scholarly journals An algorithm for approximating a common solution of some nonlinear problems in Banach spaces with an application

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdulmalik U. Bello ◽  
Monday O. Nnakwe
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Equilibrium Problem ◽  
Real Banach Space ◽  
Nonlinear Problems ◽  
Uniformly Convex ◽  
Convex Minimization Problem ◽  
Common Solution ◽  
Uniformly Smooth ◽  
Contractive Map

AbstractIn this paper, we construct a new Halpern-type subgradient extragradient iterative algorithm. The sequence generated by this algorithm converges strongly to a common solution of a variational inequality, an equilibrium problem, and a J-fixed point of a continuous J-pseudo-contractive map in a uniformly smooth and two-uniformly convex real Banach space. Also, the theorem is applied to approximate a common solution of a variational inequality, an equilibrium problem, and a convex minimization problem. Moreover, a numerical example is given to illustrate the implementability of our algorithm. Finally, the theorem proved complements, improves, and unifies some related recent results in the literature.

scholarly journals Solving split generalized mixed equality equilibrium problems and split equality fixed point problems for nonexpansive-type maps

2020 ◽  
Vol 36 (1) ◽  
pp. 119-126
Author(s):  
M. O. NNAKWE
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Real Banach Space ◽  
Equilibrium Problems ◽  
Fixed Point Problems ◽  
Strong Convergence Theorem ◽  
Uniformly Convex ◽  
Type Condition ◽  
Common Solution ◽  
Uniformly Smooth

"Let X be a 2-uniformly convex and uniformly smooth real Banach space. In this paper, an iterative algorithmofKrasnosel’skii-typeis constructed and used to approximate a common solution ofsplit generalized mixed equalityequilibrium problems(SGMEEP)andsplit equality fixed point problems(SEFPP)forquasi-ψ-nonexpansive maps.A strong convergence theorem of the sequence generated by this algorithm is proved without imposing anycompactness-type condition on either the operators or the space considered. The theorem proved improves andcomplements important recent results in the literature. "

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

2018 ◽  
Vol 62 (1) ◽  
pp. 241-257 ◽  
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.

scholarly journals Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

2016 ◽  
Vol 25 (1) ◽  
pp. 107-120
Author(s):  
T. M. M. SOW ◽  
◽  
C. DIOP ◽  
N. DJITTE ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Real Banach Space ◽  
Monotone Mappings ◽  
Uniformly Convex ◽  
Hammerstein Equation ◽  
Uniformly Smooth ◽  
Monotone Maps

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.

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

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1365
Author(s):  
Yanlai Song ◽  
Mihai Postolache
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Unique Solution ◽  
Variational Inequality Problem ◽  
Inclusion Problem ◽  
Numerical Examples ◽  
Common Solution ◽  
Variational Inclusion Problem ◽  
Uniformly Smooth

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.

Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

2016 ◽  
Vol 25 (1) ◽  
pp. 107-120
Author(s):  
T. M. M. SOW ◽  
◽  
C. DIOP ◽  
N. DJITTE ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Real Banach Space ◽  
Monotone Mappings ◽  
Uniformly Convex ◽  
Hammerstein Equation ◽  
Uniformly Smooth ◽  
Monotone Maps

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.

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

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.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
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.

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

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
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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