Q-uniformly Smooth Banach Spaces Research Articles

Variational-like inclusion problems involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p. and fixed point problems

This paper is devoted to the introduction of a new class of accretive mappings in the framework of q-uniformly smooth Banach spaces called A-maximal m-relaxed η-accretive mappings in the sense of semi-inner product of type (p) (for short, s.i.p.(p)). The resolvent operator associated with such a mapping is defined; its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed under some suitable conditions. At the same time, the problem of finding a common element of the set of fixed points of a given nearly asymptotically nonexpansive mapping and the set of solutions of a variational-like inclusion problem involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is investigated. To achieve this goal, a new iterative algorithm is constructed. Using the notions of graph convergence and resolvent operator associated with an A-maximal m-relaxed η-accretive mapping in the sense of s.i.p.(p), a new equivalence relationship between the graph convergence and the resolvent operator convergence of a sequence of A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is established. As an application of this equivalence, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is demonstrated under some suitable assumptions imposed on the parameters. The results presented in this paper unify, improve and generalize some recent works in this field.

Mann–Dotson’s algorithm for a countable family of non-self strict pseudo-contractive mappings

The aim of this paper to present some weak and strong convergence results for countable family of non-self mappings. More precisely, we employ the Mann–Dotson’s algorithm to approximate common fixed points of a countable family of non-self k-strict Pseudocontractive mappings in q-uniformly smooth Banach spaces.

Convergence Analysis for Generalized Yosida Inclusion Problem with Applications

A new generalized Yosida inclusion problem, involving A-relaxed co-accretive mapping, is introduced. The resolvent and associated generalized Yosida approximation operator is construed and a few of its characteristics are discussed. The existence result is quantified in q-uniformly smooth Banach spaces. A four-step iterative scheme is proposed and its convergence analysis is discussed. Our theoretical assertions are illustrated by a numerical example. In addition, we confirm that the developed method is almost stable for contractions. Further, an equivalent generalized resolvent equation problem is established. Finally, by utilizing the Yosida inclusion problem, we investigate a resolvent equation problem and by employing our proposed method, a Volterra–Fredholm integral equation is examined.

Convergence Analysis for Yosida Variational Inclusion Problem with Its Corresponding Yosida Resolvent Equation Problem through Inertial Extrapolation Scheme

In this paper, we study a Yosida variational inclusion problem with its corresponding Yosida resolvent equation problem. We mention some schemes to solve both the problems, but we focus our study on discussing convergence criteria for the Yosida variational inclusion problem in real Banach space and for the Yosida resolvent equation problem in q-uniformly smooth Banach space. For faster convergence, we apply an inertial extrapolation scheme for both the problems. An example is provided.

A class of Yosida inclusion and graph convergence on Yosida approximation mapping with an application

The proposed work is presented in two folds. The first aim is to deals with the new notion called generalized ?i?j-Hp(., ., ...)-accretive mappings that are the sum of two symmetric accretive mappings. It is an extension of ??-H(., .)-accretive mapping, studied and analyzed by Kazmi [18]. We define the proximalpoint mapping associated with generalized ?i?j-Hp(., ., ...)-accretive mapping and demonstrate aspects on single-valued property and Lipschitz continuity. The graph convergence of generalized ?i?j-Hp(., ., ...)- accretive mapping is discussed. Second aim is to introduce and study the generalized Yosida approximation mapping and Yosida inclusion problem. Next, we obtain the convergence on generalized Yosida approximation mappings by using the graph convergence of generalized ?i?j-Hp(., ., ...)-accretive mappings without using the convergence of its proximal-point mapping. As an application, we consider the Yosida inclusion problem in q-uniformly smooth Banach spaces and propose an iterative scheme connected with generalized Yosida approximation mapping of generalized ?i?j-Hp(., ., ...)-accretive mapping to find a solution of Yosida inclusion problem and discuss its convergence criteria under appropriate assumptions. Some examples are constructed and demonstrate few graphics for the convergence of proximal-point mapping as well as generalized Yosida approximation mapping linked with generalized ?i?j-Hp(., ., ...)-accretive mappings.

An algorithmic approach for a class of set-valued variational inclusion problems

The main goal of this paper is twofold. Our first objective is to prove the Lipschitz continuity of the proximal-point mapping associated with an H-accretive operator and to compute an estimate of its Lipschitz constant under some new appropriate conditions imposed on the parameter and mappings involved in it. Using the notion of proximal-point mapping, a new iterative algorithm is constructed for solving a new class of set-valued variational inclusion problems in the setting of q-uniformly smooth Banach spaces. As an application, the strong convergence of the sequences generated by our proposed iterative algorithm to a solution of our considered problem is proved. The second objective of this paper is to investigate and analyze the notion of ??-H((.,.), (.,.))-mixed accretive mapping introduced and studied in [S. Gupta, S. Husain, V.N. Mishra, Variational inclusion governed by ??-H((.,.), (.,.))-mixed accretive mapping, Filomat 31(20)(2017) 6529-6542]. Some comments concerning ??-H((., .), (., .))-mixed accretive mapping and related conclusions appeared in the above-mentioned paper are also pointed out.

AN ACCELERATED HALPERN-TYPE ALGORITHM FOR SOLVING VARIATIONAL INCLUSION PROBLEMS WITH APPLICATIONS

An inertial Halpern-type forward-backward iterative algorithm for approximating a zero of sum of two accretive operators is introduced and studied. Strong convergence theorem is established in a uniformly convex and q-uniformly smooth real Banach space. The convergence result obtained is applied to convex minimization and image restoration problems. Furthermore, numerical experiments are carried out on some classical test images and personal images degraded with motion blur and random noise. Finally, numerical illustrations in the Banach space, L5([−1, 1]) are presented to support the main theorem.

Set-Valued Variational Inclusion Governed by Generalized αiβj-Hp(., ., ...)-Accretive Mapping

This work is concerned with the new notion called generalized αiβj-Hp(., ., ...)-accretive mapping that is the sum of two symmetric accretive mappings. It is an extension of the generalized αβ-H(., .)-accretive mapping. The proximal point mapping linked to the generalized αiβj-Hp(., ., ...)-accretive mappings is defined, and some of its characteristics are discussed. As an application of the new proximal point mapping, we consider a set-valued variational inclusion problem in q-uniformly smooth Banach spaces. Further, we propose an iterative scheme connected with αiβj-Hp(., ., ...)-proximal point mapping to find the solution of a variational inclusion problem and discuss its convergence criteria under appropriate assumptions. Some examples are constructed in support of generalized αiβj-Hp(., ., ...)-accretive mappings.

System of Generalized Variational Inclusions Involving Cayley Operators and XOR-Operation in q-Uniformly Smooth Banach Spaces

In this paper, we consider and study a system of generalized variational inclusions involving Cayley operators and an XOR-operation in q-uniformly smooth Banach spaces. To obtain the solution of the system of generalized variational inclusions involving Cayley operators and an XOR-operation, we use some properties of Cayley operators as well as an XOR-operation. We also discuss the convergence criterion. In support of our main result, we provide an example.

Mixed Variational Inequality Problem Involving Generalized Yosida Approximation Operator in q-Uniformly Smooth Banach Space

A mixed variational inequality problem involving generalized Yosida approximation operator is considered and studied in q -uniformly smooth Banach space. We have shown that mixed variational inequality problem involving generalized Yosida approximation operator is equivalent to a fixed-point equation. The fixed-point formulation is applied to establish an algorithm to obtain the solution of mixed variational inequality problem involving generalized Yosida approximation operator. Convergence criteria are also discussed. In support of our main result, we provide an example using Matlab program together with a computation table and convergence graphs. To check the validity of mixed variational inequality problem involving generalized Yosida approximation operator and its fixed-point formulation, we construct one more example.

"A new Krasnoselskii’s type algorithm for zeros of strongly monotone and Lipschitz mappings"

"For q > 1, let E be a q-uniformly smooth real Banach space with dual space E∗. Let A : E → E∗ be a Lipschitz and strongly monotone mapping such that A^{−1}(0) ̸= ∅. For given x_1 ∈ E, let {x_n} be generated iteratively by the algorithm : x_{n+1} = x_n − λJ^{−1}(Ax_n), n ≥ 1, where J is the normalized duality mapping from E into E∗ and λ is a positive real number choosen in a suitable interval. Then it is proved that the sequence {xn} converges strongly to x∗, the unique point of A^{−1}(0). Our theorems are applied to the convex minimization problem. Futhermore, our technique of proof is of independent interest."

Inertial splitting algorithms for nonlinear operators of pseudocontractive and accretive types

ABSTRACT In this paper, inertial splitting algorithms for nonlinear operators of pseudocontractive and accretive types are proposed. Weak and strong convergence theorems are established in uniformly convex and q-uniformly smooth Banach spaces. Numerical examples are given to illustrate the effectiveness of our proposed algorithms.

Generalized Hybrid Viscosity-Type Forward-Backward Splitting Method with Application to Convex Minimization and Image Restoration Problems

Let E be a uniformly convex and q-uniformly smooth real Banach space. Let be an α- inverse strongly accretive mapping of order q, be a set-valued m- accretive mapping and be a nonexpansive mapping. In this article, a viscosity-type forward-backward splitting method for approximating a zero of (A + B) which is also a fixed point of S is introduced studied. Strong convergence theorem of the method is proved under suitable conditions. Furthermore, the convergence result obtained is applied to convex minimization and image restoration problems. Finally, numerical illustrations are presented to compare the convergence of the sequence of our algorithm and that of some recent important algorithms.

Strong convergence results for variational inclusions, systems of variational inequalities and fixed point problems using composite viscosity implicit methods

Let the VI indicate a variational inclusion, the CFPP denote a common fixed point problem of countably many nonexpansive mappings, and the SVI represent a system of variational inequalities. We introduce a composite viscosity implicit method for solving the VI and CFPP with the SVI constraint in the framework of uniformly convex and q-uniformly smooth Banach space where . Moreover, we prove the strong convergence of the sequences generated by the proposed implicit method to a solution of a certain hierarchical variational inequality (HVI). In addition, our results are also applied for solving the fixed point problem (FPP) of nonexpansive mapping, variational inequality problem, convex minimization problem and split feasibility problem in Hilbert spaces.

Modified Inertial Forward–Backward Algorithm in Banach Spaces and Its Application

In this paper, we present a new modified inertial forward–backward algorithm for finding a common solution of the quasi-variational inclusion problem and the variational inequality problem in a q-uniformly smooth Banach space. The proposed algorithm is based on descent, splitting and inertial ideas. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the abovementioned problems. Numerical examples are also given to demonstrate our results.

H-Mixed Accretive Mapping and Proximal Point Method for Solving a System of Generalized Set-Valued Variational Inclusions

In this paper, we introduce a new system of variational inclusion problems called system of generalized set-valued variational inclusion problems involving -mixed accretive mapping in q-uniformly smooth Banach spaces. By means of proximal-point mapping method, we prove the existence of solution of this system of variational inclusions. Further, we propose an iterative algorithm for finding the approximate solution of the system. Furthermore, we prove that the sequences generated by the iterative algorithm converge strongly to a solution of the system.

A Method for Solving the Variational Inequality Problem and Fixed Point Problems in Banach Spaces

The purpose of this research is to modify Halpern iteration’s process for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of a strictly pseudo contractive mapping in q-uniformly smooth Banach space. We also introduce a new technique to prove a strong convergence theorem for a finite family of strictly pseudo contractive mappings in q-uniformly smooth Banach space. Moreover, we give a numerical result to illustrate the main theorem.

Convergence and Stability of a Perturbed Mann Iterative Algorithm with Errors for a System of Generalized Variational-Like Inclusion Problems in q-uniformly smooth Banach Spaces

In this paper, we introduce a class of \({\cal H}(\cdot,\cdot)\)-\(\phi\)-\(\eta\)-accretive operators in a real \(q\)-uniformly smooth Banach space. We define the resolvent operator associated with \({\cal H}(\cdot,\cdot)\)-\(\phi\)-\(\eta\)-accretive operator and prove that it is single-valued and Lipschitz continuous. Moreover, we propose a perturbed Mann iterative method with errors for approximating the solution of the system of generalized variational-like inclusion problems and discuss the convergence and stability of the iterative sequences generated by the algorithm. Our results presented in this paper generalize and unify many known results in the literature.

Inertial Krasnoselskii–Mann Method in Banach Spaces

In this paper, we give a general inertial Krasnoselskii–Mann algorithm for solving inclusion problems in Banach Spaces. First, we establish a weak convergence in real uniformly convex and q-uniformly smooth Banach spaces for finding fixed points of nonexpansive mappings. Then, a strong convergence is obtained for the inertial generalized forward-backward splitting method for the inclusion. Our results extend many recent and related results obtained in real Hilbert spaces.

Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems

Let X be a uniformly convex and q-uniformly smooth Banach space with 1< qleq 2. In the framework of this space, we are concerned with a composite gradient-like implicit rule for solving a hierarchical monotone variational inequality with the constraints of a system of monotone variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonlinear operators {S_{n}}^{infty }_{n=0}. Our rule is based on the Korpelevich extragradient method, the perturbation mapping, and the W-mappings constructed by {S_{n}}^{infty }_{n=0}.