Mann iteration of Cesàro means for asymptotically non-expansive mappings

Let K be a nonempty closed convex subset of a uniformly convex Banach space E with a uniformly Gâteaux differentiable norm. Suppose that T:K→K is an asymptotically non-expansive mapping and for arbitrary initial value x₀∈K, we will introduce the Mann iteration of its Cesàro means:

     

Solution of nonlinear integral equations of Hammerstein type

Let E be a 2-uniformly real Banach space and F,K:E→E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u+KFu=0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator.     

Fixed point theorems for generalized set-contraction maps and their applications

A new generalized set-valued contraction on topological spaces with respect to a measure of noncompactness is introduced. Two fixed point theorems for the KKM type maps which are either generalized set-contraction or condensing ones are given. Furthermore, applications of these results for existence of coincidence points and maximal elements are deduced.     

Approximation methods for common fixed points for a countable family of nonexpansive mappings

Let E=L_p or l_p space, 1<p<∞. Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ∈(0,1), define a family of nonexpansive maps by S_i?(1−δ)I+δT_i where I is the identity map of K. Let . It is proved that the iterative sequence {x_n} defined by: x₀∈K,x_n+1=α_nu+∑_i≥1σ_i,tnS_ix_n,n≥0 converges strongly to a common fixed point of the family where {α_n} and {σ_i,tn} are sequences in (0,1) satisfying appropriate conditions, in each of the following cases: (a) E=l_p,1<p<∞, and (b) E=L_p,1<p<∞ and at least one of the maps T_i’s is demicompact. Our theorems extend the results of [P. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 325 (2007) 469-479] from Hilbert spaces to l_p spaces, 1<p<∞.     

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.     

Existence of equilibria of set-valued maps on bounded epi-Lipschitz domains in Hilbert spaces without invariance conditions

In this paper, we provide a new result of the existence of equilibria for set-valued maps on bounded closed subsets K of Hilbert spaces. We do not impose either convexity or compactness assumptions on K but we assume that K has epi-Lipschitz sections, i.e. its intersection with suitable finite dimensional spaces is locally the epigraph of Lipschitz functions. In finite dimensional spaces, the famous Brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set K to itself. Our result could be viewed as a kind of generalization of this classical result in the context of Hilbert spaces and when the function (or the set-valued map) does not necessarily map K into itself (K is not invariant under the map). Our approach is based firstly on degree theory for compact and for condensing set-valued maps and secondly on flows generated by trajectories of differential inclusions.     

Degree of convergence of modified averaged iterations for fixed points problems and operator equations

Let H be a real Hilbert space. We propose a modification for averaged mappings to approximate the unique fixed point of a mapping T:H→H such that T is boundedly Lipschitzian and −T is monotone. We not only prove strong convergence theorems, but also determine the degree of convergence. Using this result, an iteration process is given for finding the unique solution of the equation Ax=f, where A:H→H is strongly monotone and boundedly Lipschitzian.     

Factorization of delta-monotone linear mappings

For a delta-monotone linear mapping we prove that the factors in the polar decomposition are delta-monotone. Also, we prove that every delta-monotone linear mapping can be factored into a product of (1-ε)-monotone mappings for any ε∈(0,1). As an application in nonlinear case, we give a new proof of the following fact: the quasiconformality constant K(δ,n) of a δ-monotone mapping can be chosen such that K(δ,n) tends to 1 as δ tends to 1.     

Locally defined operators in Hölder’s spaces

We prove that every local operator acting between two Hölder spaces H_? and H_ψ is a Nemytskij operator and if H_??H_ψ then it is a constant map. Moreover, we show that if K is local and uniformly continuous then it is an affine mapping.     

The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices

When A∈B(H) and B∈B(K) are given, we denote by M_C the operator matrix acting on the infinite-dimensional separable Hilbert space H⊕K of the form In this paper, for given A and B, the sets and ?_C∈Inv(K,H)σ_l(M_C) are determined, where σ_l(T),B_l(K,H) and Inv(K,H) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H.     

A new topological degree theory for perturbations of the sum of two maximal monotone operators

Let X be an infinite dimensional real reflexive Banach space with dual space X^∗ and G⊂X, open and bounded. Assume that X and X^∗ are locally uniformly convex. Let T:X⊃D(T)→2^X∗ be maximal monotone and strongly quasibounded, S:X⊃D(S)→X^∗ maximal monotone, and C:X⊃D(C)→X^∗ strongly quasibounded w.r.t. S and such that it satisfies a generalized (S₊)-condition w.r.t. S. Assume that D(S)=L⊂D(T)∩D(C), where L is a dense subspace of X, and 0∈T(0),S(0)=0. A new topological degree theory is introduced for the sum T+S+C, with degree mapping d(T+S+C,G,0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S+C, as above.     

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.     

Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators

Let X be a real reflexive Banach space and be maximal monotone. Let be quasibounded, finitely continuous and generalized pseudomonotone with X′⊂D(B), where X′ is a dense subspace of X such that X′∩D(A)≠∅. Let S⊂X∗. Conditions are given under which and intS⊂intR(A+B). Results of Browder concerning everywhere defined continuous and bounded operators B are improved. Extensions of this theory are also given using the degree theory of the last two authors concerning densely defined perturbations of nonlinear maximal monotone operators which satisfy a generalized (S₊)-condition. Applications of this extended theory are given involving nonlinear parabolic problems on cylindrical domains.     

Elliptic problems with lack of compactness via a new fixed point theorem

This paper provides a new fixed point theorem for increasing self-mappings G:B→B of a closed ball B⊂X, where X is a Banach semilattice which is reflexive or has a weakly fully regular order cone X₊. By means of this fixed point theorem, we are able to establish existence results of elliptic problems with lack of compactness.     

On a Mann type implicit iteration process for continuous pseudo-contractive mappings

Let K$K$ be a nonempty closed convex subset of a Banach space E$E$, T:K→K$T:K\to K$ a continuous pseudo-contractive mapping. Suppose that {_αn}$\left\{{\alpha }_n\right\}$ is a real sequence in [0,1]$[0,1]$ satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the Mann type implicit iteration process {_xn}$\left\{x_n\right\}$ given by _xn=_αnxn−1+(1−_αn)T_xn,n≥0$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n,n\ge 0$, strongly and weakly converges to a fixed point of T$T$, respectively.     

An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings

Let H be a Hilbert space and C be a nonempty closed convex subset of H, {T_i}_i∈N be a family of nonexpansive mappings from C into H, G_i:C×C→R be a finite family of equilibrium functions (i∈{1,2,…,K}), A be a strongly positive bounded linear operator with a coefficient and -Lipschitzian, relaxed (μ,ν)-cocoercive map of C into H. Moreover, let , {α_n} satisfy appropriate conditions and ; we introduce an explicit scheme which defines a suitable sequence as follows:

     

Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces

The purpose of this paper is to study hybrid iterative schemes of Halpern types for a semigroup ℑ={T(s):s∈S} of relatively nonexpansive mappings on a closed and convex subset C of a Banach space with respect to a sequence {μ_n} of asymptotically left invariant means defined on an appropriate invariant subspace of l^∞(S). We prove that given a certain sequence {α_n} in [0,1], x∈C, we can generate an iterative sequence {x_n} which converges strongly to Π_F(ℑ)x where Π_F(ℑ)x is the generalized projection from C onto the fixed point set F(ℑ). Our main result is even new for the case of a Hilbert space.     

Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces

Let (X,?) be a partially ordered set and d be a complete metric on X. Let F,G be two set-valued mappings on X. We obtained sufficient conditions for the existence of common fixed point of F and G satisfying an implicit relation in partially ordered set X.     

Uniformly normal structure and uniformly Lipschitzian semigroups

Assume that X is a real Banach space with uniformly normal structure and C is a nonempty closed convex subset of X. We show that a κ-uniformly Lipschitzian semigroup of nonlinear self-mappings of C admits a common fixed point if the semigroup has a bounded orbit and if κ is appropriately greater than one. This result applies, in particular, to the framework of uniformly convex Banach spaces.