Coupled fixed points for (ψ-α-β)- general contractions on partially ordered partial metric spaces

Introduction

In 1994 Matthews. SG, [14] introduced the notion of a Partial Metric Space (PMS) as a part of the study of denotational Semantics of data flow networks. He also showed that the Banach Contraction Principle can be generalized to the PMSs. Many researchers have proved fixed point theorems in PMSs [2,3,5,7,8,10,11,12,17,18] . Not only in Mathematics, PMSs have vast applications in Computer Science, Information Science and Biological Sciences. Bhaskar. TG, Lakshmikantham. V [9] introduced the notion of a coupled fixed point and proved some fixed point theorems in partially ordered Metric Spaces. Later some researchers proved coupled fixed and coupled common fixed point theorems [1,4,11,16,19,20,21,22] . Matthews. SG, [13,14] , Oltra. S and Valera. O [15] and Altun et al. [6] proved some fixed point theorems in PMSs for a single map, taking generalized contractions in PMSs. Rao et al. [21] introduced the notion of (𝜓 -𝛼 -𝛽)contraction maps with respect to another map on a Partial metric space. In this paper we introduce (𝜓 -𝛼 -𝛽)general contraction map 𝐹 with respect to another map g on a partial metric space and obtain conditions for the existence of coupled coincidence points and common coupled fixed points of 𝐹 and 𝑔. Supporting examples and applications are provided.

(2.14.4) 𝜓 (𝑝(𝐹(𝑥, 𝑦), 𝐹(𝑢, 𝑣))) ≤ 𝛼(𝑚(𝑥, 𝑦, 𝑢, 𝑣)) -𝛽(𝑚(𝑥, 𝑦, 𝑢, 𝑣)) for all 𝑥, 𝑦, 𝑢, 𝑣 ∈ 𝑋, with 𝑔𝑥 ≤ 𝑔𝑢, 𝑔𝑦 ≥ 𝑔𝑣 where 𝑚(𝑥, 𝑦, 𝑢, 𝑣) = 𝑚𝑎𝑥 { 𝑝(𝑔𝑥, 𝑔𝑢), 𝑝(𝑔𝑦, 𝑔𝑣), 𝑝(𝑔𝑥, 𝐹(𝑥, 𝑦)), 𝑝(𝑔𝑦, 𝐹(𝑦, 𝑥)), 𝑝(𝑔𝑢, 𝐹(𝑢, 𝑣)), 𝑝(𝑔𝑣, 𝐹(𝑣, 𝑢)), 𝑝(𝑔𝑥,𝐹(𝑥,𝑦))𝑝(𝑔𝑦,𝐹(𝑦,𝑥)) 1+𝑝(𝑔𝑥,𝑔𝑢)+𝑝(𝑔𝑦,𝑔𝑣)+𝑝(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣)) , 𝑝(𝑔𝑢,𝐹(𝑢,𝑣))𝑝(𝑔𝑣,𝐹(𝑣,𝑢)) 1+𝑝(𝑔𝑥,𝑔𝑢)+𝑝(𝑔𝑦,𝑔𝑣)+𝑝(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣)) }.

Using the above definition Rao et al. [21] proved the following coupled fixed point result in partially ordered partial metric space. [21]: Let (𝑋, ≤) be a partially ordered set and 𝑝 be a partial metric so that (𝑋, 𝑝, ≤) is a partially ordered partial metric space. Let 𝐹: 𝑋 × 𝑋 → 𝑋 and 𝑔: 𝑋 → 𝑋 be such that (2.15.1) 𝐹 is a (𝜓 -𝛼 -𝛽)contraction with respect to 𝑔, (2.15.2) 𝐹(𝑋 × 𝑋) ⊆ 𝑔(𝑋) and 𝑔(𝑋) is a complete subspace of 𝑋 (2.15.3) 𝐹 has mixed 𝑔 monotone property (2.15.4) (a) If a non-decreasing sequence {𝑥 𝑛 } → 𝑥, then 𝑥 𝑛 ≤ 𝑥 ∀ 𝑛 (b) If a non-increasing sequence {𝑦 𝑛 } → 𝑦, then 𝑦 ≤ 𝑦 𝑛 ∀ 𝑛 If there exist 𝑥 0 , 𝑦 0 ∈ 𝑋 such that 𝑔𝑥 0 ≤ 𝐹(𝑥 0 , 𝑦 0 ) and 𝑔𝑦 0 ≥ 𝐹(𝑦 0 , 𝑥 0 ), then 𝐹 and 𝑔 have a coupled coincidence point in 𝑋 × 𝑋.

Theorem

Main Results

In this section, we introduce (𝜓 -𝛼 -𝛽)general contraction map 𝐹 with respect to another map 𝑔 on a partially ordered partial metric space and obtain conditions for the existence of coupled coincidence points and common coupled fixed points of 𝐹 and 𝑔.

Definition

Let (𝑋, 𝑝) be a PMS and (𝑋, ≤) be a partially ordered set. Let 𝐹: 𝑋 × 𝑋 → 𝑋 and 𝑔: 𝑋 → 𝑋 be two mappings. We say that 𝐹 is a (𝜓 -𝛼 -𝛽)general contraction with respect to 𝑔 if there exist 𝜓, 𝛼, 𝛽: [0, ∞) → [0, ∞) satisfying the following (3.1.1) 𝜓 is continuous and monotonically non-decreasing, 𝛼 is continuous and 𝛽 is continuous. = 𝛼(max{𝑝(𝑢, 𝑔𝑥), 𝑝(𝑣, 𝑔𝑦)}) -𝛽(max{𝑝(𝑢, 𝑔𝑥), 𝑝(𝑣, 𝑔𝑦)}) = 𝛼(max{𝑝(𝑢, 𝑢), 𝑝(𝑣, 𝑣)}) -𝛽(max{𝑝(𝑢, 𝑢), 𝑝(𝑣, 𝑣)}) = 𝛼(0) -𝛽(0) = 0 ⟹ 𝑝(𝑢, 𝐹(𝑥, 𝑦)) = 0 ⟹ 𝑢 = 𝐹(𝑥, 𝑦) ⟹ 𝑔𝑥 = 𝐹(𝑥, 𝑦) Similarly 𝑝(𝑣, 𝐹(𝑦, 𝑥)) = 0 ⟹ 𝑣 = 𝐹(𝑦, 𝑥) ⟹𝑔𝑦 = 𝐹(𝑦, 𝑥). Hence 𝐹 and 𝑔 have a coupled coincidence point in 𝑋 × 𝑋. Note: Theorem 3.2 holds even when 𝛽 is lower semi continuous.

Corollary: (Theorem 2.15)

Proof: Since m(x,y,u,v) ≤ M (x,y,u,v), the result follows in view of the note above. Now we state and prove our second main theorem.

Theorem

In addition to the hypothesis of Theorem 3.2, we suppose that (𝑥, 𝑦) and (𝑥 ′ , 𝑦 ′ ) are coupled fixed points of 𝐹 and 𝑔, such that (𝑔𝑥, 𝑔𝑦) and (𝑔𝑥 ′ , 𝑔𝑦 ′ ) are comparable. Then 𝐹(𝑥, 𝑦) = 𝑔𝑥 = 𝑔𝑥 ′ = 𝐹(𝑥 ′ , 𝑦 ′ ) And 𝐹(𝑦, 𝑥) = 𝑔𝑦 = 𝑔𝑦 ′ = 𝐹(𝑦 ′ , 𝑥 ′ ) (3.3.1) Moreover if (𝐹, 𝑔) is 𝑤 -compatible, then 𝐹 and g have a unique common coupled fixed point in 𝑋 × 𝑋, in the sense that any two comparable common coupled fixed points coincide. Proof: We may suppose that 𝑔𝑥 ≤ 𝑔𝑥 ′ And𝑔𝑦 ≥ 𝑔𝑦 ′ . Then from theorem (3.2) we have 𝜓 (𝑝(𝐹(𝑥, 𝑦), 𝐹(𝑥 ′ , 𝑦 ′ ))) ≤ 𝛼(𝑀(𝑥, 𝑦, 𝑥 ′ , 𝑦 ′ )) -𝛽(𝑀(𝑥, 𝑦, 𝑥 ′ , 𝑦 ′ ))

Where 𝑀(𝑥, 𝑦, 𝑥 ′ , 𝑦 ′ ) = max { 𝑝(𝑔𝑥, 𝑔𝑥 ′ ), 𝑝(𝑔𝑦, 𝑔𝑦 ′ ), 𝑝(𝑔𝑥, 𝑔𝑥), 𝑝(𝑔𝑦, 𝑔𝑦), 𝑝(𝑔𝑥 ′ , 𝑔𝑥 ′ ), 𝑝(𝑔𝑦 ′ , 𝑔𝑦 ′ ), Hence any two comparable common coupled fixed points coincide.

Corollary

Let (𝑋, ⪯) be a partially ordered set and 𝑝 be a partial metric such that (𝑋, 𝑝) is a complete partial metric space. Let 𝐹: 𝑋 × 𝑋 → 𝑋 be such that If there exist 𝑥 0 , 𝑦 0 ∈ 𝑋 such that 𝑥 0 ⪯ 𝐹(𝑥 0 , 𝑦 0 ) and 𝑦 0 ⪰ 𝐹(𝑦 0, 𝑥 0 ), then 𝐹 has a unique coupled fixed point in 𝑋 × 𝑋. Proof: In theorem 3.2, if we take 𝑔 as the identity map 𝐼: 𝑋 → 𝑋, the result follows.

The following example supports our results.

Example

Let 𝑋 = [0, 1]. Let ⪯ be the partially order on 𝑋 defined by ⪯ 𝑦 ⇔ 𝑥 ≥ 𝑦.

The mapping 𝐹: 𝑋 × 𝑋 → 𝑋 is defined by 𝐹(𝑥, 𝑦) =

and 𝑔 is the identity map. Take 𝑥 0 = 𝑦 0 = 1.

So that𝑔𝑥 0 ⪯ 𝐹(𝑥 0 , 𝑦 0 ).

(max{𝑟, 𝛿}) = 𝛼(max{𝑝(𝑥, 𝑢), 𝑝(𝑦, 𝑣)}) -𝛽(max{𝑝(𝑥, 𝑢), 𝑝(𝑦, 𝑣)}) ∴ 𝜓 (𝑝(𝐹(𝑥, 𝑦), 𝐹(𝑢, 𝑣))) ≤ 𝛼(max{𝑝(𝑥, 𝑢), 𝑝(𝑦, 𝑣)}) -𝛽(max{𝑝(𝑥, 𝑢), 𝑝(𝑦, 𝑣)}) Hence all conditions of theorem 3.4 hold and (0,0) Is the unique coupled fixed point of 𝐹 In 𝑋 × 𝑋.

Application

In this section we give an application of our main results.

Theorem

Let (𝑋, 𝑝) be a complete partial metric space. 𝑈 be an open subset of 𝑋 and 𝑈 ̅ be a closed subset of 𝑋 such that 𝑈 ⊆ 𝑈 ̅ , Suppose 𝐻: 𝑈 ̅ × 𝑈 ̅ × [0,1] → 𝑋 is an operator such that the following conditions are satisfied 1. 𝑥 ≠ 𝐻(𝑥, 𝑦, 𝜆) and 𝑦 ≠ 𝐻(𝑦, 𝑥, 𝜆) for each 𝑥, 𝑦 ∈ 𝜕𝑈 and 𝜆 ∈ [0,1] (here 𝜕𝑈 denotes the boundary of 𝑈 in 𝑋) 2. 𝜓 (𝑝(𝐻(𝑥, 𝑦, 𝜆), 𝐻(𝑢, 𝑣, 𝜆))) ≤ 𝛼(max{𝑝(𝑥, 𝑢), 𝑝(𝑦, 𝑣)}) -𝛽(max{𝑝(𝑥, 𝑢), 𝑝(𝑦, 𝑣)}) for all 𝑥, 𝑦 ∈ 𝑈 ̅ and 𝜆 ∈ [0,1], where 𝜓: [0, ∞) → [0, ∞) is continuous and non-decreasing and 𝛼, 𝛽: [0, ∞) → [0, ∞) are continuous with 𝜓(𝑡) -𝛼(𝑡) + 𝛽(𝑡) > 0 for 𝑡 > 0 3. There exists 𝑀 ≥ 0 such that 𝑝(𝐻(𝑥, 𝑦, 𝜆), 𝐻(𝑥, 𝑦, 𝜇)) ≤ 𝑀|𝜆 -𝜇| for all 𝑥 ∈ 𝑈 ̅ and 𝜆, 𝜇 ∈ [0, 1]. Then 𝐻(. , 𝜆 0 ) has a coupled fixed point for some 𝜆 0 ∈ [0,1] if and only if 𝐻(. , 𝜇) has a coupled fixed point for all 𝜇 ∈ [0,1] Proof: Consider the set 𝐴 = {𝜆 ∈ [0,1]: (𝑥, 𝑦) = (𝐻(𝑥, 𝑦, 𝜆), 𝐻(𝑦, 𝑥, 𝜆)) for some 𝑥, 𝑦 ∈ 𝑈} Since 𝐻(. , 𝜆 0 ) has a coupled fixed point in 𝑈, we have 𝜆 0 ∈ 𝐴, so that 𝐴 is a non-empty set We will show that 𝐴 is both open and closed in [0, 1], so by connectedness we have 𝐴 = [0,1] As a result 𝐻(. , 𝜇) has a coupled fixed point in 𝑈 for all 𝜇 ∈ [0,1]. Let 𝜇 0 ∈ 𝐴, then there exist 𝑥 0 , 𝑦 0 ∈ 𝑈 such that (𝑥 0 , 𝑦 0 ) = (𝐻(𝑥 0 , 𝑦 0 , 𝜇 0 ), 𝐻(𝑦 0 , 𝑥 0 , 𝜇 0 )) where 𝑝(𝐻(𝑥, 𝑦, 𝜇), 𝑥 0 ) = 𝑝(𝐻(𝑥, 𝑦, 𝜇), 𝐻(𝑥 0 , 𝑦 0 , 𝜇 0 )) ≤ ( 𝑝(𝐻(𝑥, 𝑦, 𝜇), 𝐻(𝑥, 𝑦, 𝜇 0 )) + 𝑝(𝐻(𝑥, 𝑦, 𝜇 0 ), 𝐻(𝑥 0 , 𝑦 0 , 𝜇 0 )) -𝑝(𝐻(𝑥, 𝑦, 𝜇 0 ), 𝐻(𝑥, 𝑦, 𝜇 0 )) ) ≤ 𝑀|𝜇 -𝜇 0 | + 𝑝(𝐻(𝑥, 𝑦, 𝜇 0 ), 𝐻(𝑥 0 , 𝑦 0 , 𝜇 0 )) Suppose |𝜇 -𝜇 0 | < 1 𝑀 𝑛 for large 𝑛 ≤ 1 𝑀 𝑛-1 + 𝑝(𝐻(𝑥, 𝑦, 𝜇 0 ), 𝐻(𝑥 0 , 𝑦 0 , 𝜇 0 )) (4.8)

Similarly 𝑝(𝐻(𝑦, 𝑥, 𝜇), 𝑦 0 ) ≤ 1 𝑀 𝑛-1 + 𝑝(𝐻(𝑦, 𝑥, 𝜇 0 ), 𝐻(𝑦 0 , 𝑥 0 , 𝜇 0 )) (4.9)

Now consider max {𝑝(𝐻(𝑥, 𝑦, 𝜇), 𝑥 0 ), 𝑝(𝐻(𝑦, 𝑥, 𝜇), 𝑦 0 )} ≤ 1 𝑀 𝑛-1 max { 𝑝(𝐻(𝑥, 𝑦, 𝜇 0 ), 𝐻(𝑥 0 , 𝑦 0 , 𝜇 0 )), 𝑝(𝐻(𝑦, 𝑥, 𝜇 0 ), 𝐻(𝑦 0 , 𝑥 0 , 𝜇 0 )) }

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