On the influence of integral perturbations to the asymptotic stability of solutions of a second-order linear differentia
- PDF / 157,591 Bytes
- 6 Pages / 593.863 x 792 pts Page_size
- 16 Downloads / 138 Views
On the influence of integral perturbations to the asymptotic stability of solutions of a second-order linear differential equation Samandar Iskandarov, Nazigai A. Abdiraiimova (Presented by F. Abdullaev) Abstract. Sufficient conditions for the asymptotic stability of the solutions of a second-order linear integro-differential equation of the Volterra type are established in the case where the solutions of the corresponding second-order linear differential equation may have no property under study. Thus, the influence of integral perturbations on the asymptotic stability of solutions of linear differential equations of the second order is revealed. For this purpose, the method of auxiliary kernels is developed. An illustrative example is given. Keywords. Linear integro-differential equation, linear differential equation, asymptotic stability of solutions, influence of integral perturbations, method of auxiliary kernels.
All used functions and their derivatives are continuous, and all relations hold for t ≥ t0 , t ≥ τ ≥ t0 ; I = [t0 , ∞); IDE and DE denote integro-differential and differential equations, respectively; the asymptotic stability of solutions of a linear second-order IDE means the convergence of its all solutions and their first derivatives to zero as t → ∞ . Problem. To find the sufficient conditions of asymptotic stability for the solutions of the linear second-order Volterra-type IDE x′′ (t) + a1 (t)x′ (t) + a0 (t)x(t) ∫t [ +
] Q0 (t, τ )x(τ ) + Q1 (t, τ )x′ (τ ) dτ = f (t),
t ≥ t0
(1)
t0
in the case where the solutions of the corresponding second-order DE x′′ (t) + a1 (t)x′ (t) + a0 (t)x(t) = f (t),
t ≥ t0
(10 )
can be asymptotically unstable. To solve this problem, we will develop the method of auxiliary kernels [1, 2]. It is worth to note that such problem for other classes of IDEs of the form (1) and with the use of other methods was considered in many works (see, e.g., [3, 4]). First, we introduce some auxiliary kernel H(t, τ ) with x′ (τ ) [1] into IDE (1) by the rule of “weight” [4, p. 114]: Q0 (t, τ )x(τ ) + Q1 (t, τ )x′ (τ ) = Q0 (t, τ )x(τ ) +[Q1 (t, τ ) − H(t, τ )]x′ (τ ) + H(t, τ )x′ (τ ).
(2)
Then we carry on the integration by parts: ∫t
′
∫t
H(t, τ )x (τ )dτ = H(t, t)x(t) − H(t, t0 )x(t0 ) − t0
′
Hτ (t, τ )x(τ )dτ.
(3)
t0
Translated from Ukrains’ki˘ı Matematychny˘ı Visnyk, Vol. 17, No. 2, pp. 188–195 April–June, 2020. Original article submitted November 03, 2019 c 2020 Springer Science+Business Media, LLC 1072 – 3374/20/2495–0733 ⃝
733
With regard for (2) and (3), we get the following loaded IDE from IDE (1): ∫t { ′ [Q0 (t, τ ) − Hτ (t, τ )]x(τ ) x (t) + a1 (t)x (t) + [a0 (t) + H(t, t)]x(t) + ′′
′
t0
} +[Q1 (t, τ ) − H(t, τ )]x′ (τ ) dτ = f (t) + H(t, t0 )x(t0 ).
(4)
In IDE (4), we now make the following nonstandard change [5]: x′ (t) + λ2 x(t) = W (t)y(t),
(5)
where λ is some auxiliary parameter, λ ∈ R, 0 < W (t) is some weight function, and y(t) is a new inknown function. Analogously to [5], we reduce IDE (4) to the equivalent system x′ (t) + λ2 x(t) = W (t)y(t), t
Data Loading...
