A hybrid iterative method for common solutions of variational inequality problems and fixed point problems for single-valued and multi-valued mappings with applications

Revie

Weak Convergence Theorem for Infinite Families of Nonlinear Mappings in Banach Spaces (Study on Nonlinear Analysis and Convex Analysis)

In this article, we prove a weak convergence theorem of Mann's type iteration for infinite families of extended generalized hybrid mappings in a Banach space satisfying Opial's condition. This theorem solves a problem posed by Hojo and Takahashi [8]. Using this result, we get well-known and new weak convergence theorems in a Banach space. In particular, we obtain a weak convergence theorem of Mann's type iteration for finite families of extended generalized hybrid mappings in a Banach space

Ergodic approximations via matrix regularization approach

AbstractIn this paper we use a matrix approach to approximate solutions of variational inequalities in Hilbert spaces. The methods studied combine new or well-known iterative methods (as the original Mann method) with regularized processes that involve regular matrices in the sense of Toeplitz. We obtain ergodic type results and convergence

Generalizations of some fixed point theorems in banach and metric spaces

A fixed point of a mapping is an element in the domain of the mapping that is mapped into itself by the mapping. The study of fixed points has been a field of interests to mathematicians since the discovery of the Banach contraction theorem, i.e. if is a complete metric space and is a contraction mapping (i.e. there exists such that for all ), then has a unique fixed point. The Banach contraction theorem has found many applications in pure and applied mathematics. Due to fixed point theory being a mixture of analysis, geometry, algebra and topology, its applications to other fields such as physics, economics, game theory, chemistry, engineering and many others has become vital. The theory is nowadays a very active field of research in which many new theorems are published, some of them applied and many others generalized. Motivated by all of this, we give an exposition of some generalizations of fixed point theorems in metric fixed point theory, which is a branch of fixed point theory about results of fixed points of mappings between metric spaces, where certain properties of the mappings involved need not be preserved under equivalent metrics. For instance, the contractive property of mappings between metric spaces need not be preserved under equivalent metrics. Since metric fixed point theory is wide, we limit ourselves to fixed point theorems for self and non-self-mappings on Banach and metric spaces. We also take a look at some open problems on this topic of study. At the end of the dissertation, we suggest our own problems for future research