Search Results (196)

This paper advances a comprehensive controllability framework for Prabhakar fractional differential systems ($\mathcal{PFDS}$s) of order $\eta \in (1,2)$ with nonlocal initial conditions, where the second-order setting requires the joint specification of both an initial state and an initial velocity. Explicit solution representations for four structurally distinct classes of second-order Prabhakar systems are derived via the Laplace transform method and Neumann series expansions, revealing that the placement of the forcing term directly in the system or under the Prabhakar fractional integral operator produces fundamentally different convolution kernels. For linear integro-differential systems, necessary and sufficient controllability conditions are established through a Gramian rank criterion with an explicit norm-bounded control law, while for nonlinear systems, sufficient conditions are obtained via the Schauder fixed-point theorem under an asymptotic growth condition. Three numerical examples validate the theory: a three-dimensional linear system and a two-dimensional nonlinear integro-differential system achieve terminal errors of order ${10}^{-12}$ and ${10}^{-7}$, respectively, and a Prabhakar fractional mass–spring–damper system with viscoelastic hereditary damping demonstrates direct physical relevance, with all theoretical conditions verified and a terminal error of $7.42\times {10}^{-5}$ confirming precise rest-position steering by the Gramian-based control law. Full article

This paper investigates the fractional calculus of distributions supported on ${\mathbb{R}}^{+}$ in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractional derivative terms with orders in the interval $(0,2)$. The existence and uniqueness of solutions for its nonlinear form are established in a space of continuous functions by applying Banach’s contraction principle, the Leray–Schauder fixed-point theorem, inverse operators, and the multivariate Mittag–Leffler function. Finally, several examples are presented, in which the values of multivariate Mittag–Leffler functions are computed to illustrate the main results. Full article

We investigate the local well-posedness of semilinear fractional integro-differential equations in Banach spaces with the Hadamard fractional derivative. The equation is ${}^{H}D_t^{\beta }u\left(t\right)=Au\left(t\right)+\phi \left(t,u\left(t\right),{\int }_1^{t}K(t,s)\rho (s,u\left(s\right))ds\right),u\left(1\right)=u_0,$ where A generates a compact $C_0$ semigroup. Using Schauder’s fixed point theorem, we prove local existence under linear growth conditions. Uniqueness is obtained via Banach’s contraction principle under Lipschitz assumptions. The main contribution is a detailed theorem for non-Lipschitz nonlinearities satisfying Carathéodory conditions and Osgood-type growth, where we prove the existence and additional regularity of mild solutions. An illustrative example with Lipschitz nonlinearities is provided. Full article

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 paper investigates the solution theory of a class of prescribed positive Q-curvature equations with power-law singularity at the origin and polynomial growth at infinity in the four-dimensional Euclidean space. We focus on the equation involving the biharmonic operator and an exponential nonlinearity, with the prescribed curvature function combining a singular term and a growth term, where a parameter characterizes the strength of the conical singularity at the origin and another parameter describes the growth rate at infinity. Under the finite total curvature constraint, we systematically analyze the asymptotic behavior of normal solutions, establish the necessary condition for existence, prove the existence and uniqueness of radially symmetric normal solutions, and give a complete characterization of the optimal admissible range of the total curvature. Our main results are as follows: (i) We derive the sharp asymptotic behavior of normal solutions both near the singular origin and at infinity, and establish the Pohozaev identity for the singular Q-curvature equation, which yields a universal necessary condition for the existence of normal solutions. (ii) We prove the existence of radially symmetric normal solutions via the Leray–Schauder fixed point theorem combined with a regularization technique, and establish the uniqueness of radial solutions with respect to the initial value at the origin by the strong maximum principle and monotonicity analysis. (iii) We prove the continuity of the total curvature with respect to the initial value via blow-up analysis and energy quantization, and determine the optimal range of the total curvature: for small growth rates, the necessary and sufficient condition for existence is that the total curvature lies between two critical values; for large growth rates, we give a sharp necessary condition and an explicit sufficient condition for the existence of radial solutions. Full article

This paper examines an m-cyclic coupled system of sequential $(k,\psi )$-Hilfer and $(k,\psi )$-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

This study addresses the existence of solutions for a class of coupled fractional differential inclusions subject to coupled boundary conditions. The analysis is developed within the framework of nonlinear functional analysis by employing Carathéodory-type assumptions, the Leray–Schauder nonlinear alternative, and the Covitz–Nadler fixed-point theorem for multivalued mappings. The proposed approach accommodates both convex and non-convex set-valued nonlinearities, thereby broadening the scope of the results. Under suitable restrictions on the problem parameters, several corollaries are established as direct consequences of the main findings. An example is included to demonstrate the practical applicability and validity of the theoretical results. Full article

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

In this paper, we investigate a class of coupled fractional differential systems involving Caputo derivatives and nonlinear $\varphi$-Laplacian operators subject to nonlocal boundary conditions. By transforming the problem into an equivalent integral system via appropriate Green’s functions, the existence of solutions is studied within a generalized Banach space framework. Using a Leray–Schauder type fixed point theorem and suitable growth conditions on the nonlinear terms, we establish the existence of at least one bounded solution. Furthermore, we prove that the solution set is compact. An illustrative example involving the p-Laplacian operator is provided to demonstrate the applicability of the obtained theoretical results. Full article

In this paper, we study a fractional Kirchhoff problem with a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient: ${\displaystyle M\left({\int \int }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}\frac{{|u\left(x\right)-u\left(y\right)|}^q}{{|x-y|}^{N+qs}}dxdy\right){(-\Delta )}^{s}u=\lambda \frac{u}{{\left|x\right|}^{2s}}+f(x,u,\nabla u)\mathrm{in}\Omega ,}$ where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain containing the origin, $s\in (0,1)$, $q\in (1,2]$ with $N>2s$, $\lambda >0$, and f is a measurable non-negative function satisfying suitable hypotheses. The main objective is to establish the existence of positive solutions for the largest possible class of nonlinearities f without imposing restrictions on $\lambda$. Two main cases areconsidered: $\left(I\right)-f(x,u,\nabla u)=u^p+\mu ,\mathrm{and}\left(II\right)-f(x,u,\nabla u)={|\nabla u|}^p+\mu g.$ Existence is proved under suitable hypotheses on $q,p$ and the data $g,\mu$. The results are new, including for the local case $s=1$. Full article

This work presents a new and efficient numerical framework for solving three-dimensional time-fractional diffusion and mobile–immobile equations in the Caputo sense. The method is formulated using four-variable Jacobi polynomials, constructed systematically via the Kronecker product of one-dimensional Jacobi bases to accurately represent the multidimensional nature of the governing equations. Within a pseudo-operational collocation formulation, these polynomials enable a highly accurate and computationally efficient approximation of the fractional operators in both temporal and spatial directions. From the theoretical standpoint, the existence and uniqueness of the approximate solution are rigorously established through Schauder’s fixed-point theorem. Furthermore, the Ulam–Hyers stability of the numerical solution is verified, demonstrating the robustness of the method with respect to perturbations in the input data. To reinforce the reliability of the approach, an explicit error bound for the residual function is derived in a Jacobi-weighted Sobolev space, offering a firm analytical basis for assessing convergence. Numerical experiments confirm that the proposed approach achieves superior accuracy and efficiency, highlighting its potential as a powerful tool for high-dimensional fractional partial differential equations. Full article

This paper investigates a class of implicit neutral fractional integro-differential equations of Volterra–Fredholm type. The equations incorporate a tempered fractional derivative in the Caputo sense, along with both retarded (delay) and advanced arguments. The problem is formulated on a time domain segmented into past, present, and future intervals and includes nonlinear mixed integral operators. Using Banach’s contraction mapping principle and Schauder’s fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions within the space of continuous functions. The study is then extended to general Banach spaces by employing Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. Ulam–Hyers–Rassias stability is also analyzed under appropriate conditions. To demonstrate the practical applicability of the theoretical framework, explicit examples with specific nonlinear functions and integral kernels are provided. Furthermore, detailed numerical simulations are conducted using MATLAB-based specialized algorithms, illustrating solution convergence and behavior in both finite-dimensional and Banach space contexts. Full article

This paper establishes new fixed-point theorems in the framework of complete p-normed spaces, where $p\in (0,1]$. By extending the classical Banach, Schauder, and Krasnosel’skii fixed-point theorems, we derive several results for the sum of contraction and compact operators acting on s-convex subsets. The analysis is further generalized to multivalued upper semi-continuous operators by employing Kuratowski and Hausdorff measures of noncompactness. These results lead to new Darbo–Sadovskii-type fixed-point theorems and global versions of Krasnosel’skii’s theorem for multifunctions in p-normed spaces. The theoretical findings are then applied to demonstrate the existence of solutions for nonlinear integral equations formulated in p-normed settings. A section on numerical applications is also provided to illustrate the effectiveness and applicability of the proposed results. Full article

In this paper, we investigate the existence of positive solutions of the eigenvalue problem for a singular tempered fractional equation with a Riemann–Stieltjes integral boundary condition and signed measures. By establishing the Green function and its properties, an eigenvalue interval for the existence of positive solutions is outlined based on Schauder’s fixed-point theorem and the upper and lower solutions method. An interesting feature of this paper is that f may be singular in both the time and space variables, and the Riemann–Stieltjes integral may involve signed measures. Full article

The paper is devoted to the study of a class of singular high-order fractional integro-differential equations with p-Laplacian operator, involving both the Riemann–Liouville fractional derivative and the Caputo fractional derivative. First, we investigate the problem by the method of reducing the order of fractional derivative. Then, by using the Schauder fixed point theorem, the existence of solutions is proved. The upper and lower bounds for the unique solution of the problem are established under various conditions by employing the Banach contraction mapping principle. Furthermore, four numerical examples are presented to illustrate the applications of our main results. Full article