On Weakly C, ψ, φ -Contractive Mappings in Ordered Partial Metric Spaces

and Applied Analysis 3 Definition 1.6 see 34 . Let X, d be a metric space and T : X → X be a mapping. Then T is called a weakly C-contractive if there exists a continuous function φ : 0, ∞ × 0, ∞ → 0, ∞ such that for all x, y ∈ X, we have d ( Tx, Ty ) ≤ 1 2 ( d ( x, Ty ) d ( Tx, y )) − φdx, Ty, dTx, y. 1.4 Harjani et al. 35 announced some fixed point results for weakly C-contractive mappings in a complete metric space endowed with a partial order. Meanwhile, Shatanawi 36 proved some fixed point and coupled fixed point theorems for a nonlinear weaklyC-contraction type mapping in metric and ordered metric spaces. In this paper, we introduce the concept of weakly C, ψ, φ -contractive mappings in ordered partial metric spaces, and we prove some existence theorems of common fixed point for such mapping in the context of complete partial metric spaces under certain conditions. 2. The Main Result We start this section with the following definitions. Definition 2.1. Suppose that X, p is a partial metric space. A mapping T : X → X is said to be continuous at x ∈ X if for every > 0, there exists δ > 0 such that T Bp x, δ ⊆ Bp Tx, . We say that T is continuous if T is continuous at all x ∈ X. It is easy to see that if X, p is a partial metric space, T : X → X is continuous, xn is a sequence in X, x ∈ X and lim n→ ∞ p xn, x p x, x , then lim n→ ∞ p Txn, Tx p Tx, Tx . 2.1 Altun and Simsek 37 introduce the notion of weakly increasing of two mappings T, S : X → X in the following way. Definition 2.2 see 37 . Let X, be a partially ordered set. Two mappings T, S : X → X are said to be weakly increasing if Tx STx and Sx TSx for all x ∈ X. For more details on weakly increasing mappings, we refer the reader to 24, 38–41 . Let φ denote all functions φ : 0,∞ × 0, ∞ → 0,∞ such that φ is continuous, φ t, s 0 if and only if t s 0. Similarly, we denote by Ψ all functions ψ : 0, ∞ → 0, ∞ such that ψ is continuous and nondecreasing, ψ t 0 if and only if t 0. Inspired the definitions above, we introduce the following definition. 4 Abstract and Applied Analysis Definition 2.3. Let X, , p be an partially ordered metric space. Then the mappings T, S : X → X are said to be weakly C, ψ, φ -contractive mappings if T and S are weakly increasing with respect to and for any comparable x and y, we have ψ ( p ( Tx, Sy )) ≤ ψ ( 1 2 ( p ( Tx, y ) p ( x, Sy )) ) − φpTx, y, px, Sy, 2.2 where φ ∈ Φ and ψ ∈ Ψ. Now, we introduce and prove our first results. Theorem 2.4. Let X, be a partially ordered set and suppose that there exists a partial metric p on X such that X, p is complete. Suppose that T, S : X → X are weakly C, ψ, φ -contractive mappings. If T and S are continuous, then T and S have a common fixed point; that is, there exists u ∈ X such that u Tu Su. Proof. Given x0 ∈ X. Set Tx0 x1 and Sx1 x2. Continuing this process, we construct sequences xn in X such that x2n 1 Tx2n and x2n 2 Sx2n 1. Using the fact that that S and T are weakly increasing with respect to , we obtain that x1 Tx0 S Tx0 Sx1 x2 · x2n 1 x2n 2· 2.3 Now, we will prove that lim n→ ∞ p xn, xn 1 0. 2.4 Since x2n 1 and x2n 2 are comparable, by 2.2 , we have ψ ( p x2n 1, x2n 2 ) ψ ( p Tx2n, Sx2n 1 ) ≤ ψ ( 1 2 ( p Tx2n, x2n 1 p x2n, Sx2n 1 ) ) − φp Tx2n, x2n 1 , p x2n, Sx2n 1 ) ψ ( 1 2 ( p x2n 1, x2n 1 p x2n, x2n 2 ) ) − φp x2n 1, x2n 1 , p x2n, x2n 2 ) . 2.5