21 research outputs found
Self-adaptive inertial algorithms for approximating solutions of split feasilbility, monotone inclusion, variational inequality and fixed point problems.
Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, we introduce a self-adaptive hybrid inertial algorithm for approximating
a solution of split feasibility problem which also solves a monotone inclusion problem
and a fixed point problem in p-uniformly convex and uniformly smooth Banach spaces.
We prove a strong convergence theorem for the sequence generated by our algorithm which
does not require a prior knowledge of the norm of the bounded linear operator. Numerical
examples are given to compare the computational performance of our algorithm with other
existing algorithms.
Moreover, we present a new iterative algorithm of inertial form for solving Monotone Inclusion
Problem (MIP) and common Fixed Point Problem (FPP) of a finite family of
demimetric mappings in a real Hilbert space. Motivated by the Armijo line search technique,
we incorporate the inertial technique to accelerate the convergence of the proposed
method. Under standard and mild assumptions of monotonicity and Lipschitz continuity
of the MIP associated mappings, we establish the strong convergence of the iterative
algorithm. Some numerical examples are presented to illustrate the performance of our
method as well as comparing it with the non-inertial version and some related methods in
the literature.
Furthermore, we propose a new modified self-adaptive inertial subgradient extragradient
algorithm in which the two projections are made onto some half spaces. Moreover, under
mild conditions, we obtain a strong convergence of the sequence generated by our proposed
algorithm for approximating a common solution of variational inequality problems
and common fixed points of a finite family of demicontractive mappings in a real Hilbert
space. The main advantages of our algorithm are: strong convergence result obtained
without prior knowledge of the Lipschitz constant of the the related monotone operator,
the two projections made onto some half-spaces and the inertial technique which speeds
up rate of convergence. Finally, we present an application and a numerical example to
illustrate the usefulness and applicability of our algorithm
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
The Split Common Fixed Point Problem and the Hybrid Method in Banach Spaces (Nonlinear Analysis and Convex Analysis)
In this article, we consider the split common fixed point problem in Banach spaces. Using the hybrid method in mathematical programming, we prove a strong convergence theorem for finding a solution of the split common fixed point problem in Banach spaces. Using this result, we get well-known and new results which are connected with the split feasibility problem and the split common null point problem in Banach spaces
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
Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem
AbstractIn this paper, we introduce an iterative method to approximate a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in real Hilbert spaces. We prove that the sequences generated by the iterative scheme converge strongly to a common solution of the split equilibrium problem, the variational inequality problem and the fixed point problem for a nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area