[Paper Review] Advances in Startpoint Theory for quasi-pseudometric spaces

Yaé Ulrich Gaba|arXiv (Cornell University)|Dec 5, 2014

Fixed Point Theorems Analysis2 references3 citations

TL;DR

This paper advances startpoint theory in quasi-pseudometric spaces by establishing new fixed point, startpoint, and endpoint theorems for multi-valued maps using generalized contraction conditions involving non-decreasing, subadditive functions φ and η. The key contribution is proving the existence of startpoints and endpoints under weaker completeness and contractive conditions than previously known, extending prior results in asymmetric metric spaces.

ABSTRACT

This paper presents some startpoint (endpoint, fixed point) theorems for mutli-valued maps that generalize recent results proved by Y. U. Gaba \cite{rico, ricoo}.

Motivation & Objective

• To generalize recent fixed point results for multi-valued maps in quasi-pseudometric spaces by introducing a broader class of contraction conditions.
• To establish the existence of startpoints and endpoints under weaker completeness assumptions, such as left/right K-completeness and bicompleteness.
• To unify and extend existing results in asymmetric metric spaces by incorporating subadditive and non-decreasing functions φ and η with specific limit superior conditions.
• To provide sufficient conditions for the existence of fixed points, startpoints, and endpoints in multi-valued mappings via a novel iterative construction and convergence argument.

Proposed method

• Introduces a generalized contraction condition using two functions φ and η, where φ(t) < η(t) and limsup_{r→t⁺} φ(r)/η(r) < 1 for all t ≥ 0.
• Defines a real-valued function f(x) = H({x}, Fx) for startpoint analysis and f(x) = H(Fx, {x}) for endpoint analysis, measuring the distance from x to its image under F.
• Constructs a sequence (xₙ) via selection of y ∈ Fx satisfying H({y}, Fy) ≤ c·d(x,y) or generalized inequalities involving φ and η.
• Proves that the sequence (xₙ) is left K-Cauchy using the decreasing property of f(xₙ) and the convergence of φ(H({xₙ}, {xₙ₊₁})).
• Establishes d⁰-convergence of (xₙ) to a limit x by showing φ(H({xₙ}, {xₙ₊ₚ})) → 0 as n → ∞, leveraging subadditivity and geometric decay.
• Applies completeness assumptions (left/right K-completeness or bicompleteness) to conclude that the limit is a startpoint, endpoint, or fixed point depending on the setting.

Experimental results

Research questions

• RQ1 Under what conditions does a multi-valued map on a quasi-pseudometric space admit a startpoint, defined by H({x}, Fx) = 0?
• RQ2 Can the existence of an endpoint (H(Fx, {x}) = 0) be guaranteed under generalized φ-η contraction conditions in asymmetric spaces?
• RQ3 What completeness conditions (left K-completeness, right K-completeness, bicompleteness) are sufficient to ensure convergence of iterative sequences to fixed points?
• RQ4 How do the properties of subadditive and non-decreasing functions φ and η influence the convergence and existence of solutions in quasi-pseudometric settings?
• RQ5 Can the results be extended from left K-completeness to right K-completeness and bicompleteness, and are the conditions symmetric across these cases?

Key findings

• A startpoint exists for any multi-valued map F: X → CB(X) on a left K-complete quasi-pseudometric space if there exists c ∈ (0,1) such that for every x ∈ X, there is y ∈ Fx with H({y}, Fy) ≤ c·d(x,y).
• An endpoint exists for T: X → B(X) on a right K-complete space if there exist functions φ and η satisfying φ(t) < η(t) and limsup_{r→t⁺} φ(r)/η(r) < 1, with H(Tx, {x}) ≤ η(H({x}, {y})) and f(y) ≤ φ(H({x}, {y})) for some y ∈ Tx.
• A fixed point exists for T: X → B(X) on a bicomplete space when the generalized contraction condition holds with η(H⁰({x}, {y})) ≤ min{H({x}, Tx), H(Tx, {x})} and f(y) ≤ min{φ(H({x}, {y})), φ(H({y}, {x}))} for some y ∈ Tx.
• The sequence (xₙ) constructed via iterative selection is left K-Cauchy and d⁰-convergent to a startpoint when the function f(xₙ) decreases geometrically with rate q ∈ (0,1).
• The results remain valid under the dual completeness conditions: right K-completeness implies endpoint existence, and left K-completeness implies startpoint existence.
• The use of subadditive and non-decreasing functions φ and η allows for a broader class of contractions than standard c-contractions, enhancing applicability to asymmetric spaces.