Common Fixed Point Theorems in Metric spaces with Applications

In this paper, we investigate the existence and uniqueness of common fixed point theorems for certain contractive type of mappings. As an application the existence and uniqueness of common solutions for a system of functional equations arising in dynamic programming are discuss by using the our results. 2010 Mathematics Subject Classifications: 49L20, 49L99, 54H25, 90C39


Introduction
Bellman and Lee [3] first introduced the basic form of the functional equations in dynamic programming is as follows: f (x) = opt y∈D H(x, y, f (T (x, y)))∀x ∈ S where opt represent sup. or inf., x, y denote the state and decicion vectors respectively, T stands for the transformation of the process and f (x) represents the optimal return function with the initial state x.Afterwards, the existence and uniqueness of fixed point solutions for several classes of contractive mappings and functional equations studied by many investigators such as Bhakta and Mitra [5], Liu [15], Liu and ume [20], Pathak and Fisher [21], Baskaran and Subhramanyam [1] and others.
Ray [22] proved two common fixed point theorems for three self mappings f ,g and h in the complete metric space using the following contractive condition: d(f x, gy) ≤ d(hx, hy) − w(d(hx, hy)), ∀x, y ∈ X (iii) The map h is said to be (f, g, h)-orbitally continuous at x 0 if it is continuous on O(x 0 , f, g, h).
Throughout in this paper, we assume that R + = [0, +∞), R = (−∞, +∞), w and N denote the set of all non-negative and positive integers respectively.
The aim of this paper is to provide the sufficient conditions for the existence and uniqueness of common fixed point for the following type of contractive mappings metric space (X, d).
As an applications, we discuss the existence and uniqueness of common solutions of the following functional equations arising in dynamic programming. and

Main Results
Theorem 1. Let f, g and h be three self maps on a metric space X satisfying: (i) either f commute with h or g commute with h.
(ii) there exists w ∈ W such that (5) hold for all x, y ∈ X.
(iii) The pair (f, g) is asymptotically regular with respect to h at x 0 . (iv)The space X is (f, g, h)-orbitally complete at x 0 and h is orbitally continuous at x 0 . Then f, g and h have a unique common fixed point in X.
Proof Since (f, g) is asymptotically respect to h at x 0 , there exists a sequence {x n } in X such that f x 2n = hx 2n+1 and gx 2n+1 = hx 2n+2 , n = 0, 1, 2, ... and d(hx n , hx n+1 ) → zero as n → ∞. Now we show that hx n is Cauchy.On contrary suppose that hx n is not Cauchy, then there exists an > 0 and positive integers m k and n k with m k < n k such that d(hx m k , hx n k ) ≥ and d(hx m k , hx n k −1 ) ≤ for all k = 0, 1, 2, ....Since d(hx m k , hx n k ) ≤ d(hx m k , hx n k −1 ) + d(hx n k −1 , hx n k ).Then we obtain d(hx m k , hx n k ) → as k → ∞. Now there are four cases: (i) m k is even and n k is odd (ii) m k is even and n k is even (iii) m k is odd and n k is even (iv) m k is odd and n k is odd. Suppose m k is even and n k is odd, we have Letting k → ∞, we obtain In the remaining cases we have a similar situation.Hence {hx n } is Cauchy.Since X is (f, g, h)orbitally complete at x 0 , it follows that there exist z ∈ X s.t. hx n → z as n → ∞. Now, again Taking k → ∞, we obtain and Since h is orbitally continuous at Taking k → ∞, we obtain a contradiction.Hence hz = z.Using (8) and (9) together with T z = z, we infer that f z = gz = hz = z.Further uniqueness of common fixed point can easily prove.
Taking ψ(t) = t and φ(t) = ht where < h < 1, we state the following Corollary 1. Let A, B and T be self maps on a metric space (X, d) such that T commutes with both A and B and the pair (A, B) is asymptotically regular w.r.to T at x 0 ∈ X, X is orbitally complete and T is orbitally continuous at x 0 and for all x, y ∈ X. Then A, B and T have unique common fixed point in X.
Theorem 2. Let (X, d) be a metric space and f , g and h be self mappings on X such that f (X) ∪ g(X) ⊆ h(X).If there exists a w ∈ W satisfying (5).Then the pair (f, h) and (g, h) have a coincidence point in X, provided that (i) X is h-asymptotically complete, (ii) h is asymptotically continuous and (iii) h is weakly commute with f and g.Further f , g and h have a unique common fixed point in X.
Corollary 2. Let (X, d) be a complete metric space and f , g and h be self mappings on X such that f (X) ∪ g(X) ⊆ h(X).If there exists a w ∈ W satisfying (5).Then the pair (f, h) and (g, h) have a coincidence point in X, provided that (i) h is continuous and (iii) h commutes with both f and g.Further f , g and h have a unique common fixed point in X.
Taking ψ(t) = t = φ(t),in cor.2 we obtain the following Corollary 3. Let (X, d) be a complete metric space.Let f, g and h be self maps on X such that f (X) ∪ g(X) ⊆ h(X) and h is continuous and commute with both f and g. If there exists a w ∈ W satisfying the following condition: for all x, y ∈ x. Then f and g have unique common fixed point in X.

An Application
Throughout in this section,let X and Y be Banach spaces S ⊆ X be the state space and D ⊆ Y be decision space.   , f a) for all (x, y) ∈ S × D; a, b ∈ B(S) and t ∈ S.Where f , g and h are defined as follows: for all x ∈ S, a i ∈ B(S) and i = {1, 2, 3} f (a 1 (x)) = opt y∈D {u(x, y) + H 1 (x, y, a 1 (T (x, y)))} g(a 2 (x)) = opt y∈D {u(x, y) + H 2 (x, y, a 2 (T (x, y)))} h(a 3 (x)) = opt y∈D {u(x, y) + H 3 (x, y, a 3 (T (x, y)))} and h is asymptotically continuous and weakly commute with both f and g.
Then the system of functional equations possess a unique common solution in B(S).