[Paper Review] An Entourage Approach to the Contraction Principle in Uniform Spaces Endowed with a Graph

Aris Aghanians, Kamal Fallahi+1 more|arXiv (Cornell University)|May 15, 2013

Fixed Point Theorems Analysis3 references8 citations

TL;DR

This paper extends the Banach contraction principle to uniform spaces endowed with a directed graph using entourages, establishing sufficient conditions for a mapping to be a Picard operator. By leveraging basic entourages and graph connectivity, it generalizes Jachymski's metric space results to uniform spaces, proving that weakly connected graphs and orbital G-continuity ensure unique fixed points under Banach G-contractions.

ABSTRACT

In this paper, we study Banach contractions in uniform spaces endowed with a graph and give some sufficient conditions for a mapping to be a Picard operator. Our main results generalize some results of [J. Jachymski, "The contraction principle for mappings on a metric space with a graph", Proc. Amer. Math. Soc. 136 (2008) 1359-1373] employing the basic entourages of the uniform space.

Motivation & Objective

• Generalize Jachymski's contraction principle from metric to uniform spaces endowed with a graph.
• Establish sufficient conditions for a mapping to be a Picard operator in uniform spaces with graph structure.
• Employ basic entourages of uniform spaces to generalize fixed point results beyond metric settings.
• Investigate the role of weak connectivity and orbital G-continuity in ensuring unique fixed points.
• Provide a unified framework combining uniform space theory and graph-theoretic contraction conditions.

Proposed method

• Use basic entourages derived from pseudometrics to define uniformity in the space, enabling metric-like analysis in general uniform structures.
• Introduce the concept of a Banach G-contraction in uniform spaces, where the contraction condition is expressed via entourages and graph edges.
• Define the equivalence relation [x]_G̃ based on weak connectivity in the underlying undirected graph to identify invariant sets.
• Apply the notion of orbital G-continuity to ensure convergence of iterates within connected components.
• Utilize sequential completeness and equicontinuity of iterates to extend convergence from dense subsets to the whole space.
• Prove that if G is weakly connected and T is a nonexpansive or equicontinuous Banach G-contraction, then T restricted to the closure of [x]_G̃ is a Picard operator.

Experimental results

Research questions

• RQ1 Under what conditions on a uniform space with a graph does a Banach G-contraction have a unique fixed point?
• RQ2 How can the classical contraction principle be generalized from metric to uniform spaces using graph structure?
• RQ3 What role does weak connectivity of the graph play in ensuring convergence of iterative sequences in uniform spaces?
• RQ4 In what way do orbital G-continuity and equicontinuity of iterates affect the fixed point behavior of mappings in uniform spaces?
• RQ5 Can nonexpansive mappings in uniformly complete spaces with graph structure be guaranteed to be Picard operators under graph connectivity?

Key findings

• If G is weakly connected and T is an orbitally G-continuous Banach G-contraction with a nonempty set of fixed points, then T is a weakly Picard operator.
• Sequential completeness and weak connectivity imply that each orbitally G-continuous Banach G-contraction with a fixed point is a Picard operator.
• A nonexpansive Banach G-contraction on a sequentially complete space is a Picard operator if the graph G is weakly connected.
• For any x ∈ X with Tx ∈ [x]_G̃, the restriction of T to the closure of [x]_G̃ is a Picard operator, provided the iterates are equicontinuous.
• If the underlying graph is not weakly connected, there exists an orbitally G-continuous Banach G-contraction with at least two fixed points.
• The convergence of iterates T^n x to a fixed point is guaranteed for all x in the weakly connected component [x]_G̃, provided the mapping is orbitally G-continuous and the space is sequentially complete.