Role of Contractive Type Mappings in Fixed Point Theory

Abstract

Last three decades the role of contractive type mappings in fixed point theory gain a considerable importance and several works have been appeared in the literature using various kind of contractive type mappings viz. contractive mappings, weak contractive mappings, rational contractive mappings, etc. In the present thesis, further developments are to be made in the same direction and several new results which ensure the existence of fixed points, common fixed points and contractive mappings are investigated. newlineThe thesis is comprised of eight chapters along with a chapter containing concluding remark and scope of future research. Chapter 1 is the introductory chapter, where brief introduction and adequate literature survey on fixed point theory and various types of spaces have been presented. In Chapter 2, we derive new common fixed point results for a pair of compatible as well as relatively asymptotically regular mappings defined on an orbitally ordered complete metric space using the orbital continuity of the involved mappings. Also immediate consequences of all theorems are established as consequence. In Chapter 3, we have presented several results on common fixed points using (and#966;_1-and#966;_2-and#966;_3 ) -contraction condition including rational expressions in orbitally 0-complete partial metric spaces. Chapter 4 deals with the new concepts of cyclic (and#945;,and#946;)-type and#947;-FG -contractive mapping and prove some fixed point theorems for such mappings in a complete b-metric space. In Chapter 5, we have extended the results those obtained in [R. P. Agarwal, E. Karapand#305;nar and A. F. Roldan-Lopez-de-Hierro, On an extension of contractivity conditions via auxiliary functions. Fixed Point Theory Appl. (2015) 2015:104] using families of control functions, here also through conditions that involve and#945;-admissibility of type S. Chapters 6, extends and generalizes the results of Mukheimer [(and#945;-and#968;-and#966;)-contractive mappings in ordered partial b-metric spaces, J. Nonlinear Sci. Appl. 7(2014), 168 179]. A new concept of (and#946;-and#968;_1-and#968;_2 )-contractive mapping usi