Search Results (95)

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators, and $(k_2,{\psi }_2)$- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ${\psi }_1$ and the parameter $\beta$. Also the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and $L^1$-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study. Full article

The two-parameter $(p,q)$-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation using two-parameter $(p,q)$-Caputo derivatives. For this new Langevin equation, equivalently, we obtain the solution structure as a post-quantum integral equation and then conduct an existence analysis via a fixed-point-based approach. The use of theorems such as the Krasnoselskii and Leray–Schauder fixed-point theorems will guarantee the existence of solutions to this equation, whose uniqueness is later proven by Banach’s contraction principle. Finally, we provide three examples in different structures and validate the results numerically. Full article

This paper investigates the existence, uniqueness, and finite-time stability of solutions to a class of nonlinear systems governed by the Hadamard fractional derivative. The analysis is carried out using two fundamental tools from fixed point theory: the Krasnoselskii fixed point theorem and the Banach contraction principle. These methods provide rigorous conditions under which solutions exist and are unique. Furthermore, criteria ensuring the finite-time stability of the system are derived. To demonstrate the practicality of the theoretical results, a detailed example is presented. This paper also discusses certain assumptions and presents corollaries that naturally emerge from the main theorems. Full article

In this study, we investigate the existence, uniqueness, and Hyers–Ulam stability of a multi-term boundary value problem involving generalized $\phi$-Riemann–Liouville operators. The uniqueness of the solution is demonstrated using Banach’s fixed-point theorem, while the existence is established through the application of classical fixed-point theorems by Krasnoselskii. We then delve into the Hyers–Ulam stability of the solutions, an aspect that has garnered significant attention from various researchers. By adapting certain sufficient conditions, we achieve stability results for the Hyers–Ulam (HU) type. Finally, we illustrate the theoretical findings with examples to enhance understanding. Full article

In this paper, we investigate the existence results of solutions for Caputo-type fractional $(p,q)$-difference equations. Using Banach’s fixed-point theorem, we obtain the existence and uniqueness results. Meanwhile, by applying Krasnoselskii’s fixed-point theorem and Leray-Schauder’s nonlinear alternative, we also obtain the existence results of non-trivial solutions. Finally, we provide examples to verify the correctness of the given results. Moreover, relevant applications are presented through specific examples. Full article

Neutral transmission line models are essential for analyzing stability and periodicity in systems influenced by nonlinear and delayed dynamics, particularly in modern smart grids. This study utilizes Krasnoselskii’s fixed-point theorem to establish sufficient conditions for the existence and asymptotic stability of periodic solutions, eliminating the need for differentiability in delay terms and coefficients. The results extend existing findings and are validated through a single test example, demonstrating the theoretical applicability of the proposed approach. These findings provide a mathematical framework for understanding the behavior of power distribution systems under nonlinear and delayed influences. Full article

The $(k,\psi )$-fractional derivative based on the k-gamma function is a more general version of the Hilfer fractional derivative. It is widely used in differential equations to describe physical phenomena, population dynamics, and biological genetic memory problems. In this article, we mainly study the $4m+2$-point symmetric integral boundary value problem of nonlinear $(k,\psi )$-fractional differential coupled Laplacian equations. The existence and uniqueness of solutions are obtained by the Krasnosel’skii fixed-point theorem and Banach’s contraction mapping principle. Furthermore, we also apply the calculus inequality techniques to discuss the stability of this system. Finally, three interesting examples and numerical simulations are given to further verify the correctness and effectiveness of the conclusions. Full article

This paper is devoted to studying the existence and uniqueness of mild solutions for semilinear fractional evolution equations with the Hilfer–Katugampola fractional derivative and under the nonlocal multi-point condition. The analysis is based on analytic semigroup theory, the Krasnoselskii fixed-point theorem, and the Banach fixed-point theorem. An application to a time-fractional real Ginzburg–Landau equation is also given to illustrate the applicability of our results. Furthermore, we determine some conditions to make the control (Bifurcation) parameter in the Ginzburg–Landau equation sufficiently small. Full article

In this manuscript, we study a class of equations with two different Riemann–Liouville-type orders of nabla difference operators. We show some new and fundamental properties of the related Green’s function. Depending on the values of the orders of the operators, we split our research into two main cases, and for each one of them, we obtain suitable conditions under which we prove that the considered problem possesses a positive solution. We consider the latter to be the main novelty in this work. Our main tool in both cases of our study is Guo–Krasnoselskii’s fixed point theorem. In the end, we give particular examples in order to offer a concrete demonstration of our new theoretical findings, as well as some possible future work in this direction. Full article

Fractional differential equations (FDEs) are employed to describe the physical universe. This article investigates the attractivity of solutions for FDEs and Ulam–Hyers–Rassias stability, involving the $\Psi$-Hilfer fractional derivative. Important results are presented using Krasnoselskii’s fixed point theorem, which provides a framework for analyzing the stability and attractivity of solutions. Novel results on the attractiveness of solutions to nonlinear FDEs in Banach spaces are derived, and the existence of solutions, stability properties, and behavior of system equilibria are examined. The application of $\Psi$-Hilfer fractional derivatives in modeling financial crises is explored, and a financial crisis model using $\Psi$-Hilfer fractional derivatives is proposed, providing more general and global results. Furthermore, we also perform a numerical analysis to validate our theoretical findings. Full article

This paper explores the mild solutions of partial impulsive fractional integro-differential systems of order $1<\alpha <2$ in a Banach space. We derive the solution of the system under the assumption that the homogeneous part of the system admits an $\alpha$-resolvent operator. Krasnoselskii’s fixed point theorem is used for the existence of solution, while uniqueness is ensured using Banach’s fixed point theorem. The stability of the system is analyzed through the framework of Hyers–Ulam stability using Lipschitz conditions. Finally, examples are presented to illustrate the applicability of the theoretical results. Full article

This study introduces a novel approach for investigating the solvability of boundary value problems for differential equations that incorporate both ordinary and fractional derivatives, specifically within the context of non-autonomous variable order. Unlike traditional methods in the literature, which often rely on generalized intervals and piecewise constant functions, we propose a new fractional operator better suited for this problem. We analyze the existence and uniqueness of solutions, establishing the conditions necessary for these properties to hold using the Krasnoselskii fixed-point theorem and Banach’s contraction principle. Our study also addresses the Ulam–Hyers stability of the proposed problems, examining how variations in boundary conditions influence the solution dynamics. To support our theoretical framework, we provide numerical examples that not only validate our findings but also demonstrate the practical applicability of these mixed derivative equations across various scientific domains. Additionally, concepts such as symmetry may offer further insights into the behavior of solutions. This research contributes to a deeper understanding of complex differential equations and their implications in real-world scenarios. Full article

This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To ensure uniqueness, the Banach Contraction Principle is applied, and the contraction condition is verified. Stability is analyzed using Ulam’s stability concept, emphasizing the resilience of solutions to perturbations and providing insights into their long-term behavior. An example is included, accompanied by graphical analysis that visualizes the solutions and their dynamic properties. Full article

The mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of mechanisms for the applications. In this paper, the expression of the Green function as well as its special properties are acquired and presented through theoretical analyses. Subsequently, on the basis of these properties of the Green function, the existence and uniqueness of positive solutions are achieved for a singular p-Laplacian fractional-order differential equation with nonlocal integral and infinite-point boundary value systems by using the method of a nonlinear alternative of Leray–Schauder-type Guo–Krasnoselskii’s fixed point theorem in cone, and the Banach fixed point theorem, respectively. Some existence results are obtained for the case in which the nonlinearity is allowed to be singular with regard to the time variable. Several examples are correspondingly provided to show the correctness and applicability of the obtained results, where nonlinear terms are controlled by the integrable functions ${\displaystyle \frac{1}{\pi {(lnt)}^{\frac{1}{2}}{(1-lnt)}^{\frac{1}{2}}}}$ and ${\displaystyle \frac{1}{\pi {(lnt)}^{\frac{3}{4}}{(1-lnt)}^{\frac{3}{4}}}}$ in Example 1, and by the integrable functions $\theta ,\overline{\theta }$ and $\phi \left(v\right),\psi \left(u\right)$ in Example 2, respectively. The present work may contribute to the improvement and application of the coupled p-Laplacian Hadamard fractional differential model and further promote the development of fractional differential equations and fractional differential calculus. Full article

In this paper, we study the initial value problem for the fractional differential equation with multiple deviating arguments. By using Krasnoselskii’s fixed point theorem, the conditions of solvability of the problem are obtained. Furthermore, we establish Ulam–Hyers and generalized Ulam–Hyers stability of the fractional functional differential problem. Finally, two examples are presented to illustrate our results, one is with a pantograph-type equation and the other is numerical. Full article