Kaleem Raza Kazmi

An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping

In this paper, we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping in real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping which is the unique solution of the variational inequality problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area.

Existence and iterative approximation of solutions of generalized mixed equilibrium problems

In this paper, we consider a generalized mixed equilibrium problem involving non-monotone set-valued mappings in real Hilbert space. We extend the notions of the Yosida approximation and its corresponding regularized operator given by Moudafi and Thera [A. Moudafi, M. Thera, Proximal and dynamical approaches to equilibrium problems, in: Lecture Notes in Econom. and Math. System, vol. 477, Springer-Verlag, Berlin, 2002, pp. 187-201] and discuss some of their properties. Further, we consider a generalized Wiener-Hopf equation problem and show its equivalence with the generalized mixed equilibrium problem. Using a fixed point formulation of the generalized Wiener-Hopf equation problem, we construct an iterative algorithm. Furthermore, we extend the notion of stability given by Harder and Hick [A.M. Harder, T.L. Hicks, Stability results for fixed-point iteration procedures, Math. Japonica 33 (5) (1998) 693-706]. We prove the existence of a solution and discuss the convergence and stability of the iterative algorithm for the generalized Wiener-Hopf equation problem. Since the generalized mixed equilibrium problems include variational inequalities as special cases, the results presented in this paper continue to hold for these problems.

Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup

PurposeIn this paper, we introduce and study an iterative method to approximate a common solution of a split generalized equilibrium problem and a fixed point problem for a nonexpansive semigroup in real Hilbert spaces.MethodsWe prove a strong convergence theorem of the iterative algorithm in Hilbert spaces under certain mild conditions.ResultsWe obtain a strong convergence result for approximating a common solution of a split generalized equilibrium problem and a fixed point problem for a nonexpansive semigroup in real Hilbert spaces, which is a unique solution of a variational inequality problem. Further, we obtain some consequences of our main result.ConclusionsThe results presented in this paper are the supplement, extension, and generalization of results in the study of Plubtieng and Punpaeng and that of Cianciaruso et al. The approach of the proof given in this paper is also different.

Some remarks on vector optimization problems

We prove the existence of a weak minimum for vector optimization problems by means of a vector variational-like inequality and preinvex mappings.

A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem ☆

Abstract In this paper, we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a system of variational inequality problems, a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real Hilbert space. Further we establish a strong convergence theorem based on this method. The results presented in this paper improves and generalizes the results given in Yao et al. [36] and Ceng et al. [7] , and some known corresponding results in the literature.

Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces

Abstract In this paper, we give the notion of P-η-proximal mapping, an extension of P-proximal mapping given by Ding and Xia [J. Comput. Appl. Math. 147 (2002) 369], for a nonconvex lower semicontinuous η-subdifferentiable proper functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a class of generalized set-valued variational-like inclusions in Banach space and show its equivalence with a class of implicit Wiener–Hopf equations using the concept of P-η-proximal mapping. Using this equivalence, we propose a new class of iterative algorithms for the class of generalized set-valued variational-like inclusions. Furthermore, we prove the existence of solution of generalized set-valued variational-like inclusions and discuss the convergence criteria and the stability of the iterative algorithm.

Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P-η-accretive mappings

In this paper, using proximal-point mapping technique of P -η-accretive mapping and the property of the fixed-point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of the solution set of a parametric generalized implicit quasi-variational-like inclusion involving P -η-accretive mapping in real uniformly smooth Banach space. Further, under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameter. The technique and results presented in this paper can be viewed as extension of the techniques and corresponding results given in [R.P. Agarwal, Y.-J. Cho, N.-J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett. 13 (2002) 19–24; S. Dafermos, Sensitivity analysis in variational inequalities, Math. Oper. Res. 13 (1988) 421–434; X.-P. Ding, Sensitivity analysis for generalized nonlinear implicit quasi-variational inclusions, Appl. Math. Lett. 17 (2) (2004) 225–235; X.-P. Ding, Parametric completely generalized mixed implicit quasi-variational inclusions involving h-maximal monotone mappings, J. Comput. Appl. Math. 182 (2) (2005) 252–269; X.-P. Ding, C.L. Luo, On parametric generalized quasi-variational inequalities, J. Optim. Theory Appl. 100 (1999) 195–205; Z. Liu, L. Debnath, S.M. Kang, J.S. Ume, Sensitivity analysis for parametric completely generalized nonlinear implicit quasi-variational inclusions, J. Math. Anal. Appl. 277 (1) (2003) 142–154; R.N. Mukherjee, H.L. Verma, Sensitivity analysis of generalized variational inequalities, J. Math. Anal. Appl. 167 (1992) 299–304; M.A. Noor, Sensitivity analysis framework for general quasi-variational inclusions, Comput. Math. Appl. 44 (2002) 1175–1181; M.A. Noor, Sensitivity analysis for quasivariational inclusions, J. Math. Anal. Appl. 236 (1999) 290–299; J.Y. Park, J.U. Jeong, Parametric generalized mixed variational inequalities, Appl. Math. Lett. 17 (2004) 43–48].

Existence of solutions for vector optimization

Abstract In this paper, we prove the existence of a weak minimum for constrained vector optimization problem by making use of vector variational-like inequality and preinvex functions.

An iterative algorithm based on M-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces

In this paper, we give the notion of M-proximal mapping, an extension of P-proximal mapping given in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369-383], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a system of generalized implicit variational inclusions in Banach spaces and show its equivalence with a system of implicit Wiener-Hopf equations using the concept of M-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational inclusions and discuss the convergence and stability analysis of the iterative algorithm.

Split nonconvex variational inequality problem

In this paper, we propose a split nonconvex variational inequality problem which is a natural extension of split convex variational inequality problem in two different Hilbert spaces. Relying on the prox-regularity notion, we introduce and establish the convergence of an iterative method for the new split nonconvex variational inequality problem. Further, we also establish the convergence of an iterative method for the split convex variational inequality problem. The results presented in this paper are new and different form the previously known results for nonconvex (convex) variational inequality problems. These results also generalize, unify, and improve the previously known results of this area.2010 MSCPrimary 47J53, 65K10; Secondary 49M37, 90C25