A Study of Common Fixed Point Theorem in Cone Metric Spaces

INTRODUCTION

This study has been divided in three sections. In the first section we have proved some results on fixed point theorem in complete cone metric space for commutative mapping which is an extension.The idea for the above has been taken from the work done. Second section consists of two theorems. First theorem has been proved by using orbitally continuous mapping and finite number of function which is studied. Second theorem has been proved by using four mappings and finite number of function. In the third section we have proved two theorems in cone rectangular metric space. First theorem is an extended work of Azam Akbar, Arshad and Beg. In the second theorem we have established a result by using rational type contractive condition which is studied by Jaleli Mohamed and SametBessem.

Definition 1.1. If E be a real Banach space and a nonempty set X. Suppose that the mapping satisfies Then distance d is called a cone metric on X and set X with cone metric d is called cone metric space (X, d) .

Two self-mapping f and g of a set X are said to be weakly compatible if they commute at their coincidence point, that is, if Let f and g be weakly compatible self-mapping of a set X. If f and g have a unique point of coincidence, that is , then w is the unique common fixed point of f and g .

Definition 1.1: E is a nonempty and real Banach space. Let P be a subset of E . The subset P be called a cone iff, Now applying the condition for the normality of cone we get from (1.11)

CONCLUSION

The fixed point theory for multivalue operators in metric spaces was used in a number of works published in specialist literature, starting with a multivalued-view version of the Banach-Caccioppoli contraction principle demonstrated by S. B. Nadler Jr. of 1969. The development of this theory led to the creation of different applications in many fields, including: optimization theory, integral and differential equations and inclusions, fractal theory, econometrics, etc. One with remarkable applications is also the Avramescu-Markin-Nadler theorem, fixed point theorems for Kasahara, fixed point function theorems for Darboux, etc. Fixed point theorems in fudge metric spaces, which are included in a large number of works published in the specialty literature, are other significant results with many applications in fixed point theory. Fixed-point theories are important not only to the theory of differential equations, integrated equations, differential integrations and inclusions, but also in economic and managerial sciences, in the fields of computer science and in other areas.A new direction of research includes a new operator type, namely the α-to contractive operator. Published works include, for example, w-distance conditions. The problem here is that concrete examples for the w-distance are difficult to find since it is a more abstract concept. Asl, Rezapour, and Shahzad (2012) created the concept of α-t-al-purpose multinationals and showed how fixed point results could be achieved for this new type. Guran and Bota (2015) studied the existence of a fixed point of the α-to–constractive type of operator on KST space, its uniqueness and generalized Ulam-Hyers stability. A new problem is the establishment of the conditions in which there would be and would be uniquely fixed α-técontractive type operators, which is the problem of this type of vector contractions. The theory of fixed points in the last decades has been widely applied. It applies highly to optimisation theory, gaming theories, conflicts, but also to mathematical quality modeling and its management. These applications are useful and interesting.

REFERENCES

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Research Scholar, Department of Mathematics, OPJS University, Churu, Rajasthan