114 research outputs found
A general iterative method for obtaining an infinite family of strictly pseudo-contractive mappings in Hilbert spaces
AbstractIn this work, we consider a general composite iterative method for obtaining an infinite family of strictly pseudo-contractive mappings in Hilbert spaces. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of fixed points, which solves the variational inequality 〈(γf−μF)q,p−q〉≤0,forp∈∩i=1∞F(Ti). Our results improve and extend corresponding ones announced by many others
New Hybrid Steepest Descent Algorithms for Equilibrium Problem and Infinitely Many Strict Pseudo-Contractions in Hilbert Spaces
We propose an explicit iterative scheme for finding a common
element of the set of fixed points of infinitely many strict pseudo-contractive mappings and
the set of solutions of an equilibrium problem by the general iterative method, which solves
the variational inequality. In the setting of real Hilbert spaces, strong convergence theorems
are proved. The results presented in this paper improve and extend the corresponding
results reported by some authors recently. Furthermore, two numerical examples are given
to demonstrate the effectiveness of our iterative scheme
Iterative methods for approximating solutions of certain optimization problems and fixed points problems.
Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.Abstract available in PDF file
Iterative schemes for approximating common solutions of certain optimization and fixed point problems in Hilbert spaces.
Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, we introduce a shrinking projection method of an inertial type with
self-adaptive step size for finding a common element of the set of solutions of Split Gen-
eralized Equilibrium Problem (SGEP) and the set of common fixed points of a countable
family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step
size incorporated helps to overcome the difficulty of having to compute the operator norm
while the inertial term accelerates the rate of convergence of the propose algorithm. Under
standard and mild conditions, we prove a strong convergence theorem for the sequence
generated by the proposed algorithm and obtain some consequent results. We apply our
result to solve Split Mixed Variational Inequality Problem (SMVIP) and Split Minimiza-
tion Problem (SMP), and present numerical examples to illustrate the performance of
our algorithm in comparison with other existing algorithms. Moreover, we investigate the
problem of finding common solutions of Equilibrium Problem (EP), Variational Inclusion
Problem (VIP)and Fixed Point Problem (FPP) for an infinite family of strict pseudo-
contractive mappings. We propose an iterative scheme which combines inertial technique
with viscosity method for approximating common solutions of these problems in Hilbert
spaces. Under mild conditions, we prove a strong theorem for the proposed algorithm and
apply our results to approximate the solutions of other optimization problems. Finally,
we present a numerical example to demonstrate the efficiency of our algorithm in comparison with other existing methods in the literature. Our results improve and complement
contemporary results in the literature in this direction
Iterative algorithms for approximating solutions of variational inequality problems and monotone inclusion problems.
Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban, 2017.In this work, we introduce and study an iterative algorithm independent of the operator
norm for approximating a common solution of split equality variational inequality prob-
lem and split equality xed point problem. Using our algorithm, we state and prove a
strong convergence theorem for approximating an element in the intersection of the set
of solutions of a split equality variational inequality problem and the set of solutions of
a split equality xed point problem for demicontractive mappings in real Hilbert spaces.
We then considered nite families of split equality variational inequality problems and
proposed an iterative algorithm for approximating a common solution of this problem and
the multiple-sets split equality xed point problem for countable families of multivalued
type-one demicontractive-type mappings in real Hilbert spaces. A strong convergence re-
sult of the sequence generated by our proposed algorithm to a solution of this problem was
also established. We further extend our study from the frame work of real Hilbert spaces
to more general p-uniformly convex Banach spaces which are also uniformly smooth. In
this space, we introduce an iterative algorithm and prove a strong convergence theorem for
approximating a common solution of split equality monotone inclusion problem and split
equality xed point problem for right Bregman strongly nonexpansive mappings. Finally,
we presented numerical examples of our theorems and applied our results to study the
convex minimization problems and equilibrium problems
Approximation methods for solutions of some nonlinear problems in Banach spaces.
Doctor of Philosophy in Mathematics. University of KwaZulu-Natal, Durban 2016.Abstract available in PDF file
Theory and Application of Fixed Point
In the past few decades, several interesting problems have been solved using fixed point theory. In addition to classical ordinary differential equations and integral equation, researchers also focus on fractional differential equations (FDE) and fractional integral equations (FIE). Indeed, FDE and FIE lead to a better understanding of several physical phenomena, which is why such differential equations have been highly appreciated and explored. We also note the importance of distinct abstract spaces, such as quasi-metric, b-metric, symmetric, partial metric, and dislocated metric. Sometimes, one of these spaces is more suitable for a particular application. Fixed point theory techniques in partial metric spaces have been used to solve classical problems of the semantic and domain theory of computer science. This book contains some very recent theoretical results related to some new types of contraction mappings defined in various types of spaces. There are also studies related to applications of the theoretical findings to mathematical models of specific problems, and their approximate computations. In this sense, this book will contribute to the area and provide directions for further developments in fixed point theory and its applications
General Iterative Methods for System of Equilibrium Problems and Constrained Convex Minimization Problem in Hilbert Spaces
We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of solutions of system of equilibrium problems and a constrained convex minimization problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by Tian and Liu (2012) and many others. Furthermore, we give numerical example to demonstrate the effectiveness of our iterative scheme
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