2-uniformly Smooth Banach Space Research Articles

Directional Differentiability of the Metric Projection Operator in Uniformly Convex and Uniformly Smooth Banach Spaces

Directional Differentiability of the Metric Projection Operator in Uniformly Convex and Uniformly Smooth Banach Spaces

Inertial iterative method for a generalized mixed equilibrium, variational inequality and a fixed point problems for a family of quasi-ϕ-nonexpansive mappings

We introduce an inertial type gradient projection hybrid iterative method for finding a common solution of generalized mixed equilibrium, variational inequality and fixed point problems in a twouniformly convex and uniformly smooth Banach space. Next, we analyze the strong convergence for a common solution of problem. Furthermore, we carry out some consequences and present a numerical example to show and tell the applicability of main theorem. Our result improves, unifies, generalizes and extends ones from several earlier works.

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 nonlinear variational-like inclusion problems in 2-uniformly smooth Banach spaces

In this manuscript, we introduce and study the existence of a solution of a system of generalized nonlinear variational-like inclusion problems in 2-uniformly smooth Banach spaces by using $H(.,.)$-$eta$-proximal mapping. The method used in this paper can be considered as an extension of methods for studying the existence of solutions of various classes of variational inclusions considered and studied by many authors in 2-uniformly smooth Banach spaces. Some important results, theorems and the existence of solution of the proposed system of generalized nonlinear variational-like inclusion problems have been derived.

A theorem for solving Banach generalized system of variational inequality problems and fixed point problem in uniformly convex and 2-uniformly smooth Banach space

In this paper, we consider a Banach generalized system of variational inequality problems by using the concept of Kangtunyakarn (Fixed Point Theory Appl 2014:123, 2014) and showed the equivalence between a Banach generalized system of variational inequality problems and fixed point problems. And also, using modified viscosity iterative method, we prove a strong convergence theorem for finding a common solution of a Banach generalized system of variational inequality problems and fixed point problems for a nonexpansive mapping. The main theorem presented in this paper extend the corresponding result of variational inequality problems introduced by Aoyama et al. (Fixed Point Theory Appl 2006:35390, https://doi.org/10.1155/FPTA/2006/35390 , 2006). Moreover, we give some numerical examples for supporting our main theorem.

Remarks on a recent paper titled: \u201cOn the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces\u201d

In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition 0<kappa < frac{1}{sqrt{2}}, as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is E=l_{p}, 2leq p<infty . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition 0<kappa leq frac{1}{sqrt{2}}, then E is necessarily l_{2}, a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

On a Convergence Theorem for the General Noor Iteration Process in Uniformly Smooth Banach Spaces

In this paper, two different classes of mappings namely, uniformly continuous asymptotically nonexpansive and uniformly continuous asymptotically demicontractive mappings are considered on the general modified Noor iteration process with errors and proved to converge strongly to the fixed point of uniformly continuous asymptotically demicontractive mappings in uniformly smooth Banach spaces. The new result can be viewed as an improvement to a multitude of results in the fixed point theory especially those of Xu and Noor [8], Owojori and Imoru [5] and also the results of Owojori [6].

Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces

In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.

Boundary Point Method and Mann-Dotson’s Algorithm for Nonself Strict Pseudo-contractive Mappings in Uniformly Smooth Banach Spaces

Let C be a closed, convex and nonempty subset of a q-uniformly smooth Banach space X. Let be a strict pseducontractive inward mapping. We consider the boundary point map depending on C and T defined by , for all . Then for a suitable step-by-step construction of the control coefficients by using the function , we show the convergence of the Mann-Dotson algorithm to a fixed point of T. We obtain strong convergence if and weak convergence if .

New semi-implicit midpoint rule for zero of monotone mappings in Banach spaces

Let E be a nonempty closed uniformly convex and 2-uniformly smooth Banach space with dual E∗ and A : E∗ → E be Lipschitz continuous monotone mapping with A− 1(0) ≠ ∅. A new semi-implicit midpoint rule (SIMR) with the general contraction for monotone mappings in Banach spaces is established and proved to converge strongly to x∗ ∈ E, where Jx∗ ∈ A− 1(0). Moreover, applications to convex minimization problems, solution of Hammerstein integral equations, and semi-fixed point of a cluster of semi-pseudo mappings are included.

J-variational inequalities and zeroes of a family of maximal monotone operators by sunny generalized nonexpansive retraction

In this paper, using generalized resolvents and retractions onto shrinking closed subsets, we present new extragradient algorithms for finding the solution of a J-variational-like inequality, the zero points of a family of maximal monotone operators, and the solution set of mixed equilibrium problems for a J- $$\alpha $$ -inverse-strongly monotone-like operator in a 2-uniformly convex and 2-uniformly smooth Banach space. By introducing the new definitions, we prove strong convergence of generated iterates in the extragradient methods. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in the literature to illustrate the usability of our results.

A splitting algorithm in a uniformly convex and 2-uniformly smooth Banach space

A splitting algorithm in a uniformly convex and 2-uniformly smooth Banach space

Implicit and explicit iterative algorithms for hierarchical variational inequality in uniformly smooth Banach spaces

The purpose of this paper is to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms which converge strongly to a unique solution of the hierarchical variational inequality problem. Our results improve and extend the corresponding results announced by some authors.

System of nonlinear variational inclusion problems with $(A,\eta)$-maximal monotonicity in Banach spaces

This paper deals with a new system of nonlinear variational inclusion problems involving $(A,\eta)$-maximal relaxed monotone and relative $(A,\eta)$-maximal monotone mappings in 2-uniformly smooth Banach spaces. Using the generalized resolvent operator technique, the approximation solvability of the proposed problem is investigated. An iterative algorithm is constructed to approximate the solution of the problem. Convergence analysis of the proposed algorithm is investigated. Similar results are also proved for other system of variational inclusion problems involving relative $(A,\eta)$-maximal monotone mappings and $(H,\eta)$-maximal monotone mappings.

Iterative algorithms for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces

The purpose of this paper is to study some iterative algorithms for finding a common element of the set of solutions of systems of variational inequalities for inverse-strongly accretive mappings and the set of fixed points of an asymptotically nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach space or uniformly convex and q-uniformly smooth Banach space. Strong convergence theorems are obtained under suitable conditions. We also give some numerical examples to support our main results. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.

Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

For q &gt; 1 and p &gt; 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

Implicit Variational Inclusions and Algorithms Involving (A, η)-monotone Operators in 2-uniformly Smooth Banach Spaces

Implicit Variational Inclusions and Algorithms Involving (A, η)-monotone Operators in 2-uniformly Smooth Banach Spaces

Strong convergence of some algorithms for $$\lambda $$ λ -strict pseudo-contractions in Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of common fixed points of an infinite family of strictly pseudo-contractive mappings in a real 2-uniformly smooth Banach space. Then we proved a strong convergence theorem under some suitable conditions. Our results generalize and improve several recent results.

Iterative Methods for Nonconvex Equilibrium Problems in Uniformly Convex and Uniformly Smooth Banach Spaces

We suggest and study the convergence of some new iterative schemes for solving nonconvex equilibrium problems in Banach spaces. Many existing results have been obtained as particular cases.

Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

The present paper is devoted to the study of the generalized projectionπK:X∗→K, whereXis a uniformly convex and uniformly smooth Banach space andKis a nonempty closed (not necessarily convex) set inX. Our main result is the density of the pointsx∗∈X∗having unique generalized projection over nonempty close sets inX. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.