A FIXED POINT THEOREM FOR A PAIR OF MAPS SATISFYING A GENERAL CONTRACTIVE CONDITION OF INTEGRAL TYPE

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1 A FIXED POINT THEOREM FOR A PAIR OF MAPS SATISFYING A GENERAL CONTRACTIVE CONDITION OF INTEGRAL TYPE P. VIJAYARAJU, B. E. RHOADES, AND R. MOHANRAJ Received 8 October 4 and in revised form July 5 We give a general condition which enables one to easily establish fixed point theorems for a pair of maps satisfying a contractive inequality of integral type. Branciari [] obtained a fixed point result for a single mapping satisfying an analogue of Banach s contraction principle for an integral-type inequality. The second author [3] proved two fixed point theorems involving more general contractive conditions. In this paper, we establish a general principle, which makes it possible to prove many fixed point theorems for a pair of maps of integral type. Define Φ ={ϕ : ϕ : R + R} such that ϕ is nonnegative, Lebesgue integrable, and satisfies Let ψ : R + R + satisfy that (i ψ is nonnegative and nondecreasing on R +, (ii ψ(t <tfor each t>, (iii n= ψ n (t < for each fixed t>. Define Ψ ={ψ : ψ satisfies (i (iii}. ɛ ϕ(tdt > foreach ɛ >. ( Lemma. Let S and T be self-maps of a metric space (X,d. Suppose that there exists a sequence {x n } X with x X, x n+ := Sx n, x n+ := Tx n+, such that {x n } is complete and there exists a k [, such that d(sx,ty ( d(x,y ϕ(tdt ( for each distinct x, y {x n } satisfying either x = Ty or y = Sx,whereϕ Φ, ψ Ψ. Copyright 5 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 5:5 ( DOI:.55/IJMMS.5.359

2 36 Fixed point theorem for a pair of maps Then, either (a S or T has a fixed point in {x n } or (b {x n } converges to some point p X and d(xn,p i (d for n>, (3 where d(x,x d := ϕ(tdt. (4 Proof. Suppose that x n+ = x n for some n. Thenx n = x n+ = Sx n,andx n is a fixed point of S. Similarly, if x n+ = x n+ for some n,thenx n+ is a fixed point of T. Now assume that x n x n+ for each n.withx = x n, y = x n+,(becomes d(xn+,x n+ ( d(xn,x n+ ϕ(tdt. (5 Substituting x = x n, y = x n,(becomes d(xn+,x n Therefore, for each n, d(xn,x n+ ( d(xn,x n ( d(xn,x n Let m,n N, m>n. Then, using the triangular inequality, It can be shown by induction that Using (7and(9, d(xn,x m d(xn,x m d ( m x n,x m ϕ(tdt m ϕ(tdt. (6 ϕ(tdt ψ n (d. (7 d ( x i,x i+. (8 d(xi,x i+ ϕ(tdt. (9 i (d ψ i (d. ( Taking the limit of (asm,n and using condition (iii for ψ, it follows that {x n } is Cauchy, hence convergent, since X is complete. Call the limit p. Taking the limit of ( as m yields (3.

3 P. Vijayaraju et al. 36 Theorem. Let (X,d be a complete metric space, and let S, T be self-maps of X such that for each distinct x, y X, d(sx,ty ( M(x,y ϕ(tdt, ( where k [,, ϕ Φ, ψ Ψ,and { [ ] } d(x,ty+d(y,sx M(x, y:= max d(x, y,d(x,sx,d(y,ty,. ( Then S and T have a unique common fixed point. Proof. We will first show that any fixed point of S is also a fixed point of T,andconversely. Let p = Sp.Then and (becomes { M(p, p = max d(p,tp,,d(p,tp, d(p,tp ( d(p,tp which, from (, implies that p = Tp. Similarly, p = Tpimplies that p = Sp. We will now show that S and T satisfy (. { M(x,Sx = max d(x,sx,d(x,sx,d(sx,tsx, } = d(p,tp, (3 ϕ(tdt, (4 [ d(x,tsx+ ] }. (5 From the triangular inequality, [ ] d(x,tsx d(x,sx+d(sx,tsx max { d(x,sx,d(sx,tsx }. (6 Thus, (becomes d(sx,tsx d(sx,tsx ϕ(tdt k ϕ(tdt, (7 a contradiction to (. Therefore, for all x X, M(x,Sx = d(x,sx, and ( is satisfied. If condition (a of Lemma is true, then S or T has a fixed point. But it has already been shown that any fixed point of S is also a fixed point of T, andconversely.thuss and T have a common fixed point. Suppose that conclusion (b of Lemma is true. Then, from (3, d(sxn,tp ( d(xn,p ϕ(tdt, (8 which implies, since X is complete, that limd(sx n,tp =.

4 36 Fixed point theorem for a pair of maps Therefore, d(p,tp d ( p,sx n + d ( Sxn,Tp, (9 and p is a fixed point of T, hence a fixed point of S. Condition ( clearly implies uniqueness of the fixed point. Every contractive condition of integral type automatically includes a corresponding contractive condition not involving integrals, by setting ϕ(t overr +. There are many contractive conditions of integral type which satisfy (. Included among these are the analogues of the many contractive conditions involving rational expressions and/or products of distances. We conclude this paper with one such example. Corollary 3. Let (X,d beacompletemetricspace,sand T self-maps of X such that, for each distinct x, y X, d(sx,ty n(x,y ϕ(tdt k ϕ(tdt, ( where ϕ Φ, k [,,and { [ ] } d(y,ty +d(x,sx n(x, y:= max,d(x, y. ( +d(x, y Then S and T have a unique common fixed point. Proof. n(x,sx = max { d(sx,tsx,d(x,sx }. ( As in the proof of Theorem, it is easy to show that any fixed point of S is also a fixed point of T,andconversely. If n(x,sx = d(sx,tsx, then an argument similar to that of Theorem leads to a contradiction. Therefore n(x,sx = d(x,sx, and either S or T has a common fixed point, or (3 is satisfied. In the latter case, with limx n = p, n(p, p =, so that, from (, p is a fixed point of S, hence of T. Uniqueness of p is easily established. Corollary 3 is also a consequence of Lemma. We now provide an example, kindly supplied by one of the referees, to show that Lemma is more general than [, Theorem 3.]. Example 4. Let X :={/n : n N {}} with the Euclidean metric and S, T are self-maps of X defined by ( S = n if n is odd, n + if n is even, n + ifn =, ( T = n if n is even, n + if n is odd, n + ifn =. (3

5 P. Vijayaraju et al. 363 For each n,definex n+ = Sx n, x n+ = Tx n+.withx =, let O( denote the orbit of x = ; that is, O( ={,/,/3,...} and O( = O( {} =X. Forx, y O(, y = /m, m even and x = /n = Ty= /(m +,Sx = /(m +, so that m + m + = m + m + = (m +(m +, d(x, y = n m = m + n = m (4 m + = m(m +. Thus Also d(sx,ty d(x, y = m. (5 m + d(sx,ty sup =, (6 n N d(x, y so that there is no number c [, such that d(sx,ty cd(x, y forx, y O( and x = Ty. Therefore, [, Theorem 3.] cannot be used. On the other hand, the hypotheses of Lemma are satisfied. To see this, it will be shown that condition ( is satisfied for some ϕ Φ. We will first show that for any x = /n, y = /m O( satisfying either x = Ty or y = Sx, d(sx,ty n + m +. (7 There are four cases. Case. y = /m, m even, x = /n = Ty= /(m +,andsx = /(m +.Then m + m + = n + m +. (8 Case. y = /m, m odd, x = /n = Ty= /(m +,andsx = /(m +3.Then m +3 m + = m + m +3 m + m +3 = n + (9 m +. Case 3. x = /n, n even, y = /m = Sx = /(n +,andty= /(n +3.Then n + n +3 = n + n +3 n + n +3 = n + (3 n +3.

6 364 Fixed point theorem for a pair of maps Case 4. x = /n, n odd, y = /m = Sx = /(n +,andty= /(n +.Then n + n + = n + m +. (3 Thus in all cases, (issatisfied. Define ϕ by ϕ(t = t / [ logt]fort>andϕ( =. Then, for any τ>, and ϕ Φ. Using [, Example 3.6], τ ϕ(tdt = τ /τ, (3 d(sx,ty ϕ(tdt d(sx,ty /d(sx,ty n + m + n m / (/n+ (/m+ / (/n (/m = d(x, y /d(x,y for each x, y as in Lemma, and condition ( is satisfied with ψ(t = t/. (33 Acknowledgment The authors thank each of the referees for careful reading of the manuscript. References [] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 9 (, no. 9, [] S. Park, Fixed points and periodic points of contractive pairs of maps, Proc. College Natur. Sci. SeoulNat.Univ.5 (98, no., 9. [3] B.E.Rhoades,Two fixed-point theorems for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 3 (3, no. 63, P. Vijayaraju: Department of Mathematics, Anna University, Chennai-6 5, India address: vijay@annauniv.edu B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN , USA address: rhoades@indiana.edu R. Mohanraj: Department of Mathematics, Anna University, Chennai-6 5, India address: vrmraj@yahoo.com