Abstract:
A new fixed point theorem is obtained for theclass of cyclic weak -contractions on partially metric spaces. It is proved thata self-mapping on a complete partial metric space has a fixed point if itsatisfies the cyclic weak -contraction principle.

Abstract:
The probabilistic metric space as one of the important generalization of metric space was introduced by K. Menger in 1942. In this paper, we briefly discuss the historical developments of contraction mappings in probabilistic metric space with some fixed point results. Keywords : Fixed point; Distribution function; t-norm; PM space; contraction mapping. DOI: http://dx.doi.org/ 10.3126/kuset.v7i1.5425 KUSET 2011; 7(1): 79-91

Abstract:
In this paper, we introduce cyclic weak $phi-$contractions in fuzzy metric spaces and utilize the same to prove some results on existence and uniqueness of fixed point in fuzzy metric spaces. Some related results are also proved besides furnishing illustrative examples.

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In this article, we present some fixed point theorems in partially ordered G-metric space using the concept of $(\psi,\phi)$- weak contraction which extend many existing fixed point theorems in such space. We also give some examples to show that if we transform a metric space into a G-metric space our results are not equivalent to the existing results in metric space.

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We prove common fixed point theorem for coincidentally commuting nonself mappings satisfying generalized contraction condition of iri type in cone metric space. Our results generalize and extend all the recent results related to non-self mappings in the setting of cone metric space.

Abstract:
We prove common fixed point theorem for coincidentally commuting nonself mappings satisfying generalized contraction condition of iri type in cone metric space. Our results generalize and extend all the recent results related to non-self mappings in the setting of cone metric space.

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Let A and B be two nonempty subsets of a Banach space X. A mapping T : is said to be cyclic relatively nonexpansive if T(A) and T(B) and for all ( ) . In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X. It is shown that if (A, B) is a nonempty, weakly compact, and convex pair and (A, B) has seminormal structure, then a cyclic relatively nonexpansive mapping T : has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings. 1. Introduction Let be a Banach space and . Recall that a mapping is nonexpansive provided that for all . A closed convex subset of a Banach space has normal structure in the sense of Brodskii and Milman [1] if for each bounded, closed, and convex subset of which contains more than one point, there exists a point which is not a diametral point; that is, where is the diameter of . The set is said to have fixed point property (FPP) if every nonexpansive mapping has a fixed point. In 1965, Kirk proved the following famous fixed theorem. Theorem 1 (see [2]). Let be a nonempty, weakly compact, and convex subset of a Banach space . If has normal structure, then has the FPP. We mention that every compact and convex subset of a Banach space has normal structure (see [3]) and so has the FPP. Moreover, every bounded, closed, and convex subset of a uniformly convex Banach space has normal structure (see [4]) and then by Theorem 1 has the FPP. It is interesting to note that there exists a weakly compact and convex subset of which does not have the fixed point property (see [5] for more information). In particular, cannot have normal structure. In the current paper, we introduce a geometric notion of seminormal structure on a nonempty, closed, and convex pair of subsets of a Banach space and present a new fixed point theorem which is an extension of Kirk’s fixed point theorem. We also study the stability of fixed points by using this geometric property. Finally, we establish a best proximity point theorem for a new class of mappings. 2. Preliminaries In [6], Kirk et al. obtained an interesting extension of Banach contraction principle as follows. Theorem 2 (see [6]). Let and be nonempty closed subsets of a complete metric space . Suppose that is a cyclic mapping, that is, and . If for some and for all , , has a unique fixed point in . An interesting feature

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In this paper, we prove that the Banach contraction principle proved by S. G. Matthews in 1994 on 0--complete partial metric spaces can be extended to cyclical mappings. However, the generalized contraction principle proved by D. Ili\'{c}, V. Pavlovi\'{c} and V. Rako\u{c}evi\'{c} in "Some new extensions of Banach's contraction principle to partial metric spaces, Appl. Math. Lett. 24 (2011), 1326--1330" on complete partial metric spaces can not be extended to cyclical mappings. Some examples are given to illustrate the effectiveness of our results. Moreover, we generalize some of the results obtained by W. A. Kirk, P. S. Srinivasan and P. Veeramani in "Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (1) (2003),79--89". Finally, an Edelstein's type theorem is also extended in case one of the sets in the cyclic decomposition is 0-compact.

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In this paper, we present the generalization of B-contraction and C-contraction due to Sehgal and Hicks respectively. We also study some properties of C-contraction in probabilistic metric space.

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Cyclic weaker-type contraction conditions involving a generalized control function (with two variables) are used for mappings on 0-complete partial metric spaces to obtain fixed point results, thus generalizing several known results. Various examples are presented showing how the obtained theorems can be used and that they are proper extensions of the known ones. 1. Introduction The celebrated Banach contraction principle has been generalized in several directions and widely used to obtain various fixed point results, with applications in many branches of mathematics. Cyclic representations and cyclic contractions were introduced by Kirk et al. [1] and further used by several authors to obtain various fixed point results. See, for example, papers [2–9]. Note that while a classical contraction has to be continuous, cyclic contractions might not be. On the other hand, Matthews [10] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. In partial metric spaces, self-distance of an arbitrary point need not be equal to zero. Several authors obtained many useful fixed point results in these spaces–-we just mention [11–27]. Several results in ordered partial metric spaces have been obtained as well [28–36]. Some results for cyclic contractions in partial metric spaces have been very recently obtained in [37–41]. Khan et al. [42] addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function. This idea was further used in many papers, such as Choudhury [43] where generalized control functions were used. This approach has been very recently used in [44, 45] to obtain fixed point results in partial metric spaces. In this paper, we extend these results further, considering cyclic weaker-type contraction conditions involving a generalized control function (with two variables) for mappings on -complete partial metric spaces (Romaguera [16]). We obtain fixed point theorems for such mappings, thus generalizing several known results. Various examples are presented showing how the obtained results can be used and that they are proper extensions of the known ones. 2. Preliminaries In 2003, Kirk et al. introduced the following notion of cyclic representation. Definition 1 (see [1]). Let be a nonempty set, , and let be a self-mapping. Then is a cyclic representation of with respect to if(a) are nonempty subsets of ;(b) , , . They proved the following fixed point result. Theorem 2 (see [1]). Let be a complete metric