[Paper Review] Coincidence and Common Fixed Point Results for Contraction Type Maps in Partially Ordered Metric Spaces

Hassen Aydi|arXiv (Cornell University)|Feb 27, 2011

Fixed Point Theorems Analysis9 references22 citations

TL;DR

This paper establishes new coincidence and common fixed point theorems for three self-mappings satisfying a generalized contraction condition in partially ordered metric spaces, where the contraction is controlled by a function β ∈ S satisfying β(tₙ) → 1 implies tₙ → 0. The main contribution is a sufficient condition for the existence of a common fixed point under compatibility, weak increasingness, and continuity assumptions, with an application to proving the existence of a common solution to a system of integral equations via Corollary 2.8.

ABSTRACT

We present coincidence and common fixed point results of selfmappings satisfying a contraction type in partially ordered metric spaces. As an application, we give an existence theorem for a common solution of integral equations.

Motivation & Objective

• To extend Geraghty-type fixed point results to three self-mappings in partially ordered metric spaces.
• To establish conditions under which three mappings f, g, and H have a common coincidence point.
• To prove the existence of a unique common fixed point under compatibility, weak increasingness, and continuity.
• To apply the theoretical results to prove the existence of a common solution for a system of integral equations.

Proposed method

• Define a contraction condition using a function β ∈ S, where β(tₙ) → 1 implies tₙ → 0.
• Construct iterative sequences {xₙ} and {yₙ} via mappings f, g, and H with fX ⊆ HX and gX ⊆ HX.
• Use weak increasingness with respect to H to generate monotonic sequences {Hxₙ} and {yₙ}.
• Prove that {d(yₙ₊₁, yₙ)} is decreasing and converges to 0 using the contraction and properties of β.
• Establish that {Hxₙ} is a Cauchy sequence in the complete metric space (X, d), hence convergent.
• Use continuity and compatibility of pairs {f, H} and {g, H} to show that the limit is a common fixed point.

Experimental results

Research questions

• RQ1 Under what conditions do three self-mappings f, g, and H on a partially ordered metric space have a common coincidence point?
• RQ2 How does weak increasingness with respect to H contribute to the convergence of iterative sequences in the fixed point construction?
• RQ3 What role does the class S of functions β play in ensuring convergence and uniqueness of the fixed point?
• RQ4 Can the theoretical fixed point results be applied to prove existence of solutions for systems of integral equations?
• RQ5 What conditions ensure the uniqueness of the common fixed point in the proposed framework?

Key findings

• The sequence {d(yₙ₊₁, yₙ)} converges to 0, implying that the distances between consecutive iterates vanish.
• The sequence {Hxₙ} is a Cauchy sequence in the complete metric space (X, d), hence convergent.
• The limit of {Hxₙ} is a common fixed point of f, g, and H, satisfying fu = gu = Hu.
• The common fixed point is unique under the assumption that any two elements in X have a common comparable element.
• The application to integral equations shows that under continuity and contraction conditions, a common solution exists in C([0, T]).
• By choosing β(t) = √(log(t² + 1))/t, the contraction condition is satisfied, and Corollary 2.8 guarantees a common fixed point for f and g.