Abstract
We introduce a qϱ-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation U=B+∑i=1mAi*G(U)Ai, where B is an n×n Hermitian positive definite matrix, A1, A2, … Am are n×n matrices, and G is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot.
Highlights
It is well known that Banach Contraction Principle (BCP) from 1922 was a starting point for the development of an important area called metric fixed point theory, which has an enormous field of application
Motivated by the concepts of quasi partial metric space and binary relation, in Section 3, we introduce a modified implicit relation of [19] with some illustrations
Investigations of fixed point problems in so-called metric fixed point theory have been broadened to problems formulated in terms of q$ -quasi implicit contractive condition for a self-mapping on a relational quasi partial metric space
Summary
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