scholarly journals On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxedη−αMonotone Mapping

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Marwan Amin Kutbi ◽  
Wutiphol Sintunavarat
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Existence Of Solutions ◽  
Monotone Mapping ◽  
Existence Of Solution ◽  
Monotone Mappings ◽  
Solution Existence ◽  
Reflexive Banach Spaces ◽  
New Concepts ◽  
The Solution Existence

We introduce two new concepts of weakly relaxedη-αmonotone mappings and weakly relaxedη-αsemimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxedη-αmonotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxedη-αsemimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.

scholarly journals Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohammed Ali Alghamdi ◽  
Naseer Shahzad ◽  
Habtu Zegeye
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Finite Family ◽  
Monotone Mappings ◽  
Reflexive Banach Spaces ◽  
Convergence Theorems

We study a strong convergence for a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in the framework of real reflexive Banach spaces. As a consequence, convergence for a common fixed point of a finite family of Bergman relatively nonexpansive mappings is discussed. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common solution of a finite family equilibrium problem and a common zero of a finite family of maximal monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

scholarly journals Variational-Like Inequalities and Equilibrium Problems with Generalized Monotonicity in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
N. K. Mahato ◽  
C. Nahak
Keyword(s):  
Equilibrium Problem ◽  
Banach Spaces ◽  
Equilibrium Problems ◽  
Existence Of Solutions ◽  
Existence Of Solution ◽  
Generalized Monotonicity ◽  
Reflexive Banach Spaces ◽  
Strong Pseudomonotonicity ◽  
Pseudomonotone Maps ◽  
Pseudomonotone Mappings

We introduce the notion of relaxed (ρ-θ)-η-invariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps. Using the KKM technique, we establish the existence of solutions for variational-like inequality problems with relaxed (ρ-θ)-η-invariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (ρ-θ)-pseudomonotonicity for bifunctions, and we consider some examples to show that (ρ-θ)-pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (ρ-θ)-pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique.

scholarly journals An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
H. Zegeye ◽  
N. Shahzad
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Variational Inequality Problem ◽  
Monotone Mapping ◽  
Common Point ◽  
Finite Family ◽  
Monotone Mappings ◽  
Nonlinear Operators ◽  
Asymptotically Nonexpansive

We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for a monotone mapping and fixed point of uniformly Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme that converges strongly to a common zero of finite family of monotone mappings under suitable conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

scholarly journals Existence results for second order functional differential and integrodifferential inclusions in Banach spaces

2001 ◽  
Vol 32 (4) ◽  
pp. 315-325
Author(s):  
M. Benchohra ◽  
S. K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Second Order ◽  
Initial Value ◽  
Functional Differential ◽  
Condensing Maps

In this paper we investigate the existence of solutions on a compact interval to second order initial value problems for functional differential and integrodifferential inclusions in Banach spaces. We shall make use of a fixed point theorem for condensing maps due to Martelli.

Biconnected Multifunctions of Trees which have an end Point as Fixed Point or Coincidence

1971 ◽  
Vol 23 (3) ◽  
pp. 461-467 ◽  
Author(s):  
Helga Schirmer
Keyword(s):  
Fixed Point ◽  
Monotone Mapping ◽  
Monotone Mappings ◽  
Point Sets ◽  
End Point ◽  
Continuous Use ◽  
Fixed Point Sets

It was proved almost forty years ago that every mapping of a tree into itself has at least one fixed point, but not much is known so far about the structure of the possible fixed point sets. One topic related to this question, the study of homeomorphisms and monotone mappings of trees which leave an end point fixed, was first considered by G. E. Schweigert [6] and continued by L. E. Ward, Jr. [8] and others. One result by Schweigert and Ward is the following: any monotone mapping of a tree onto itself which leaves an end point fixed, also leaves at least one other point fixed.It is further known that not only single-valued mappings, but also upper semi-continuous (use) and connected-valued multifunctions of trees have a fixed point [7], and that two use and biconnected multifunctions from one tree onto another have a coincidence [5].

An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces

2018 ◽  
Vol 9 (3) ◽  
pp. 167-184 ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Ferdinard Udochukwu Ogbuisi ◽  
Oluwatosin Temitope Mewomo
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Monotone Operators ◽  
Convergence Result ◽  
Reflexive Banach Spaces ◽  
Strongly Monotone Operators ◽  
System Of Equilibrium Problems

Abstract In this paper, we propose an iterative algorithm for approximating a common fixed point of an infinite family of quasi-Bregman nonexpansive mappings which is also a solution to finite systems of convex minimization problems and variational inequality problems in real reflexive Banach spaces. We obtain a strong convergence result and give applications of our result to finding zeroes of an infinite family of Bregman inverse strongly monotone operators and a finite system of equilibrium problems in real reflexive Banach spaces. Our result extends many recent corresponding results in literature.

scholarly journals Split equality common null point problem for Bregman quasi-nonexpansive mappings

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 3917-3932
Author(s):  
Ali Abkar ◽  
Elahe Shahrosvand
Keyword(s):  
Fixed Point ◽  
Equilibrium Problem ◽  
Banach Spaces ◽  
Optimization Problem ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Fixed Point Problem ◽  
Null Point ◽  
Point Problem ◽  
Reflexive Banach Spaces

In this paper, we introduce a new algorithm for solving the split equality common null point problem and the equality fixed point problem for an infinite family of Bregman quasi-nonexpansive mappings in reflexive Banach spaces. We then apply this algorithm to the equality equilibrium problem and the split equality optimization problem. In this way, we improve and generalize the results of Takahashi and Yao [22], Byrne et al [9], Dong et al [11], and Sitthithakerngkiet et al [21].

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