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Optimal pair of fixed points: existence, uniqueness and approximation.

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Title: Optimal pair of fixed points: existence, uniqueness and approximation.
Authors: Safari-Hafshejani, A.1 (AUTHOR) asafari@pnu.ac.ir, Gabeleh, M.2 (AUTHOR) Gabeleh@abru.ac.ir
Source: Fixed Point Theory & Algorithms for Sciences & Engineering. 2/10/2026, Vol. 2026 Issue 1, p1-20. 20p.
Subjects: Fixed point theory, Existence theorems, Metric spaces, Banach spaces, A posteriori error analysis
Abstract: In this paper, we introduce a novel class of mappings, referred to as noncyclic generalized θ-contractions. By employing the geometric concept of W U C property in metric spaces, we establish new existence and convergence theorems for the fixed points associated with these mappings. The results presented herein generalize and improve several existing fixed point theorems related to generalized φ-contractions. Furthermore, we address the issue of error estimation and derive both a priori and a posteriori error bounds for the fixed points obtained via the Picard iterative process applied to a noncyclic generalized θ-contraction mapping defined on a uniformly convex Banach space. A distinctive feature of our analysis lies in avoiding the use of geometric progression techniques. Consequently, the resulting error estimates hold unconditionally in uniformly convex Banach spaces, thereby removing the need for any restrictive power-type condition on the modulus of convexity. We then present a comprehensive example to illustrate and validate the applicability and robustness of the main theoretical results. Finally, we apply the existence and convergence results for optimal pairs of fixed points to a system of differential equations. [ABSTRACT FROM AUTHOR]
Copyright of Fixed Point Theory & Algorithms for Sciences & Engineering is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: In this paper, we introduce a novel class of mappings, referred to as noncyclic generalized θ-contractions. By employing the geometric concept of W U C property in metric spaces, we establish new existence and convergence theorems for the fixed points associated with these mappings. The results presented herein generalize and improve several existing fixed point theorems related to generalized φ-contractions. Furthermore, we address the issue of error estimation and derive both a priori and a posteriori error bounds for the fixed points obtained via the Picard iterative process applied to a noncyclic generalized θ-contraction mapping defined on a uniformly convex Banach space. A distinctive feature of our analysis lies in avoiding the use of geometric progression techniques. Consequently, the resulting error estimates hold unconditionally in uniformly convex Banach spaces, thereby removing the need for any restrictive power-type condition on the modulus of convexity. We then present a comprehensive example to illustrate and validate the applicability and robustness of the main theoretical results. Finally, we apply the existence and convergence results for optimal pairs of fixed points to a system of differential equations. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Fixed Point Theory & Algorithms for Sciences & Engineering is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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