Search Results (107)

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

Fractional stochastic differential Equations (FSDEs) with time delays and non-instantaneous impulses describe dynamical systems whose evolution relies not only on their current state but also on their historical context, random fluctuations, and impulsive effects that manifest over finite intervals rather than occurring instantaneously. This combination of features offers a more precise framework for capturing critical aspects of many real-world processes. Recent findings demonstrate the existence, uniqueness, and Ulam–Hyers stability of standard fractional stochastic systems. In this study, we extend these results to include systems characterized by FSDEs that incorporate time delays and non-instantaneous impulses. We prove the existence and uniqueness of the solution for this system using Krasnoselskii’s and Banach’s fixed-point theorems. Additionally, we present findings related to Ulam–Hyers stability. To illustrate the practical application of our results, we develop a population model that incorporates memory effects, randomness, and non-instantaneous impulses. This model is solved numerically via the Euler–Maruyama method, and graphical simulations effectively depict the dynamic behavior of the system. Full article

The current manuscript is concerned with the uniqueness and existence of a solution for a new class of $\Phi$-Hilfer fractional neutral functional integro-differential equations ($\Phi$-HFNFIDEs) with terminal conditions. Firstly, employing Babenko’s approach, we convert the aforesaid equation under consideration into an analogous integral equation. More precisely, using the multivariate Mittag-Leffler function, Banach contraction principle, and Krasnoselskii’s fixed-point theorem, we derive some conditions that guarantee the uniqueness and the existence of the solutions. For an illustration of our results in this manuscript, two examples are provided as well. Full article

This paper investigates a coupled system of nonlinear implicit fractional differential equations of order $\alpha \in (1,2]$ subject to anti-periodic boundary conditions. The analysis is conducted using the $\psi$-Caputo fractional derivative, a generalized operator that incorporates several well-known fractional derivatives. The system features implicit coupling, where each equation depends on both unknown functions and their first derivatives, as well as an implicit dependence on the fractional derivatives themselves. The boundary value problem is transformed into an equivalent system of integral equations. Sufficient conditions for the existence and uniqueness of solutions are established using Banach’s and Krasnoselskii’s fixed-point theorems in an appropriately chosen Banach space. Furthermore, the Ulam–Hyers stability of the system is analyzed. The applicability of the theoretical results is demonstrated through a detailed example of a coupled system where all hypotheses are verified. Full article

This work analyses the existence, uniqueness, and Ulam-type stability of neutral fractional functional differential equations with recursively defined state-dependent delays. Employing the Caputo fractional derivative of order $\alpha \in (0,1)$ with respect to a strictly increasing function $\phi$, the analysis extends classical results to nonuniform memory. The neutral term and delay chain are defined recursively by the solution, with arbitrary continuous initial data. Existence and uniqueness of solutions are established using Krasnosel’skii’s fixed point theorem. Sufficient conditions for Ulam–Hyers stability are obtained via the Volterra-type integral form and a $\phi$-fractional Grönwall inequality. Examples illustrate both standard and nonlinear time scales, including a Hopfield neural network with iterated delays, which has not been previously studied even for integer-order equations. Fractional neural networks with iterated state-dependent delays provide a new and effective model for the description of AI processes—particularly machine learning and pattern recognition—as well as for modelling the functioning of the human brain. Full article

This paper deals with a new coupled system of integro-differential equations involving the generalized proportional Caputo derivatives equipped with nonlocal four-point boundary conditions. Sufficient criteria for the existence and uniqueness of solutions for the studied system are derived based on Krasnoselskii’s and Banach fixed-point theorems, respectively. Applications are constructed with three different cases to illustrate the main results. Full article

This paper investigates the uniform continuity and strong continuity of the semigroups of the fractional integral operators of power functions. Using the Krasnoselskii’s fixed-point theorem, we have studied the non-local problem related to fractional differential equations involving power functions with multi-point integral boundary conditions and obtain the existence of the solution. Full article

Fractional differential equations offer a natural framework for describing systems in which present states are influenced by the past. This work presents a nonlinear Caputo-type fractional differential equation (FDE) with a nonlocal initial condition and attempts to describe a model of memory-dependent behavioral adaptation. The proposed framework uses a fractional-order derivative $\eta \in (0,1)$ to discuss the long-term memory effects. The existence and uniqueness of solutions are demonstrated by Banach’s and Krasnoselskii’s fixed-point theorems. Stability is analyzed through Ulam–Hyers and Ulam–Hyers–Rassias benchmarks, supported by sensitivity results on the kernel structure and fractional order. The model is further employed for behavioral despair and learned helplessness, capturing the role of delayed stimulus feedback in shaping cognitive adaptation. Numerical simulations based on the convolution-based fractional linear multistep (FVI–CQ) and Adams–Bashforth–Moulton (ABM) schemes confirm convergence and accuracy. The proposed setup provides a compact computational and mathematical paradigm for analyzing systems characterized by nonlocal feedback and persistent memory. Full article

Fractional differential equations are used to model complex systems where present dynamics depend on past states. In this work, we study a linear fractional model with two Caputo orders that captures long-term memory together with short-term adaptation. The existence and uniqueness of solutions are established using Banach and Krasnoselskii’s fixed-point theorems. A parameter study isolates the roles of the fractional orders and coefficients, yielding an explicit stability region in the $(\alpha ,\beta )$–plane via computable contraction bounds. For computation, we implement the Adams–Bashforth–Moulton (ABM) and fractional linear multistep (FLM) methods, comparing accuracy and convergence. As an application, we model animal learning in which proficiency evolves under memory effects and pulsed stimuli. The results quantify the impact of feedback timing on trajectories within the admissible region, thereby illustrating the suitability of dual-order fractional models for memory-driven behavior. Full article

The current work studies difference problems including two different nabla operators coupled with general summation boundary conditions that depend on a parameter. After we deduce the Green’s function, we obtain an interval of the parameter, where it is strictly positive. Then, we establish a lower and upper bound of the related Green’s function and we impose suitable conditions of the nonlinear part, under which, using the classical Guo–Krasnoselskii fixed point theorem, we deduce the existence of at least one positive solution of the studied equation. After that, we impose more restricted conditions on the right-hand side and we obtain the existence of n positive solutions again using fixed point theory, which is the main novelty of this research. Finally, we give particular examples as an application of our theoretical findings. Full article

Let $\mathcal{E}$ and $\mathcal{F}$ be nonempty disjoint subsets of a metric space $(M,d)$. For a non-self-mapping $\phi :\mathcal{E}\to \mathcal{F}$, which is fixed-point free, a point $\varkappa \in \mathcal{E}$ is said to be a best proximity point for the mapping $\phi$ whenever the distance of the point $\varkappa$ to its image under $\phi$ is equal to the distance between the sets, $\mathcal{E}$ and $\mathcal{F}$. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations. Full article

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of solutions. Stability is analyzed in the Ulam–Hyers (UHS), generalized Ulam–Hyers (GUHS), and Ulam–Hyers–Rassias (UHRS) senses. A modified Adomian decomposition method (MADM) is introduced to derive explicit solutions without linearization, preserving the problem’s original structure. The first numerical example validates the theoretical findings on existence, uniqueness, and stability, supplemented by graphical results obtained via the MADM. Further examples illustrate fuzzy solutions by varying the uncertainty level (r), the variable (x), and both parameters simultaneously. The numerical results align with the theoretical analysis, demonstrating the efficacy and applicability of the proposed method. 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

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