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Abstract In this paper, the existing theorems and methods for finding solutions of systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces are studied. To overcome the difficulties, due to the presence of a proper convex lower semi-continuous function, φ and a mapping g, which appeared in the considered problem, we have used some applications of the resolvent operator technique. We would like to point out that although many authors have proved results for finding solutions of the systems of nonlinear set-valued (mixed) variational inequalities problems, it is clear that it cannot be directly applied to the problems that we have considered in this paper because of φ and g. 2000 AMS Subject Classification: 47H05; 47H09; 47J25; 65J15.

Abstract In this article, a new concept of mixed monotone-generalized contraction in partially ordered probabilistic metric spaces is introduced, and some coupled coincidence and coupled fixed point theorems are proved. The theorems Presented are an extension of many existing results in the literature and include several recent developments. Mathematics Subject Classification: Primary 54H25; Secondary 47H10.

Abstract This article discusses a more general contractive condition for a class of extended (p ≥ 2) -cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are non-empty, closed and convex, then all the iterates converge to a unique closed limiting finite sequence which contains the best proximity points of adjacent subsets and reduces to a unique fixed point if all such subsets intersect.

Abstract In this paper, an implicit iterative algorithm with errors is considered for two families of generalized asymptotically nonexpansive mappings. Strong and weak convergence theorems of common fixed points are established based on the implicit iterative algorithm. Mathematics Subject Classification (2000) 47H09 · 47H10 · 47J25

Abstract We present some fixed point theorems for the sum of a weakly-strongly continuous map and a nonexpansive map on a Banach space . Our results cover several earlier works by Edmunds, Reinermann, Singh, and others.

The purpose of this paper is to introduce and consider a general implicit iterative process which includes Schu's explicit iterative processes and Sun's implicit iterative processes as special cases for a finite family of generalized asymptotically quasi-nonexpansive mappings. Strong convergence of the purposed iterative process is obtained in the framework of real Banach spaces.

Abstract We present new fixed point theorems for ws-compact operators. Our fixed point results are obtained under Sadovskii, Leray-Schauder, Rothe, Altman, Petryshyn, and Furi-Pera type conditions. An example is given to show the usefulness and the applicability of our results.

We present new Leray-Schauder alternatives, Krasnoselskii and Lefschetz fixed point theory for multivalued maps between Fréchet spaces. As an application we show that our results are directly applicable to establish the existence of integral equations over infinite intervals.

We glance at recent advances to the general theory of maximal (set-valued) monotone mappings and their role demonstrated to examine the convex programming and closely related field of nonlinear variational inequalities. We focus mostly on applications of the super-relaxed ( )-proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal ( )-monotonicity. Investigations highlighted in this communication are greatly influenced by the celebrated work of Rockafellar (1976), while others have played a significant part as well in generalizing the proximal point algorithm considered by Rockafellar (1976) to the case of the relaxed proximal point algorithm by Eckstein and Bertsekas (1992). Even for the linear convergence analysis for the overrelaxed (or super-relaxed) ( )-proximal point algorithm, the fundamental model for Rockafellar's case does the job. Furthermore, we attempt to explore possibilities of generalizing the Yosida regularization/approximation in light of maximal ( )-monotonicity, and then applying to first-order evolution equations/inclusions.

A new applicable Leray-Schauder alternative is presented for weakly-strongly sequentially continuous maps. This result is then used to establish a general existence principle for operator equations.