Common Fixed Point Theorem of Two Mappings Satisfying a Generalized Weak Contractive Condition

Existence of common fixed point for two mappings which satisfy a generalized weak contractive condition is established. As a consequence, a common fixed point result for mappings satisfying a contractive 
condition of integral type is obtained. Our results generalize, extend, and unify several well-known comparable results in literature.


Introduction and Preliminaries
Let X be a metric space and T : C → C a mapping.Recall that T is contraction if d Tx, Ty ≤ kd x, y for all x, y ∈ X, where 0 ≤ k < 1.A point x ∈ C is a fixed point of T provided Tx x.If a map T satisfies F T F T n for each n ∈ N, where F T denotes the set of all fixed points of T, then it is said to have property P. Banach contraction principle which gives an answer on existence and uniqueness of a solution of an operator equation Tx x is the most widely used fixed point theorem in all of analysis.Branciari 1 obtained a fixed point theorem for a mapping satisfying an analogue of Banach's contraction principle for an integral type inequality.Akgun and Rhoades 2 have shown that a map satisfying a Meir-Keeler type contractive condition of integral type has a property P. Rhoades and Abbas 3 extended 4, Theorem 1 for mappings satisfying contractive condition of integral type.They also studied several results for maps which have property P, defined on a metric space satisfying generalized contractive conditions of integral type.Rhoades 5 proved two fixed point theorems involving more general contractive condition of integral type see, also 6, 7 .If maps S and T satisfy F S ∩ F T F S n ∩ F T n for each n ∈ N, then they are said to have property Q.Jeong and Rhoades 8 studied the property Q for pairs of maps satisfying a number of contractive conditions.

International Journal of Mathematics and Mathematical Sciences
Recently Dutta and Choudhury 9 gave a generalization of Banach contraction principle, which in turn generalize 4, Theorem 1 and corresponding result of 10 .Sessa 11 defined the concept of weakly commuting to obtain common fixed point for pairs of maps.Jungck generalized this idea, first to compatible mappings 12 and then to weakly compatible mappings 13 .There are examples that show that each of these generalizations of commutativity is a proper extension of the previous definition.The aim of this paper is to present a common fixed point theorem for weakly compatible maps satisfying a generalized weak contractive condition which is more general than the corresponding contractive condition of integral type.Our results substantially extend, improve, and generalize comparable results in literature 3, 14, 15 .
The following definitions and results will be needed in the sequel.

A Common Fixed Point Theorem
Set {φ : R → R : φ is a Lebesgue integrable mapping which is summable and nonnegative and satisfies ε 0 φ t dt > 0, for each ε > 0} and G {ψ : 0, ∞ → 0, ∞ : ψ is continuous and nondecreasing mapping with ψ t 0 if and only if t 0}.The following is the main result of this paper.for all x, y ∈ X, where ψ, ϕ ∈ G.If range of g contains the range of f and g X is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover if f and g are weakly compatible, f and g have a unique common fixed point.
Proof.Let x 0 be an arbitrary point of X. Choose a point x 1 in X such that f x 0 g x 1 .This can be done, since the range of g contains the range of f.Continuing this process, having chosen x n in X, we obtain x n 1 in X such that f x n g x n 1 , n 0, 1, 2, . . . .Suppose for any n, g x n / g x n 1 , since, otherwise, f and g have a point of coincidence.From 2.1 , we have ψ d q, fp 0, and f p q. Hence q is the point of coincidence of f and g.Assume that there is another point of coincident r in X such that r / q.Then there exists s in X such that f s g s r.Using 2.
x y, if at least one of x or y / ∈ 0, 1 , x / y, 0, if x y.

2.18
Then X, d is a complete metric space 17 .Consider f : X → X, and ψ, ϕ ∈ G as given in 9 : fx

2.21
Let g : X → X be defined as

International Journal of Mathematics and Mathematical Sciences
Assume that x > y and discuss the following cases.When x ∈ 0, 1 , then

2.30
Note that x 1 is the unique coincidence point of f and g, and f and g are commuting at x 1. Hence all conditions of Theorem 2.1 are satisfied.Moreover, x 1 is the unique common fixed point of f and g.
Following theorem can be viewed as generalization and extension of 3, Theorem 3 .

International Journal of Mathematics and Mathematical Sciences
Proof.Existence and uniqueness of fixed point of f follows from Corollary 2.2.Now we prove that f has property P. Let u ∈ F f n .We shall always assume that n > 1, since the statement for n 1 is trivial.We claim that fu u.If not, then, by 2.31 ,

2.32
Continuing this process we arrive at

Theorem 2 . 1 .
Let f, g be two self maps of a metric space X, d satisfying ψ d fx, fy ≤ ψ d gx, gy − ϕ d gx, gy 2.1

Theorem 2 . 5 .
Let f be a self map of a complete metric space X, d satisfying for all x, y ∈ X, where φ ∈ and ψ, ϕ ∈ G. Then f has a unique fixed point.Moreover f has property P .

Remarks 2 . 6 .
contradiction.Hence the result follows.Existence and uniqueness of fixed point of f in above theorem also follows from 9, Theorem 1 .Remarks 2.7. a It is noted that if maps f and g involved in Theorem 2.1 are commuting, then they have property Q. b Suzuki 18 observed that Branciari 1, Theorem 1 is a particular case of Meir-Keeler fixed point theorem 19 .We pose an open problem to see if a link exists between the contractive conditions 2.15 and the Meir-Keeler condition.
Definition 1.1.Let X be a set, and f, g selfmaps of X.A point x in X is called a coincidence point of f and g if and only if fx gx.We will call w fx gx a point of coincidence of f and g.
Lemma 1.3 see 16 .Let f and g be weakly compatible self maps of a set X.If f and g have a unique point of coincidence w (say), then w is the unique common fixed point of f and g.
It follows that {d gx n , gx n 1 } is monotone decreasing sequence of numbers and consequently there exists r ≥ 0 such that d gx n , gx n 1 → r as n → ∞.We may assume that n k are even and m k are odd and that d gx n k , gx m k > ε for all k.Putr k min m k : d gx n k , gx m k > ε .International Journal of Mathematics and Mathematical Sciencesgives d gx n k 1 , gx r k 1 → ε , as k → ∞.Therefore Hence {gx n } is a Cauchy sequence.From completeness of g X , there exists a point q in g X such that gx n → q as n → ∞.Consequently, we can find p in gx n , gx n−1 < ψ d gx n , gx n−1 , which on taking limit as n → ∞ yieldsψ r ≤ ψ r − ϕ r < ψ r , 2.5which is a contradiction.Therefore r 0. Now we prove that {gx n } is a Cauchy sequence.If not, then there exist some ε > 0 and subsequences{gx n k } and {gx m k } of {gx n } with k < n k < m k such that d gx n k , gx m k ≥ 3ε for each k.As d gx n k 1 , gx n k → 0 as k → ∞,for large enough k, we have d gx n k 1 , gx n k < ε and d gx m k 1 , gx m k < ε.Thus we obtain d gx n k 1 , gx m k ≥ d gx n k , gx m k − d gx n k 1 , gx n k > ε, d gx n k 1 , gx m k −1 ≥ d gx n k , gx m k − d gx m k −1 , gx m k − d gx n k 1 , gx n k > ε. k → ε as k → ∞.Furthermore d gx n k , gx r k − d gx n k , gx n k 1 − d gx r k , gx r k 1 ≤ d gx n k 1 , gx r k 1 ≤ d gx n k , gx r k d gx n k , gx n k 1 d gx r k , gx r k 1 2.9 n , gp − ϕ d gx n , gp 2.12 on taking limit as n → ∞ implies Φ • ψ and ϕ 1 Φ • ϕ ∈ G. Clearly ψ 1 , ϕ 1 ∈ G.Hence by Theorem 2.1 f and g have unique common fixed point.Now we present two examples in the support of Theorem 2.1.
for all x, y ∈ X, where φ ∈ and ψ, ϕ ∈ G.If range of g contains the range of f and g X is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover if f and g are weakly compatible, f and g have a unique common fixed point.