Controllability of fractional measure evolution systems with state-dependent delay and nonlocal condition

In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

We consider the controllability problem for a class of fractional evolution systems of mixed type in an infinite dimensional Banach space. The existence of mild solutions and controllability results are discussed by a new estimation technique of the measure of noncompactness and a fixed point theorem with respect to a convex-power condensing operator. However, the main results do not need any restrictive conditions on estimated parameters of the measure of noncompactness. Since we do not assume that the semigroup is compact and other conditions are more general, the outcomes we obtain here improve and generalize many known controllability results. An example is also given to demonstrate the applications of our main results. MSC:26A33, 34B15, 93B05, 93C25.

This paper investigates a new class of non-autonomous second-order measure evolution systems involving state-dependent delay and non-instantaneous impulses. We introduce a stronger concept of exact controllability called total controllability. The existence of mild solutions and controllability for the considered system are obtained by applying strongly continuous cosine family and the Mönch fixed point theorem. Finally, an example is used to verify the practical application of the conclusion.

This paper investigated the existence and stability of solutions for boundary value problems involving Caputo fractional differential equations of variable order. Unlike constant-order models, variable-order equations allow the fractional order to change over time, enabling more flexible and accurate modeling of complex systems with evolving dynamics and memory. Using Darbo's fixed point theorem and the Kuratowski measure of noncompactness, we established new existence results for solutions within a Banach space of continuous functions. Our approach treated the variable order as piecewise constant, transforming the problem into a sequence of more manageable constant-order subproblems. Furthermore, we demonstrated Ulam-Hyers stability of the solutions, ensuring that small perturbations in the system did not lead to significant deviations in the results. To validate the theoretical findings, we provided a detailed example supported by numerical simulations. These results offered a solid foundation for future applications in science and engineering where system dynamics evolve over time.

The main scope of this paper is to focus the approximate controllability of second order (q∈(1,2]) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. The existence of mild solutions is derived by using Schauder fixed point theorem. Sufficient conditions for the approximate controllability are established by under the assumptions that the corresponding linear system is approximately controllable and it is checked by using Lebesgue dominated convergence theorem. The main results are completly based on the results that the existence and approximate controllability of the fractional stochastic system of order 1<q≤2 and are derived by using stochastic analysis theory, fixed point technique, q-order cosine family {<i>C_a</i>(<i>t</i>)}_<i>t</i>≥0, new set of novel sufficient conditions and methods adopted directly from deterministic fractional equations for the second order nonlinear impulsive fractional nonlocal stochastic differential systems with state-dependent delay and Poisson jumps in Hildert space H. Finally an example is added to illustrate the main results.

"By using the fixed-point theorems, we give sufficient conditions for the existence and uniqueness of solutions for the nonlocal fractional boundary value problem of nonlinear Riesz-Caputo differential equation. The boundedness assumption on the nonlinear term is replaced by growth conditions or by a continuous function. Finally, some examples are presented to illustrate the applications of the obtained results. Keywords: Fractional boundary value problem, Riesz-Caputo fractional derivative, existence and uniqueness, fixed point, nonlocal conditions. "

In this article, we studied the approximate controllability of Ψ-Caputo fractional differential systems. We prove the sufficient conditions for an abstract Cauchy problem invloving infinite delay, impulsive and nonlocal conditions. The result is shown by means of the infinitesimal operator, semigroup theory, fractional calculus, and Schauder’s fixed point theorem. First, we prove the existence of the mild solution and demonstrate that the Ψ-Caputo fractional system is approximately controllable. Finally, an example is given to analyse the obtained results.

In this paper we prove the exact controllability of a time varying semilinear system considering non-instantaneous impulses, delay, and nonlocal conditions occurring simultaneously. It is done by using the Rothe’s fixed point theorem together with some sub-linear conditions on the nonlinear term, the impulsive functions, and the function describing the nonlocal conditions. Furthermore, a control steering the semilinear system from an initial state to a final state is exhibited.

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results.

In this article, we prove the existence of solutions for the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions. It allows applying the noncompactness measure of Hausdorff, fractional calculus theory, and the nonlinear alternative for Kakutani maps fixed point theorem to obtain the existence results under the assumptions that the nonlocal item is compact continuous and Lipschitz continuous and multifunction is compact and Lipschitz, respectively. Our results extend the existence theorems for the classical Bagley-Torvik inclusion and some related models.

The existence of mild solutions and approximate controllability for Riemann-Liouville fractional neutral evolution systems with nonlocal conditions of a fractional order is investigated. The Laplace transform and semigroup theory are the tools used to prove the existence. In turn, approximate controllability is proved on the basis of a Nemytskii operator, a Mittag-Leffler function and certain hypotheses using fixed point theorems, as well as the construction of a Cauchy sequence. An example is provided to highlight the main results.

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results.

Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls

We prove new existence results of mild solutions to fractional differential equa- tions with nonlocal conditions in Banach spaces. The nonlocal item is only assumed to be continuous. This generalizes some recent results in this area.

In this work, we prove the existence and uniqueness of mild solution of the fractional conformable Cauchy problem with nonlocal condition. We obtained these results by applying the fixed point theorems precisely to the fixed point theorem of Krasnoselskii and Banach’s fixed point theorem. At the end, we provide application.