A Unique Common Fixed Point Theorem Using ψ − φ Condition in a Partial Metric Space Using an ICS Mapping

The ICS mapping was introduced by K.P.Chi [On a fixed point theorem for certain class of maps satisfying a contractive condition depended on an another function, Lobachevskii J. math., 30(4), 2009, 289 291.] In this paper, we obtain a unique common fixed point theorem in partial metric spaces by using ICS mapping and also introduced supported example to our main theorem.


Introduction and Preliminaries
The notion of partial metric space was introduced by Matthews [10] as a part of the study of denotational semantics of data flow networks. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation and domain theory in computer science (see e.g. [12,13,14,15,16,17,18,19,20]).
First we recall some basic definitions and lemmas which play crucial role in the theory of partial metric spaces. Definition 1.1 (See [10,11]) A partial metric on a nonempty set X is a function p : X × X → R + such that for all x, y, z ∈ X: (p 1 ) x = y ⇔ p(x, x) = p(x, y) = p(y, y), The pair (X, p) is called a partial metric space (PMS).
Clearly p(x, y) = 0 implies x = y and x ̸ = y implies p(x, y) > 0. If p is a partial metric on X, then the function d p : 2 A Unique Common Fixed Point Theorem Using ψ − ϕ Condition in a Partial Metric Space Using an ICS Mapping Example 1.2 (See e.g. [11,7,1]) Consider X = [0, ∞) with p(x, y) = max{x, y}. Then (X, p) is a partial metric space. It is clear that p is not a (usual) metric. Note that in this case d p (x, y) = |x − y|.
Each partial metric p on X generates a T 0 topology τ p on X which has as a base the family of open p-balls We now state some basic topological notions (such as convergence, completeness, continuity) on partial metric spaces (see e.g. [10,11,2,1,7,9] p(x n , x m ).

A mapping
We need the following lemmas in PMS( [10,11,1,2,7,9]  In this paper, we obtain a unique common fixed point theorem for two mappings using ICS mapping in Partial metric spaces. Our result generalizes the recent several known results.
Recently [5] introduced the concept of ICS mapping as follows.
Definition 1.6 [5] Let (X, d) be a metric space. A mapping T : X → X is said to be ICS if T is injective, continuous and has the property : for every sequence

MAIN RESULT
Let Ψ denote the set of all continuous and monotonically increasing functions ψ : Theorem 2.1 Let (X, p) be a partial metric space and T : X → X be an ICS mapping and F, G : X → X be satisfying ∀x, y ∈ X, where ψ ∈ Ψ and ϕ ∈ Φ. Then F and G have a unique common fixed point in X.
Universal Journal of Applied Science 1(1): 1-7, 2013 3 Thus α is a common fixed point of F and G.
If β is another common fixed point of F and G, then T α ̸ = T β.
Thus α is the unique common fixed point of F and G. Case(b) : Assume that y n ̸ = y n+1 for all n. Denote p n = p(y n , y n+1 ).
Letting k → ∞ and then using ( Hence, we have lim k→∞ p(y 2n k +1 , y 2m k ) = ϵ 2 . (2.10) Letting k → ∞ and then using (2.7) and (2.4) in Letting k → ∞ and then using (2.11) and (2.4) in (2.14) Now, Universal Journal of Applied Science 1(1): 1-7, 2013 5 Letting k → ∞ and then using (2.10), (2.12), (2.2), (2.14) and (2.8), we get Hence {y 2n+1 } is Cauchy. Thus {y n } is a Cauchy sequence in (X, d p ). Hence, we have lim Since X is complete and {y n } is a Cauchy sequence in complete metric space (X, d p ). Thus lim n→∞ d p (y n , T z) = 0 for some T z ∈ X Also T is an ICS mapping and {y n } = {T x n } is convergent, it follows that {x n } is convergent to some z ∈ X.
Since T is continuous, from above we have lim Letting n → ∞ and using Lemma 1.5 and (2.17), we get It is a contradiction. Hence T F z = T z.
Since T is injective, we have F z = z. As in case(a), z is the common fixed point of F and G.
6 A Unique Common Fixed Point Theorem Using ψ − ϕ Condition in a Partial Metric Space Using an ICS Mapping Also ψ(p(T F x, T Gy)) = max{ F x 2 , Gy Clearly 0 is unique common fixed point of F and G.

Corollary 2.3
Let (X, p) be complete partial metric space and T : X → X be an ICS mapping and F, G : X → X be satisfying Then F and G have a unique common fixed point in X.
It follows from Theorem 2.1 if we put ψ(t) = t and ϕ(t) = t − φ(t) in Theorem 2.1. If we take F = G in Corollary 2.3, we get

})
Clearly 0 is unique fixed point of F .