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Editorial

Trends in Fixed Point Theory and Fractional Calculus

by
Boško Damjanović
1 and
Pradip Debnath
2,*
1
Department of Mathematics, University Union—Nikola Tesla, 11158 Belgrade, Serbia
2
Department of Mathematical Sciences, Tezpur University, Assam 784028, India
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(9), 660; https://doi.org/10.3390/axioms14090660
Submission received: 15 August 2025 / Accepted: 27 August 2025 / Published: 28 August 2025
(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

1. Introduction

This Special Issue of Axioms, titled “Trends in Fixed Point Theory and Fractional Calculus”, presents a collection of ten high-quality papers reflecting the latest developments in two intertwined areas of modern mathematical analysis. Fixed-point theory serves as a cornerstone of nonlinear analysis, with far-reaching applications to functional equations, variational inequalities, and operator theory. Fractional calculus, by extending the concept of differentiation and integration to non-integer orders, has proven to be a versatile tool for modeling memory effects and hereditary properties in complex physical and engineering systems.
The synergy between these areas has become increasingly evident in the study of differential and integral equations, stability problems, and iterative algorithms. The contributions in this Special Issue span both theoretical advances, such as new contractive conditions, generalized spaces, and stability criteria, and practical applications, including integral equations and fractional differential systems.

2. Overview of the Published Papers

  • Fixed-Point Theorems in Branciari Distance Spaces (Seong-Hoon Cho) introduces σ -Caristi and generalized σ -contraction maps, establishing fixed-point results that extend Caristi’s theorem and Banach’s contraction principle and clarifying the relationships among various contraction conditions.
  • m-Isometric Operators with Null Symbol and Elementary Operator Entries (Bhagwati Prashad Duggal) investigates strict ( m , X ) -isometric operator pairs on Banach spaces, offering structural insights relevant to functional analysis and operator theory.
  • Relational Almost ( φ , ψ ) -Contractions and Applications to Nonlinear Fredholm Integral Equations (Fahad M. Alamrani et al.) presents new fixed-point results under relational strict almost ( φ , ψ ) -contractions, with applications to the solvability of nonlinear Fredholm integral equations.
  • Fixed-Point Results of F-Contractions in Bipolar p-Metric Spaces (Nabanita Konwar and Pradip Debnath) develops Banach-type and Reich-type theorems for F-contractions in bipolar p-metric spaces, supported by illustrative examples.
  • Fixed Point Results in Modular b-Metric-like Spaces with an Application (Nizamettin Ufuk Bostan and Banu Pazar Varol) introduces modular b-metric-like spaces, defines notions of ξ -convergence and ξ -Cauchy sequences, and proves fixed-point theorems with practical applications.
  • Enriched Z-Contractions and Fixed-Point Results with Applications to IFS (Ibrahim Alraddadi et al.) initiates a broad class of enriched ( d , Z ) -contractions on Banach spaces, establishing uniqueness and existence theorems and applying them to iterative function systems.
  • Nonlinear Contractions Employing Digraphs and Comparison Functions with an Application to Singular Fractional Differential Equations (Doaa Filali et al.) extends Jachymski’s contraction principle via digraphs to study fixed points in graph metric spaces, applying the results to singular fractional differential equations.
  • Stability of Fixed Points of Partial Contractivities and Fractal Surfaces (María A. Navascués) examines a wide class of contractions in b-metric spaces, including Banach and Matkowski maps, providing convergence and stability results for Picard iterations with implications for fractal geometry.
  • Three Existence Results in the Fixed Point Theory (Alexander J. Zaslavski) offers three new existence theorems for fixed points of nonexpansive and set-valued mappings, generalizing known results on F-contractions and set-valued contractions.
  • Fixed-Point Results of Generalized ( φ , Ψ ) -Contractive Mappings in Partially Ordered Controlled Metric Spaces with an Application to a System of Integral Equations (Mohammad Akram et al.) proves multiple fixed-point and coincidence point results, applying them to solve a system of integral equations.

3. Concluding Remarks

The contributions gathered here demonstrate both the diversity and the depth of current research in fixed-point theory and fractional calculus. From abstract generalizations in metric and Banach space settings to concrete applications in integral and fractional differential equations, these works collectively advance the frontiers of the field.
We thank all the authors for their valuable contributions, the reviewers for their careful evaluations, and the Axioms editorial team for their support in producing this Special Issue. We hope that these papers will serve as a source of inspiration for future research, fostering new connections between theoretical exploration and applied problem-solving.

Conflicts of Interest

The authors declare no conflicts of interest.

List of Contributions

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MDPI and ACS Style

Damjanović, B.; Debnath, P. Trends in Fixed Point Theory and Fractional Calculus. Axioms 2025, 14, 660. https://doi.org/10.3390/axioms14090660

AMA Style

Damjanović B, Debnath P. Trends in Fixed Point Theory and Fractional Calculus. Axioms. 2025; 14(9):660. https://doi.org/10.3390/axioms14090660

Chicago/Turabian Style

Damjanović, Boško, and Pradip Debnath. 2025. "Trends in Fixed Point Theory and Fractional Calculus" Axioms 14, no. 9: 660. https://doi.org/10.3390/axioms14090660

APA Style

Damjanović, B., & Debnath, P. (2025). Trends in Fixed Point Theory and Fractional Calculus. Axioms, 14(9), 660. https://doi.org/10.3390/axioms14090660

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