Investigation of some fixed point theorems for Various types of mappings in abstract spaces
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Date
2022-03-01
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University of Dar es Salaam
Abstract
Fixed point theory is a fundamental tool in nonlinear functional analysis. It has many
applications e.g. in Approximation theory, Optimisation theory, Variation inequalities,
Game theory and Economics etc. The fixed point theory is a powerful tool to determine
existence and uniqueness of the solutions of Differential equations, Integral equations,
Partial differential equations, Fractional differential equations, Matrix equations and
Functional equations. A fixed point problem can be stated as follows: Let X be a
non-empty set and T : X !X be a mapping. A point x 2 X is a fixed point or invariant
point of the mapping T if Tx = x. Does a fixed point exist for every map? Moreover,
if such a point exists, is it unique, and how can we find it? We can distinguish three
major approaches in fixed point theory: metric approach, topological approach, and
discrete approach. Historically, these approaches were initiated by the discovery of
major theorems: Brouwer fixed point theorem, Banach fixed point theorem, and Tarski
fixed point theorem. In this thesis, we are concerned with the second approach, the
metric fixed point theory. Fixed point theory and Banach contraction principle have
been studied and generalised in different spaces, and various fixed-point theories were
developed. Hence, this study investigated the fixed point theorems for various types
of constructive mappings in various abstract spaces. The Banach contraction method
has been used to obtain the fixed point theorems and their applications to ordinary and
fractional differential equations. This study showed several ways to construct, extend,
formulate, prove, and generalise fixed point theorems in abstract spaces using various
maps, i.e., single valued maps, multivalued maps, hybrid maps and implicit maps. Also,
the generalisation is done by considering relatively large classes of abstract spaces;
Cone metric space, b-metric space, partial b-metric spaces, metric-like spaces, partial
metric spaces, quasi partial Sb-metric-like spaces, and G-metric spaces. Finally, the
proofs of the results are established by finding coincidence points or common fixed
points.
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This Theses was Published by University of Dar es salaam in 2022
Keywords
Investigation of some fixed point theorems