Metric Fixed Point Theorems for Locally and Globally Contractions

Abstract

"Fixed point theorems agreement with the gurantee that the functional equation w = Sw has at least one solution. The solution of mapping S are name as x points. A great variation of issues of analysis and applied mathematics examine to realize the solutions of nonlinear functional equations which can be derived in terms of detecting x points of a nonlinear mappings. FP theory is very useful instument to detect the solution of mathematical modeling by using techique e.g science, engineering, medical sciences, economics etc. has a great deal of applications in our FPT." "In 1922, S. Banach established remarkable xed point theorem known as the Banach Contraction Principle(BCP)which is one of the most important result of analysis and considered to be the main source of metric xed point theory. It is the most widely applied xed point result in many branches of mathematics because it requires the structure of complete metric space with contractive condition on the map which is easy to test in this setting. BCP has genralized in di⁄erent directions. In fact, there is vast amount of literature dealing with extensions/generalizations of this remarkable theorem." "Amultivaluedfunctionisafunctionwhichtakesvaluesasaset. Inlastfewyears, conjecture of set value functions has purpose in di⁄erent ways. In 1969, comprehensively examine of BCP class xpoint theorems of set value maps was started with poineer work of Nadler [106], who evidence that a set value map in the framework of complete Y metric space into a class of closed bounded subsets of Y has xpoints." "In 2012, Samet et al. [114] introduced the concept of -admissible mappings and suggested a very interesting class of mapping, - -contraction mappings, to investigate the existence and uniqueness of a xed point. Further Mohammadi et al. [99] extended some results on xed points of - -Ciric generalized multifunctions. Asl et al. [57] launched a notion of -contractive multifunctions and inagurated xed point result for multifunctions. Recently, Hussain et al. [59] established certain new xed point results for multi-valued as well as singlevalued mappings satisfying an - -contractive conditions in complete metric space. The notion of an -admissible mapping has been characterized in many directions, for more details see [16, 30, 31, 73, 99, 108, 109, 112, 118, 121]." "In 2012, Wrdowski [123] construct a new breed of contraction called F-contraction and built a new xpoint theorem using F-contraction. Wardowski et. al [124] introduced the notion of F-weak contraction, which generalizes some known results from the literature. He gave an interesting generalization of Banach contraction principle. Afterwards Secelean [115], proved xed point theorems consisting of F-contractions by iterated function systems. Piri et al. [107] extended the result of Wardowski by applying some weaker conditions on the selfmap in a complete metric space. Cosentino and Vetro [49] presented some xed point results of Hardy-Rogers-type for self-mappings on complete metric spaces and complete ordered metric spaces." "Abbas et al. [8] further generalized the concept of F-contraction and proved certain xed and common xed point results. Hussain and Salimi [67] introduced an -GF-contraction with respect to a general family of functions G and established Wardowski type xed point results in metric and ordered metric spaces. Further Altun et al. [10] extended multivalued mappings with -Distance and established xed point results in complete metric spaces. Lately, Acar et al. [11] founded an idea of generalized set value map of F-contraction and created a xed point result, which gave proper extension of few multivalued xed point theorems including Nadlers result in [106]. Currently, Minak [97] showed few new xpoint results for cirik class outcome genralized F-contractions on complete metric spaces. Sgroi and Vetro [117] established the results to procure xpoint of set value maps as a generalization of Nadlers result [106]. Very recently, Abbas et al. [5] introduced the concept of multivalued f-almost F-contraction which generalizes the class of multivalued almost contraction mapping and obtained coincidence point results. Naturally, many authors have started to investigate the existence and uniqueness of a xed point theorem via F-contraction mappings and variations of the concept of F-contractive type mappings, for more details see [19, 45, 86]." "Azam et al. [32] proved a signicant result concerning the existence of xed points of a mapping satisfying contractive conditions on a closed ball of a complete metric space. Shoaib et al. [27] introduced a new concept of closed ball in dislocated metric space to approximate the unique solution of nonlinear functional equations. Shoaib also establised xed point, common xed point theorems for two or more contractive dominated mappings on a closed ball in an ordered dislocated metric space, for more details see [24, 28]." "This thesis made up of four chapters. Each chapter start with a detail introduction." "Chapter 1 consists of some basic denitions, few fundamentals results which are helpful in the upcoming chapters." "Chapter 2 is allocate to enquiry of existence of xpoint, by use of general contraction for obtaining common xed points. The aim of this chapter is to upgrade a idea of Geraghty contraction and create few new result for -admissible mappings with respect to ; satisfying modied ()-contractive condition in the framework of complete metric spaces. We prove new xed point theorems in complete metric space by using variety of -Geraghty contraction and rational -Geraghty contraction maps." "Chapter 3 inspect single and multi-valued F-contraction mappings. We built a concept of · Ciri·c type --GF-contraction and certied few new xpoint theorems for single and multivalued mappings in the setting of complete metric spaces. We extended the concept of multivalued --F-contraction and --F-contraction and secured class of new Wardowski type xpoint theorems in structure of complete metric spaces." "Chapter 4 aims to introduce the notion of generalized dynamic process for generalized (f;L)-almost F-contraction mappings and to obtain coincidence and common xed point results for such process. We discuss applications of our theorem and obtain the existence and uniqueness of common solution of system of functional equations in dynamical programing and the existence and uniqueness of common solution of system of Volterra type integral equations." "Chapter 5 deals with locally F-contraction and initiate a idea of F-contraction on closed ball. We established some newly results for F-contraction and GF-contraction involving closed ball in complete metric spaces." "I desire to convey my heartfelt recognition to my worthy supervisor Prof. Dr. Muhammad Arshad without whose heartfelt counselling and antique counsellor this dissertation is not achievable. I am grateful to Prof. Dr. Muhammad Arshad, Dean, Faculty of Basic & Applied Sciences, chairman department of mathematics & statistics for his altruistic cooperation in every part of my research. During my Ph.D. education, a great deal of encouragement from the faculty at International Islamic University, Islamabad, Pakistan, and specially department of Mathematics motivate and support to me for which I am thankful. Concluding, I thank you of my family members for their love and support round my research."