Editorial for the Special Issue of Mathematics “Fractional Differential Equations, Inclusions and Inequalities with Applications II”

1. Introduction

Within the field of fractional calculus, defined fractional derivatives and integrals within complex domains and their practical applications are frequently explored, and in recent years, this field has seen a significant rise in research interest. Fractional differential equations, in particular, have garnered widespread attention in academic literature due to their effectiveness in modeling various real-world phenomena across disciplines such as physics, engineering, design, materials science, fluid dynamics, probability theory, image analysis, optimal control, and more. These equations are also considered superior to traditional integer-order differential equations when it comes to representing the hereditary characteristics of different materials and dynamic systems.

2. Special Issue Overview

This Special Issue, entitled “Fractional Differential Equations, Inclusions, and Inequalities with Applications II” emphasizes new developments in both the conceptual basis and applied aspects of fractional differential equations, inclusions, and inequalities, as well as the systems involving these elements. The scope includes, but is not limited to, the following topics:

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Oscillation for fractional-order differential equations, Contribution 1;

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$(p,q)$-difference equations, Contribution 2;

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q-difference inclusions, Contribution 5;

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Integral inequalities, Contributions 4, 10, 18;

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Fractional integro-differential equations, Contribution 6;

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Sequential fractional differential equations, Contribution 7;

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Sequential fractional differential inclusions, Contribution 3;

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Numerical fractional differential equations, Contribution 8;

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$\mathsf{\Lambda }$-fractional differential equations, Contribution 9;

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Positive solutions for fractional boundary value problems, Contributions 13, 19;

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Stochastic fractional differential equations, Contribution 14;

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Fractional-order systems, Contribution 17;

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Caputo fractional differential equations, Contributions 12, 15;

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Delay difference equations, Contribution 16;

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Black–Scholes fractional equations, Contribution 11.

A total of eighty manuscripts were received, out of which nineteen papers authored by sixty-eight researchers were accepted for publication. The contributing authors represent twenty-one different countries:

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Algeria (1)

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Azerbaijan (1)

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China (8)

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Egyp (3)

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Greece (3)

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India (2)

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Indonesia (2)

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Iraq (2)

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Iran (7)

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Jordan (3)

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Malaysia (2)

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Pakistan (2)

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Poland (1)

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Portugal (1)

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Romania (1)

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Saudi Arabia (9)

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Taiwan (1)

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Thailand (6)

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Turkiye (3)

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UAE (3)

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USA (7)

This Special Issue features 19 articles spanning various areas within the field of fractional differential equations. The key findings of these contributions are summarized below.

• In Contribution 1, the authors provide an in-depth overview of recent progress in the study of oscillatory behavior associated with fractional difference equations. Various forms of these equations are examined through the application of both nabla and delta operators.
• The existence and uniqueness of solutions for a fractional $(p,q)$-difference equation with separated local boundary conditions are explored in Contribution 2. Uniqueness is established via Banach’s contraction principle, while existence is proven using Krasnosel’skiĭ’s fixed point theorem and the Leray–Schauder alternative. Several illustrative examples are included to support the main results.
• In Contribution 3, the authors focus on establishing theoretical existence results for a new class of problems that incorporates a sequential Caputo-type term within a hybrid integro-differential framework. The boundary conditions are expressed as hybrid constraints involving multiple orders of integro-differential operators. The study begins by deriving fundamental inequalities related to the corresponding integral equation. To address the problem, the authors employ recently developed analytical techniques involving product operators in Banach algebras, along with special function-based tools such as $\alpha -\psi$-contractions and $\alpha$-admissible mappings. These methods are used to establish existence criteria for the proposed class of mixed sequential hybrid boundary value problems. Key properties—including the approximate endpoint property, the $C_{\alpha }$-property, and compactness—are essential for the analysis. The paper concludes with two illustrative examples that demonstrate the applicability of the theoretical results.
• In Contribution 4, the authors introduce the notion of $(g,h)$-convexity, a generalized form of coordinated convexity defined in realtion to a pair of functions on the coordinates. The fundamental properties of this new convexity concept are investigated in detail. Additionally, the study derives new Hermite–Hadamard and Ostrowski-type inequalities corresponding to this form of coordinated convexity. These results extend and generalize several known inequalities in the existing literature. The paper concludes with various mathematical examples and graphical representations that illustrate and validate the newly developed inequalities.
• In Contribution 5, the authors analyze the existence and topological configuration of the solution set associated with a q-fractional differential inclusion in a Banach space setting. Analytical techniques are drawn from set-valued analysis, the Kuratowski non-compactness measure, and Darbo’s fixed point principle. To validate the theoretical results, an illustrative example is presented.
• In Contribution 6, the authors establish sufficient conditions for the existence of both local and global solutions for a general class of nonlinear fractional integro-differential equations in two dimensions. The uniqueness of these solutions is also proven. To enable numerical approximation, the study utilizes operational matrices derived from two-variable shifted Jacobi polynomials within a collocation method framework, thereby converting the original equations into a system that can be efficiently solved. Error estimates for the proposed numerical approach are provided. A set of five test problems is presented, and the corresponding numerical results demonstrate the accuracy, efficiency, and practical effectiveness of the method.
• Contribution 7 introduces and explores a novel class of boundary value problems characterized by a mixed-type fractional differential equation involving both the ${\psi }_1$-Hilfer and ${\psi }_2$-Caputo fractional derivatives, coupled with non-local integro-differential boundary conditions. The uniqueness of the solutions is demonstrated via the Banach contraction method and ensures existing results via the Leray–Schauder nonlinear alternative approach. Numerical examples are included to illustrate and validate the theoretical results.
• Contribution 8 aims to present a novel numerical method for solving fractional boundary value problems. This method relies on two numerical schemes: A fractional central scheme is considered for approximating the Caputo derivative of order $\alpha$, along with a newly developed central formula for approximating the Caputo derivative of order $2\alpha$, where $0<\alpha \le 1$. The first method is re-examined, while the second is derived using the generalized Taylor expansion. The stability of the proposed approach is examined through well-established theoretical frameworks. Additionally, several numerical examples are provided to showcase the method’s accuracy and practical utility.
• The study in Contribution 9 investigates wave propagation in solids using the framework of inherently non-local $\mathsf{\Lambda }$-fractional analysis. Beginning with the fundamental equations of $\mathsf{\Lambda }$-fractional continuum mechanics, the corresponding $\mathsf{\Lambda }$-fractional wave equations are formulated. Due to the global nature of the variational formulation in this setting, the analysis allows for the representation of discontinuities in strain or stress. The model is further employed to examine impact-induced phase transitions in composite materials exhibiting both elastic and viscoelastic behavior.
• The central focus of Contribution 10 lies in developing a generalized and original identity related to the Caputo–Fabrizio fractional operator. Building on this new identity, the authors derive a series of fractional integral inequalities related to exponentially convex functions. Moreover, the paper presents applications of these results to specific special means.
• In Contribution 11, the study investigates methods for solving the modified fractional Black–Scholes equation. Given the crucial role of option pricing theory in financial markets, call and put options help investors theoretically determine the optimal times to buy, sell, or hold stocks to maximize returns. However, the classical Black–Scholes model, based on the assumption of normally distributed returns, often yields option pricing formulas that do not accurately capture real market dynamics. Hence, modifying the model is necessary for more realistic option valuations. To solve the modified fractional Black–Scholes equation, this work implements a hybrid technique merging the finite difference method with the fractional differential transformation method. The results indicate that this combined technique offers a highly accurate approximation of the solution.
• As noted in several studies, the equivalence between Caputo-type fractional differential equations and their corresponding integral formulations may fail outside absolutely continuous function spaces, including within certain Hölder spaces. To address this limitation, Contribution 12 introduces a novel fractional integral operator that acts as the right inverse of the Caputo derivative in specific Hölder spaces with critical orders of less than 1. The paper provides multiple illustrative examples and counterexamples to emphasize the importance of this development. As an application, the method is used to analyze the boundary value problem for the Langevin fractional differential equation ${\displaystyle \frac{d_{\psi }^{\beta ,\mu }}{d{t}^{\beta }}}\left({\displaystyle \frac{d_{\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}+\lambda \right)u\left(t\right)=h(t,u\left(t\right)),t\in [x_1,y_1],\lambda \in \mathbb{R},$ where $h\in C([x_1,y_1]\times \mathbb{R})$ and the critical fractional orders $\beta ,\alpha \in (0,1),$ subject to appropriate initial or boundary conditions. The analysis is further extended to a wider class of $\psi$ -tempered Hilfer problems involving $\psi$ -tempered fractional derivatives. Additionally, boundary value problems for Bagley–Torvik type fractional differential equations are examined.
• The study of Contribution 13 investigates the existence and uniqueness of solutions to boundary fractional difference equations framed within the context of Riemann–Liouville operators. The study begins by revisiting the general solution of the related homogeneous problem involving fractional operators. Next, the Green’s function for the fractional boundary value problem is constructed by applying homogeneous boundary conditions to determine the unknown constants. The existence of solutions is then established through fixed-point theorems applied to this Green’s function. Moreover, uniqueness results are derived under conditions involving the Lipschitz constant.
• In Contribution 14, a novel numerical method is introduced for fractional stochastic differential equations featuring neutral delays. The method is based on a stepwise collocation technique combined with Jacobi poly-fractonomials to effectively treat the unknown stochastic components. To implement this, the original delay differential equations are transformed into equivalent delay-free systems, making them amenable to the collocation technique. The iterative application of the method leads to a system of nonlinear equations at each step. A comprehensive analysis of the method’s convergence is provided. The accuracy and computational efficiency of the proposed technique are validated through a series of numerical experiments.
• Contribution 15 addresses comments on sequential Caputo fractional differential equations governed by fractional initial and boundary conditions.
• Contribution 16 examines a system of inhomogeneous second-order difference equations featuring linear terms with noncommutative matrix coefficients. A closed-form solution is obtained by introducing novel delayed matrix sine and cosine functions, employing the Z-transform alongside a determining function. This framework facilitates the analysis of iterative learning control by integrating suitable update rules and establishing sufficient conditions to guarantee asymptotic convergence in tracking performance.
• Contribution 17 investigates barrier function-based safety control within fractional-order dynamical systems, an area less explored compared to their integer-order counterparts. While barrier functions have been extensively used to guarantee safety in integer-order systems—including nonlinear, hybrid, and linear models—this study extends their application to fractional-order systems. The authors introduce two novel constructs: the Caputo reciprocal barrier function and the Caputo zeroing barrier function. They establish theorems demonstrating that these functions ensure uniform asymptotic or exponential stability while preserving safety. Furthermore, the paper proposes a new concept of input-to-state safety for Caputo fractional-order systems, accompanied by two criteria derived from the newly introduced barrier functions. These contributions lay a foundational framework for advancing safety control in fractional-order systems.
• In Contribution 18, new theoretical advances concerning the well-posedness and Ulam–Hyers stability of fractional systems are presented, with a focus on Caputo–Katugampola fractional stochastic delay integro-differential equations. The authors develop a generalized version of Grönwall’s inequality, which is then used to establish Ulam–Hyers stability in the ${\mathcal{L}}^p$ space. These results extend existing theories by incorporating the Caputo–Katugampola fractional derivative framework, thereby enriching the mathematical foundation of fractional differential equations. An illustrative example is provided to demonstrate and validate the theoretical findings.
• Contribution 19 examines the positive solution existence and non-existence for a non-linear Riemann–Liouville fractional boundary value problem of order $\alpha +2n$ where $\alpha \in (m-1,m],$ with $m\ge 3$ and $m,n$ are natural numbers. The boundary conditions are based on Lidstone-type formulations.The non-linear term involves a positive parameter, and the authors establish conditions on this parameter that dictate the existence or non-existence of positive solutions. By convolving the Green functions from a lower-order problem and its conjugate counterpart, a Green’s function is constructed and applied along with the Guo–Krasnosel’skiĭ fixed-point theorem. Illustrative examples highlight the parameter values for which solutions exist or fail to exist.

We hope this Special Issue, along with the ideas and publications it contains, will engage readers and inspire further research on fractional differential equations, inclusions, and inequalities.