMechaiekh 2015
Received: 9 February 2015
Accepted: 17 May 2015
Published: 12 June 2015
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.
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 conesMSC
49J40 47H10 47N10 47H04 47H19 49J241 Introduction
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^{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.^{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.^{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,^{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.
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 ⇉.

Generalized quasivariational inequalities
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 \(\phi=p\).

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^{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 ^{8} variational inequality \(\exists\hat{u}\in X\), \(\langle\nabla f(\hat{u}),v\hat{u}\rangle>0\) implies that \(f(\hat{u})=\min_{X}f(v)\).

Multivalued complementarity problems

A 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 \(\operatorname{graph}(s)\subset W[\operatorname{graph}(\Phi)]\).
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^{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.^{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.^{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.^{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^{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 ^{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
 (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.
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.^{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
 (i)
P ωhomotopy dominates X, and
 (ii)
\({\mathcal{E}}(P)\) is well defined and nontrivial.
Proof
 (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 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
 (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 \(\Psi(u)\), \(\forall(u,y)\in X\times Y\).
Proof
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.
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
 (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])
 (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.
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
 (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.
 (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 \(\Psi (u)\).
Proof
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.
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
 (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
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
 (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
 (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 \(\Psi(u)\).
 (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 thatthen hypothesis (vi) is satisfied. We thus obtain a generalization of a result in [50].$$\lim_{\Vert u\Vert\rightarrow\infty,u\in\Psi(u)}\max_{y\in\Phi (u)}\bigl\langle \theta(u,y), \eta(v_{0},u)\bigr\rangle < 0, $$
 (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
 (i)
\(\langle\theta(x,y),\eta(x,x)\rangle\geq0\), \(\forall(x,y)\in \operatorname{graph}(\Phi)\).
 (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 setis ∞proximally connected in \(X_{\rho}\).$$M(u,y)=\Bigl\{ v\in\Psi_{\rho}(u);\bigl\langle \theta(u,y),\eta(v,u)\bigr\rangle =\inf_{w\in\Psi_{\rho}(u)}\bigl\langle \theta(u,y),\eta(w,u)\bigr\rangle \Bigr\} $$
 (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
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 \(\inf_{y\in\Phi (u)}\langle y,uv\rangle\geq\phi(v)\phi(u)\).
 (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 \(\phi(u+v)\leq\phi(u)\), \(\forall(u,v)\in X\times X\).
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 \(\{\frac{1}{t}(Xx^{\prime })\}_{t>0,x^{\prime}\in X}\).
\(T_{X}^{C}(x)\) 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 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., \(y_{0}\in N_{X}^{C}(x_{0})\).
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
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^{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
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 \(\Vert r(x)x\Vert =\operatorname{dist}(x,X)\), \(\forall x\in U\).
Notes
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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