Search Results (74)

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior. Full article

This work develops an analytical framework for nonlinear fractional partial differential equations that combine Kirchhoff-type terms, double-phase operators, and $\psi$-Hilfer fractional derivatives. This paper investigates two classes of problems involving variable-exponent growth conditions. The first problem analyzes general nonlinear sources and formulates the solution as a fixed point of a nonlinear operator. Precisely, by proving that the functional energy is coercive, hemicontinuous, and strictly monotone, we establish the existence and the uniqueness of weak solutions via monotone operator theory. The second problem incorporates a convection-type nonlinearity, which breaks variational structure and requires the more robust theory of pseudomonotone operators. Under suitable growth and mixed-order assumptions on the nonlinearity, we prove the existence of at least one weak solution. The main tools are grounded in variable-exponent Lebesgue and Musielak–Orlicz–Sobolev spaces, with compact embeddings, modular estimates, and fractional integral identities playing a key role in the proofs. We note that the results contribute to the mathematical modeling of phenomena involving nonlocal elasticity, viscoelastic materials, phase-transition media, and fractional dynamical systems where the stiffness of the medium depends on the total deformation (Kirchhoff effect) and the energy density alternates between distinct growth regimes (double-phase). The $\psi$-Hilfer derivative enhances the scope by enabling models with tunable memory and hereditary effects. Full article

This paper establishes a rigorous analysis of a coupled hybrid fractional differential system involving a generalized Hilfer operator under integral and antiperiodic boundary conditions. The existence and uniqueness of solutions are proved using Dhage’s fixed point theorem for existence and the Banach contraction principle for uniqueness. Furthermore, we establish Ulam–Hyers stability by deriving the following explicit and computable bound estimate: $\left(\begin{array}{c}\parallel \widehat{u}-u\parallel \\ \parallel \widehat{v}-v\parallel \end{array}\right)\le {(I-\mathit{\chi })}^{-1}\left(\begin{array}{c}{C}_1{ϵ}_1\\ C_2{ϵ}_2\end{array}\right)$, where $C_1$ and $C_2$ are positive constants depending on the system parameters, ${ϵ}_1,{ϵ}_2$ denote the perturbation bounds, and $\mathit{\chi }$ is the associated Lipschitz matrix. This formulation provides a more detailed stability description than scalar criteria, as it captures the interactions among the system components through the entries of $\mathit{\chi }$, leading to a more informative stability estimate. Two illustrative examples confirm the theoretical results and demonstrate their potential applicability for modeling real-world phenomena where memory effects are present. Full article

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 study presents findings on the existence, uniqueness, averaging principle, and numerical solutions for fractional stochastic systems influenced by both Brownian motion and Poisson jumps within the pth-moment framework. While most existing results for fractional stochastic differential equations are derived using the mean-square approach, this research offers results in the more general pth-moment framework, enhancing applicability. The results are derived using the $\gamma$-Hilfer fractional derivative, a generalized operator defined in relation to another function. This operator enables the memory effect to vary according to a nonlinear time scale. The main motivation behind this work is that there is no research work on $\gamma$-Hilfer fractional stochastic systems with standard Brownian motion and Poisson jumps regarding existence, uniqueness, and averaging principles in the pth-moment. Full article

This paper investigates non-fractional operators, a type of nonlocal operator, within the framework of Orlicz spaces. Using inclusions between certain function spaces, we prove the continuity and/or compactness of generalized operators in Orlicz spaces and show that solutions exist for integral equations of fractional order. We also introduce a generalized Hilfer-type derivative and examine the equivalence of differential and integral problems. Finally, we relate these results to the study of compositional p-Laplacian fractional problems involving generalized Hilfer fractional derivatives. Among other things, we prove the existence of solutions to such problems in Orlicz and Orlicz–Sobolev spaces. Full article

This paper develops a generalized Laplace transform framework on weighted function spaces $C_{{\delta }_{\psi },\gamma }^n[a,b]$, establishing a symmetry between integer-order ${\delta }_{\psi }$ operators and fractional $\psi$-Hilfer derivatives at the level of transform representations. Explicit transformation formulas are derived for the nth-order ${\delta }_{\psi }$-derivative and the $\psi$-Hilfer fractional derivative of order $\alpha \in (m-1,m)$, with $m\le n$. These results form an analytical basis for the treatment of higher-order hybrid fractional Cauchy problems that systematically couple integer-order and fractional operators subject to mixed initial conditions. The general solution is expressed in closed form using a bivariate Mittag–Leffler function. To illustrate the utility of the approach, a representative second-order hybrid model is studied and compared numerically with its classical integer-order counterpart. The simulations reveal significant differences in the dynamical response, including variations in amplitude, damping behavior, and long-term evolution. Full article

In this article, we investigate a nonlinear $\psi$-Hilfer fractional order Volterra integro-delay differential equation ($\psi$-Hilfer FRVIDDE) and a nonlinear $\psi$-Hilfer fractional Volterra delay integral equation ($\psi$-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the $\psi$-Hilfer FRVIDDE and $\psi$-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered $\psi$-Hilfer FRVIDDE and $\psi$-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature. Full article

This paper establishes a rigorous analytical framework for a nonlinear multi-order fractional differential system governed by the generalized $\sigma$-Hilfer operator in weighted Banach spaces. In contrast to existing studies that often treat specific kernels or fixed fractional orders in isolation, our approach provides a unified treatment that simultaneously handles multiple fractional orders, a tunable kernel $\sigma \left(\varsigma \right)$, weighted integral conditions, and a nonlinearity depending on a fractional integral of the solution. By converting the hierarchical differential structure into an equivalent Volterra integral equation, we derive sufficient conditions for the existence and uniqueness of solutions using the Banach contraction principle and Mönch’s fixed-point theorem with measures of non-compactness. The analysis is extended to Ulam–Hyers stability, ensuring robustness under modeling perturbations. A principal contribution is the systematic classification of the system’s symmetric reductions—specifically the Riemann–Liouville, Caputo, Hadamard, and Katugampola forms—all governed by a single spectral condition dependent on $\sigma \left(\varsigma \right)$. The theoretical results are illustrated by numerical examples that highlight the sensitivity of solutions to the memory kernel and the fractional orders. This work provides a cohesive analytical tool for a broad class of fractional systems with memory, thereby unifying previously disparate fractional calculi under a single, consistent framework. Full article

This study addresses the existence and approximate controllability of a type of higher-order Hilfer fractional evolution differential (HOHFED) system with time delays in Banach spaces. Using the properties of the Mittag–Leffler function, cosine families, and Hilfer-type fractional differential operators, we first demonstrate the existence and uniqueness of mild solutions using a fixed-point method. Furthermore, a sequential technique is proposed to establish adequate conditions for approximate controllability. A detailed example is provided to illustrate the applicability and effectiveness of the theoretical results. Full article

This paper develops a generalized Laplace transform theory within weighted function spaces tailored for the analysis of fractional differential equations involving the $\psi$-Hilfer derivative. We redefine the transform in a weighted setting, establish its fundamental properties—including linearity, convolution theorems, and action on ${\delta }_{\psi }$ derivatives—and derive explicit formulas for the transforms of $\psi$-Riemann–Liouville, $\psi$-Caputo, and $\psi$-Hilfer fractional operators. The results provide a rigorous analytical foundation for solving hybrid fractional Cauchy problems that combine classical and fractional derivatives. As an application, we solve a hybrid model incorporating both ${\delta }_{\psi }$ derivatives and $\psi$-Hilfer fractional derivatives, obtaining explicit solutions in terms of multivariate Mittag-Leffler functions. The effectiveness of the method is illustrated through a capacitor charging model and a hydraulic door closer system based on a mass-damper model, demonstrating how fractional-order terms capture memory effects and non-ideal behaviors not described by classical integer-order models. Full article

This paper analyzes the Bagley–Torvik fractional-order equation with generalized fractional Hilfer derivatives of two orders for functions in Banach spaces under conditions expressed in the language of weak topology. We develop a comprehensive theory of fractional-order differential equations of various orders. Our focus is on the equivalence results (or the lack thereof) of this new class of fractional-order Hilfer operators and on maximizing the regularity of the solution. To this end, we examine the equivalence of differential problems involving pseudo-derivatives and integral problems involving Pettis integrals. Our results are novel, even within the context of integer-order differential equations. Another objective is to incorporate fractional-order problems into the growing research field that uses weak topology and function spaces to study vector-valued functions. The auxiliary results obtained in this article are general and applicable beyond its scope. Full article

In this article, a novel type of equation, namely the $\mathsf{\Phi }$-Hilfer fractional-order integro-differential delay system ($\mathsf{\Phi }$-HFOIDDS), is proposed. Here, we study the existence and Hyers–Ulam–Mittag–Leffler (H-U-M-L) stability of the aforementioned equation which are obtained by using the multivariate Mittag–Leffler function, Banach contraction principle, and Picard operator method as well as generalized Gronwall inequality. Finally, we conclude this paper by constructing a suitable example to illustrate the applicability of the principal outcomes. Full article

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article

In this work, we address a nonlinear $\psi$-Hilfer fractional-order Volterra integro-differential equation that incorporates n-multiple-variable time delays. Employing the $\psi$-Hilfer fractional derivative operator, we investigate the existence of a unique solution, as well as the Ulam–Hyers–Rassias stability, semi-Ulam–Hyers–Rassias stability, and Ulam–Hyers stability of the proposed $\psi$-Hilfer fractional-order Volterra integro-differential equation through the fixed-point approach. In this study, we enhance and generalize existing results in the literature on $\psi$-Hilfer fractional-order Volterra integro-differential equations, both including and excluding single delay, by establishing new findings for nonlinear $\psi$-Hilfer fractional-order Volterra integro-differential equations involving n-multiple-variable time delays. This study provides novel theoretical insights that deepen the qualitative understanding of fractional calculus. Full article