Qualitative analysis of integro-differential equations with variable retardation

Abstract

The paper is concerned with a class of nonlinear time-varying retarded integro-differential equations (RIDEs). By the Lyapunov-Krasovskiȋ functional method, two new results with weaker conditions related to uniform stability (US), uniform asymptotic stability (UAS), integrability, boundedness, and boundedness at infinity of solutions of the RIDEs are given. For illustrative purposes, two examples are provided. The study of the results of this paper shows that the given theorems are not only applicable to time-varying linear RIDEs, but also applicable to time-varying nonlinear RIDEs.

Stability tests and solution estimates for non-linear differential equations

“…In this article, the unperturbed nonlinear system of DDEs (5) with variable delay, the perturbed nonlinear system of DDEs (3) with variable delay and the system of ODEs ( 6) were taken into consideration. Here, five new results, i.e., Theorem 1 -Theorem 5, which are dealt with the qualitative behaviors of trajectories of solutions called UAS, instability and integrability of solutions of the unperturbed system of DDEs ( 5), the boundedness of solutions of the perturbed system of DDEs (3) and the exponentially stability of solutions of the system of ODEs (6), were proved using the LKF method for the delay systems (3), ( 5) and the second method of Lyapunov for the system of ODEs (6), respectively. In the proof of the boundedness result, i.e., Theorem 5, it was not needed to use the Gronwall's inequality.…”

“…A comparison between the systems of ODEs (13) and ODEs (6) gives that BF (x(t − h(t))) ≡ 0.Next, A(t), G(x(t))and H(t, x(t)) are the same as in Example 1. The estimates for the functions A(t), G(x(t)) and H(t, x(t)) remain the same and correct.…”

“…From this point of view, we see that the LF ∆ 2 (t, x) is positive definite. The derivative of the LF ∆ 2 (t, x) of (12) along the system of ODEs (6) gives that…”

“…There are many publications on fundamental properties of solutions of FDEs, ODEs and so on. We cite here the papers [1][2][3][4][5][6], [7][8][9], [10], and the books of ( [32], [33][34][35][36][37][38][39]) fully or partially devoted to fundamental motions of trajectories of solutions of these classes of equations. In particular, UAS and boundedness of solutions at the infinity describe long time behaviors of solutions.…”

Solvability of Gripenberg's equations of fractional order with perturbation term in weighted LpL_p-spaces on R+{\mathbb{R}}^+

“…Recall that, the idea of the measure of noncompactness were utilized to demonstrate the presence of continuous or L p -solutions of the integral equations on compact or noncompact domains in [3, 5, 13, 15-17, 21, 25, 33] and for qualitative properties of the outcomes (cf. [9,10,26,32]). Further, in [8] the authors examined the solutions of the equation x(τ ) = h(τ, x(τ )) + p 1 (τ ) + ) and these results were extended to an unbounded interval in [11].…”