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23 pages, 1287 KB

Open AccessArticle

Reliability Analysis of a Hardware–Software Series Repairable System with Multiple Vacations of a Repairman

by Qi Tu and Xue Feng

Mathematics 2026, 14(9), 1524; https://doi.org/10.3390/math14091524 - 30 Apr 2026

This paper develops a reliability analysis model for a class of computer systems composed of hardware and software in series, considering a repairman taking multiple vacations. The system follows a series failure rule: hardware can be repaired to be as good as new [...] Read more.

This paper develops a reliability analysis model for a class of computer systems composed of hardware and software in series, considering a repairman taking multiple vacations. The system follows a series failure rule: hardware can be repaired to be as good as new after failure; software undergoes minor repairs to maintain operability after the first

N - 1

failures with an increasing failure rate, and is overhauled to be as good as new with cycle reset after the N-th failure. Based on the principle of probability conservation and the supplementary variable method, the state probability evolution equations of the system are derived. A Banach space is constructed, and a linear operator is defined, whose denseness, dissipativity, and closedness are verified. It has been proven that the operator generates a positive contractive

C_{0}

-semigroup, thus rigorously establishing the well-posedness of the model and the existence of a unique positive dynamic solution. Further spectral analysis verifies that zero belongs to the continuous spectrum rather than the point spectrum of the system operator.This indicates that the investigated system admits no time-invariant constant steady-state probability distribution,and only presents slowly decaying quasi-stationary dynamic behavior. The results can provide theoretical support for the reliability design and maintenance strategy optimization of hardware–software series repairable systems. Full article

14 pages, 307 KB

Open AccessArticle

Coupled System of Variable-Order Fractional Differential Equations

by Amjad E. Hamza, Mostefa Seghier, Kadda Maazouz, Zineb Bellabes, Abdelkader Moumen and Mohamed Bouye

Fractal Fract. 2026, 10(5), 305; https://doi.org/10.3390/fractalfract10050305 - 29 Apr 2026

Abstract

This work explores the growing field of fractional calculus, with particular emphasis on the complexities and opportunities associated with variable-order derivatives. We critically assess existing definitions, identifying those that are consistent with the established principles of constant-order fractional calculus. Based on this analysis, [...] Read more.

This work explores the growing field of fractional calculus, with particular emphasis on the complexities and opportunities associated with variable-order derivatives. We critically assess existing definitions, identifying those that are consistent with the established principles of constant-order fractional calculus. Based on this analysis, we introduce new formulations derived from the Grünwald–Letnikov and Liouville approaches, together with a novel variable-order Mittag–Leffler function. The core of our study is devoted to investigating the existence and uniqueness of solutions for a coupled system of variable-order fractional differential equations subject to initial conditions. Using Schauder’s fixed-point theorem and the Banach contraction principle, we establish new results that contribute to strengthening the theoretical foundation of such dynamical systems. Full article

(This article belongs to the Section General Mathematics, Analysis)

17 pages, 342 KB

Open AccessArticle

Existence, Uniqueness and Ulam-Hyers Stability for a Coupled System of Sequential Hilfer Fractional Differential Equations with Nonlocal Coupled Boundary Conditions

by Mihoub Bouderbala, Souad Ayadi, Meltem Erden Ege, Ozgur Ege and Mohammed Rabih

Fractal Fract. 2026, 10(5), 302; https://doi.org/10.3390/fractalfract10050302 - 29 Apr 2026

Abstract

This paper investigates the existence, uniqueness, and stability of solutions for a new class of coupled systems of sequential fractional differential equations involving the Hilfer fractional derivative. Generalizing previous works based on Caputo derivatives, we employ the Hilfer operator, which interpolates between Riemann–Liouville [...] Read more.

This paper investigates the existence, uniqueness, and stability of solutions for a new class of coupled systems of sequential fractional differential equations involving the Hilfer fractional derivative. Generalizing previous works based on Caputo derivatives, we employ the Hilfer operator, which interpolates between Riemann–Liouville and Caputo derivatives. The nonlinear terms are fully coupled, and the boundary conditions are nonlocal and coupled. The main results are established using the Banach Contraction Principle and Schaefer’s Fixed Point Theorem, with rigorous, detailed proofs for each step, addressing specific methodological requirements regarding operator invariance and space completeness. Furthermore, we provide a comprehensive analysis of the Ulam–Hyers stability of the proposed system, with explicitly tracked stability constants. An illustrative example with numerical verification is provided to validate the theoretical findings. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point Theory and Applications to Fractional Differential Equations)

13 pages, 277 KB

Open AccessArticle

On the Mild Solutions of Second-Order Θ-Caputo Fractional Boundary Value Problems

by Mouataz Billah Mesmouli, Abdelouaheb Ardjouni, Loredana Florentina Iambor and Taher S. Hassan

Mathematics 2026, 14(9), 1434; https://doi.org/10.3390/math14091434 - 24 Apr 2026

Viewed by 96

Abstract

In this paper, we study a class of second-order fractional boundary value problems involving

Θ

-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the

Θ

-fractional framework. Existence [...] Read more.

In this paper, we study a class of second-order fractional boundary value problems involving

Θ

-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the

Θ

-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results. Full article

(This article belongs to the Special Issue Advances in Fractional Differential Equations and Applications)

24 pages, 417 KB

Open AccessArticle

Existence, Stability, and Circular Interactions in m-Cyclic Coupled Systems of Sequential (k,ψ)-Hilfer and (k,ψ)-Caputo Type with Boundary Conditions

by F. Gassem, Mohammed Almalahi, Khaled Aldwoah, Arafa Dawood, Alawia Adam, Amer Alsulami and L. M. Abdalgadir

Fractal Fract. 2026, 10(5), 288; https://doi.org/10.3390/fractalfract10050288 - 24 Apr 2026

Viewed by 130

Abstract

This paper examines an m-cyclic coupled system of sequential

(k, ψ)

-Hilfer and

(k, ψ)

-Caputo fractional differential equations with boundary conditions. The nonlinearities follow a cyclic pattern: for [...] Read more.

This paper examines an m-cyclic coupled system of sequential

(k, ψ)

-Hilfer and

(k, ψ)

-Caputo fractional differential equations with boundary conditions. The nonlinearities follow a cyclic pattern: for

j = 1, \dots, m - 1

,

f_{j}

depends on

x_{j}

and

x_{j + 1}

and

f_{m}

depends on

x_{m}

and

x_{1}

, forming a closed loop of interactions. We convert the system into an equivalent integral equation and establish existence and uniqueness results using four fixed-point theorems: the Banach contraction principle, Schaefer’s theorem, Krasnosel’skiĭ’s theorem, and the Leray–Schauder alternative. A thorough Ulam–Hyers stability analysis is presented with explicit stability constants. Numerical examples illustrate the applicability of the theoretical findings. Full article

44 pages, 30545 KB

Open AccessArticle

A Novel Inertial-Type Iteration Algorithm: Convergence, Data Dependence, and Applications in Image Deblurring and Fractal Generation

by Kadri Doğan, Faik Gürsoy and Emirhan Hacıoğlu

Mathematics 2026, 14(9), 1433; https://doi.org/10.3390/math14091433 - 24 Apr 2026

Viewed by 128

Abstract

This study introduces a novel inertial-type iteration algorithm based on the Normal S iteration for the class of almost contraction mappings in Banach spaces. Traditional fixed point iterations often suffer from slow convergence and high computational cost; to address these limitations, the proposed [...] Read more.

This study introduces a novel inertial-type iteration algorithm based on the Normal S iteration for the class of almost contraction mappings in Banach spaces. Traditional fixed point iterations often suffer from slow convergence and high computational cost; to address these limitations, the proposed framework incorporates an adaptive inertial-type parameter. We establish strong convergence of the algorithm and derive explicit a posteriori error estimates under weak contractive conditions. In addition, we demonstrate the asymptotic equivalence of the NS inertial-type trajectories with the classical Normal S iteration, provide a comprehensive weak

w^{2} -

stability analysis, and obtain sharp upper bounds for the data dependence problem. The practical performance of the algorithm is evaluated in two distinct computational domains: image deblurring via wavelet-based

ℓ_{1}

regularization and the generation of complex fractal patterns, including Julia and Mandelbrot sets. Numerical results show that the proposed inertial-type iteration algorithm significantly outperforms existing methods—such as Picard, Mann, Ishikawa, and standard Normal S iterations—achieving faster convergence, higher PSNR values in image restoration, and more stable basins of attraction in fractal visualizations. These findings highlight the effectiveness and versatility of the NS inertial-type iteration algorithm approach for both theoretical analysis and real-world applications. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications, 3rd Edition)

19 pages, 397 KB

Open AccessArticle

On a Class of Nonlocal Integro-Delay Problems with Generalized Tempered Fractional Operators

by Marwa Ennaceur, Mohammed S. Abdo, Osman Osman, Amel Touati, Amer Alsulami, Neama Haron and Khaled Aldwoah

Fractal Fract. 2026, 10(4), 272; https://doi.org/10.3390/fractalfract10040272 - 21 Apr 2026

Viewed by 494

Abstract

This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential [...] Read more.

This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential tempering, a delayed state, a lower-order distributed fractional memory term, and multipoint nonlocal initial data, rather than introducing a new fractional operator. The resulting problem can be viewed as a rigorous well-posedness prototype for hereditary systems with delayed feedback, tempered memory, and nonlocal initialization. First, an equivalent Volterra integral equation is derived. Then, the existence and uniqueness of solutions are obtained by the Banach contraction principle in a suitable Banach space of continuous functions. Next, a Picard successive approximation procedure is introduced and shown to converge uniformly to the unique solution, together with an explicit a priori error estimate. Moreover, a continuous dependence result is proved with respect to perturbations in the initial constants, the multipoint coefficients, and the nonlinear term. Finally, the main results are illustrated with two examples enhanced by graphs of explicit Picard approximations and convergence tables. Full article

(This article belongs to the Section General Mathematics, Analysis)

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35 pages, 474 KB

Open AccessReview

Developments in Modular Space Fixed Point Theory

by Wojciech M. Kozlowski

Mathematics 2026, 14(7), 1234; https://doi.org/10.3390/math14071234 - 7 Apr 2026

Viewed by 358

Abstract

This survey article offers a snapshot view of the present state of fixed point theory within modular spaces, highlighting fundamental principles and their applications. The discussion primarily revolves around operators and their semigroups that satisfy pointwise asymptotic nonexpansive and contractive conditions in the [...] Read more.

This survey article offers a snapshot view of the present state of fixed point theory within modular spaces, highlighting fundamental principles and their applications. The discussion primarily revolves around operators and their semigroups that satisfy pointwise asymptotic nonexpansive and contractive conditions in the modular sense, and the results can also be applied directly to Banach spaces. Utilizing the framework of regular and super-regular modular spaces, our research generalizes several established results concerning fixed points of nonlinear operators, applicable to both Banach spaces and modular function spaces. The study seeks to identify and discuss current challenges, knowledge gaps, and unresolved questions, providing insights into the potential of future research opportunities. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Applications)

19 pages, 327 KB

Open AccessArticle

Well-Posedness and Ulam-Hyers Stability of Coupled Deformable Fractional Differential Systems via Perov and Leray-Schauder Approaches

by Khelifa Daoudi, Abdeldjabar Bourega, Mohammed Rabih, Osman Abdalla Osman and Muntasir Suhail

Fractal Fract. 2026, 10(4), 236; https://doi.org/10.3390/fractalfract10040236 - 1 Apr 2026

Viewed by 316

Abstract

In this paper, we investigate the well-posedness and stability of a class of coupled systems of deformable fractional differential equations in Banach spaces. The deformable fractional derivative, which interpolates continuously between a function and its classical derivative through a single scalar parameter, provides [...] Read more.

In this paper, we investigate the well-posedness and stability of a class of coupled systems of deformable fractional differential equations in Banach spaces. The deformable fractional derivative, which interpolates continuously between a function and its classical derivative through a single scalar parameter, provides a flexible and tractable framework for modeling complex dynamical phenomena with memory effects. By employing Perov’s fixed-point theorem under matrix contractive conditions, we establish the existence and uniqueness of solutions for the considered coupled system. The existence of at least one solution under broader growth conditions is then proved via the nonlinear alternative of Leray–Schauder type. Furthermore, the continuous dependence of solutions on initial data is rigorously established, confirming the well-posedness of the system. Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability results are also derived, providing quantitative estimates relevant to numerical approximation and applied analysis. Three illustrative examples are presented to demonstrate the applicability and effectiveness of the theoretical results. Full article

(This article belongs to the Section Numerical and Computational Methods)

28 pages, 383 KB

Open AccessArticle

Reduction of Implicit Caputo-Hadamard Fractional Systems to Compact Fixed-Point Operators Under Nonlocal Integral Constraints

by Muath Awadalla and Dalal Alhwikem

Mathematics 2026, 14(7), 1156; https://doi.org/10.3390/math14071156 - 30 Mar 2026

Viewed by 295

Abstract

This paper develops an operator-reduction framework for a class of coupled implicit Caputo-Hadamard fractional differential systems subject to nonlocal Hadamard integral constraints. The system, involving fractional derivatives in both state and auxiliary variables, is resolved through a pointwise contraction argument that eliminates the [...] Read more.

This paper develops an operator-reduction framework for a class of coupled implicit Caputo-Hadamard fractional differential systems subject to nonlocal Hadamard integral constraints. The system, involving fractional derivatives in both state and auxiliary variables, is resolved through a pointwise contraction argument that eliminates the auxiliary components and reduces the problem to a two-dimensional fixed-point operator acting on a Banach space of continuous functions. This reduction overcomes the compactness obstruction that arises in direct multi-component formulations. Under explicit growth and smallness conditions, the existence of at least one solution is established via Mönch’s fixed-point theorem. By imposing strengthened Lipschitz hypotheses, the reduced operator becomes a strict contraction on an invariant ball, yielding uniqueness and Ulam-Hyers stability with explicit constant

C_{U H} = 1 / (1 - Λ)

. A fully computed example demonstrates the verifiability of the theoretical assumptions and illustrates how the smallness condition

Λ < 1

governs both existence and stability. The results establish a systematic operator-based approach for implicit Caputo-Hadamard systems with nonlocal integral constraints. Full article

(This article belongs to the Special Issue Mathematical Analysis of Fractional Differential Equations and Dynamical Systems)

25 pages, 1221 KB

Open AccessArticle

Solvability and Stability Analysis of Three-Dimensional ABC Fractional Systems in Locally Compact Hausdorff Spaces: Applications to Chaotic and Fluid Systems

by Hasan N. Zaidi, Osman Osman, Arafa Dawood, Amin Saif, Amira S. Awaad, Khaled Aldwoah and L. M. Abdalgadir

Fractal Fract. 2026, 10(4), 214; https://doi.org/10.3390/fractalfract10040214 - 25 Mar 2026

Viewed by 278

Abstract

This paper considers a general three-dimensional ABC fractional dynamical system formulated in a bounded locally compact Hausdorff space. The locally compact Hausdorff structure ensures that the compact-open topology on the space of continuous functions coincides with the topology induced by the supremum norm, [...] Read more.

This paper considers a general three-dimensional ABC fractional dynamical system formulated in a bounded locally compact Hausdorff space. The locally compact Hausdorff structure ensures that the compact-open topology on the space of continuous functions coincides with the topology induced by the supremum norm, providing an appropriate Banach space framework for the analysis of the system. Within this setting we study the continuity, boundedness, and Lipschitz properties of the nonlinear operators associated with the fractional model. Based on these properties, the existence of solutions is established using Schaefer fixed point theorem, while uniqueness is obtained through Banach contraction principle under suitable conditions. Furthermore, the Hyers–Ulam stability of the system is investigated, showing that small perturbations lead to small deviations in the corresponding solutions. Finally, the theoretical results are applied to the fractional Lorenz system and the two-dimensional fractional Euler system, illustrating the applicability of the proposed framework to models arising in chaotic dynamics and fluid mechanics. Full article

(This article belongs to the Section Complexity)

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10 pages, 247 KB

Open AccessArticle

Periodic Solutions to Matrix Delay Difference Systems Under Exponential Dichotomy Conditions

by Mouataz Billah Mesmouli, Loredana Florentina Iambor and Taher S. Hassan

Mathematics 2026, 14(7), 1101; https://doi.org/10.3390/math14071101 - 25 Mar 2026

Viewed by 289

Abstract

This paper studies the existence and uniqueness of periodic solutions to a class of nonlinear neutral matrix difference systems with multiple delays. The analysis is based on the construction of a suitable Green operator combined with fixed-point methods under exponential dichotomy assumptions. The [...] Read more.

This paper studies the existence and uniqueness of periodic solutions to a class of nonlinear neutral matrix difference systems with multiple delays. The analysis is based on the construction of a suitable Green operator combined with fixed-point methods under exponential dichotomy assumptions. The existence of periodic solutions is established using Krasnoselskii’s fixed-point theorem, while uniqueness is demonstrated under a natural contraction condition via Banach’s principle. The results extend previous contributions on neutral difference systems and provide discrete analogues of related differential models. Examples are included to illustrate the applicability of the theory. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions, 3rd Edition)

18 pages, 289 KB

Open AccessFeature PaperArticle

The New Bipolar Intuitionistic Fuzzy Metric Space (NBIFM-Space) with Applications

by Bratislav Iričanin, Tatjana Došenović, Nebojša M. Ralević and Biljana Carić

Axioms 2026, 15(3), 239; https://doi.org/10.3390/axioms15030239 - 23 Mar 2026

Viewed by 351

Abstract

This paper introduces the New Bipolar Intuitionistic Fuzzy Metric Space (NBIFM-space)—a mathematical framework that extends intuitionistic and previously proposed bipolar intuitionistic structures by providing a complete three-component formulation based on positive similarity, negative similarity, and indeterminacy. Unlike earlier bipolar intuitionistic models, [...] Read more.

This paper introduces the New Bipolar Intuitionistic Fuzzy Metric Space (NBIFM-space)—a mathematical framework that extends intuitionistic and previously proposed bipolar intuitionistic structures by providing a complete three-component formulation based on positive similarity, negative similarity, and indeterminacy. Unlike earlier bipolar intuitionistic models, the NBIFM-space employs normalized metric components and coordinated triangular norms denoted by t-norm/t-conorm interactions, yielding a fully consistent topological and analytic setting. We have developed the basic properties of this structure and have demonstrated its effectiveness in image processing, where the explicit separation of attraction, repulsion, and uncertainty leads to robust edge-preserving filtering. Furthermore, a Banach-type fixed point theorem is established in the full NBIFM framework. Full article

(This article belongs to the Special Issue Advances in Fuzzy Logic with Applications)

22 pages, 504 KB

Open AccessArticle

Approximate Controllability and Existence Results of the Sobolev-Type Fractional Stochastic Differential Equation Driven by a Fractional Brownian Motion

by Sadam Hussain, Muhammad Sarwar, Syed Khayyam Shah, Kamaleldin Abodayeh and Manuel De La Sen

Fractal Fract. 2026, 10(3), 203; https://doi.org/10.3390/fractalfract10030203 - 20 Mar 2026

Viewed by 291

Abstract

In this article, we investigate the existence and approximate controllability of a class of Sobolev-type fractional stochastic differential equations of order

1 < δ < 2

with infinite delay. The analysis is carried out in an abstract Hilbert space framework, incorporating fractional dynamics [...] Read more.

In this article, we investigate the existence and approximate controllability of a class of Sobolev-type fractional stochastic differential equations of order

1 < δ < 2

with infinite delay. The analysis is carried out in an abstract Hilbert space framework, incorporating fractional dynamics together with stochastic perturbations. By employing techniques from fractional calculus, semigroup theory, and fixed point theory, particularly the Banach contraction principle along with compactness arguments, we establish the existence of mild solutions for the proposed system. Subsequently, sufficient conditions for approximate controllability are derived by combining operator-theoretic methods with stochastic analysis. The novelty of this work lies in extending controllability results to Sobolev-type fractional stochastic systems of order

1 < δ < 2

, where both the higher-order fractional structure and stochastic effects are treated simultaneously within a unified framework. This generalizes and complements several existing results in the literature that mainly address deterministic systems or fractional differential equations of order

0 < δ \leq 1

. Finally, an illustrative example is presented to demonstrate the applicability and effectiveness of the theoretical findings. Full article

(This article belongs to the Special Issue Stochastic Dynamical Systems with Fractional Derivative: Theoretical Analysis and Numerical Simulation)

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25 pages, 447 KB

Open AccessArticle

Stability and Controllability of Coupled Neutral Impulsive ϱ-Fractional System with Mixed Delays

by F. Gassem, Mohammed Almalahi, Mohammed Rabih, Manal Y. A. Juma, Amira S. Awaad, Ali H. Tedjani and Khaled Aldwoah

Fractal Fract. 2026, 10(3), 192; https://doi.org/10.3390/fractalfract10030192 - 13 Mar 2026

Cited by 1 | Viewed by 481

Abstract

This study examines a comprehensive class of coupled nonlinear

ϱ

-Hilfer fractional neutral impulsive integro-differential systems with mixed delays and non-local initial conditions. The primary contribution of this study is the creation of a unified analytical framework that encompasses coupled interactions, neutral-type dependencies, [...] Read more.

This study examines a comprehensive class of coupled nonlinear

ϱ

-Hilfer fractional neutral impulsive integro-differential systems with mixed delays and non-local initial conditions. The primary contribution of this study is the creation of a unified analytical framework that encompasses coupled interactions, neutral-type dependencies, and impulsive disturbances, which have been studied separately by researchers. We utilize the Banach contraction principle and Krasnoselskii’s fixed-point theorem to provide suitable conditions for the existence and uniqueness of solutions within the product space of piecewise continuous weighted functions. In addition to existence, we examine Ulam–Hyers–Rassias (UHR) stability using a generalized Gronwall inequality, which guarantees the system’s robustness against functional perturbations. We also develop a controllability framework and a feedback control law that steer the system towards the desired terminal states. The theoretical results are supported by a numerical simulation using a complex kernel, implemented via a modified predictor-corrector algorithm, which validates the practical effectiveness of the proposed control and stability outcomes. Full article

(This article belongs to the Section Complexity)

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