Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

For q > 1 and p > 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.

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A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000366 ◽

2018 ◽

Vol 62 (1) ◽

pp. 241-257 ◽

Cited By ~ 4

Author(s):

C. E. Chidume ◽

M. O. Uba ◽

M. I. Uzochukwu ◽

E. E. Otubo ◽

K. O. Idu

Keyword(s):

Banach Space ◽

Variational Inequality ◽

Strong Convergence ◽

Real Banach Space ◽

Maximal Monotone ◽

Variational Inequality Problems ◽

Iterative Sequence ◽

Uniformly Convex ◽

Uniformly Smooth ◽

Monotone Maps

AbstractLetEbe a uniformly convex and uniformly smooth real Banach space, and letE* be its dual. LetA : E→ 2E*be a bounded maximal monotone map. Assume thatA−1(0) ≠ Ø. A new iterative sequence is constructed which convergesstronglyto an element ofA−1(0). The theorem proved complements results obtained on strong convergence ofthe proximal point algorithmfor approximating an element ofA−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.

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Strong convergence theorems for strongly monotone mappings in Banach spaces

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.37655 ◽

2021 ◽

Vol 39 (1) ◽

pp. 169-187

Author(s):

Mathew O. Aibinu ◽

Oluwatosin Mewomo

Keyword(s):

Strong Convergence ◽

Banach Spaces ◽

Lyapunov Functions ◽

Dual Space ◽

Real Banach Space ◽

Monotone Mappings ◽

Uniformly Convex ◽

Strongly Monotone ◽

Uniformly Smooth ◽

Strongly Monotone Mappings

Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : E\rightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)\neq \emptyset$. Inspired by Alber \cite{b1}, we introduce Lyapunov functions and use the new geometric properties of Banach spaces to show the strong convergence of an iterative algorithm to the solution of $Ax=0$.

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Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces

ISRN Mathematical Analysis ◽

10.5402/2011/576108 ◽

2011 ◽

Vol 2011 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Yekini Shehu

Keyword(s):

Banach Space ◽

Fixed Point ◽

Strong Convergence ◽

Iterative Process ◽

Real Banach Space ◽

Finite Family ◽

Uniformly Convex ◽

Multivalued Maps ◽

Convergence Theorems ◽

Uniformly Convex Banach Spaces

We introduce a new iterative process to approximate a common fixed point of a finite family of multivalued maps in a uniformly convex real Banach space and establish strong convergence theorems for the proposed process. Furthermore, strong convergence theorems for finite family of quasi-nonexpansive multivalued maps are obtained. Our results extend important recent results.

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Approximation of solutions of Hammerstein equations with monotone mappings in real Banach spaces

Carpathian Journal of Mathematics ◽

10.37193/cjm.2019.03.05 ◽

2019 ◽

Vol 35 (3) ◽

pp. 305-316

Author(s):

C. E. CHIDUME ◽

◽

A. ADAMU ◽

L. C. OKEREKE ◽

◽

...

Keyword(s):

Real Banach Space ◽

Maximal Monotone ◽

General Setting ◽

Computational Time ◽

Monotone Mappings ◽

Uniformly Convex ◽

Lp Spaces ◽

Hammerstein Equation ◽

Uniformly Smooth ◽

Number Of Iterations

Let E be a uniformly convex and uniformly smooth real Banach space with dual space, E∗. Let F : E → E∗, K : E∗ → E be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u+KF u = 0. This theorem is a significant improvement of some important recent results which were proved in Lp spaces, 1 < p ≤ 2 under the assumption that F and K are bounded. This restriction on K and F have been dispensed with even in the more general setting considered here. Finally, a numerical experiment is presented to illustrate the convergence of the sequence of the algorithm which is found to be much faster, in terms of the number of iterations and the computational time than the convergence obtained with existing algorithms.

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Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2013/328740 ◽

2013 ◽

Vol 2013 ◽

pp. 1-18

Author(s):

Lu-Chuan Ceng ◽

Abdul Latif ◽

Abdullah E. Al-Mazrooei

Keyword(s):

Banach Space ◽

Variational Inequality ◽

Strong Convergence ◽

Banach Spaces ◽

Nonexpansive Mappings ◽

Approximation Methods ◽

Uniformly Convex ◽

Viscosity Approximation ◽

Uniformly Smooth ◽

Viscosity Approximation Methods

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications

Journal of Applied Mathematics ◽

10.1155/2012/185104 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 2

Author(s):

Uamporn Witthayarat ◽

Yeol Je Cho ◽

Poom Kumam

Keyword(s):

Banach Space ◽

Strong Convergence ◽

Banach Spaces ◽

Iterative Algorithm ◽

Hilbert Spaces ◽

Accretive Operator ◽

Uniformly Convex ◽

Common Solution ◽

Uniformly Smooth ◽

Convex Feasibility

The aim of this paper is to introduce an iterative algorithm for finding a common solution of the sets(A+M2)−1(0) and(B+M1)−1(0), where M is a maximal accretive operator in a Banach space and, by using the proposed algorithm, to establish some strong convergence theorems for common solutions of the two sets above in a uniformly convex and 2-uniformly smooth Banach space. The results obtained in this paper extend and improve the corresponding results of Qin et al. 2011 from Hilbert spaces to Banach spaces and Petrot et al. 2011. Moreover, we also apply our results to some applications for solving convex feasibility problems.

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On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces

Journal of Mathematics ◽

10.1155/2019/1356918 ◽

2019 ◽

Vol 2019 ◽

pp. 1-15 ◽

Cited By ~ 2

Author(s):

M. De la Sen ◽

Mujahid Abbas

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Iterative Process ◽

Iterative Scheme ◽

Convex Banach Space ◽

The Self ◽

Uniformly Convex ◽

Cyclic Structure ◽

Best Proximity Points ◽

Uniformly Convex Banach Spaces

This paper proposes a generalized modified iterative scheme where the composed self-mapping driving can have distinct step-dependent composition order in both the auxiliary iterative equation and the main one integrated in Ishikawa’s scheme. The self-mapping which drives the iterative scheme is a perturbed 2-cyclic one on the union of two sequences of nonempty closed subsets Ann=0∞ and Bnn=0∞ of a uniformly convex Banach space. As a consequence of the perturbation, such a driving self-mapping can lose its cyclic contractive nature along the transients of the iterative process. These sequences can be, in general, distinct of the initial subsets due to either computational or unmodeled perturbations associated with the self-mapping calculations through the iterative process. It is assumed that the set-theoretic limits below of the sequences of sets Ann=0∞ and Bnn=0∞ exist. The existence of fixed best proximity points in the set-theoretic limits of the sequences to which the iterated sequences converge is investigated in the case that the cyclic disposal exists under the asymptotic removal of the perturbations or under its convergence of the driving self-mapping to a limit contractive cyclic structure.

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

Journal of Function Spaces ◽

10.1155/2015/478437 ◽

2015 ◽

Vol 2015 ◽

pp. 1-7 ◽

Cited By ~ 4

Author(s):

Messaoud Bounkhel

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Convex Set ◽

Variational Problems ◽

Generalized Projection ◽

Uniformly Convex ◽

Uniformly Smooth ◽

Nonconvex Sets ◽

Generalized Projections ◽

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.

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Strong Convergence Theorems for Families of Weak Relatively Nonexpansive Mappings

Abstract and Applied Analysis ◽

10.1155/2011/251612 ◽

2011 ◽

Vol 2011 ◽

pp. 1-19

Author(s):

Yekini Shehu

Keyword(s):

Strong Convergence ◽

Convergence Theorem ◽

Real Banach Space ◽

Hybrid Methods ◽

Nonexpansive Mappings ◽

Strong Convergence Theorem ◽

Monotone Mappings ◽

Relatively Nonexpansive Mappings ◽

Uniformly Smooth ◽

Relatively Nonexpansive

We construct a new Halpern type iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalizedf-projection operator. Using this result, we discuss strong convergence theorem concerning generalH-monotone mappings. Our results extend many known recent results in the literature.

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