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Some Coupled Fixed Point Results on Partial Metric Spaces
Hassen Aydi
TL;DR
The paper extends coupled fixed point results to complete partial metric spaces, where points may have nonzero self-distance. It establishes uniqueness under stated contractive conditions, while an example shows that a key condition cannot be omitted.
Problem
The paper addresses extending coupled fixed point results to partial metric spaces, where a point's distance from itself may be nonzero.
Source: §1 · p. 1Method
Results
When k and l are nonnegative and k + l < 1, F has a unique coupled fixed point.
Source: §2 · p. 2Takeaways & Limitations
The paper's results include a unique-coupled-fixed-point conclusion under the stated contraction condition.
Source: §2 · p. 2The condition k < 1 in Corollary 2.2, and hence k + l < 1 in Theorem 2.1, cannot be omitted for the stated uniqueness results.
Source: §2 · p. 5
Abstract
from arXiv · showhide
In this paper we give some coupled fixed point results for mappings satisfying different contractive conditions on complete partial metric spaces.
1. INTRODUCTION AND PRELIMINARIES
The paper extends coupled fixed point results to partial metric spaces, where a point can have nonzero distance from itself. It introduces the relevant concepts and recalls how partial-metric completeness relates to an associated metric.
1. INTRODUCTION AND PRELIMINARIES
- A pair (x, y) is a coupled fixed point of F when F(x, y) = x and F(y, x) = y.Source: §1 · p. 1
- The paper extends coupled fixed point results to partial metric spaces, adapting results previously obtained in cone metric spaces.The cited cone-metric theorem assumes k + l < 1 and concludes that F has a unique coupled fixed point.Source: §1 · p. 1
- A partial metric allows p(x, x) to be nonzero, while retaining symmetry, a self-distance bound, and a modified triangle inequality.Source: §1 · pp. 1–2
- The associated function p_s(x, y) = 2p(x, y) − p(x, x) − p(y, y) is a metric on X.Source: §1 · p. 2
- In a partial metric space, convergence and completeness are expressed through self-distance limits, and Cauchy sequences and completeness agree with those for p_s.Source: §1 · p. 2
2. MAIN RESULTS
The main results establish unique coupled fixed points for mappings on complete partial metric spaces under several contractive conditions. The paper also gives an example showing that a strict contraction bound cannot be omitted.
Theorem 2.1
- Theorem 2.1 establishes a unique coupled fixed point in a complete partial metric space under its stated contractive condition.Its proof iterates x_(n+1) = F(x_n, y_n) and y_(n+1) = F(y_n, x_n), then shows the limits form the unique coupled fixed point.Source: §2 · pp. 2–4
Corollary 2.2
- When the constants in Theorem 2.1 are equal, Corollary 2.2 gives a unique coupled fixed point for 0 ≤ k < 1.The example applies this corollary to obtain (0, 0) as the unique coupled fixed point.Source: §2 · p. 4
Example 2.3
- When the contractive bound is relaxed to k = 1, the example has both (0, 0) and (1, 1) as coupled fixed points, so uniqueness fails.The paper uses this to show that the strict bound in Corollary 2.2 and Theorem 2.1 cannot be omitted.Source: §2 · pp. 4–5
Theorem 2.4
- Under (2.13), a mapping on a complete partial metric space has a unique coupled fixed point when k and l are nonnegative and k + l < 1.The proof constructs two interleaved iterative sequences, establishes their convergence, and verifies existence and uniqueness.Source: §2 · pp. 5–6
Theorem 2.5
- Under (2.21), a mapping on a complete partial metric space has a unique coupled fixed point when k and l are nonnegative and k + 2l < 1.The proof derives Cauchy sequences using δ = k + l, then establishes the coupled fixed point.Source: §2 · pp. 3, 6, 7
Corollaries 2.6 and 2.7
- Corollaries 2.6 and 2.7 give unique coupled fixed points under equal-constant versions of Theorems 2.4 and 2.5, respectively.Corollary 2.6 requires 0 ≤ k < 1; Corollary 2.7 follows from Theorem 2.5's condition.Source: §2 · pp. 7–8
Remarks 2.8 and 2.9
- Theorems 2.1 and 2.4 extend Theorems 2.2 and 2.5 of to partial metric spaces.For Theorem 2.4's contractive condition, the paper also reduces the general case to the equal-constants case.Source: §2 · p. 8