common fixed point and complete metric space.

Sesa 5 Introduced a concept of weakly commuting mappings and proved some common fixed point theorems in complete Metric Space.PatilS.T. 6 Proved some common fixed point theorems for weakly commuting mappings satisfying a contractive conditions in complete Metric space. In 1986 G. Jungck 7 defined notion of compatible mappings and proved some common fixed point theorems in complete Metric space. Also he proved weak commuting mappings are compatible.

Jungck [6], proved a common fixed point theorem for commuting mappings in 1976, that generalizes the Banach’s [1] fixed point theorem in a complete metric space. This result was generalized and extended in various ways by Iseki and Singh [5], Park [15], Das and Naik [2], Singh [20], Singh and Singh [21], Fisher [3], Park and Bae [16]. Recently, some common fixed point theorems of three and four commuting mappings were proved by Fisher [3], Khan and Imdad [12], Kang and Kim [10] and Lohani and Badshah [13].The result of Jungck [6] has so many applications but it requires the continuity of mappings. Sessa [17], introduced the concept

Sesa [ 8 ] Introduced a concept of weakly commuting mappings and obtained some common fixed point theorems in complete Metric space . S.T. Patil [ 7 ] Proved some common fixed point theorems for weakly commuting mappings satisfying a contractive conditions in complete Metric space. In 1986 G. Jungck [ 4 ] defined compatible mappings and proved some common fixed point theorems in complete Metric space. Also he proved weak commuting mappings are compatible.

“The purpose of this article is to prove some common fixed point theorem using two and four pair of weakly commuting mappings in complete metric space using the result of R.P.Pant[1], Adrian Constantin [3], Shin-sen change[2], H.K.Pathak, Y.J.Cho and S.M.Kang[8], S.Sessa[9] and as a special case”.

Jungck [6], proved a common fixed point theorem for commuting mappings in 1976, which generalizes the Banach’s [1] fixed point theorem in complete metric space. This result was generalized and extended in various ways by Iseki and Singh [5], Das and Naik [2], Singh [18], Singh and Singh [19], Fisher [3], Park and Bae [14]. Recently, some common fixed point theorems of three and four commuting mappings were proved by Fisher [3], Khan and Imdad [12], Kang and Kim [10] and Lohani and Badshah [13].The result of Jungck [6] has so many applications but it requires the continuity of mappings. S. Sessa [15], introduced the concept of

Jungck [6], proved a common fixed point theorem for commuting mappings in 1976, which generalizes the Banach’s [1] fixed point theorem in complete metric space. This result was generalized and extended in various ways by Iseki and Singh [5], Park [12], Das and Naik [2], Singh [17], Singh and Singh [18], Fisher [3], Park and Bae [13]. Recently, some common fixed point theorems of three and four commuting mappings were proved by Fisher [4], Khan and Imdad [10], Kang and Kim [9] .The result of Jungck [6] has so many applications but it has certain limitation that it requires the continuity of one of the mapping. S. Sessa [14], introduced the concept of weak commutativity and proved a common fixed point theorem for weakly commuting maps. In general, commuting mappings are weakly commuting but the converse is not necessarily true.

Abstract —In this paper, we consider the existence of common fixed points for a compatible pair of self maps of Gregus type defined on a complete linear metric space. We also establish the existence of com- mon fixed points for these pair of compatible maps of type (B) and consequently type (A). The results are generalisations and extensions of the results of several authors.

In this paper new fixed point theorem in complete quasi metric space has been proved for mappings satisfying some new type of rational contraction conditions.. A complete metric space is[r]

Abstract. We establish a common fixed point theorem for quadruple weakly compatible mappings satisfying construction modulus in complete metric space. Presented theorem generalize and extend the results of [Parvaneh Vahid, Some common fixed point theorem in complete metric space, Int. J. Pure Appl. Math. 76 (2012), 1-8]. We also provide an example which shows that our result is a proper generalization of the existing one.

i.e every contraction map on a complete metric space has fixed point. Inequality (1.1) implies continuity of . This theorem is very popular and effective tool in solving existence problems in many branches of mathematical analysis and engineering. There are a lot of generalizations of this principle has been obtained in several directions, such as ordered Banach

We must point out that they have given most of their attention to the class of fuzzy sets with nonempty compact α-cut sets in the metric space X , but few of their attention to the class of fuzzy sets with nonempty bounded closed α-cut sets. However, it is known that all compact sets are bounded closed sets in a general metric space and the converse is not always true. In this paper, based on the Hausdorff fuzzy metric H M , we introduce a suitable notion for the

Suppose ( , ) is a complete S-metric space and {f } is a sequence of self maps of X such that there is an ∈ A for which (2.18)’ S(f x, f x, f y) ≤ α(S(x, x, y), S(f x, f x, x), S f y, f y, y ) holds for all x, y ∈ X and for all ≠ . Then for any ∈ the sequence { } defined by x = f x for n ≥ 1 converges to a point ∈ and this is the unique common fixed point of the sequence { }.

In 1963, Gahler [1] [2] introduced 2-metric spaces and claimed them as gene- ralizations of metric spaces. But many researchers proved that there was no rela- tion between these two spaces. These considerations led Dhage [3] to initiate a study of general metric spaces called D-metric spaces. As a probable modifica- tion to D-metric spaces, Shaban Sedghi, Nabi Shobe and Haiyun Zhou [4] have introduced D* -metric spaces. In 2006, Zead Mustafa and Brailey Sims [5] in- itiated G -metric spaces. Several researchers proved many common fixed point theorems on G -metric spaces.

Theorem 3.1Let  X M N , , , ,    be a complete  – chainable intuitionistic fuzzy metric spacewith continuous t-norm  and continuous t-conorm  definedby a  a  a and  1– a   1–  a    1– a  for all a    0, 1 . Let A B P Q S , , , , and T be six self-mappings on X , satisfying the following conditions: (3.1.1) A X    TP X   and B X    SQ X   ,

In this paper, we introduce the concept of α-compatible mappings of type (P) in metric space (X, d), which is equivalent to the concept of α-compatible mappings and as well as α-compatible mappings of type (A) under certain conditions. We prove a common α-fixed point theorem for four α-compatible mappings of type (P) on a complete metric spaces.

Abstract: In the present paper, we prove a common fixed point theorem for six weakly compatible selfmaps of a complete G-metric space.. As an illustration, we give an example.[r]

which implies d(Au, z) = 1, i.e Au = z = Iu. Since A(X ) ⊂ J(X ), implies that z ∈ J(X), Let v ∈ J −1 (z). Then Jv = z. Again using this argument it can easily show that Bv = z yielding there by Jv = Bv = z. The remaining two case certain essentially to the previous case. Indeed if B(X ) is complete then z ∈ B(X) ⊂ I(X ) and if A(X ) is complete then z ∈ A(X ) ⊂ J(X ).

Singh and Jain [10] proved some common fixed point theorems for four self maps on a complete fuzzy metric space satisfying an implicit relation. These results have been proved for semicompatible, compatible and weakly compatible maps and being one of the maps continuous. Since the concept of weakly compatible maps is most general among all the commutativity concepts in this field as every pair of compatible maps is weakly compatible but reverse is not always true (see [10], Ex. 2.16) and similarly, every pair of semi compatible maps is weakly compatible but reverse is not always true (see [10], Ex. 2.14). In this paper, our objective is to prove a common fixed point theorem for weakly compatible maps on a non complete fuzzy metric space satisfying an implicit relation. Also, our result does not require the continuity of the maps. Our result generalizes the results of Singh and Jain ([10], Theorem 3.1, Corollary 3.2, Theorem 3.5, Corollary 3.6).

The concept of fuzzy sets was introduced by Zadeh [14] in 1965. Since then, to use this concept in topology and analysis many authors have expansively developed the theory of fuzzy sets and applications. Fixed point theorems in fuzzy mathematics are emerging with vigorous hope and vital trust. It appears that Kramosil and Michalek’s study of fuzzy metric spaces paves a way for very soothing machinery to develop fixed point theorems for contractive type maps. Kramosil and Michalek [6] introduced the concept of fuzzy metric space and modified by George and Veeramani [2]. Many authors [4, 7] have proved fixed point theorems in fuzzy metric space. Recently Sedghi and Shobe [12] introduced D* - metric space as a probable modification of the definition of D - metric introduced by Dhage [1], and prove some basic properties in D* - metric spaces. Using D* - metric concepts, Sedghi and Shobe define M – fuzzy metric space and proved a common fixed point theorem in it. In this paper we prove some common fixed point theorems for generalized contraction mappings in complete M – fuzzy metric space. Also we prove some common fixed point theorems for two weakly compatible mappings in M – fuzzy metric space.

The fixed point theory is an important and major topic of the nonlinear functional analysis and finds some nice applications to other branches of mathematics. For proving the theorems on selfmaps and some nonlinear problems that arise in various physical and biological process.The various topological spaces such as metric, Banach, Hilbert, locally convex and Housdorff spaces etc are used in formulating the fixed point theorems. The famous French Mathematician H. Poincare (1854-1912) first recognized the importance of study of the nonlinear problems and predicated that the future Mathematics will mainly deal only with non-linearity and since then several methods have been developed and employed in the study of nonlinear equations.The fixed point technique is one of the most important and powerful tools that has been generally used for solving self maps and fixed point theorems which is major core part of the nonlinear functional analysis.It is known that the D-metric is a continuous function in the topology of D-Metric convergence. The generalization of a D-metric up to n variables appear in Dhage and Dolhare (2003). Also Das and Gupta (1975), Edelstien (1962), Poincare (1886), Rakotch (1962), Gu Seman (1970) etc. Worked on Fixed point theory and proved fixed fixed point theorems. We need the following fixed point theorem for selfmaps proved in Dolhare and Dhage (2003) and Rhoades (1977), Caccippoli (1930) in the sequel K.Iseki proved the following theorem for selfmap in complete metric space. It is well known that the notion of metric function is generalization of the idea of the distance function.