Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).

Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences{xn}are introduced for an infinite family of asymptotically nonexpansive mappingsTii=1∞in this paper. Under some appropriate conditions, we prove that the iterative sequences{xn}converge strongly to a common fixed point of the mappingsTii=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

Viscosity Approximation Methods for Two Accretive Operators in Banach Spaces

We introduced a viscosity iterative scheme for approximating the common zero of two accretive operators in a strictly convex Banach space which has a uniformly Gâteaux differentiable norm. Some strong convergence theorems are proved, which improve and extend the results of Ceng et al. (2009) and some others.

On the computational content of convergence proofs via Banach limits

This paper addresses new developments in the ongoing proof mining programme, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method of extracting rates of metastability (as defined by Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all that is known) rely on the axiom of choice. In this paper, we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X . Combined with our method of eliminating Banach limits, this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type with Lipschitz and Bounded Nonlinear Operators

Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm and F, K : be Lipschitz accretive maps with Suppose that the Hammerstein equation has a solution. An explicit iteration method is shown to converge strongly to a solution of the equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorems are significant improvements on important recent results (e.g., (Chiume and Djitte, 2012)).

An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

LetEbe a real reflexive Banach space with a uniformly Gâteaux differentiable norm. LetKbe a nonempty bounded closed convex subset ofE,and every nonempty closed convex bounded subset ofKhas the fixed point property for non-expansive self-mappings. Letf:K→Ka contractive mapping andT:K→Kbe a uniformly continuous pseudocontractive mapping withF(T)≠∅. Let{λn}⊂(0,1/2)be a sequence satisfying the following conditions: (i)limn→∞λn=0; (ii)∑n=0∞λn=∞. Define the sequence{xn}inKbyx0∈K,xn+1=λnf(xn)+(1−2λn)xn+λnTxn, for alln≥0. Under some appropriate assumptions, we prove that the sequence{xn}converges strongly to a fixed pointp∈F(T)which is the unique solution of the following variational inequality:〈f(p)−p,j(z−p)〉≤0, for allz∈F(T).

Mixed Approximation for Nonexpansive Mappings in Banach Spaces

The mixed viscosity approximation is proposed for finding fixed points of nonexpansive mappings, and the strong convergence of the scheme to a fixed point of the nonexpansive mapping is proved in a real Banach space with uniformly Gâteaux differentiable norm. The theorem about Halpern type approximation for nonexpansive mappings is shown also. Our theorems extend and improve the correspondingly results shown recently.

Strong Convergence of Viscosity Iteration Methods for Nonexpansive Mappings

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterative scheme to a solution of a ceratin variational inequality is established.

Strong Convergence of Viscosity Methods for Continuous Pseudocontractions in Banach Spaces

We define a viscosity method for continuous pseudocontractive mappings defined on closed and convex subsets of reflexive Banach spaces with a uniformly Gâteaux differentiable norm. We prove the convergence of these schemes improving the main theorems in the work by Y. Yao et al. (2007) and H. Zhou (2008).

Strong convergence and control condition of modified Halpern iterations in Banach spaces

LetCbe a nonempty closed convex subset of a real Banach spaceXwhich has a uniformly Gâteaux differentiable norm. LetT∈ΓCandf∈ΠC. Assume that{xt}converges strongly to a fixed pointzofTast→0, wherextis the unique element ofCwhich satisfiesxt=tf(xt)+(1−t)Txt. Let{αn}and{βn}be two real sequences in(0,1)which satisfy the following conditions:(C1)lim⁡n→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1. For arbitraryx0∈C, let the sequence{xn}be defined iteratively byyn=αnf(xn)+(1−αn)Txn,n≥0,xn+1=βnxn+(1−βn)yn,n≥0. Then{xn}converges strongly to a fixed point ofT.