Search

Degree theory has been developed as a tool for checking the solution existence of nonlinear equations. Hu and Parageorgiou [S.C. Hu, N.S. Parageorgiou, Generalisation of Browders degree theory, Trans. Amer. Math. Soc. 347 (1995), pp. 233–259] generalized the results of Browder [F.E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. 9 (1983), pp. 1–39] on the degree theory to mappings of the form f + T + G, where f is a bounded and demicotinuous mapping of class (S)+ from a bounded open set in a reflexive Banach space X into its dual X*, T is a maximal monotone mapping with 0 ∈ T(0) from X into X*, and G is an u.s.c. compact set-valued mapping from X into X*. In this article we continue to generalize and extend the results of Browder on the degree theory to mappings of the form f + T + G. By enlarging the class of maximal monotone mappings and pseudo-monotone homotopies we obtain some new results of the degree theory for such mappings. As an application, an existence result of solu...

In this paper, we consider stochastic affine variational inequality problems with nonlinear perturbation (for short, SVIPP). Firstly, we formulate SVIPP as an ERM problem that minimizes the expected residual of the so-called gap function. Furthermore, we study some properties of the ERM problem under some suitable conditions. By means of quasi-Monte Carlo method, we obtain a discrete approximation of ERM problem. Finally, we consider the convergence of optimal solutions and stationary points of the approximation problem as sample size increases.

In this paper, some characterizations for the solution sets of a class of set-valued vector mixed variational inequalities to be nonempty and bounded are presented in real reflexive Banach spaces. An equivalence relation between the solution sets of the vector mixed variational inequalities and the weakly efficient solution sets of the vector optimization problems is shown under some suitable assumptions. By using some known results for the vector optimization problems, several characterizations for the solution sets of the vector mixed variational inequalities are obtained in real reflexive Banach spaces. Furthermore, some stability results for the vector mixed variational inequality are given when the mapping and the constraint set are perturbed by two different parameters. Finally, the upper semicontinuity and the lower semicontinuity of the solution sets are given under some suitable assumptions which are different from the ones used in [7, 11, 22]. Some examples are also given to illustrate our results.

In this paper, we consider CVaR-based formulation and approximation method pro- posed by Chen and Lin (5) for a class of stochastic variational inequality problems (for short, SVIP). Different from the work mentioned above, we regard the regularized gap function for SVIP as a loss function for SVIPs and obtain a restrained deterministic minimization reformula- tion for SVIPs. We show that the reformulation is a convex program for a wider class of SVIPs than that in (5). Furthermore, by using the smoothing techniques and Monte Carlo method, we get an approximation problem of the minimization reformulation and consider the convergence of optimal solutions and stationary points of the approximation problems. Finally we apply our proposed model to solve the migration equilibrium problem under uncertainty.

In this paper, a coincidence theorem for a compact RC-mapis proved in an abstract convex space. Several more general coincidencetheorems for noncompact RC-maps are derived in abstract convex spaces.Some examples are given to illustrate our coincidence theorems. As ap-plications, an alternative theorem concerning the existence of maximalelements, an alternative theorem concerning equilibrium problems anda minimax inequality for three functions are proved in abstract convexspaces. 1. IntroductionMany problems in nonlinear analysis can be solved by showing the intersec-tion of certain family of subsets of an underlying set is nonempty. The firstremarkable result on the nonempty intersection was the celebrated Knaster-Kuratowski-Mazurkiewicz theorem (simply, the KKM principle) in 1929 [11],which concerns with certain types of maps called KKM maps later.At the beginning, the KKM theory was mainly devoted to the study ofconvex subsets of topological vector spaces. Later it has been extended toconvex spaces by Lassonde [12], to C-spaces (or H-spaces) by Horvath [8, 9],and to generalized convex (G-convex) spaces by Park and Kim [24] and Park[21, 22, 23]. Recently, Park [16] introduced a new concept of abstract convexspaces which include convex subsets of topological vector spaces, convex spaces,C-spacesand G-convexspacesasspecialcases. Park[16] alsointroducedcertainbroad classes ROand RCof maps (having the KKM property), which includesthe well-known classKKM(X,Y) introduced by Chang and Yen [5] as a specialcase. With these new concepts, some coincidence theorems and fixed point

In this paper, a relationship between a vector variational inequality and a vector optimization problem is given on a Hadamard manifold. An existence of a weak minimum for a constrained vector optimization problem is established by an analogous to KKM lemma on a Hadamard manifold.

In this paper, characterizations of the set of solutions for VVI are presented by using scalarization approaches. The set of solutions of VVI is shown to be the set of weak sharpness for gap functions of some scalarization of VVI and for gap functions of VVI under semi-strong monotonicity. Some examples are given to illustrate these results.

In this paper, firstly a modified particle swarm optimization algorithm (MPSO) is developed, in which the mean value of past optimal positions for each particle and the mutation operation are considered for avoiding premature. In the optimization test, MPSO performs better than particle swarm optimization algorithm (PSO). Then MPSO is applied to solve four portfolio optimization models with the real data from the Hong Kong Stock Market, and optimal values are obtained when the number of swarm n = 80 , 160 , respectively. Finally, actual return rates of these models are calculated in numerical experiments, and it is illustrated from these graphs of actual return rates that when considering higher return, Cai’s model performs better in short-term investment.

In this paper, well-posedness of generalized quasi-variational inclusion problems and of optimization problems with generalized quasi-variational inclusion problems as constraints is introduced and studied. Some metric characterizations of well-posedness for generalized quasi-variational inclusion problems and for optimization problems with generalized quasi-variational inclusion problems as constraints are given. The equivalence between the well-posedness of generalized quasi-variational inclusion problems and the existence of solutions of generalized quasi-variational inclusion problems is given under suitable conditions.

In this paper, a differential vector variational inequality is introduced and studied in finite-dimensional Euclidean spaces. The existence of a Caratheodory weak solution for the differential vector variational inequality is presented under some suitable conditions. Furthermore, the upper semicontinuity and the lower semicontinuity of the solution sets for the differential variational inequality are established when both the mapping and the constraint set are perturbed by two different parameters.

In this paper, combining the new splitting method proposed by Eckstein and Svaiter and the forcing strong convergence method of Solodov and Svaiter, we propose a splitting proximal projection method for the sum of two maximal monotone operators. We prove that the proposed method is well-defined whether the solution set of the problem is nonempty or not and the sequence generated by the method converges strongly to an extended solution of the problem.

In this paper, we establish the equivalence between the half-space representation and the vertex representation of a smooth parametric semiclosed polyhedron. By virtue of the smooth representation result, we prove that the solution set of a smooth parametric piecewise linear program can be locally represented as a finite union of parametric semiclosed polyhedra generated by finite smooth functions. As consequences, we prove that the corresponding marginal function is differentiable and the solution map admits a differentiable selection.

We study the strong convergence of two viscosity iteration processes for pseudo-contractive semigroup and for ϕ -strongly pseudo-contractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. As special cases, we get strong convergence of two viscosity iteration processes for approximating common fixed points of nonexpansive semigroups in certain Banach spaces. The results presented in this paper extend and generalize previous results.

In this paper, we study the mean square asymptotic stability of a class of generalized nonlinear neutral stochastic differential equations with variable time delays by using fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved which improves and generalizes some well-known results. Finally, two examples are given to illustrate our results.

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511---520, 2010), Chen and Li (Pac J Optim 6:141---152, 2010), Chen et al. (J Glob Optim 45:309---318, 2009), Cheng and Zhu (J Glob Optim 32:543---550, 2005), Gong (J Optim Theory Appl 139:35---46, 2008), Li and Fang (J Optim Theory Appl 147:507---515, 2010), Li et al. (Bull Aust Math Soc 81:85---95, 2010) and Peng et al. (J Optim Theory Appl 152(1):256---264, 2011).

In this article, we propose an easily implementable algorithm method for solving mixed variational inequalities. The proposed method combines two strategies: inexact projected subgradient methods and regularization techniques. We prove that the sequence generated by the method is strongly convergent to a solution of the problem in Hilbert spaces, whenever it exists. The results presented in this article generalize and improve some recent results.

This paper is devoted to the existence of solutions for the variational-hemivariational inequalities in reflexive Banach spaces. Using the notion of the stable $${\phi}$$ -quasimonotonicity and the properties of Clarke's generalized directional derivative and Clarke's generalized gradient, some existence results of solutions are proved when the constrained set is nonempty, bounded (or unbounded), closed and convex. Moreover, a sufficient condition to the boundedness of the solution set and a necessary and sufficient condition to the existence of solutions are also derived. The results presented in this paper generalize and improve some known results.

In this paper, we investigate the proximal point algorithm (in short PPA) for variational inequalities with pseudomonotone vector fields on Hadamard manifolds. Under weaker assumptions than monotonicity, we show that the sequence generated by PPA is well defined and prove that the sequence converges to a solution of variational inequality, whenever it exists. The results presented in this paper generalize and improve some corresponding known results given in literatures.

In this paper, we study the solution stability for a class of parametric generalized vector quasiequilibrium problems. By virtue of the parametric gap function, we obtain a sufficient and necessary condition for the Hausdorff lower semicontinuity of the solution mapping to the parametric generalized vector quasiequilibrium problem. The results presented in this paper generalize and improve some main results of Chen et al. (2010) [34], and Zhong and Huang (2011) [35].

In this paper, a generalized vector equilibrium problem is introduced and studied. A scalar characterization of weak efficient solutions for the generalized vector equilibrium problem is obtained. By using the scalarization result, the existence of the weak efficient solutions and the connectedness of the set of weak efficient solutions for the generalized vector equilibrium problem are proved in locally convex spaces.

This article introduces a new concept of an exceptional family of elements for a generalized set-valued variational inequality in Banach spaces. By using this concept and the degree theory for the generalized set-valued variational inequality introduced by Wang and Huang [Zh.B. Wang and N.J. Huang, Degree theory for a generalized set-valued variational inequality with an application in Banach spaces, J. Glob. Optim. 49 (2011), pp. 343–357], some solvability results for the generalized set-valued variational inequality and its special cases are given in Banach spaces under suitable conditions.

A projected subgradient method for solving a class of set-valued mixed variational inequalities (SMVIs) is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.

The concept of pseudomonotone vector field on Hadamard manifold is introduced. A variant of Korpelevich's method for solving the variational inequality problem is extended from Euclidean spaces to constant curvature Hadamard manifolds. Under a pseudomonotone assumption on the underlying vector field, we prove that the sequence generated by the method converges to a solution of variational inequality, whenever it exists. Moreover, we give an example to show the effectiveness of our method.

In this paper, we study the economic model proposed by Anderlini and Canning, a parameterized class of generalized abstract fuzzy economies together with an associated abstract rationality function, and show that the structural stability of this model implies its robustness to @e-equilibria.

In this paper, we consider an extension of well-posedness for a minimization problem to a class of variational–hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed variational–hemivariational inequality and give some conditions under which the variational–hemivariational inequality is strongly well-posed in the generalized sense. Under some mild conditions, we also prove the equivalence between the well-posedness of variational–hemivariational inequality and the well-posedness of corresponding inclusion problem.

In this paper, a concept of graph convergence concerned with the H (·, ·)-accretive operator is introduced in Banach spaces and some equivalence theorems between of graph-convergence and resolvent operator convergence for the H (·, ·)-accretive operator sequence are proved. As an application, a perturbed algorithm for solving a class of variational inclusions involving the H (·, ·)-accretive operator is constructed. Under some suitable conditions, the existence of the solution for the variational inclusions and the convergence of iterative sequence generated by the perturbed algorithm are also given.

In this paper, a projection-proximal point method for solving a class of generalized variational inequalities is considered in Hilbert spaces. We investigate a general iterative algorithm, which consists of an inexact proximal point step followed by a suitable orthogonal projection onto a hyperplane. We prove the convergence of the algorithm for a pseudomonotone mapping with weakly upper semicontinuity and weakly compact and convex values. We also analyze the convergence rate of the iterative sequence under some suitable conditions.

In this paper, a class of generalized implicit inclusion problems is introduced, which can be regarded as a generalization of variational inequality problems, equilibrium problems, optimization problems and inclusion problems. Some existence results of solutions for such problems are obtained on noncompact subsets of Hausdorff topological vector spaces using the famous FKKM theorem. As applications, some existence results for vector equilibrium problems and vector variational inequalities on noncompact sets of Hausdorff topological vector spaces are given.

In this paper, a key assumption similar to that of Li and Chen is introduced by virtue of a gap function for a class of parametric set-valued weak vector variational inequalities in Banach spaces. By using this key assumption, sufficient and necessary conditions of the continuity and Hausdorff continuity of the solution set mapping for such parametric set-valued weak vector variational inequalities are given in Banach spaces when the image space is infinite dimensional. The results presented in this paper generalize and improve some main results of Li and Chen.

The purpose of this paper is to investigate the nonemptiness and boundedness of solution set for a generalized mixed variational inequality with strict feasibility in reflexive Banach spaces. A concept of strict feasibility for the generalized mixed variational inequality is introduced, which recovers the existing concepts of strict feasibility for variational inequalities and complementarity problems. By using the equivalence characterization of nonemptiness and boundedness of the solution set for the generalized mixed variational inequality due to Zhong and Huang (J. Optim. Theory Appl. 147:454–472, 2010), it is proved that the generalized mixed variational inequality problem has a nonempty bounded solution set is equivalent to its strict feasibility.

In this article, constrained continuous and combinatorial vector optimization problems (VOPs) are considered in the setting of finite-dimensional Euclidean spaces. Equivalence results between constrained integer and continuous VOPs are established by virtue of that between a constrained VOP and its penalized problem. Finally, one of the established equivalences is applied to derive necessary optimality conditions for a constrained integer VOP.

In this paper, some properties of the generalized f-projection operator are proved in Banach spaces. Using these results, the strong convergence theorems for relatively nonexpansive mappings are studied in Banach spaces. As applications, the strong convergence of general H-monotone mappings in Banach spaces is also given. The results presented in this paper generalize and improve the main results of Matsushita and Takahashi (2005) [9].

In this paper, the notion of weak sharp minima is employed to the investigation of set-valued vector variational inequalities. The gap function φ T for set-valued strong vector variational inequalities (for short, SVVI) is proved to be less than the gap function ϕ T for set-valued weak vector variational inequalities (for short, WVVI) under certain conditions, which implies that the solution set of SVVI is equivalent to the solution set of WVVI. Moreover, it is shown that weak sharp minima for the solution sets of SVVI and WVVI hold for min 1 ⩽ i ⩽ n p T i and for gap functions φ T and ϕ T under the assumption of strong pseudomonotonicity, where p T i is a gap function for i-th component of SVVI and WVVI. As an application, the weak Pareto solution set of vector optimization problems (for short, VOP) is proved to be weak sharp minimum for min 1 ⩽ i ⩽ n p ∇ g i when each component g i of objective function is strongly convex.

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566–2574, 2008) and He (J. Math. Anal. Appl. 330:352–363, 2007).

In this paper, by using the semi-order method, two new exis- tence theorems of coupled quasi-solutions for a class of nonlinear operator equations in Banach spaces are proved under some suitable conditions.

In this paper, we introduce and study a class of differential mixed variational inequalities in finite dimensional Euclidean spaces. Under various conditions, we obtain linear growth and bounded linear growth of the solution set for the mixed variational inequalities. Moreover, we present some conclusions which enrich the literature on the mixed variational inequalities and generalize the corresponding results of [4] . In particular we prove existence theorems for weak solutions of a differential mixed variational inequality in the weak sense of Caratheodory by using a result on differential inclusions involving an upper semicontinuous set-valued map with closed convex values. Also by employing the results from differential inclusions we establish a convergence result on Euler time-dependent procedure for solving initial-value differential mixed variational inequalities.

In this paper, vector variational inequalities (VVI) with matrix inequality constraints are investigated by using the image space analysis. Linear separation for VVI with matrix inequality constraints is characterized by using the saddle-point conditions of the Lagrangian function. Lagrangian-type necessary and sufficient optimality conditions for VVI with matrix inequality constraints are derived by utilizing the separation theorem. Gap functions for VVI with matrix inequality constraints and weak sharp minimum property for the solutions set of VVI with matrix inequality constraints are also considered. The results obtained above are applied to investigate the Lagrangian-type necessary and sufficient optimality conditions for vector linear semidefinite programming problems as well as VVI with convex quadratic inequality constraints.

In this paper, a kind of Ishikawa type iterative scheme with errors for approximating a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings is introduced and studied in convex metric spaces. Under some suitable conditions, the convergence theorems concerned with the Ishikawa type iterative scheme with errors to approximate a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings were proved in convex metric spaces. The results presented in the paper generalize and improve some recent results of Wang and Liu (C. Wang, L.W. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Anal., TMA 70 (2009), 2067-2071).

In this paper, we introduce and study a new class of (H,@f)[email protected] operators in Banach spaces. We define a proximal mapping associated with the (H,@f)[email protected] operator and show its Lipschitz continuity. As an application, we consider the solvability for a class of variational inclusions involving the (H,@f)[email protected] operators in Banach spaces. By using the technique of proximal mapping, an iterative algorithm for solving such class of variational inclusions is constructed in Banach spaces. Under some suitable conditions, we prove the convergence of iterative sequence generated by the algorithm.

In this paper, a new notion of a generalized H-η-accretive operator is introduced and studied, which provides a unifying framework for the generalized m-accretive operator and the H-η-monotone operator in Banach spaces. A resolvent operator associated with the generalized H-η-accretive operator is defined, and its Lipschitz continuity is shown. As an application, the solvability for a class of variational inclusions involving the generalized H-η-accretive operators in Banach spaces is considered. By using the technique of the resolvent mapping, an iterative algorithm for solving the variational inclusion in Banach spaces is constructed. Under some suitable conditions, it is proven that the solution for the variational inclusion and the convergence of the iterative sequence generated by the algorithm exist.

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 new class of variational inclusions in Banach spaces. For solving such class of variational inclusions, we introduce a new notion of B-monotone operator and prove the Lipschitz continuity of the proximal mapping associated with the B-monotone operator. By using the proximal mapping, an iterative algorithm for solving such class of variational inclusions is constructed in Banach spaces. Under some suitable conditions, we prove the convergence of iterative sequence generated by the algorithm.

In this paper, we study some new stochastic inequalities involving the Itntegral and give some estimates for the solutions of controlled stochastic differential equations. As applications, we utilize the stochastic integral inequalities presented in this paper to show an existence theorem of the solution for a class of stochastic differential equations and to give necessary conditions that make the solution for a class of stochastic differential equations be a diffusion process.

(Communicated by J. Pec) Abstract. In this paper, we consider a general composed convex optimization problem with in- equality systems involving a finite number of convex constraints. We establish the strong duality between the primal problem and the Fenchel-Lagrange dual problem by a conjugate duality ap- proach. Moreover, we obtain some new Farkas-type results for this problem by using weak and strong duality theorems. Our results contain some recent results as special cases.

In this paper, we study the Levitin-Polyak type well-posedness of variational inequalities and optimization problems with variational inequality constraints in Banach spaces. We derive some criteria and characterizations for these Levitin-Polyak well-posedness. We also investigate conditions under which the existence and uniqueness of solution is equivalent to the Levitin-Polyak well-posedness of the problem.

In this paper, we further study some properties of the generalized f -projection operator introduced by Wu and Huang. By employing the properties of the generalized f -projection operator and the well-known KKM and Kakutani–Fan–Glicksberg theorems, we establish some new existence theorems of solutions for two classes of generalized set-valued variational and quasi-variational inequalities in reflexive and smooth Banach spaces under some suitable conditions.

In this paper, we introduce a total step method for solving a system of linear complementarity problems with perturbations and interval data. It is applied to two interval matrices [A] and [B] and two interval vectors [b] and [c]. We prove that the sequence generated by the total step method converges to ([x^*],[y^*]) which includes the solution set for the system of linear complementarity problems defined by any fixed [email protected]?[A],[email protected]?[B],[email protected]?[b] and [email protected]?[c]. We also consider a modification of the method and show that, if we start with two interval vectors containing the limits, then the iterates contain the limits. We close our paper with two examples which illustrate our theoretical results.

In this paper, we study the mean square asymptotic stability of a generalized half-linear neutral stochastic differential equation with variable delays applying fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton, Zhang and Luo. Two examples are given to illustrate our results.

In this paper, we establish some new nonempty intersection theorems for generalized L-KKM mappings and prove some new fixed point theorems for set-valued mappings under suitable conditions in topological spaces. As applications, an existence theorem for an equilibrium problem with lower and upper bounds and two existence theorems for a quasi-equilibrium problem with lower and upper bounds are obtained in topological spaces. Our results generalize some known results in the literature.

In this paper, we extend the extremality results for the variational inequality to a higher order evolution hemivariational inequality. More precisely, we give an existence theorem of solution for the higher order evolution hemivariational inequality by using the sub–super-solution method. We prove the compactness of the solution set within an order interval formed by the sub-solution and super-solution. We also show an existence theorem of the extremal solution for the higher order evolution hemivariational inequality under consideration.