Fixed points on proximinal sets and one step iterative scheme

Abstract

In Chapter (1), we discuss some basic definitions, theorems about fixed and coincidence point. Also, we give a brief discussion about different types of contraction, nonexpansive, mean nonexpansive and p-mean nonexpansive mappings. In Chapter (2), we study different iterative schemes and show how they are very useful to find a fixed point for single valued and multivalued mappings. In Chapter (3), we present some of our new results as: Section 3.1: in this section, we found fixed point result for three φ-weak contraction mappings. Section 3.2: in this section, we proved fixed point result for finite φ-weak contraction mappings. In Chapter (4), we discuss the following: Section 4.1: in this section, we give a brief introduction about the modular metric space. We also study some basic definitions, examples and fixed point theorems in the case of single valued and multivalued mappings. Section 4.2: in this section, we show how P. L. Chebyshev discovered the best approximation and proximinal sets. Section 4.3: in this section, we study the Volterra and Fredholm integral equations. Section 4.4: in this section, we introduce proximinality, conjoint F-contraction and conjoint F-contraction of Hardy-Rogers type for two multivalued mappings on proximinal sets in a regular modular metric space. We also use the new definitions to find common fixed point results for two multivalued mappings. Moreover, we enhance our results by giving an application in integral equations. Section 4.5: in this section, we generalize our previous results by finding common fixed point results for several multivalued mappings on proximinal sets in a regular modular metric space. Furthermore, we enhance our results by giving an application in integral equations. Finally, In Chapter 5, we get fixed point results on a countably normed space.