Search Results (84)

We study an inverse source problem for a semilinear diffusion equation involving a Caputo-type time-fractional derivative whose order is a function of time. The equation is considered in a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$, $d\ge 1$, and is supplemented with homogeneous Dirichlet boundary conditions. The source term is taken to be separable, $h\left(t\right)f\left(x\right)$, where the temporal component $h\left(t\right)$ is unknown. This quantity is to be identified from spatially localized measurements $m\left(t\right)$ of the solution. In this setting, we establish existence and uniqueness results in suitable function spaces, thereby demonstrating the well-posedness of the corresponding inverse source problem. Full article

This paper focuses on the Caputo-Hadamard fractional uncertain differential equations (CH-FUDEs) governed by Liu processes, which combine the Caputo–Hadamard fractional derivative with uncertain differential equations to describe dynamic systems involving memory characteristics and uncertain information. Within the framework of uncertain theory, this Liu process serves as the counterpart to Brownian motion. We establish some new Bihari type fractional inequalities that are easy to apply in practice and can be considered as a more general tool in some situations. As applications of those inequalities, we establish the well-posedness of a proposed class of equations under specified non-Lipschitz conditions. Building upon this result, we establish the notions of stability in distribution and stability in measure solutions to CH-FUDEs, deriving sufficient conditions to ensure these stability properties. Finally, the theoretical findings are verified through two numerical examples. Full article

In this work, we first establish a new connection between adjoint Bernoulli’s polynomials and gamma function as a new sequence of linear positive operators denoted by ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$. Further, convergence results for these sequences of operators, i.e., ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ are derived in various functional spaces with the aid of the Korovkin theorem, the Voronovskaja-type theorem, the first order of the modulus of continuity, the second order of the modulus of continuity, Peetre’s K-functional, the Lipschitz condition, etc. In the last section, we focus our research on the bivariate extension of these sequences of operators; their uniform rate of approximation and order of approximation are investigated in different functional spaces. Full article

This paper investigates a novel class of fractional Langevin equations, which introduces a time-varying p(t)-Laplacian operator and parameterized anti-periodic boundary conditions. This approach overcomes the limitations of traditional models characterized by constant diffusion exponents and fixed boundary locations. Under non-compactness conditions, the existence of solutions is established by applying Schaefer’s fixed-point theorem, which significantly relaxes the conventional constraints on the nonlinear term. Moreover, by imposing a Lipschitz condition on the nonlinear term, a Ulam–Hyers-type stability criterion for the coupled system is derived. This work not only extends the relevant stability theory but also provides a rigorous theoretical foundation for error control in practical applications. The effectiveness of the theoretical results is validated through numerical examples. Full article

This study has developed a unified framework for modeling economic growth through Caputo fractional differential equations. The framework has established the existence and uniqueness of solutions by employing a generalized fixed-point approach. In particular, the analysis has introduced and utilized new classes of symmetric operators, including symmetric Lipschitz-type mappings, symmetric Kannan-type contractions, and symmetric Chatterjea-type contractions. These mappings are based on a refined symmetric Lipschitz condition that enables the examination of the behavior of their iterative sequences. The study has focused on several forms of symmetric contractions defined on metric spaces endowed with a binary relation, providing a setting that generalizes and unifies various existing fixed-point theorems. This framework has extended classical results by Goebel and Sims, Goebel and Japon-Pineda, and others. Finally, to illustrate the practical significance of the theoretical findings, the developed results have been applied to demonstrate the existence of solutions for fractional models of economic growth and a related Fredholm integral equation. Full article

We establish the existence and uniqueness of solutions to a class of second-order nonlinear boundary value problems involving impulses, delay, and possible singularities. The approach leverages the recent notion of paired-Chatterjea-type contractions. Under a smallness condition ensuring the associated integral operator is a Banach contraction with constant $\mu <{\displaystyle \frac{1}{3}}$, we show that it is also a Chatterjea, and hence, a paired-Chatterjea contraction. By the fixed point theorem of Chand, this guarantees at most two fixed points; a supplementary uniqueness argument then ensures a unique solution in the Banach space $P{C}^1\left([a,b]\right)$. Full article

The performance of iterative schemes used to solve nonlinear operator equations is strongly influenced by the initial guess. Therefore, it is essential to accurately determine convergence radii and develop theoretical strategies to broaden the region where convergence is guaranteed in order to enhance the reliability and efficiency of these methods. A crucial tool for this purpose is local convergence analysis, which investigates behavior near the true solution to establish convergence criteria. This work is dedicated to extending the convergence region of a parameter-based iteration scheme of the fifth-order. We carry out a comprehensive local convergence study within the framework of Banach spaces and derive precise formulas for the convergence radius, error estimates, and convergence zones associated with the method. A notable advantage of our approach is that it relies solely on the first derivative and avoids the need for additional conditions, making it easier to apply and significantly expanding the convergence region relative to earlier approaches. The theoretical contributions are further validated through a series of numerical experiments applied to diverse classes of nonlinear equations. Furthermore, the examination of the basins of attraction and their symmetry provides a deeper understanding of the method’s dynamic characteristics, robustness, and effectiveness in tackling complex-valued polynomial equations. Full article

The purpose of this paper is to investigate a class of novel symmetric coupled fuzzy fractional partial differential equation system involving the Caputo–Katugampola (C-K) generalized Hukuhara (gH) derivative. Within the framework of C-K gH differentiability, two types of gH weak solutions are defined, and their existence is rigorously established through explicit constructions via employing Schauder fixed point theorem, overcoming the limitations of traditional Lipschitz conditions and thereby extending applicability to non-smooth and nonlinear systems commonly encountered in practice. A typical numerical example with potential applications is proposed to verify the existence results of the solutions for the symmetric coupled system. Furthermore, we introduce Ulam–Hyers stability (U-HS) theory into the analysis of such symmetric coupled systems and establish explicit stability criteria. U-HS ensures the existence of approximate solutions close to the exact solution under small perturbations, and thereby guarantees the reliability and robustness of the systems, while it prevents significant deviations in system dynamics caused by minor disturbances. We not only enrich the theoretical framework of fuzzy fractional calculus by extending the class of solvable systems and supplementing stability analysis, but also provide a practical mathematical tool for investigating complex interconnected systems characterized by uncertainty, memory effects, and spatial dynamics. Full article

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here, the relations between linear Caputo fractional differential equations and their corresponding linear integral equations are investigated, and the results demonstrate that a proper Lipschitz-type condition is needed for studying nonlinear Caputo fractional differential equations. Then, an existence and uniqueness result is established in some vector subspaces by Banach’s fixed-point theorem and ${\parallel ·\parallel }_e$ norm. In addition, two examples are presented to illustrate the theoretical conclusions. Full article

We investigate the solvability and stability properties of a class of nonlinear stochastic delay differential equations (SDDEs) driven by Wiener noise and incorporating discrete time delays. The equations are formulated within a Banach space of continuous, adapted sample paths. Under standard Lipschitz and linear growth conditions, we construct a solution operator and prove the existence and uniqueness of strong solutions using a fixed-point argument. Furthermore, we derive exponential mean-square stability via Lyapunov-type techniques and delay-dependent inequalities. This framework provides a unified and flexible approach to SDDE analysis that departs from traditional Hilbert space or semigroup-based methods. We explore a Banach space fixed-point approach to SDDEs with multiplicative noise and discrete delays, providing a novel functional-analytic framework for examining solvability and stability. Full article

This paper investigates weighted Milne-type ($\mathcal{M}-\mathfrak{t}$) inequalities within the context of Riemann–Liouville ($\mathfrak{R}-\mathfrak{L}$) fractional integrals. We establish multiple versions of these inequalities, applicable to different function categories, such as convex functions with differentiability properties, bounded functions, functions satisfying Lipschitz conditions, and those exhibiting bounded variation behavior. In particular, we present integral equalities that are essential to establish the main results, using non-negative weighted functions. The findings contribute to the extension of existing inequalities in the literature and provide a deeper understanding of their applications in fractional calculus. This work highlights the advantage of the established inequalities in extending classical results by accommodating a broader class of functions and yielding sharper bounds. It also explores potential directions for future research inspired by these findings. Full article

This paper introduces a new class of Newton-type inequalities (NTIs) within the framework of extended fractional integral operators. This study begins by establishing a fundamental identity for generalized fractional Riemann–Liouville (FR-L) operators, which forms the basis for deriving various inequalities under different assumptions on the integrand. In particular, fractional counterparts of the classical $1/3$ and $3/8$ Simpson rules are obtained when the modulus of the first derivative is convex. The analysis is further extended to include functions that satisfy a Lipschitz condition or have bounded first derivatives. Moreover, an additional NTI is presented for functions of bounded variation, expressed in terms of their total variation. In all scenarios, the proposed results reduce to classical inequalities when the fractional parameters are specified accordingly, thus offering a unified perspective on numerical integration through fractional operators. Full article

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

This study presents a comprehensive analysis of the semilocal convergence properties of a high-order iterative scheme designed to solve nonlinear equations in Banach spaces. The investigation is carried out under the assumption that the first derivative of the associated nonlinear operator adheres to a generalized Lipschitz-type condition, which broadens the applicability of the convergence analysis. Furthermore, the research demonstrates that, under an additional mild assumption, the proposed scheme achieves a remarkable ninth-order rate of convergence. This high-order convergence result significantly contributes to the theoretical understanding of iterative schemes in infinite-dimensional settings. Beyond the theoretical implications, the results also have practical relevance, particularly in the context of solving complex systems of equations and integral equations that frequently arise in applied mathematics, physics, and engineering disciplines. Overall, the findings provide valuable insights into the behavior and efficiency of advanced iterative schemes in Banach space frameworks. The comparative analysis with existing schemes also demonstrates that the ninth-order iterative scheme achieves faster convergence in most cases, particularly for smaller radii. Full article