Search Results (49)

This work addresses the Ulam–Hyers stability of mild solutions of fractional noninstantaneous evolution nonlinear equations subject to noninstantaneous impulses in the Banach space through an approach based on the measure of noncompactness. Mild solutions are formulated via operators generated by a closed linear operator and a probability density function. Existence results are derived by applying the fixed point theorem for k-set contractive operators. Furthermore, Ulam–Hyers stability is established under certain hypotheses, and an illustrative example is provided to validate the derived results. Full article

This paper studies mild solutions and Bolza optimal control for Riemann–Liouville fractional stochastic integro-differential systems incorporating fourth-order diffusion, time-varying control, non-instantaneous impulses, and infinite delay. Based on our self-developed $(\mu ,\nu ,\xi ,e,k)$-resolvent family, we derive the mild solution formulation and prove its existence via the Krasnoselskii–Schaefer fixed-point theorem. Using the Arzel$\acute{a}$-Ascoli theorem, Mazur’s lemma, and Balder’s lower semicontinuity principle, we further establish the existence of optimal control pairs. A numerical example from Euler–Bernoulli beam dynamics illustrates the theoretical results. Full article

This paper examines the sufficient conditions that guarantee the existence of solutions and anti-periodic solutions to five classes of fractional differential equations and inclusions involving the weighted generalized Atangana–Baleanu differential operator of order $\delta \in (1,2)$ under non-local conditions and with instantaneous or non-instantaneous impulses in Banach spaces whose dimension is infinite. First, we deduce some novel properties of this differential operator, then derive the formula for the solutions and anti-periodic solutions, and investigate their existence for the problems presented. Our method relies on certain properties of the Atangana–Baleanu differential operator, which we will obtain, as well as the fixed-point theorems that can be applied to the functions and multi-valued functions. Our work generalizes recently published results. In the final section, we present some examples to illustrate how our theoretical results can be applied. Full article

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

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

This study deals with P-type iterative learning control (ILC) techniques for switched impulsive systems governed by composite fractional derivatives. The systems considered incorporate non-instantaneous impulses and an initial state offset, with the objective of accurately tracking time-varying reference trajectories over a finite time interval using a finite number of iterations. By implementing a P-type learning law integrated with an initial iteration mechanism, we derive sufficient conditions that guarantee the convergence of the tracking error. The effectiveness and robustness of the proposed control concepts are validated through a comprehensive illustrative example. Full article

In this paper, we consider a class of fractional advection–dispersion equations involving instantaneous and non-instantaneous impulses. The existence of nontrivial solutions is established via Bonanno and D’Aguì’s critical point theorem. Under suitable conditions, we further prove the nonexistence of nontrivial solutions, which is the new result. Additionally, the application of our main results is demonstrated through two examples. Full article

This paper investigates non-instantaneous impulsive Hilfer fractional stochastic evolution equations. To obtain a more accurate convergence rate, an equivalent form of the above equation is derived by the time-scale separation method. Then, we prove that the solution of the equivalent equation converges to that of the averaged equation. Furthermore, we estimate the convergence rate between the exact and approximate solutions of the equation. Finally, we provide an example to justify our result. Full article

This paper aims to explore sufficient conditions for the existence of mild solutions to two types of nonlocal, non-instantaneous, impulsive semilinear differential inclusions involving a conformable fractional derivative, where the linear part is the infinitesimal generator of a $C_0$-semigroup or a sectorial operator and the nonlinear part is a multi-valued function with convex or nonconvex values. We provide a definition of the mild solutions, and then, by using appropriate fixed-point theorems for multi-valued functions and the properties of both the conformable derivative and the measure of noncompactness, we achieve our findings. We did not assume that the semigroup generated by the linear part is compact, and this makes our work novel and interesting. We give examples of the application of our theoretical results. Full article

Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the model may suddenly change due to outside influences. Occasionally, such changes start off impulsively and continue to exist for specific amounts of time. Non-instantaneous impulses are used in the creation of the models for this kind of scenario. In this paper, a new class of non-instantaneous impulsive $\psi$-Caputo fractional stochastic differential equations under integral boundary conditions driven by the Rosenblatt process was examined. Semigroup theory, stochastic theory, the Banach fixed-point theorem, and fractional calculus were applied to investigating the existence of piecewise continuous mild solutions for the systems under consideration. The impulsive Gronwall’s inequality was employed to establish the unique stability conditions for the system under consideration. Furthermore, we examined the controllability results of the proposed system. Finally, some examples were provided to demonstrate the validity of the presented work. Full article

In this paper, we present the existence of a mild solution for a class of a neutral stochastic integro-differential system over a Hilbert space. Such systems are influenced by both multiplicative and fractional noise, alongside non-instantaneous impulses, with a Hurst index H in the interval $({\displaystyle \frac{1}{2}},1)$. Additionally, the systems under consideration feature state-dependent delays (SDDs). To address this, we develop an approach to reformulate the neutral stochastic integro-differential system, incorporating SDDs and non-instantaneous impulses, into an equivalent fixed-point (FP) problem via an appropriate integral operator. By integrating stochastic analysis with the theory of resolvent operators, we employ Banach’s FP theorem to establish both the existence and uniqueness of the solution. Furthermore, we explore the Ulam–Hyers–Rassias stability of the system. Lastly, we provide illustrative examples to demonstrate the practical applicability of our results. Full article

The purpose of this study is to provide some criteria for the existence and multiplicity of solutions for a class of fractional advection–dispersion coupled systems with nonlinear Sturm–Liouville conditions and instantaneous and non-instantaneous impulses. Specifically, the existence is derived through the Nehari manifold method, and the proof of multiplicity is based on Bonanno and Bisci’s critical point theorem, which does not require proof that the functional satisfies the Palais–Smale condition. Finally, to illustrate the obtained results, an example is provided. Full article

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the $p$-proportional $\omega$-weighted $\kappa$-Hilfer derivative includes an exponential function, $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$, and then we consider a non-instantaneous impulse differential inclusion containing $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$ with order $\sigma \in (1,2)$ and of kind $\rho \in [0,1]$ in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the $(p,\omega ,\kappa )$-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the $(p,\omega ,\kappa )$-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. Full article

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of $\beta$-Banach spaces. Moreover, we examine the $\beta$–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained. Full article

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, $\mathrm{\Phi }$-Hilfer, fractional derivative of order $\mu \in (1,2)$ with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted $\mathrm{\Phi }$-Laplace transform, w-weighted $\psi$-convolution and the measure of non-compactness. Since the operator, the w-weighted $\mathrm{\Phi }$-Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results. Full article