Continuous selections and fixed points of multi-valued mappings on noncompact or nonmetrizable spaces

In this paper, we obtain several new continuous selection theorems for multi-valued mappings on completely regular spaces and fixed point theorems for multi-valued maps on nonmetrizable spaces. They, in particular, provide a partial solution of a conjecture of X. Wu.

     

Multi-valued variational inequalities with K-pseudomonotone operators

In this paper, we first employ the 1961 celebrated Fan lemma to derive a very general existence result for multi-valued variational inequalities involving multi-valued K-pseudomonotone operators. It will be seen that this result improves and unifies existence results of variational inequalities for monotone operators. Next, we establish some uniqueness results for multi-valued variational inequalities by introducing the concepts of strict, , and strong K-pseudomonotonicity of multi-valued operators, respectively. These uniqueness results appear to be new even if the underlying space is finite-dimensional.This work was partially supported by the National Science Council Grant NSC 82-0208-M-110-023. The author would like to express his sincere thanks to the referees for their valuable comments and suggestions that improved this paper substantially.     

Krasnoselskii type fixed point theorems for multi-valued mappings with weakly sequentially closed graph

In this work, using an analogue of Sadovskii’s fixed point result for multi-valued mappings with weakly sequentially closed graph, we prove new multi-valued analogues of Krasnoselskii fixed point theorem for mappings with weakly sequentially closed graph and under weak topology features. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness.     

Solvability of Unilateral Parabolic Problems and Variational Inequalities

Sufficient conditions are given for the existence of solutions of strong parabolic variational inequalities with nonlinear and multi-valued operators. Pseudomonotone operators are used. A new multi-valued analogue of the acute-angle lemma is used for the localization. The theory can be applied to problems with distributed parameters. We also consider conditions for the existence of strong generalized solutions of parabolic equations with unilateral boundary conditions with applications to control problems.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 250–303, 2004.     

Common fixed point theorem for cyclic generalized multi-valued contraction mappings

In this paper, we extend a multi-valued contraction mapping to a cyclic multi-valued contraction mapping. We also establish the existence of common fixed point theorem for a cyclic multi-valued contraction mapping. Our results extend, generalize and unify Nadler’s multi-valued contraction mapping and many fixed point theorems for multi-valued mappings.     

Banach空间中多值Φ-增生和多值Φ-伪压缩映象的收敛定理

在任意Banach空间中引入比重要的多值-强增生映象和多值-强伪压缩映象更一般的多值Φ-增生映象和多值Φ-伪压缩映象,研究多值Φ-增生映象方程的解和多值Φ-伪压缩映象不动点的具随机误差的Ishikawa迭代逼近问题.这些结论推广和改进了最新文献中相应的结果.     

Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings

A Mann-type iterative scheme which converges strongly to a fixed point of a multi-valued nonexpansive non-self mapping T is constructed in a real Hilbert space H. We also constructed a Mann-type sequence which converges to a fixed point of a multi-valued quasi-nonexpansive non-self mapping under appropriate conditions. In addition, an Ishikawa-type iterative scheme which approximates the fixed points of multi-valued Lipschitz pseudocontractive non-self mappings is constructed in Banach spaces. The results obtained in this paper improve and extend the known results in the literature.

     

Some results on multi-valued weakly jungck mappings in <Emphasis Type="Italic">b</Emphasis>-metric space

In this paper, the concept of multi-valued weak contraction of Berinde and Berinde [8] for the Picard iteration in a complete metric space is extended to the case of multi-valued weak contraction for the Jungck iteration in a complete b-metric space. While our main results generalize the recent results of Berinde and Berinde [8], they also extend, improve and unify several classical results pertainning to single and multi-valued contractive mappings in the fixed point theory. Our results also improve the recent results of Daffer and Kaneko [16].      

Convergence of an iterative method for relatively nonexpansive multi-valued mappings and equilibrium problems in Banach spaces

In this paper, an iterative sequence for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two relatively nonexpansive multi-valued mappings is introduced. This iterative scheme can be viewed as a multi-valued version of the corresponding one introduced by Zhang et al. (Comput Math Appl 61, 262–276, 2011) for two relatively nonexpansive multi-valued mappings. Finally, strong convergence of this sequence is studied in Banach spaces.     

Common coincidence- and fixed-point theorems for a class of nonlinear hybrid mappings

The concepts of multiple quasicompatibility for single-valued and multi-valued mappings are defined. Some common fixed- and coincidence-point theorems for single-valued and multi-valued mappings satisfying a ?-type condition are obtained by an iteration scheme. The conditions are not assumed to be of a contractive type.