Existence of Unique Common Fixed Point for Pairs of Mappings in Complete Metric Spaces

Banach (1922) proved a theorem which ensures the existence and uniqueness of a fixed point under appropriate conditions. His result is called Banach‘s fixed point theorem or the Banach contraction principle. This theorem is also applied to show the existence and uniqueness of the solutions of differential and integral equations and many other applied mathematics. Many authors have extended, generalized and improved Banach‘s fixed point theorem in different ways. Some fixed point theorems for two metric spaces have been proved by Brain Fisher [1], V. Popa [3], P.P. Murthy et al [4], R.K. Namdeo [5] and Luljeta Kikina et al [7]. Now our aim is to generalize and extend result of [3].

a positive integer such that

Definition 1.3 A metric space (X, d) is said to be complete if and only if Cauchy sequence in X converge to a point of X. The following theorem was proved by V. Popa [3]. Theorem 1.1 Let (X, d) and (y, p) be complete metric spaces. If T: X → Y and S:Y→X satisfying the inequalities. and , where , then has a unique fixed point point and has a unique fixed point . Further, and .

Theorem 2.1 Let (X, d1) and (Y, d2) be complete metric spaces. Let A, B: X → Y and C, D:Y→X satisfying the inequalities. and where . If one of mappings A, B, C, and D is continuous then CA and DB have a unique common fixed point and BC and AD have a unique common fixed point . Further and Proof. Let x be an arbirary point in X. Let

and in general let, Using inequality (2.1), we get, Now If and by using inequality (2.2), we have, If , it follows that and since , is a Cauchy sequence with limit and is a Cauchy sequence with limit . Now suppose that A is a continuous. Then, and so Now using inequality (4), we get, Letting , we have, So, Using inequality (2.1), we get, Letting we get, The same result of course add if one of mapping is continuous instead of .

Suppose that has a second fixed point . Then by inequality (2.1) and (2.2), we have, By using inequality (2.2), we get, Since the uniqueness of z follows.Similarly z is the unique fixed point of CA and w is the unique fixed point of BC and AD. This complete the proof of the theorem. Corollary 2.1 Let (X, d1) be complete metric space. Let A, B, C, D: X→X satisfying the inequality, and D is continuous then CA and DB have a unique common fixed point z and BC and AD have a unique common fixed point w. Further,

Corollary 2.2 Let (X,d1) and (Y, d2) be complete metric spaces. Let A, B: X→Y and C, D:Y→X satisfying the inequalities, and where and then CA and DB have a unique common fixed point and BC and AD have a unique common fixed point . Further,

Let satisfying the inequality, where and and then CA and DB have a unique common fixed point z and BC and AD have a unique common fixed point w. Further,

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Assistant Professor of Mathematics, Pt. C. L. S. Govt. College, Sector – 14, Karnal, Haryana