[Paper Review] Fixed point theorems for a class of mappings depending of another function and defined on cone metric spaces

E R. Morales, Edixon Rojas|ArXiv.org|Jun 11, 2009

Fixed Point Theorems Analysis26 references3 citations

TL;DR

This paper establishes fixed point theorems for a generalized class of mappings, denoted $ D_T(a,b) $, defined on complete cone metric spaces, where the contraction condition depends on an auxiliary function $ T $. By extending the $ T $-contraction framework to cone metric spaces, the authors prove the existence and uniqueness of fixed points under mild conditions—such as continuity, injectivity, and sequential convergence of $ T $—without requiring continuity of the mapping $ S $. The key contribution is a unified fixed point result that generalizes Banach, Kannan, and Reich-type theorems in cone metric settings.

ABSTRACT

In this paper we study the existence and uniqueness of fixed points of a class of mappings defined on complete, (sequentially compact) cone metric spaces, without continuity conditions and depending on another function.

Motivation & Objective

• To extend the $ T $-contraction framework of Beiranvand et al. and Moradi to cone metric spaces.
• To establish existence and uniqueness of fixed points for a class of mappings $ D_T(a,b) $ without requiring continuity of the mapping $ S $.
• To generalize known fixed point results—such as Banach, Kannan, and Reich contractions—for cone metric spaces using a function-dependent contraction condition.
• To unify and extend prior results in cone metric spaces by introducing a parameterized contraction condition involving a function $ T $.

Proposed method

• Define a new class of mappings $ D_T(a,b) $ on cone metric spaces, where the contraction condition involves a function $ T $, with parameters $ a,b \geq 0 $, and satisfies $ d(TSx,TSy) \leq a d(Tx,Ty) + b [d(Tx,TSx) + d(Ty,TSy)] $ for all $ x,y \in M $.
• Utilize the structure of cone metric spaces, where the metric takes values in a real Banach space with a cone $ P $, and define convergence via the order induced by $ P $.
• Assume $ T $ is continuous, injective, and either subsequentially or sequentially convergent to ensure convergence of iterates.
• Prove that under the condition $ a + 2b < 1 $, the mapping $ S $ has a unique fixed point in a complete cone metric space.
• Apply the results to recover and generalize known theorems: when $ T = \text{id} $, the result reduces to the classical $ D(a,b) $ class; when $ a = 0 $, it generalizes $ T $-Kannan contractions.
• Use the cone metric space framework to extend metric space results to more general settings, particularly with normal or regular cones.

Experimental results

Research questions

• RQ1 Can the $ T $-contraction framework be extended to cone metric spaces to generalize fixed point theorems?
• RQ2 What conditions on the function $ T $ and parameters $ a,b $ ensure the existence and uniqueness of a fixed point for $ S $ in a cone metric space?
• RQ3 How does the $ D_T(a,b) $ class unify and generalize known contraction types such as Banach, Kannan, and Reich contractions in cone metric spaces?
• RQ4 What role does the sequential convergence of $ T $ play in ensuring convergence of the iterative sequence $ (S^n x_0) $ to the fixed point?
• RQ5 Under what conditions does the mapping $ S $ have a unique fixed point when $ T \neq \text{id} $, and how does this generalize prior results?

Key findings

• The mapping $ S $ has a unique fixed point in a complete cone metric space if $ T $ is continuous, injective, and subsequentially convergent, and $ a + 2b < 1 $, with $ d(TSx,TSy) \leq a d(Tx,Ty) + b [d(Tx,TSx) + d(Ty,TSy)] $.
• If $ T $ is sequentially convergent, then for any $ x_0 \in M $, the sequence $ (S^n x_0) $ converges to the unique fixed point of $ S $.
• When $ T = \text{id} $, the result reduces to the known $ D(a,b) $ class on cone metric spaces, recovering Theorem 2.7 of Khan and Samanipour (2008).
• For $ a = 0 $ and $ b \in [0, \frac{1}{2}) $, the result generalizes the $ T $-Kannan contraction, extending Moradi’s work to cone metric spaces.
• When $ b = 0 $ and $ a < 1 $, the result generalizes the $ T $-Banach contraction, extending the classical Banach contraction principle to cone metric spaces.
• The framework unifies and generalizes multiple known fixed point theorems, including those of Guang and Zhang, Khan and Samanipour, and Rhoades, in the cone metric setting.