This paper deals with some nonlinear integral inequalities that involve the maximum of the unknown scalar function of one variable. The considered inequalities are generalizations of the classical integral inequality of Gronwall–Bellman. The importance of these integral inequalities is due to their wide applications in qualitative investigations of differential equations with “maxima”, and it i...

In this paper, we study boundary value problems for differential equations involving Caputo derivative of order α ∈ (2, 3) in Banach spaces. Some sufficient conditions for the existence and uniqueness of solutions are established by virtue of fractional calculus, a special singular type Gronwall inequality and fixed point method under some suitable conditions. Examples are given to illustrate t...

We discuss the exponential stability in mean square of mild solution for neutral stochastic partial functional differential equations with impulses. By applying impulsive Gronwall-Bellman inequality, the stochastic analytic techniques, the fractional power of operator, and semigroup theory, we obtain some completely new sufficient conditions ensuring the exponential stability in mean square of ...

Abstract. In this paper we study nonlinear control systems with multiple delays on controls and states. To obtain asymptotic stability, we impose Hölder-type assumptions on the perturbing function, and show a Gronwall-type inequality for difference equations with delay. We prove that a nonlinear control system can be stabilized if its linear control system can be stabilized. Some examples are i...

We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability convergence are proved with aids discrete Gronwall inequality, numerical examples presented to support theoretical analysis.

In this paper, we investigate the existence and Ulam-Hyers-Rassias stability of solutions for stochastic differential equations with random impulses. Based on Krasnoselskii?s fixed point theorem, perform investigations to system We apply integral inequality Gronwall type study their stability.

Fractional differential equations and fractional integral equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines. Gronwall-Bellman inequalities are important tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of Fractional differential equations and fracti...

Abstract This paper addresses the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives. Using Mittag–Leffler function, Laplace transform, and generalized Gronwall inequality, a sufficient stability condition is derived for such systems. Numerical examples illustrate theoretical results.

Fractional optimal control problems via a wide class of fractional operators with general analytic kernel are introduced. Necessary optimality conditions Pontryagin type for the considered problem obtained after proving Gronwall inequality as well results on continuity and differentiability perturbed trajectories. Moreover, Mangasarian sufficient global condition is proved. An illustrative exam...