In this paper, we present a new type of Gronwall inequality and discuss some particular cases. We apply these results to investigate the uniqueness δ-Ulam–Hyers–Rassias stability mild solutions fractional differential equation with non-instantaneous impulses in Pδ-normed Banach space. sense, an example, order elucidate one discussed.

The purpose of this paper is to study nonlinear implicit differential equations with the Caputo–Katugampola fractional derivative. By using Gronwall inequality and Banach fixed-point theorem, existence solution equation proved, relevant conclusions about stability Ulam–Hyers are obtained. Finally, correctness verified by an example.

Integral inequalities are very important in applied sciences. Chebyshev's integral inequality is widely used mathematics. First of all, some necessary definitions and results regarding conformable derivative given this article. Then we give Chebyshev for simultaneously positive (or negative) functions using the fractional derivative. We Gronwall to prove our results, unlike other studies litera...

It is well known that differential equations, integral equations and integral-differential equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physical, mechanics, chemistry, aerodynamics, and the electrodynamics of complex mediums, etc. Gronwal...

Abstract We establish a discrete Gronwall–Halanay-type inequality with infinite delay, which is not covered in the existing literature. As an application, new criterion obtained for asymptotic stability of zero solutions class Volterra difference equations. A concrete example also given to illustrate efficiency general results.

This paper treats the concept of practical uniform $h$-stability for such perturbed dynamical systems as an extension exponential stability. We present a converse Lyapunov theorem and we give sufficient conditions that guarantee time-varying system using Gronwall-Bellman inequality Lyapunov's theory. Some examples are introduced to illustrate applicability main results.

Keywords: Fractional order with delay Generalized Gronwall inequality Iterative learning control a b s t r a c t In this paper, we discuss a P-type iterative learning control (ILC) scheme for a class of fractional order nonlinear systems with delay. By introducing the k-norm and using Gronwall inequality, the sufficient condition for the robust convergence of the tracking errors is obtained. Ba...

This paper aims to investigate the robustness of exponential synchronization in complex dynamic networks (CDNs) with time-varying delays and random disturbances. Via Gronwall–Bellman lemma partial inequality methods, by calculating transcendental equations, limits maximum disturbance size CDNs are estimated. means that perturbed system achieves if strength is within our estimation range. The th...

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, existence and uniqueness theorem a solution for obtained using Lipschitz condition Banach contraction principle. Then, expression derived by mathematical induction. On basis, we obtain via Gronwall–Bellman inequality. Finally, two examples equations are given to explain our main results.