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In this paper, we propose two inertial projection methods to solve mixed variational inequalities without assuming monotonicity on the cost operator. Consequently, we show that the sequence of iterates generated by our proposed methods converge globally to a solution of the mixed variational inequality under the condition that the set of solutions of the dual mixed variational inequality is nonempty. Our convergence results are obtained without assuming any further stringent condition imposed in other related results in the literature. Moreover, our results improve on many important related results in the literature. Some numerical illustrations are given to show the efficiency of our proposed methods.

In this paper, we employ an asymptotic assumption to show that the maximum of finitely many convex polynomials is an asymptotically well behaved function. We also give two examples to illustrate our main result. As an application, we reproduce some known results appeared recently in the literature.

In this paper, we first recall a class of parametric variational–hemivariational inequalities (PVHIs) introduced in Jiang et al. (2020). Then, based on the properties of the Clarke generalized gradient, we establish the Holder continuity of the solution mapping for PVHIs in terms of regularized gap functions under some assumptions imposed on the data of PVHIs. Finally, an example is given to illustrate our main results.

In the present paper the method of random integral guiding functions is applied to study the periodic problem for random differential inclusions with causal multioperators. At first we consider the case when the multioperator is closed and convex-valued. Then the case of a nonconvex-valued and lower semicontinuous right-hand part is considered. Thereafter the theory is extended to the case of nonsmooth guiding functions.

In this paper, we study the convergence of the Newton-type methods for solving the square inverse singular value problem with possible multiple and zero singular values. Comparing with other known results, positivity assumption of the given singular values is removed. Under the nonsingularity assumption in terms of the (relative) generalized Jacobian matrices, quadratic/superlinear convergence properties (in the root-convergence sense) are proved. Moreover, numerical experiments are given to demonstrate our theoretic results.

In this paper we consider a class of feedback control systems described by an evolution hemivariational inequality involving history-dependent operators. Under the mild conditions, first, we prove a priori estimates of the solutions to the feedback control system. Then, an existence theorem for the feedback control system is obtained by using the well-known Bohnenblust-Karlin fixed point theorem. Moreover, we consider an optimal control problem driven by the feedback control system, and establish its solvability. Finally, a parabolic partial differential system with Clarke subgradient term is considered to illustrate the applicability of the theoretical results.

Based on an approximation to Cayley transform, we propose an approximate Cayley transform method and its inexact version for solving inverse eigenvalue problems, which has the advantage over other known methods in the sense that it avoids solving systems in obtaining the approximate eigenvectors. Under the nonsingular condition used in D. Sun and J. Sun [29] , we show that the proposed methods converge at least superlinearly. Moreover, numerical experiments are given which illustrate that, comparing with the Cayley transform methods, our methods need much less inner iterations and CPU time.

We propose several approximate Gauss–Newton methods, i.e., the truncated, perturbed, and truncated-perturbed GN methods, for solving underdetermined nonlinear least squares problems. Under the assumption that the Frechet derivatives are Lipschitz continuous and of full row rank, Kantorovich-type convergence criteria of the truncated GN method are established and local convergence theorems are presented with the radii of convergence balls obtained. As consequences of the convergence results for the truncated GN method, convergence theorems of the perturbed and truncated-perturbed GN methods are also presented. Finally, numerical experiments are presented where the comparisons with the standard inexact Gauss–Newton method and the inexact trust-region method for bound-constrained least squares problems [23] are made.

This paper deals with the existence and regularity of solutions to an optimal control problem governed by semilinear elliptic equations with mixed pointwise constraints in which the controls act in the domain and on the boundary. We give some criteria under which the optimal solutions do exist and are Holder continuous.

We consider the convergence problem of some Newton-type methods for solving the inverse singular value problem with multiple and positive singular values. Under the nonsingularity assumption of the relative generalized Jacobian matrices at the solution c * , a convergence analysis for the multiple and positive case is provided and the superlinear or quadratical convergence properties are proved. Moreover, numerical experiments are given in the last section and comparisons are made.

The recovery properties of nonconvex regularized M -estimators are analysed, under the general sparsity assumption on the true parameter. In the statistical aspect, the recovery bound for any stationary point of the nonconvex regularized M -estimator is established under some regularity conditions. In the computational aspect, the proximal gradient method is used to solve the nonconvex optimization problem and is proved to achieve a linear convergence rate, by virtue of a slight decomposition of the objective function. In particular, for commonly-used regularizers such as SCAD and MCP, a simpler decomposition is applicable thanks to the assumption on the regularizer, which helps to construct the estimator with better recovery performance. In the aspect of application, theoretical consequences are obtained on the corrupted error-in-variables linear regression model by verifying the required conditions. Finally, statistical and computational results as well as advantages of the assumptions are demonstrated by several numerical experiments. Simulation results show remarkable consistency with the theory under high-dimensional scaling.

We establish the porosity results on the existence and the topological structure of fixed points for nonexpansive set-valued maps in hyperbolic spaces, which in particular extend and/or improve some known results in this direction.

In this paper, we introduce and analyze a new unified hybrid iterative method to compute the approximate solution of the general optimization problem defined over the set D=Fix(T)@[email protected][GMEP(@F,@J,@f)], where Fix(T) is the set of common fixed points of a family T={T(t):[email protected]?t

The purpose of this paper is to introduce and analyze an extragradient method with regularization for finding a common element of the solution set @C of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. Combining the regularization method and the extragradient method due to Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191-201], we propose an iterative algorithm for finding an element of Fix(S)@[email protected] We prove that the sequences generated by the proposed algorithm converge weakly to an element of Fix(S)@[email protected] under mild conditions.

In this paper, we consider the split feasibility problem (SFP) in infinite-dimensional Hilbert spaces, and study the relaxed extragradient methods for finding a common element of the solution set Γ of SFP and the set Fix ( S ) of fixed points of a nonexpansive mapping S . Combining Mann’s iterative method and Korpelevich’s extragradient method, we propose two iterative algorithms for finding an element of Fix ( S ) ∩ Γ . On one hand, for S = I , the identity mapping, we derive the strong convergence of one iterative algorithm to the minimum-norm solution of the SFP under appropriate conditions. On the other hand, we also derive the weak convergence of another iterative algorithm to an element of Fix ( S ) ∩ Γ under mild assumptions.

We explore in arbitrary Banach spaces the Frechet type e -subdifferentials and the limiting subdifferentials for the perturbed distance function d S J ( ⋅ ) determined by a closed subset S and a lower semicontinuous function J defined on S . In particular, upper and lower estimates for the Frechet type e -subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions J and δ S .

The present paper is divided into two parts. In the first part, we introduce implicit and explicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. We establish results on the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping. Such a fixed point is also a solution of a variational inequality defined on the set of fixed points. In the second part, we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem. Such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping. The results of this paper extend and improve several results presented in the literature in the recent past.

In this paper, we suggest and analyze some new relaxed extragradient iterative methods for finding a common element of the solution set of a variational inequality, the solution set of a general system of variational inequalities and the set of fixed points of a strictly pseudo-contractive mapping defined on a real Hilbert space. Strong convergence of the proposed methods under some mild conditions is established.

A set-constrained optimization problem and a mathematical programming problem are considered. We assume that the sublevel sets of the involving functions are convex only at the point under question and hence these functions are not assumed quasiconvex. Using the two star subdifferentials and the adjusted subdifferential, we establish optimality conditions for usual minima and strict minima. Our results contain and improve some recent ones in the literature. Examples are provided to explain the advantages of each of our results.

Assume that X is a real Banach space with uniformly normal structure and C is a nonempty closed convex subset of X . We show that a κ -uniformly Lipschitzian semigroup of nonlinear self-mappings of C admits a common fixed point if the semigroup has a bounded orbit and if κ is appropriately greater than one. This result applies, in particular, to the framework of uniformly convex Banach spaces.

Very recently, Yao, Chen and Yao [20] proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition β n → 0 is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space.

We consider a semilinear functional differential inclusion with infinite delay and impulse characteristics in a Banach space assuming that its linear part is a non-densely defined Hille–Yosida operator. We assume that the multivalued nonlinearity of upper Caratheodory or almost lower semicontinuous type satisfies a regularity condition expressed in terms of the measures of noncompactness. We apply the theory of integrated semigroups and the theory of condensing multivalued maps to obtain local and global existence results. The application to an optimization problem for an impulsive feedback control system is given.

In this paper, we consider and study a general system of generalized nonlinear mixed composite-type equilibria in Hilbert spaces. First, we prove the existence and uniqueness of the solution for this system of generalized nonlinear mixed composite-type equilibria. Second, the Mann iterative method with errors is extended to develop some new iterative algorithms for finding approximate solutions for this system of generalized nonlinear mixed composite-type equilibria. We also derive the strong convergence of the sequences generated by these iterative algorithms in Hilbert spaces.

Motivated by the subsmoothness of a closed set introduced by Aussel et al. (2005) [8] , we introduce and study the uniform subsmoothness of a collection of infinitely many closed subsets in a Banach space. Under the uniform subsmoothness assumption, we provide an interesting subdifferential formula on distance functions and consider uniform metric regularity for a kind of multifunctions frequently appearing in optimization and variational analysis. Different from the existing works, without the restriction of convexity, we consider several fundamental notions in optimization such as the linear regularity, CHIP, strong CHIP and property (G) for a collection of infinitely many closed sets. We establish relationships among these fundamental notions for an arbitrary collection of uniformly subsmooth closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of closed convex sets to the nonconvex setting.

We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator ΠCg and the relative projection operator PCg in terms of the D-approximate (weak) compactness for a nonempty closed set C in a Banach space X. We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a Banach space X. Our results extend the corresponding results of [H.H. Bauschke, X.F. Wang, J. Ye and X. M. Yuan, Bregman distances and Chebyshev sets, J. Approx. Theory 159 (2009) 3–25] to infinite dimensional spaces.

By constructing some suitable examples, Jeyakumar and Yen (2004) [1] have shown that the Robinson metric regularity (Rmr) and the Lipschitz-like property (Llp) of implicit multifunctions are not equivalent. This paper clarifies relationships between the two properties of implicit multifunctions. It turns out that the (reasonable) sufficient conditions for having (Rmr) ⇒ (Llp) are quite different from those for the validity of the reverse implication. The implicit function theorem due to Yen and Yao (2009) [2] serves as a tool for our analysis of (Rmr) and (Llp).

In this paper, we introduce a new and interesting system of generalized mixed quasi-variational-like inclusions with (A,@h,m)-accretive operators and relaxed cocoercive mappings which contains variational inequalities, variational inclusions, systems of variational inequalities, systems of variational-like inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (A,@h,m)-accretive operators, we prove the existence of solutions and the convergence of a new p-step iterative algorithm for this system of generalized mixed quasi-variational-like inclusions in real q-uniformly smooth Banach spaces. The results in this paper unifies, extends and improves some known results in the literature.

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a mixed variational inequality problem in a Banach space. We establish some metric characterizations of the well-posedness by perturbations. We also show that under suitable conditions, the well-posedness by perturbations of a mixed variational inequality problem is equivalent to the well-posedness by perturbations of a corresponding inclusion problem and a corresponding fixed point problem. Also, we derive some conditions under which the well-posedness by perturbations of a mixed variational inequality is equivalent to the existence and uniqueness of its solution.

In this paper, we introduce and study a general iterative method with strongly positive operators for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. The explicit and implicit iterative algorithms are proposed by virtue of the general iterative method with strongly positive operators. Under two sets of quite mild conditions, we prove the strong convergence of these explicit and implicit iterative algorithms to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively.

This paper is devoted to the study of the pseudo-Lipschitz property of Pareto solution map for the parametric linear semi-infinite vector optimization problem (LSVO). We establish new sufficient conditions for the pseudo-Lipschitz property of the Pareto solution map of (LSVO) under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. Examples are given to illustrate the results obtained.

This paper is devoted to the study of the pseudo-Lipschitz property of the efficient (Pareto) solution map for the perturbed convex semi-infinite vector optimization problem (CSVO). We establish sufficient conditions for the pseudo-Lipschitz property of the efficient solution map of (CSVO) under continuous perturbations of the right-hand side of the constraints and functional perturbations of the objective function. Examples are given to illustrate the obtained results.

In this paper, we introduce a new viscosity approximation scheme based on the extragradient method for finding a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions to the variational inequality for a monotone, Lipschitz continuous mapping. Several convergence results for the sequences generated by these processes in Hilbert spaces were derived.

We establish the existence and uniqueness results for variational inequality problems on Riemannian manifolds and solve completely the open problem proposed in [S.Z. Nemeth, Variational inequalities on Hadamard manifolds, Nonlinear Anal. 52 (2003) 1491–1498]. Also the relationships between the constrained optimization problem and the variational inequality problems as well as the projections on Riemannian manifolds are studied.

Let E be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, D a nonempty closed convex subset of E , and T : D → K ( E ) a nonself multimap such that F ( T ) ≠ 0 and P T is nonexpansive, where F ( T ) is the fixed point set of T , K ( E ) is the family of nonempty compact subsets of E and P T ( x ) = { u x ∈ T x : ‖ x − u x ‖ = d ( x , T x ) } . Suppose that D is a nonexpansive retract of E and that for each v ∈ D and t ∈ ( 0 , 1 ) , the contraction S t defined by S t x = t P T x + ( 1 − t ) v has a fixed point x t ∈ D . Let { α n } , { β n } and { γ n } be three real sequences in ( 0 , 1 ) satisfying approximate conditions. Then for fixed u ∈ D and arbitrary x 0 ∈ D , the sequence { x n } generated by x n ∈ α n u + β n x n − 1 + γ n P T ( x n ) , ∀ n ≥ 0 , converges strongly to a fixed point of T .

The purpose of this paper is to study the asymptotic behavior of solutions of the first initial boundary value problem for hyperbolic systems in an infinite cylinder with the base containing conical points. Some results on the asymptotics of the solutions of an elliptic problem depending on a parameter are given.

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems in several topics of applied sciences [see for example: El Naschie MS. Iterated function systems and the two-slit experiment of quantum mechanics. Chaos, Solitons & Fractals 1994;4:1965–8; Iovane G. Cantorian spacetime and Hilbert space: Part I-Foundations. Chaos, Solitons & Fractals 2006;28:857–78; Iovane G. Cantorian space-time and Hilbert space: Part II-Relevant consequences. Chaos, Solitons & Fractals 2006;29:1–22; Fedeli A. On chaotic set-valued discrete dynamical systems. Chaos, Solitons & Fractals 2005;23:13814; Shi Y, Chen G. Chaos of discrete dynamical systems in complete metric spaces. Chaos, Solitons & Fractals 2004;22:55571]. The purpose of this paper is twofold. First, some existence and uniqueness results for the self-similar sets of a mixed iterated function systems are given. Then, using the concept of well-posed fixed point problem, the well-posedness of the self-similarity problem for some classes of iterated multifunction systems is also studied. Well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals.

In this paper, we suggest and analyze a relaxed viscosity iterative method for a commutative family of nonexpansive self-mappings defined on a nonempty closed convex subset of a reflexive Banach space. We prove that the sequence of approximate solutions generated by the proposed iterative algorithm converges strongly to a solution of a variational inequality. Our relaxed viscosity iterative method is an extension and variant form of the original viscosity iterative method. The results of this paper can be viewed as an improvement and generalization of the previously known results that have appeared in the literature.

We define a generalized KKM mapping, called T -KKM mapping, and the corresponding generalized KKM property, which include many counterparts existing in the literature. KKM-type theorems, coincidence theorems and geometric section theorems are established to generalize recent known results.

In this paper, we study some non-traditional schemes of proximal point algorithm for nonsmooth convex functionals in a Banach space. The proximal approximations to their minimal points and/or their minimal values are considered separately for unconstrained and constrained minimization problems on convex closed sets. For the latter we use proximal point algorithms with the metric projection operators and first establish the estimates of the convergence rate with respect to functionals. We also investigate the perturbed projection proximal point algorithms and prove their stability. Some results concerning the classical proximal point method for minimization problems in a Banach space is also presented in this paper.

Let X and Y be real Banach spaces, K be a nonempty convex subset of X, and C:K->2^Y be a multifunction such that for each [email protected]?K, C(u) is a proper, closed and convex cone with intC(u) [email protected]?, where intC(u) denotes the interior of C(u). Given the mappings T:K->2^L^(^X^,^Y^), A:L(X,Y)->L(X,Y), f"1:L(X,Y)xKxK->Y, f"2:KxK->Y, and g:K->K, we introduce and consider the generalized implicit vector equilibrium problem: Find u^*@?K such that for any [email protected]?K, there is s^*@?Tu^* satisfying f"1(As^*,v,g(u^*))+f"2(v,g(u^*))@?-intC(u^*). By using the KKM technique and the well-known Nadler's result, we prove some existence theorems of solutions for this class of generalized implicit vector equilibrium problems. Our theorems extend and improve the corresponding results of several authors.

It is proved that the modified Mann iteration process: x n + 1 = ( 1 − α n ) x n + α n T n x n , n ∈ N , where { α n } is a sequence in (0, 1) with δ ≤ α n ≤ 1 − κ − δ for some δ ∈ ( 0 , 1 ) , converges weakly to a fixed point of an asymptotically κ -strict pseudocontractive mapping T in the intermediate sense which is not necessarily Lipschitzian. We also develop CQ method for this modified Mann iteration process which generates a strongly convergent sequence.

In this paper, a new iterative scheme based on the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Applications to optimization problems are given.

We obtain some point-based sufficient conditions for the metric regularity in Robinson’s sense of implicit multifunctions in a finite-dimensional setting. The new implicit function theorem (which is very different from the preceding results of Ledyaev and Zhu [Yu.S. Ledyaev, Q.J. Zhu, Implicit multifunctions theorems, Set-Valued Anal. 7 (1999) 209–238], Ngai and Thera [H.V. Ngai, M. Thera, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal. 12 (2004) 195–223], Lee, Tam and Yen [G.M. Lee, N.N. Tam, N.D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008) 11–22]) can be used for analyzing parametric constraint systems as well as parametric variational systems. Our main tools are the concept of normal coderivative due to Mordukhovich and the corresponding theory of generalized differentiation.

We introduce a new composite iterative scheme to approximate a zero of an m -accretive operator A defined on uniform smooth Banach spaces and a reflexive Banach space having a weakly continuous duality map. It is shown that the iterative process in each case converges strongly to a zero of A . The results presented in this paper substantially improve and extend the results due to Ceng et al. [L.C. Ceng, H.K. Xu, J.C. Yao, Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators, Taiwanese J. Math. (in press)], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643]. Our work provides a new approach for the construction of a zero of m -accretive operators.

Let X be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. First purpose of this paper is to introduce a modified viscosity iterative process with perturbation for a continuous pseudocontractive self-mapping T and prove that this iterative process converges strongly to x∗ ∈ F(T) ≔ {x ∈ X∣x = T(x)}, where x∗ is the unique solution in F(T) to the following variational inequality: 〈 f ( x ∗ ) - x ∗ , j ( v - x ∗ ) 〉 ⩽ 0 for all v ∈ F ( T ) . Second aim of the paper is to propose two modified implicit iterative schemes with perturbation for a continuous pseudocontractive self-mapping T and prove that these iterative schemes strongly converge to the same point x∗ ∈ F(T). Basically, we show that if the perturbation mapping is nonexpansive, then the convergence property of the iterative process holds. In this respect, the results presented here extend, improve and unify some very recent theorems in the literature, see [L.C. Zeng, J.C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal. 64 (2006) 2507–2515; H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291; Y.S. Song, R.D. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. (2006)].

Let E be a reflexive Banach space with the dual space E ∗ and K be a nonempty closed convex subset of E . Let us have Ψ : K × K × E ∗ → R and A : E ∗ → E ∗ . We introduce the class of generalized α -monotone multifunctions T : K → 2 E ∗ with respect to Ψ and A where α : E × E → R . By using the KKM technique and the concept of the Hausdorff metric, we establish some existence results for generalized variational-like inequalities with generalized monotone multivalued mappings in E .

In this paper, a degree theory for finite dimensional generalized variational inequalities is built and employed to prove some results on solution existence and solution stability.

In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E ∗ , a multifunction T : K → 2 E ∗ , and a vector b ∈ E ∗ , find x ∈ K such that there exists u ∈ T ( x ) satisfying 〈 u − b , y − x 〉 ≥ 0 , for all y ∈ K . By using generalized projection and the well-known Fan–KKM Theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.