Search Results (461)

Fractional Langevin models are found to be useful in the study of physical phenomena such as diffusion processes, gait variability, etc. Langevin equations involving different fractional–order operators and boundary conditions have been addressed by many researchers. In this article, we formulate a new Langevin model consisting of a coupled system of Riemann–Liouville and Hadamard–type nonlinear fractional differential equations and coupled multipoint–integral boundary conditions. We present the existence and Ulam–Hyers stability criteria for solutions of the given model problem. Our study is based on the tools of the fixed–point theory. Numerical examples with graphical representations of solutions are offered to demonstrate the application of the obtained results. Our work is novel and useful in the given configuration, and specializes to some new results. 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

The article considers nonlinear fractional differential equations with cotangent derivative. The boundary conditions are multipoint and integral specified, and the nonlinear terms are in Orlicz function spaces. Several existence theorems for solutions of such boundary value problems are obtained by different fixed-point methods. Illustrative examples serve to illustrate the theoretical parts. Full article

The aim of the present work is to study a class of nabla fractional problems with two different nabla Riemann–Liouville operators and three-point parameter-dependent boundary conditions. First, we derive the expression of the Green’s function; then, we deduce a few useful inequalities with it, and we establish an interval for the parameter in which the Green’s function is always positive. Using these properties, we manage to prove some non-existence, existence and multiplicity results using different fixed-point theorems. At the end, we give a few examples that verify and clarify the applications of our results. Full article

This paper investigates a class of coupled $\psi$-Hilfer fractional pantograph–Langevin equations with nonlocal integral boundary conditions. By reformulating the problem as an equivalent fixed point equation and employing Mönch’s fixed point theorem together with the Kuratowski measure of noncompactness, we establish sufficient conditions for the existence of at least one solution. Under additional Lipschitz-type assumptions, we prove Ulam–Hyers stability on a suitable closed ball and derive explicit, computable stability constants. A concrete numerical example is presented in which all hypotheses are verified and the stability constants are explicitly computed (e.g., $K_1\approx 3.811$, $K_2\approx 2.761$), illustrating the applicability of the theoretical results. The study contributes additional qualitative results to the analysis of fractional pantograph–Langevin systems within the unified framework of $\psi$-Hilfer fractional derivatives. Full article

This work focuses on the analysis of a sequential fractional boundary value problem involving coupled Erdélyi–Kober and Caputo fractional differential operators, together with nonlocal boundary conditions of fractional type. The well-posedness of the problem is addressed by deriving conditions that ensure the existence and uniqueness of solutions. Uniqueness is obtained through an application of Banach’s contraction principle, whereas existence is established by employing Krasnosel’skiĭ’s fixed point approach and the nonlinear alternative of Leray–Schauder. Several numerical examples are presented to demonstrate and support the theoretical findings. Full article

In this paper, we study a nonlinear system of sequential Caputo fractional differential equations equipped with coupled mixed multi-point boundary conditions. In particular, the boundary conditions involve the values of the unknown functions at the endpoints expressed as linear combinations of their values at several interior points, forming a closed system of relations. The existence of solutions is established by applying the Leray–Schauder alternative, while uniqueness is proved using Banach’s contraction principle. In addition, we investigate the Hyers–Ulam stability of the proposed system. Several examples are included to demonstrate the applicability of the theoretical results. Some special cases of the general problem are also discussed. Full article

We introduce and study a new class of nonlinear operators on metric spaces, called rational $(a,p)-$quasicontractions. Within this framework, we establish Greguš-type fixed-point theorems for closed, convex subsets of Banach spaces. The results establish the existence and uniqueness of fixed points, as well as the convergence of the Picard iteration for every initial guess. We show that rational $(a,p)-$quasicontractions strictly extend several classical contractive classes, including Hardy-Rogers, Kannan, Chatterjea, and rational contractions, and we provide explicit examples exhibiting the properness of these inclusions. As an application, we consider a nonlocal boundary value problem for a Caputo fractional differential equation of order $\alpha \in (1,2)$ with distributed delay and mixed nonlocal boundary conditions. By rewriting the problem as a Hammerstein-Volterra integral equation on a cone, and imposing natural growth and rational Lipschitz conditions on the delayed nonlinearity, we show that the associated Hammerstein operator is a rational $(a,p)-$quasicontraction. This yields the existence, uniqueness, and global attractivity of a positive solution. Two model fractional nonlinearities with delayed feedback are discussed in detail, along with a numerical scheme that illustrates the predicted geometric convergence of the discrete Picard iteration in the Caputo fractional setting. Full article

In this paper, we develop an accelerated three-step iterative scheme for the approximation of fixed points of contraction mappings in Banach spaces, with a particular focus on applications to fractional models. Strong convergence of the proposed iteration is established under standard contraction assumptions, together with stability and data dependence results. A refined rate of convergence analysis shows that the new scheme achieves a smaller effective contraction factor and converges faster than several classical two- and three-step iterative methods, including the Picard, Mann, Ishikawa, and S-iteration processes. The theoretical results are applied to Caputo-type fractional differential equations by reformulating the associated boundary value problems as fixed-point equations. Existence and uniqueness of solutions follow from the Banach contraction principle, while the accelerated convergence of the proposed iteration leads to improved numerical efficiency. Extensive numerical experiments, including fractional differential equations and nonlinear contraction mappings on the real line, are presented to validate the theoretical findings. The results demonstrate that the proposed three-step iteration provides an effective and reliable computational tool for fractional and non-local models. Full article

Mathematical modeling of heat and mass transfer processes in porous media using fractional derivative equations is of great practical importance. Within the framework of such models, obtaining analytical solutions to the corresponding initial–boundary value problems is generally difficult. In this work, we numerically investigate quasilinear parabolic problems involving Caputo time-fractional derivatives. First, the well-posedness and existence of weak solutions are discussed. Then, we construct and implement a finite-difference scheme that is fourth-order accurate in space and second-order accurate in time. Convergence in the maximum norm is proven. Numerical experiments confirm the accuracy and efficiency of the proposed approach. Full article

In this paper, we investigate a class of nonlinear fractional boundary value problems involving the Caputo fractional derivative under two-point boundary conditions. By combining the Green function of the associated linear problem with a generalized Gronwall inequality, we derive pointwise estimates for solutions expressed explicitly in terms of the Mittag–Leffler function. In contrast to existing Lyapunov-type inequalities, which are mainly restricted to linear equations and rely on global supremum norm estimates, our approach preserves the nonlinear structure of the problem and captures the local behavior of solutions. These pointwise estimates lead to a Lyapunov-type inequality for nonlinear fractional equations, extending the classical result of Jleli and Samet beyond the linear framework. Moreover, we show that the obtained Lyapunov condition serves not only as a necessary condition for the existence of nontrivial solutions, but also as a sufficient criterion ensuring Hyers–Ulam stability and uniqueness. An illustrative example is provided to demonstrate the applicability of the theoretical results. Full article

This study employs the Atangana–Baleanu–Caputo fractional derivative within the Moore–Gibson–Thompson heat conduction model to analytically investigate the thermoelastic vibrations in solid medium-containing voids. The ABC–MGT formulation incorporates a non-singular Mittag–Leffler memory kernel, facilitating the modeling of tempered hereditary relaxation in voided thermoelastic media, thereby producing more realistic attenuation and phase lag characteristics in transient responses than conventional integer-order models. Specifically, our novelty lies in developing a coupled thermoelastic–void formulation within an ABC–MGT heat conduction framework, deriving the full governing system and boundary-value solution in the Laplace domain, and providing a systematic parametric analysis showing how the ABC order changes attenuation, phase lag, and stress/void interactions. This approach enables a precise analytical resolution of the problem. The analysis indicates that the presence and size of voids substantially impact the system response variables, with smaller apertures yielding reduced magnitudes. Thus, this analytical investigation introduces a novel methodology for addressing the complex challenges associated with advanced functional materials and high-performance engineering structures. Full article

This paper introduces polynomial F-contractions, a novel category of contractive mappings within metric spaces. This concept synthesizes two powerful generalizations of the Banach contraction principle: the F-contractions originally developed by Wardowski and the polynomial-type contractions studied very recently by Jleli et al. We formulate fixed point theorems for this new class of mappings in complete metric spaces, which extends and unifies several established theorems in fixed point theory. We first prove our main result for continuous mappings and then extend it to a broader class of mappings that are not necessarily continuous but satisfy the Picard continuity condition. The significance and novelty of our results are highlighted through illustrative examples and further supported by applications to a fractional boundary value problem. Full article

This paper studies mixed impulsive boundary value problems involving tempered $\mathsf{\Psi }$-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The generalized tempered $\mathsf{\Psi }$-operator unifies several existing fractional derivatives, offering enhanced flexibility for modeling complex dynamical phenomena. Impulsive effects and integral boundary conditions are incorporated to describe processes subject to sudden changes and historical dependence. The problem is reformulated as a Volterra integral equation, and fixed-point theory is employed to establish analytical results. Existence and uniqueness of solutions are proven using the Banach Contraction Mapping Principle, while the Leray–Schauder nonlinear alternative ensures existence in non-contractive cases. The proposed framework provides a rigorous analytical basis for modeling phenomena characterized by both fading memory and sudden perturbations, with potential applications in physics, control theory, population dynamics, and epidemiology. A numerical example is presented to illustrate the validity and applicability of the main theoretical results. Full article

The maximum principle is a widely used qualitative property of linear (and not only) elliptic boundary value problems. A natural goal for developing numerical methods is for the approximate solution to have a similar property. In this case, we say that a discrete maximum principle holds. In many cases, such a requirement is critical to ensuring the reliability of computational models. Here, we consider multidimensional linear elliptic problems with diffusion and reaction terms. Such problems have been studied and analyzed for many decades. Since relatively recently, scientists have faced conceptually new challenges when considering anomalous (fractional) diffusion. In the present paper, we concentrate on the case of spectral fractional diffusion. Discretization was carried out using the finite difference method and the finite element method with a lumped mass matrix. In large-scale multidimensional problems, the computational complexity of dense matrix operations is critical. To overcome this problem, BURA (best uniform rational approximation) methods were applied to find the efficient numerical solutions of emerging dense linear systems. Thus, along with the need to satisfy the discrete maximum principle associated with the mesh method applied for discretization of the differential operator, the issue of the monotonicity of BURA numerical solution arises. The presented results are three-fold and include the following: (i) maximum principles for fractional diffusion–reaction problems; (ii) sufficient conditions for discrete maximum principles; and (iii) sufficient conditions for monotonicity of the investigated BURA- or BURA-like approximation methods. A novel, systematic theoretical analysis is developed for sub-diffusion with a fractional power $\alpha \in (1/2,1)$ and a constant reaction coefficient. The theoretical findings are further supported by numerical examples. Full article