Fixed point theorems of multivalued monotone mappings in ordered metric spaces

In this work, some new fixed point and coupled fixed point theorems for multivalued monotone mappings in ordered metric spaces are proved.     

Tangency conditions for multivalued mappings

We prove that interiority conditions imply tangency conditions for two multivalued mappings from a topological space into a normed vector space. As a consequence, we obtain the lower semicontinuity of the intersection of two multivalued mappings. An application to the epi-upper semicontinuity of the sum of convex vector-valued mappings is given.     

Carathéodory selections for multivalued mappings

We prove the existence of a Carathéodory selection for a set-valued mapping satisfying standard conditions. Also, an application to random fixed point for random operators is established.     

Viscosity approximation methods for nonexpansive mappings and monotone mappings

Viscosity approximation methods for nonexpansive mappings are studied. Consider the iteration process {xn}, where x₀∈C is arbitrary and xn+1=αnf(xn)+(1−αn)SPC(xn−λnAxn), f is a contraction on C, S is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. It is shown that {xn} converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly-monotone mapping which solves some variational inequality.     

The homotopies of admissible multivalued mappings

Certain properties of homotopies of admissible multivalued mappings shall be presented, along with their applications as the tool for examining the acyclicity of a space.     

Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.     

The fixed point property for multivalued nonexpansive mappings

We show some properties concerning geometrical constants of Banach spaces which imply the existence of fixed points for multivalued nonexpansive mappings and we study the relationship between these properties.     

Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems

In this paper, we introduce some new iterative sequences for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of nonspreading mappings and a finite family of nonexpansive multivalued mappings in Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative process. The results obtained in this paper extend and improve some recent known results.     

Complementarity problems for multivalued monotone operator in Banach spaces

We utilize S. Park's maximal element theorem in this paper to prove the existence theorems of solutions of the complementarity problems for multivalued monotone operator in Banach spaces.     

Banach ultrapowers and multivalued nonexpansive mappings

The Banach ultrapower construction is applied in fixed point theory for multivalued mappings. We introduce the notion of ultra-asymptotic centers and use it to remove the separability assumption from the results of Domínguez Benavides, Lorenzo Ramírez (2004) and Dhompongsa, Kaewcharoen, Kaewkhao (2006).     

A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings

In this paper, we introduce an general iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the two sets. Using this results, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. The results of this paper extended and improved the results of Iiduka and Takahashi (Nonlinear Anal. 61:341–350, 2005).     

Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings

In this paper, we prove strong convergence theorems to a zero of monotone mapping and a fixed point of relatively weak nonexpansive mapping. Moreover, strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions.