Fixed Point Theorem for Cyclic Weakly Generalized Contraction Mapping of Ciric Type

In this article, the concept of cyclic weakly generalized contraction mapping of Ciric type has been introduced and the existence of a fixed point for such mappings in the setup of complete metric spaces has been established. Result obtained extends and improves some fixed point results in the literature. Example is also given to show that class of contraction mappings introduced in the paper is strictly larger class than the class of mappings used in the literature and thus ensures wider applicability of the result by producing the solutions to new problems.

then T is said to be a generalized contraction. Ciric [1] proved that every generalized contraction in a complete metric space has a fixed point. * Corresponding author: sujatagoyal184@gmail.com The class of generalized contractions include the Banach's contractions and the contractions introduced by Kannan [3] and Chatterjea [4]. Alber and Guerre-Delabriere [6] introduced the concept of weakly contractive type mapping and proved the existence of fixed point for such mappings in Hilbert Spaces. Weakly contractive mappings include contractions as a special case. Many fixed point results were proved using the concept of weakly contractions [3][4][5][6][7][8][9][10][11][12][13]. Kirk et al. [2] introduced the concept of cyclic contractions. Definition 1.2: [2] Let X be a nonempty set and T: X X be a map.
In this case T is said to be cyclic contraction. Karapinar et al. [5] introduced generalized cyclic weakly Chatterjea type contractions.
be a metric space and m be a natural number.
is called a generalized cyclic weakly Chatterjea type contraction if (  Tx  y  d  Ty  x  d  Ty  y  d  Tx  x  d   Tx  y  d  Ty  x  d  Ty  y  d  Tx  x  d  Ty  Tx Karapinar et al. [5] proved the following theorem. In this paper, cyclic weakly generalized contraction mappings of Ciric type have been introduced and the existence of fixed point for such maps has been proved. Example is also given to show that our result is a generalization of the result in Karapinar et al. [5].

Main Results
The concept of cyclic weakly generalized contractions of Ciric type has been introduced and a fixed point result for such contractions in the framework of complete metric space has been derived. Let  denote all monotonically increasing continuous functions and  denote the set of all lower semi-continuous maps .    Then T is said to be a cyclic weakly generalized contraction mapping of Ciric type.
is a cyclic weakly generalized contraction mapping of Ciric type. Then T has a unique fixed then n x will be the fixed point and hence the result. Therefore, it is assumed that Since  is monotonically increasing function, for all n = 1, 2, ..., therefore ,  ). , Hence the claim is proved. Now the sequence ) ( n x will be proved to be Cauchy Let 0   be given.
By previous lemma, we can find Using (2.1.9) and (2.1.10) in eq. (2.1.11), Since Y is closed in X . Therefore Y is also complete and there exists Y x such that Which is a contradiction unless 1 Hence the main result is proved Now consider the following example,  which is true. Hence T is cyclic weakly generalized contraction of Ciric type. However, it is obvious that every generalized cyclic weakly Chatterjea type contraction is cyclic weakly generalized contraction of Ciric type. Thus the class of cyclic weakly generalized contraction of Ciric type is actually a strictly larger class of mappings than the class of generalized cyclic weakly Chatterjea type contraction.

Conclusion
In this paper, we introduced cyclic weakly generalized contraction mapping of Ciric type and presented fixed point for such mapping in complete metric spaces. We gave an   example to show that the class of these contraction mappings is strictly larger than the class of generalized cyclic weakly Chatterjea type contractions. Results obtained in this paper can be expanded. Also, a new more general condition can be achieved. There is also possibility of extending the results to other spaces.