Existence and Uniqueness of Solutions for Nonlinear Functional Equations

Abstract

In 1922, the Polish mathematician Stefan Banach established a signicant xed point theorem known as the Banach Contraction Principle, BCP which is one of the most prominent results 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 a contractive condition on the map which is easy to test in this setting. The BCP has been expanded in many di⁄erent directions. In fact, there is a huge amount of literature dealing with extensions/generalizations of this important theorem. A multivalued function is the one which takes values as a set. In the last forty years, the theory of multivalued functions has progressed in a number of ways. In 1969, the systematic analysis of Banach type xed theorems involving multivalued mappings began with the work of Nadler [78], who declared that a multivalued contractive mapping of a complete metric space @ into the family of closed bounded subsets of @ has a xed point. The analysis of metric spaces (MS) proved a most important tool for many elds both in pure and applied sciences such as biology, medicine, physics and computer science (see [62], [97]). Some generalizations of a MS have been suggested by some writers, such as rectangular MS, semi MS, pseudo MS, probabilistic MS, fuzzy MS, quasi MS, quasi semi MS, D-MS, and cone MS (see [3, 35, 40, 89, 90]) . Branciari [28] brought forward the idea of a generalized MS replacing the triangle inequality by a rectangular type inequality. Branciari advanced BCP in such spaces. In 1994, Matthews [65] initiated partial MS and got di⁄erent xed point theorems. Actually, he expressed that the BCP can be generalized to the partial metric context for applications in program verication. Romaguera [91] initiated the idea of 0-Cauchy sequences (C-seq) and 0-complete partial MS (PMS) and proved some characterizations of PMS in terms of completeness and 0-completeness. Mustafa and Sims [69] initiated the @GMS as a generalization of the notion of MS. Mustafa and Sims acquired some xed point theorems for mappings satisfying di⁄erent contractive conditions for more xed point results on @G-MS (see [69]-[76]). In 2014, Aghajani et al.[9] presented the notion of @GbMS and proved that the class of @GbMS is practically greater than that of @G-MS given in [69]. In 2012, Wardowski [103] initiated a new type of contraction called Fcontraction and proved a new xed point (FP) theorem concerning Fcontraction. Wardowski generalized the BCP. Afterwards Secelean [96] proved FP theorems consisting of Fcontractions by Iterated function systems. Piri et al.[84] proved a FP result for FSuzuki contractions for some weaker conditions on the self map which generalizes the result of Wardowski. Later on, Acar et al. [8] initiated the idea of generalized multivalued Fcontraction mappings. Altun et al. [7] extended multivalued mappings with distance and established FP results in complete MS. Sgroi et al. [98] developed FP theorems for multivalued Fcontractions and achieved the solution of a few functional and integral equations, which was a suitable generalization of several multivalued FP theorems including Nadlers. Lately Ahmad et al. [12, 18, 46] revised the concept of Fcontraction to attain some FP, and CFP results in the discourse of complete MS. We divide this thesis into ve chapters. Each chapter begins with a brief introduction which proceeds as a summery to the material there in. Chapter 1 is an overview aimed at explaining the terminology to be used to recall basic denitions and facts. Chapter 2 is focused to the new concepts called (g F) contractions and generalized MizoguchiTakahashi contractions for complete @GMS and developed some new coincidence points and CFP results. Also, we prove some FP theorems of JS @Gcontraction in the setting of generalized MS, and to prove some FP results on @Gb-complete MS for a new contraction. Most of these theorems are already known from the literature, we include new alternative proofs and some general investigations regarding the underlying spaces. Chapter 3 is devoted to single and multivalued F-contraction mappings. We introduce the idea of generalized F-contraction and establish several new FP theorems for single and multivalued mappings in the setting of complete MS. We extend the concept of fuzzy FPs to common -fuzzy FP of generalized F-contraction in the setting of complete MS. Our results unify and generalize di⁄erent known comparable results from the current literature. Chapter 4 is devoted to introduce Frational cyclic contraction on PMS and to present new FP results for such cyclic contraction in 0-complete PMS. We establish a CFP theorem for a pair of multivalued F proximinal mappings satisfying Ciric-wardowski type contraction in PMS. We discuss applications of our theorem and obtain the existence and uniqueness of common solution of system of integral equations. Chapter 5 is focused on the concept of Hausdor⁄ metric on the family of closed bounded subsets of a dualistic PMS (DPMS) and establishes a CFP theorem of a pair of multivalued mappings satisfying Mizoguchi and Takahashis contractive conditions. Furthermore, we apply the concept of dislocated MS to obtain theorems asserting the existance of CFPs for a pair of mappings satisfying new generalized rational contractions in such spaces.