Fixed point theory has a beautiful mixture of analysis, topology, and geometry. Since

Transcription

1 Chapter 1 Introduction Fixed point theory has a beautiful mixture of analysis, topology, and geometry. Since from 1922 the theory of fixed points has been revealed as a very powerful and important technique for solving a variety of applied problems in mathematical sciences and engineering. In particular fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, game theory, and physics i.e. quantum mechanics. Also in the method of successive approximations for proving existence and uniqueness of solutions of differential equations. This method is associated with the names of such celebrated nineteenth century mathematicians as Cauchy, Liouville, Lipschitz, Peano, and specially Picard. In 1922 Banach [30] established well known result called Banach Contraction Principle in complete metric spaces. The Banach theorem seems somewhat limited. It clear that any continuous function mapping the unit interval into itself will have a fixed point, but the Banach theorem applies only to functions f that satisfy f (x) β for some β < 1. An elementary example of this is the function f(x) = 1 x, which has an obvious fixed point at x = 1/2, but whose derivative satisfies f (x) = 1 everywhere, therefore d(f(x), f(y)) = d(x, y) so f is not a contraction and the Banach fixed point theorem does not apply to f. The fixed point theorem due to Brouwer covers this case as well as many 1

2 2 others that the Banach theorem fails to cover because the relevant functions are not contractions. Banach contraction principle is extended by various authors in metric spaces and generalized metric spaces. For some of the generalizations of Banach s results, details see in [57, 60, 55, 41, 42, 43, 84, 58, 95, 37, 141]. But Kannan [88] first proved fixed point theorems for map satisfying contractive condition that did not required continuity at each points. Then number of fixed points theorems for different kinds of contractive conditions added in literature of fixed point theory, some details are given in [126, 45, 146, 129]. G. Jungck [61] established first common fixed point theorem for a pair of two commutating maps in a complete metric space. Then, flow of common fixed point results have appeared in the literature, some of details see in [64, 51, 65, 66, 68, 92, 46, 130, 33, 48, 50, 96]. The notation of weakly commuting maps was initiated by Sessa [131] infect it is a generalization of commuting mapping and generalized results of J. Danes [50] and K. M. Das, K. V. Naik [51]. Jungck [63, 66] generalized weak commutative by introducing the pair of compatible mappings, then huge of different fixed point theorems have been seen in the literature. For details please see in [6, 21, 22, 24, 25, 28, 67, 68, 23, 143]. In these years, M. S. Khan et al [97] introduced concept of altering distance in complete metric spaces. Al-Thagafi and Shahzad [105] defined the concept of occasionally weakly compatible mappings which is more general than the concept of weakly compatible maps. G. Jungck, B.E. Rhoades, [70] proved fixed point theorems for occasionally weakly compatible mappings. Many authors are successfully generalize the concept of compatible mappings in different ways, see details in [110, 111, 112, 77, 78]. But G.Jungck, B. E. Rhoades [69] coined notion of pair of weakly compatible mappings. It is most interesting generalization of compatible mappings and till many authors have been adding new common fixed point theorems using this concept, see in [54, 52, 64, 68]. Alber and Guerre-Delabriere, [13] first defined the concept of the weak contraction. In this paper they discussed the existence of the fixed point for a single valued φ-contractive in Hilbert spaces. Then B. E. Rhoades,

3 3 [26] extended the work of Alber and Guerre-Delabriere in Banach space. In recent years, there has been increasing interest in the study of fixed points and common fixed points of mappings satisfying contractive conditions of integral type. In 2006, Aliouche [18] proved a fixed point theorem using a general contractive condition of integral type in symmetric spaces. In 2007, Djoudi and Aliouche [56] obtained common fixed point theorems for two pairs of weakly compatible mappings satisfying contractive conditions of integral type. In 2009, Pathak [121] bore out a general common fixed point theorem of integral φ-type for two pairs of weakly compatible mappings satisfying certain integral type implicit relations in symmetric spaces. In 2006, Mustafa in collaboration with Sims introduced a new notion of generalized metric space called G-metric space, [154]. Based on the notion of generalized metric spaces, Mustafa et.al. [156, 157, 155] obtained some fixed point theorems for mappings satisfying different contractive conditions. W. Shatanawi, [151] proved fixed point theorem for contractive mappings satisfying φ-maps in G-metric spaces. Also proved fixed point theory for semicompatibility, unbounded G-metric spaces. In 1992, Matthews [134] introduced the notion of a partial metric space in which d(x, x) are not necessarily zero. For more, see [2, 147, 134, 59, 133, 109, 91, 75, 81]. Ayse Snmez,[8] proved fixed point theorems in partial Cone metric Spaces. Recently, M.R.Ahmadi Zand and A.Dehghan Nezhad, [102] in 2011 introduce generalization of partial metric space. In 2007, Huang and Zhang [72] generalized the notion of metric spaces by replacing the real numbers by ordered Banach space and defined cone metric spaces. They have proved Banach contraction mapping theorem and some other fixed point theorems of contractive type mappings in cone metric spaces. Abbas and Jungck [3] had generalized the results of Huang and Zhang [72] and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. Presently, number of researchers in the world are working on fixed points

4 4 in cone metric space see in [3, 4, 123, 5, 89, 128]. Recently C.T.Aage and J.N. Salunke [12] proved fixed point theorem for expansion onto mappings on cone metric spaces. Azam, Arshad and Beg [7] introduced the notion of cone rectangular metric spaces by replacing the triangular inequality of a cone metric space by a rectangular inequality. A fixed point theorem of multifunctions applied in cone metric spaces see in [125, 6, 31, 124]. The cone metric space with w-distance was introduced by Osama Kada, Tomonari Suzuki, Wataru Takahashi and Nakoshi Shioji [107, 106] and some generalization by H.Lakzian F.Arabyani see in [76]. After Matthews [134] introduction of partial metric space and this concept generalized by Ayse Sonmez, [8] in partial cone metric spaces. Now we introduced generalized partial cone metric space. Fuzzy metric spaces is a generalization of Menger spaces. Fuzzy set was defined by Zadeh [94] in Kramosil and Michalek [108] introduced fuzzy metric space, George and Veermani [17] modified the notion of fuzzy metric spaces with the help of continuous t-norms. Many researchers have obtained common fixed point theorems for mappings satisfying different types of commutativity conditions. In 1994 Pant [110], introduced the concept of R-weakly commutating maps in metric space, Vasuki [122] proved fixed point theorems for R-weakly commutating mappings. Pant [112, 113, 115, 114, 111], introduced new concept reciprocally continuous mappings and established some common fixed point theorems, noncompactible maps. Balasubramaniam et.al. [38] proved the Rhoades [22] open problem on the existence of contractive definition which generates a fixed point but does not force the mappings to be continuous at the fixed point posses an affirmative answer. Pant and Jha [115] obtained some anologus results proved by Balasubramaniam et al. Recently Amari and Moutawakil [99] introduce the property (E.A.) literature in fixed point in fuzzy metric spaces is notated in [9, 10, 11, 98, 144, 132]. The very first chapter is introduction and survey of literature. The second chapter consists of two sections. In the first section we have proved

5 5 fixed point result in generalized metric spaces with T-orbitally complete metric space and second section consists introduction for an integral type of the Banach contraction principle with occasionally weakly compatible (owc) mappings satisfying a contractive condition of integral type. In the third chapter we have studied some fixed point theorem of G-Metric space. In the first section proves fixed point theorems for mappings satisfying contractive conditions of integral type and the second section analyze fixed point theorem for an unbounded G- metric spaces. Our results are supported with examples. The fourth chapter having two sections. In the first section we have studied some fixed point theorems in partial metric space and in second section we have proved some results on generalized partial metric spaces. Our results are supported by an examples. Some fixed point theorem in cone metric spaces and cone rectangular metric spaces are studied in the fifth chapter. In cone metric space includes five subsections. Section (5.1.2) is devoted to the basic operations related to the cone, cone metric space and some basic properties of cone metric spaces with w-distance and some generalization are recalled in this section (5.1.3). A fixed point theorem of multifunctions applied in cone metric spaces result in section (5.1.4). In section (5.1.5) applying notion of weakly compatible maps we have proved fixed point theorem in cone metric spaces. The second section entitled with cone rectangular metric spaces is an expansion of cone metric space. In which there are five subsections. The first section is an introduction and preliminaries. In third and fourth subsections we have introduced the concept of Banach and Kannans contraction mapping principles in a complete normal cone rectangular metric spaces and some fixed point theorem for expansion mappings. In chapter sixth, we have studied the fixed point theorems in a partial cone metric spaces as well as new invented concept i.e. generalized partial cone metric spaces (GPCMS) with supportive examples.

6 6 In the last chapter, we studied the concept in fuzzy metric space with weakly compatible maps satisfying integral type contractive conditions and fixed point theorems for reciprocally continuous maps.