Abstract Fixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach’s fixed point theorem. There exits a vast literature on the topic and this is a very active field of research at present. A selfmap T of a metric space X is said to have a fixed point x if Tx =x. Theorems concerning the existence and properties of fixed points are known as fixed point theorems. Such theorems are very important tools for proving the existence and uniqueness of the solutions to various mathematical models (differential, integral and partial differential equations and variational inequalities etc.) representing phenomena arising in different fields, such as steady state temperature distribution, chemical equations, neutron transport theory, economic theories, epidemics and flow of fluids. They are also used to study the problems of optimal control related to these systems. Fixed point theorems of ordered Banach spaces provide us exact or approximate solutions of boundary value problems. For details, one can refer to H. Amann [34], Collatz [28], Franklin [31], Karamardian [43], Lions [49], Martin [51], Mercier [52], Peitgen and Walther [64], Robinson [66], Smart [68], Swaminathan [69], Tartar [86] and Waltman [95]. During the last few years, many branches of science have been benefited from the theory and many generalizations are emerging from it. In 1988, B. E. Rhoades [25] surveyed some of the contractive definitions. Though he could not have the complete coverage of the definitions, yet his paper was a good survey. The fixed point theory was extended to multivalued mappings in 1941 with the fixed point theorems of S. Nadler (1969) and R. Fraser (1969). Recently, compatibility idea was introduced by G. Jungck [33]. The topological fixed point theory is of fundamental importance in the broader context of fixed point theory. The generalizations of existing ideas is very important tool for the advancement in the field of science and the present thesis has been written on the same basis. In 1963, S. Ghaler introduced 2metric spaces and after that there was a spat of papers dealing with this generalized space. Several papers are available in semi metric spaces and Quasi semi metric spaces. In 1984, B. C. Dhage further generalized 2metric spaces to Dmetric spaces. In this thesis, fixed point theorems and related results in metric spaces, 2metric spaces, quasi semi 2metric spaces and Dmetric spaces have been discussed. and many concepts are generalized. The fixed point theory for nonexpansive mappings includes the contractions and strictly contractive mappings. Moreover, it contains all isometries including the identity. It may be noted that nonexpensive mappings may be fixed point free, and when such a mapping has a fixed point it needs not be unique, for example, the identity metric. The theory of nonexpansive mappings is different from that of contraction mappings. Even if a nonexpansive mapping has a fixed point, the iterative method generally fails to converge. In chapter 1, we study some basic definitions and fundamental results of the theory. The main aim of this chapter is to keep the thesis in sequence. The notations and terminology used in the thesis are also fixed. Proof of the famous Banach Contraction Principle is given in section 1.2. The generalizations are established in Section 1.1 whereas Section 1.2 is devoted to discuss multivalued mappings. The multivalued version of Banach’s Fixed Point Theorem and its proof given by S. Nadler is also stated. In section 1.3, basic results in 2metric spaces are noted. The results given in this section include singlevalued as well as multivalued mappings. In section 1.5, fundamental results in Dmetric spaces are stated and notations / terminology is fixed. The proof of Triangle Contraction Principle is also given. In section 1.6, compact Dmetric spaces are defined. In chapter 2, some interesting results in metric spaces are obtained. M. S. Khan [54] used expansive square root condition to obtain an identity mapping. However, it is shown that if we impose the conditions that fix(S) and fix(T) are nonempty and the space is connected then we can obtain a common fixed point. Counter examples are given to support the results. B. Ahmed and F. U. Rehman in [7] proved several fixed point theorems in 2metric spaces, the generalization of metric spaces. Some fixed point theorems of M. S. Khan [53] in the setting of 2metric spaces have been proved. We also prove some fixed point theorems of [56] in the setting of 2metric spaces. Finally, we refer to K. Iseki [46, theorem 1] and H. Murakami [39, theorem 4] for certain fixed point theorems in metric spaces with different contraction conditions. We improve the same for 2metric spaces. The concept of a proximinal set in 2metric spaces is introduced and certain results [80] for multivalued mappings are extended. We establish two results which classify 2metric spaces and satisfy a rational inequality. The results of this chapter have been submitted to The Aligarh Bulletin of Mathematics. In chapter 4, Quasi Semi 2metric spaces are discussed. G. Jungck, B. E. Rhoades, H. Kaneko [33, 24, 37] worked on fixed point theorems for weakly commuting and compatible maps and obtained several fixed point theorems. K. J. Chang, T. Taniguchi, B. E. Rhoades, H. Kaneko [44, 90, 34, 35] worked on fixed point theorems for contractive, generalized contractive and contractive type setvalued mapping and obtained several interesting results. M. Telsi and K. Tas [61] obtained several results concerning p'contraction mappings in quasi semi metric spaces. Analogous to these mappings in quasi semi metric spaces, Ф' contraction mappings are introduced in quasi semi 2metric spaces and hence the results of M. S. Khan [56] and M. Telsi and K. Tas [61] have been generalized in quasi semi 2metric spaces. Necessity of various conditions used therein is explained with the help of remarks and examples. In section 4.4, fixed point theorems in quasi semi 2metric spaces are proved for expansion mappings and subjective selfmaps. The contents of this chapter have been published in the International Journal of Pure and Applied Mathematics Vol. 7 No.2(2003), 137146. Chapter 5 is devoted to Dmetric spaces. In section 5.2, we introduce expansive mappings in the setting of Dmetric spaces. We extend the results of P. Z. Daffer and H. Kaneko [63] for a pair of expansive mappings in Dmetric spaces. The results of this section have been published in Indian Journal of Pure and Applied Math., 32(10), Oct. 2001,15131518. In section 5.3, we continue investigating properties of Dmetric spaces using expansion mappings. Initially, we prove fixed point theorems for selfmaps in Dmetric spaces. We show that if a selfmap of the Euclidean space R" satisfies the expansive condition, then surjectivity and continuity coincide over R". Some results for compact Dmetric spaces are also proved. In an important paper T. L. Hicks and L. M. Saliga [88] proved some fixed point theorem for nonself maps motivated by the interesting work of S. Z. Wang, B. Y. Li, Z. M. Gao and K. Iseki [96] on expansion self maps. They showed that the first four theorems of K. Iseki et al [96] still hold even if the mapping f maps C, a closed subset of complete metric space X, onto X or f maps C, into X with the property that C c f(C). Motivated by the work of T. L. Hicks and L. M. Saliga [88] for nonselfmaps in metric spaces, we extend the results in selfmaps to nonself maps in Dmetric spaces. The results of B. Ahmad, M. Ashraf and B. E. Rhoades [9] for selfmaps in Dmetric spaces has been extended to nonselfmaps in Dmetric spaces. The work in this section has been published in Southeast Asian Bulletin of Mathematics No. 27 (2004),769780 In section 5.4 we generalize the results of H. Murakami and C. C. Yeh [39], and M. S. Khan [56] in the setting of Dmetric spaces. Analogous conditions are defined in Dmetric spaces. These results have been submitted in Southeast Asian Bulletin of Mathematics. In section 5.5, we generalize the Triangle Contraction Principle to setvalued mappings in Dmetric spaces by defining Hausdorff D metric in a different way. In his paper [12], Dhage defined the generalized Hausdorff Dmetric in such a way that one set A is fixed and sup over the elements of two sets Band C is taken. However, by fixing the two sets and taking the sup over the elements of the third set is more rational and compatible with metric spaces as is done in this section. In view of the above, the definitions have been reframed and applied successfully to obtain the multivalued version of the Triangle Contraction Principle. In this case, some basic definitions and ideas for setvalued mappings of metric spaces to Dmetric 'spaces have been generalized. The notions of Hausdorff Dmetric, strongly regular orbit, and lower semi continuous mappings in the setting of Dmetric spaces are defined in a different way and certain fixed point theorems are obtained. Consequently, the results so obtained are interesting and different from those of Dhage [12]. The results of H. Kaneko [35, 36] and B. C. Dhage, A. M. Pathan and B. E. Rhoades [14] for multivalued mappings are extended to Dmetric spaces. These results have been accepted for publication in the International Journal of Pure and Applied Mathematics (2005).
