Nonspreading Mappings Research Articles

THE FIXED POINT PROPERTY AND UNBOUNDED SETS IN BANACH SPACES

Let $E$ be a smooth, strictly convex and reflexive Banach space, let $J$ be the duality mapping of $E$ and let $C$ be a nonempty closed convex subset of $E$. Then, a mapping $S:C\to C$ is said to be nonspreading \cite{koh-tak3} if \[ \phi(Sx, Sy)+\phi(Sy, Sx)\le\phi(Sx, y)+\phi(Sy, x) \] for all $x, y\in C$, where $\phi(x,y)= \|x\|^2-2\langle x, Jy\rangle +\|y\|^2$ for all $x, y\in E$. In this paper, we prove that every nonspreading mapping of $C$ into itself has a fixed point in $C$ if and only if $C$ is bounded. This theorem extends Ray's theorem \cite{ray} in a Hilbert space to that in a Banach space.

Fixed Point Theorems and Weak Convergence Theorems for Generalized Hybrid Mappings in Hilbert Spaces

In this paper, we first consider a broad class of nonlinear mappings containing the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings in a Hilbert space. Then, we deal with fixed point theorems and weak convergence theorems for these nonlinear mappings in a Hilbert space.

Weak Convergence Theorems for a System of Mixed Equilibrium Problems and Nonspreading Mappings in a Hilbert Space

We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping. The result obtained in this paper improves and extends the recent ones announced by Moudafi (2009),Iemoto and Takahashi (2009), and many others. Using this result, we improve and unify several results in fixed point problems and equilibrium problems.

Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces

We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonspreading mapping in a Hilbert space. Then, we prove a strong convergence theorem which is connected with the work of S. Takahashi and W. Takahashi (2007) and Iemoto and Takahashi (2009).

Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space

In this paper, we obtain some fundamental properties for nonspreading mappings in a Hilbert space. Further, we study the approximation of common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space.

Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces

In this paper, the class of nonspreading mappings in Banach spaces is introduced. This class contains the recently introduced class of firmly nonexpansive type mappings in Banach spaces and the class of firmly nonexpansive mappings in Hilbert spaces. Among other things, we obtain a fixed point theorem for a single nonspreading mapping in Banach spaces. Using this result, we also obtain a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces.