D. R. Sahu

A Generalized Hybrid Steepest-Descent Method for Variational Inequalities in Banach Spaces

The hybrid steepest-descent method introduced by Yamada (2001) is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi (1996) introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamadas hybrid steepest-descent and Lehdili and Moudafis algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions. Our strong convergence theorems extend and improve certain corresponding results in the recent literature.

The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces

It is known, by Rockafellar (SIAM J Control Optim 14:877–898, 1976), that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi (Optimization 37:239–252, 1996) introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with error sequence is also discussed.

Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.

Strong convergence theorems for approximating common fixed points of families of nonexpansive mappings and applications

An implicit algorithm for finding common fixed points of an uncountable family of nonexpansive mappings is proposed. A new inexact iteration method is also proposed for countable family of nonexpansive mappings. Several strong convergence theorems based on our main results are established in the setting of Banach spaces. Both algorithms are applied for finding zeros of accretive operators and for solving convex minimization, split feasibility and equilibrium problems.

Variable KM-like algorithms for fixed point problems and split feasibility problems

AbstractThe convergence analysis of a variable KM-like method for approximating common fixed points of a possibly infinitely countable family of nonexpansive mappings in a Hilbert space is proposed and proved to be strongly convergent to a common fixed point of a family of nonexpansive mappings. Our variable KM-like technique is applied to solve the split feasibility problem and the multiple-sets split feasibility problem. Especially, the minimum norm solutions of the split feasibility problem and the multiple-sets split feasibility problem are derived. Our results can be viewed as an improvement and refinement of the previously known results. MSC:47H10, 65J20, 65J22, 65J25.

A unified extragradient method for systems of hierarchical variational inequalities in a Hilbert space

In this paper, we introduce and analyze a multistep Mann-type extragradient iterative algorithm by combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method, Mann’s iteration method, and the projection method. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings and a strict pseudocontraction, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions and the solution set of a variational inequality problem (VIP), which is just a unique solution of a system of hierarchical variational inequalities (SHVI) in a real Hilbert space. The results obtained in this paper improve and extend the corresponding results announced by many others.MSC:49J30, 47H09, 47J20, 49M05.

Semilocal Convergence Analysis of S-iteration Process of Newton---Kantorovich Like in Banach Spaces

In the present article, we establish a semilocal convergence theorem for the S-iteration process of Newton–Kantorovich like in Banach space setting for solving nonlinear operator equations and discuss its semilocal convergence analysis. We apply our result to solve the Fredholm-integral equations.

Chapter 5 – Extragradient Methods for Some Nonlinear Problems

In this chapter, we discuss several extragradient iterative algorithms for some nonlinear problems, namely fixed point problems, variational inequality problems, hierarchical variational inequality problems and split feasibility problems. We also present and analyze extragradient iterative algorithms for finding a common solution to a fixed point problem and the variational inequality problem.

Hierarchical Minimization Problems and Applications

In this chapter, several iterative methods for solving fixed point problems, variational inequalities, and zeros of monotone operators are presented. A generalized mixed equilibrium problem is considered. The hierarchical minimization problem over the set of intersection of fixed points of a mapping and the set of solutions of a generalized mixed equilibrium problem is considered. A new unified hybrid steepest descent-like iterative algorithm for finding a common solution of a generalized mixed equilibrium problem and a common fixed point problem of uncountable family of nonexpansive mappings is presented and analyzed.

A general inexact iterative method for monotone operators, equilibrium problems and fixed point problems of semigroups in Hilbert spaces

AbstractLet H be a real Hilbert space. Consider on H a nonexpansive family T={T(t):0≤t<∞} with a common fixed point, a contraction f with the coefficient 0 < α < 1, and a strongly positive linear bounded self-adjoint operator A with the coefficient γ̄>0. Assume that 0<γ<γ̄/α and that S={St:0≤t<∞} is a family of nonexpansive self-mappings on H such that F(T)⊆F(S) and has property (A) with respect to the family . It is proved that the following schemes (one implicit and one inexact explicit): xt=btγfxt+I-btAStxt and x0∈H,xn+1=αnγfxn+βnxn+1-βnI-αnAStnxn+en,n≥0 converge strongly to a common fixed point x*∈F(T), where F(T) denotes the set of common fixed point of the nonexpansive semigroup. The point x* solves the variational in-equality 〈(γf −A)x*, x−x*〉 ≤ 0 for all x∈F(T). Various applications to zeros of monotone operators, solutions of equilibrium problems, common fixed point problems of nonexpansive semigroup are also presented. The results presented in this article mainly improve the corresponding ones announced by many others.2010 Mathematics Subject Classification: 47H09; 47J25.