An implicit iterative scheme for monotone variational inequalities and fixed point problems

In this paper we introduce an implicit iterative scheme for finding a common element of the set of common fixed points of N$N$ nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The implicit iterative scheme is based on two well-known methods: extragradient and approximate proximal. We obtain a weak convergence theorem for three sequences generated by this implicit iterative scheme. On the basis of this theorem, we also construct an implicit iterative process for finding a common fixed point of N+1$N+1$ mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other N$N$ mappings are nonexpansive.     

A new iteration process for finite families of generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings

A new iteration process is introduced and proved to converge strongly to a common fixed point for a finite family of generalized Lipschitz nonlinear mappings in a real reflexive Banach space E$E$ with a uniformly Gâteaux differentiable norm if at least one member of the family is pseudo-contractive. It is also proved that a slight modification of the process converges to a common zero for a finite family of generalized Lipschitz accretive operators defined on E$E$. Results for nonexpansive families are obtained as easy corollaries. Finally, the new iteration process and the method of proof are of independent interest.     

Path convergence,approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces

Let E$E$ be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K$K$ be a nonempty closed convex subset of E$E$, and let T:K?E$T:K?E$ be a continuous pseudocontraction which satisfies the weakly inward condition. For f:K?K$f:K?K$ any contraction map on K$K$, and every nonempty closed convex and bounded subset of K$K$ having the fixed point property for nonexpansive self-mappings, it is shown that the path x→_xt,t∈[0,1)$x\to x_t,t\in [0,1)$, in K$K$, defined by _xt=tT_xt+(1−t)f(_xt)$x_t=tT{x}_t+(1-t)f\left(x_t\right)$ is continuous and strongly converges to the fixed point of T$T$, which is the unique solution of some co-variational inequality. If, in particular, T$T$ is a Lipschitz pseudocontractive self-mapping of K$K$, it is also shown, under appropriate conditions on the sequences of real numbers {_αn},{_μn}$\left\{{\alpha }_n\right\},\left\{{\mu }_n\right\}$, that the iteration process: _z1∈K$z_1\in K$, _zn+1=_μn(_αnT_zn+(1−_αn)_zn)+(1−_μn)f(_zn),n∈N$z_{n+1}={\mu }_n({\alpha }_{n}T{z}_n+(1-{\alpha }_n)z_n)+(1-{\mu }_n)f\left(z_n\right),n\in \mathbb{N}$, strongly converges to the fixed point of T$T$, which is the unique solution of the same co-variational inequality. Our results propose viscosity approximation methods for Lipschitz pseudocontractions.     

Strong convergence theorem for approximation of solutions of equations of Hammerstein type

Let H$H$ be a real Hilbert space. Let K,F:H→H$K,F:H\to H$ be bounded, continuous and monotone mappings. Suppose that u^∗∈H$u^{\ast }\in H$ is a solution to the Hammerstein equation u+KFu=0$u+KFu=0$. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the Hammerstein equation. Furthermore, we give some examples to show that our result is interdisciplinary in nature, covers a large variety of areas and should be of much interest to a wide audience.     

Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are Lipschitz ?$?$-strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Contractive mappings and existence of cycle times for a monotone and homogeneous function

Given a monotone and homogeneous self-mapping f$f$ of the n$n$-dimensional positive cone, a family of contractive mappings is used to define an equivalence relation in the index set, as well as a total order among the equivalence classes. Then, it is shown (i) that the cycle times are well-defined at each index belonging to the maximal and minimal classes, and (ii) that the cycle times of f$f$ exist at every index whenever a weak convexity condition is satisfied.     

Approximation of solutions of Hammerstein equations with bounded strongly accretive nonlinear operators

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are bounded strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. A new explicit coupled iteration process is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

A topological property of the common fixed points set of two multivalued operators

Let (X,d)$(X,d)$ be a complete metric space and absolute retract for metric spaces. We prove that the common fixed points set of two multivalued operators defined on X$X$, which have the selection property and satisfy a contraction type condition, is an absolute retract for metric spaces.     

Convergence of iterative algorithms for multivalued mappings in Banach spaces

We show strong and weak convergence for Mann iteration of multivalued nonexpansive mappings T$T$ in a Banach space. Furthermore, we give a strong convergence of the modified Mann iteration which is independent of the convergence of the implicit anchor-like continuous path _zt∈tu+(1−t)T_zt$z_t\in tu+(1-t)T{z}_t$.     

Generalized quasi-variational-like hemivariational inequalities

In this paper, we introduce and study a new class of generalized quasi-variational-like hemivariational inequalities with multi-valued η$\eta$-pseudomonotone operators in Banach spaces. Some new existence theorems of solutions for this class of generalized quasi-variational-like hemivariational inequalities are proved. The results presented in this paper generalize and extend some known results.     

Implicit iteration process for a finite family of asymptotically hemi-contractive mappings in Banach spaces

The purpose of this paper is to study the strong convergence of a process of implicit iteration to a common fixed point for a finite family of asymptotically hemi-contractive mappings. Our results extend a recent result of M.O. Osilike and B.G. Akuchu [Common fixed points of a finite family of asymptotically pseudocontractive maps, Fixed Point Theory Appl. 2 (2004) 81–88] from Hilbert spaces to p$p$-uniformly convex Banach spaces with p>1$p>1$.