Smooth Banach Space Research Articles

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.

Generalized $ (f, \lambda) $-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces

<abstract><p>In this paper, we expanded from the convex case to the nonconvex case in the setting of reflexive smooth Banach spaces, the concept of the $ f $-generalized projection $ \pi^{f}_S:X^*\to S $ initially introduced for convex sets and convex functions in <sup>[<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b20">20</xref>]</sup>. Indeed, we defined the $ (f, \lambda) $-generalized projection operator $ \pi^{f, \lambda}_S:X^*\to S $ from $ X^* $ onto a nonempty closed set $ S $. We proved many properties of $ \pi^{f, \lambda}_S $ for any closed (not necessarily convex) set $ S $ and for any lower semicontinuous function $ f $. Our principal results broaden the scope of numerous theorems established in <sup>[<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b20">20</xref>]</sup> from the convex setting to the nonconvex setting. An application of our main results to solutions of nonconvex variational problems is stated at the end of the paper.</p></abstract>

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 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.

Convergence theorem for split feasibility problem, equilibrium problem and zeroes of sum of monotone operators

The main purpose of this paper is to introduce a parallel iterative algorithm for approximating the solution of a split feasibility problem on the zero of monotone operators, generalized mixed equilibrium problem and fixed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating a common element in the set of solutions of a problem of finding zeroes of sum of two monotone operators,generalized mixed equilibrium problem and fixed point problem for a finite family of $\eta$-demimetric mappings in the frame work of a reflexive, strictly convex and smooth Banach spaces. We also give a numerical experiment applying our main result. Our result improves, extends and unifies other results in this direction in the literature.

An iterative method involving a class of quasi-phi-nonexpansive mappings for solving split equality fixed point problems

"A new inertial iterative algorithm for approximating solution of split equality fixed point problem (SEFPP) for quasi-$\phi$- nonexpansive mappings is introduced and studied in $p$-uniformly convex and uniformly smooth real Banach spaces, $p>1$. A strong convergence theorem is proved without imposing any compactness-type condition on the mappings. Our theorems complement several important recent results that have been proved in 2-uniformly convex and uniformly smooth real Banach spaces. It is well known that these spaces do not include $L_p, l_p $ and the Sobolev spaces ${W^m}_p(\Omega)$, for $2< p < \infty$. Our theorems, in particular, are applicable in these spaces. Furthermore, application of our theorem to split equality variational inclusion problem is presented. Finally, numerical examples are presented to illustrate the convergence of our algorithms."

An inertial self-adaptive iterative algorithm for finding the common solutions to split feasibility and fixed point problems in specific Banach spaces

In this paper, we propose a new self adaptive iterative algorithm with an inertial technique for solving split feasibility problems and fixed point problems of relative nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. Under some appropriate assumptions imposed on the parameters and operators, we prove that the iterative sequence generated by our proposed algorithm converges strongly to a common solution of split feasibility problems and fixed point problems. Our algorithm does not require a prior knowledge of the norm of the bounded linear operator as we use a new self adaptive step size. Moreover, we give some numerical examples to show the effectiveness and feasibility of our algorithm.

Self-adaptive forward–backward splitting algorithm for the sum of two monotone operators in Banach spaces

In this work, we prove the weak convergence of a one-step self-adaptive algorithm to a solution of the sum of two monotone operators in 2-uniformly convex and uniformly smooth real Banach spaces. We give numerical examples in infinite-dimensional spaces to compare our result with some existing algorithms. Finally, our results extend and complement several existing results in the literature.

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.

A new system of generalized nonlinear variational inclusion problems in semi-inner product spaces

In this work we reflect a new system of generalized nonlinear variational inclusion problems in 2-uniformly smooth Banach spaces. By using resolvent operator technique, we offer an iterative algorithm for figuring out the approximate solution of the said system. The motive of this paper is to review the convergence analysis of a system of generalized nonlinear variational inclusion problems in 2-uniformly smooth Banach spaces. The proposition used in this paper can be considered as an extension of propositions for examining the existence of solution for various classes of variational inclusions considered and studied by many authors in 2-uniformly smooth Banach spaces.

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.

Self-Adaptive Extragradient Methods for Solving Variational Inequalities and Fixed Point Problems in 2-Uniformly Convex and Uniformly Smooth Banach Spaces

In this paper, we mainly introduce a new and different self adaptive extragradient algorithm with an inertial technique for solving variational inequality problems of pseudomonotone operators and fixed point problems of relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth Banach spaces. Under some appropriate assumptions imposed on the parameters and operators, we prove that the iterative sequence generated by our proposed algorithm converges strongly to a common solution of variational inequality problems and fixed point problems. Moreover, we give some numerical examples to show the effectiveness and feasibility of our algorithm.

Geometric inequalities in real Banach spaces with applications

"In this paper, new geometric inequalities are established in real Banach spaces. As an application, a new iterative algorithm is proposed for approximating a solution of a split equality fixed point problem (SEFPP) for a quasi-$\phi$-nonexpansive semigroup. It is proved that the sequence generated by the algorithm converges {\it strongly} to a solution of the SEFPP in $p$-uniformly convex and uniformly smooth real Banach spaces, $p>1$. Furthermore, the theorem proved is applied to approximate a solution of a variational inequality problem. All the theorems proved are applicable, in particular, in $L_p$, $l_p$ and the Sobolev spaces, $W_p^m(\Omega)$, for $p$ such that $2<p<\infty.$ "

On Birkhoff–James orthogonality preservers between real non-isometric Banach spaces

We present a short proof for the fact that if smooth real Banach spaces of dimension three or higher have isomorphic Birkhoff–James orthogonality structures, then they are (linearly) isometric to each other. This generalizes results of Koldobsky and of Wójcik. Moreover, in an arbitrary dimension, we construct examples of non-isometric pairs of non-smooth real Banach spaces that admit norm preserving homogeneous bicontinuous Birkhoff–James orthogonality preservers among them.

Split common fixed point problem for demimetric mappings and Bregman relatively nonexpansive mappings

In this paper, we study the split common fixed point problem for a finite family of demimetric mappings and a finite family of Bregman relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We prove a strong convergence theorem of Halpern's type iteration for finding a solution of the split common fixed point problem. The iterative scheme does not require prior knowledge of operator norm. The obtained result of this paper complements many recent results in this direction.

Littlewood-Paley-Stein Theory and Banach Spaces in the Inverse Gaussian Setting

In this paper we consider Littlewood-Paley functions defined by the semigroups associated with the operator mathcal {A}=-frac {1}{2}{Delta }-xnabla in the inverse Gaussian setting for Banach valued functions. We characterize the uniformly convex and smooth Banach spaces by using L^{p}(mathbb R^{n},gamma _{-1})- properties of the mathcal {A}-Littlewood-Paley functions. We also use Littlewood-Paley functions associated with mathcal {A} to characterize the Köthe function spaces with the UMD property.

A Generalized Explicit Iterative Method for Solving Generalized Split Feasibility Problem and Fixed Point Problem in Real Banach Spaces

In this paper, we propose a generalized explicit algorithm for approximating the common solution of generalized split feasibility problem and the fixed point of demigeneralized mapping in uniformly smooth and 2-uniformly convex real Banach spaces. The generalized split feasibility problem is a general mathematical problem in the sense that it unifies several mathematical models arising in (symmetry and non-symmetry) optimization theory and also finds many applications in applied science. We designed the algorithm in such a way that the convergence analysis does not need a prior estimate of the operator norm. More so, we establish the strong convergence of our algorithm and present some computational examples to illustrate the performance of the proposed method. In addition, we give an application of our result for solving the image restoration problem and compare with other algorithms in the literature. This result improves and generalizes many important related results in the contemporary literature.

"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."

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.

The convex hull and the Carathéodory number of a set in terms of the metric projection operator

We prove that each point of the convex hull of a compact set $M$ in a smooth Banach space $X$ can be approximated arbitrarily well by convex combinations of best approximants from $M$ to $x$ (values of the metric projection operator $P_M(x)$), where $x \in X$. As a corollary, we show that the Carathéodory number of a compact set $M \subset X$ with at most $k$-valued metric projection $P_M$ is majorized by $k$, that is, each point in the convex hull of $M$ lies in the convex hull of at most $k$ points of $M$. Bibliography: 26 titles.