Asymptotic behaviours of the solutions of neutral type Volterra integro-differential equations and some numerical solutions via differential transform method

Abstract

( ) [ ( ) ( ) ( ( ))] ( ) ( ) ( ) ( ( )) ( , ) ( ( )) t t t d u t c t u t t a t u b t u t t k t s f u s ds dt ξ ξ μ ξ − + − = − − − + ∫ ( , ( ), ( ( )), 0, h t u t u t t t ξ + − ≥ (1) MJEN MANAS Journal of Engineering, Volume 9 (Special Issue 1) © 2021 www.journals.manas.edu.kg Y. Altun / MANAS Journal of Engineering 9 (2021) 49-57 50 where 0 0 ( ), ( ), ( ) :[ , ) [0, ), 0, a t b t c t t t ∞ → ∞ ≥ are continuous and ( ) c t is differentiable with 0 0 ( ) 1 ( constant) c t c c ≤ < − ; , and h k f with , 0 ) 0 ( = f are real-valued and continuous functions on their respective domains. The varying delay argument 0 ( ) :[0, ) [0, ] t ξ ξ ∞ → is continuous and differentiable and satisfying 0 0 ( ) , 0 ( ) 1, t t ξ ξ ξ δ ′ ≤ ≤ ≤ ≤ < (2) where 0 and ξ δ are positive constants. For each solution ( ) u t of equation (1), we assume the following existence initial condition: 0 0 0 ( ) ( ), [ , ], u t t θ φ θ θ ξ = ∈ − where 0 0 0 ([ , ], ). C t t φ ξ ∈ − R Define 1 0 ( ) , 0 ( ) (0) , 0. u u u u d u dt μ μ μ − \uf8f1 ≠ \uf8f4 = \uf8f2 = \uf8f4\uf8f3 (3) Hence, from (1) and (3), we have 0 ( ) [ ( ) ( ) ( ( ))] ( ) ( ) ( ) ( ) ( ( )) ( , ) ( ( )) t t t d u t c t u t t a t u u t b t u t t k t s f u s ds dt ξ ξ μ ξ − + − = − − − + ∫ ( , ( ), ( ( )), 0. h t u t u t t t ξ + − ≥ (4) 2. Main problem In this section, before proceeding further, we will state some assumptions for main result. 2.1. Assumptions (A1) There exists a positive constant , β such that ( , ) , k t s β ≤ for all 0 . t ≥ (A2) There is an 0, M > such that , u z M ≤ imply that, ( ) ( ) and (0) 0. f u f z u z f − ≤ − = (A3) Let ( , , ( ( )) h t u u t t ξ − ∈R be a non-linear uncertainty such that MJEN MANAS Journal of Engineering, Volume 9 (Special Issue 1) © 2021 www.journals.manas.edu.kg Y. Altun / MANAS Journal of Engineering 9 (2021) 49-57 51 1 2 1 2 ( , , ( ( )) ( ) ( ( )) , , 0 . h t u u t t q u t q u t t q q ξ ξ − ≤ + − ≥ (A4) μ\uf025 be a positive constant such that 0 1 ( ) u μ μ ≤ ≤ \uf025 for all . u∈R Theorem 1: Assumptions (A1)–(A4) are satisfied. Then, the zero solution of NVIDE (4) is asymptotically stable, if there exists a constant 0 c such that 0 ( ) 1 c t c ≤ < and ( ) 0, jk Λ = Λ < (5) where Λ is a 3 3 × symmetric matrix with the elements ( ) 11 1 0 0 2 2 ( ) 2 ( 1) 1, a t q c q β ξ Λ = − + + + + + + ( ) ( ) 12 0 13 22 2 0 0 1 2 23 33 0 ( ) ( ) , 0, (1 ) 2 ( ) 2 , 0, (1 ). c a t b t q c b t q q c δ βξ β β δ Λ = − + Λ = Λ = − − + − + + + Λ = Λ = + − − Proof. Consider the legitimate Lyapunov functional as 2 2 2 ( ) ( ) ( ) [ ( ) ( ) ( ( ))] ( ) ( ( )) . t t t t t t t s W t u t c t u t t u s ds f u v dvds ξ ξ ξ − − = + − + + ∫ ∫ ∫ From the calculation of the time derivative of the Lyapunov functional , W we have 0 2[ ( ) ( ) ( ( ))][ ( ) ( ) ( ) ( ) ( ( )) dW u t c t u t t a t u u t b t u t t dt ξ μ ξ = + − − − − 2 ( ) ( , ) ( ( )) ( , ( ), ( ( ))] ( ) t t t k t s f u s ds h t u t u t t u t ξ ξ − + + − + ∫ 2 2 2 ( ) ( ) (1 ( )) ( ( )) ( ( )) (1 ( )) ( ( )) t t t t t t t u t t f u t ds t f u v dv ξ ξ ξ ξ ξ − − ′ ′ − − − + − − ∫ ∫ 2 0 ( ) 2 ( ) ( ) ( ) 2 ( ) ( ) ( ( )) 2 ( ) ( , ) ( ( )) t t t a t u u t b t u t u t t u t k t s f u s ds ξ μ ξ − = − − − + ∫ 0 2 ( ) ( , ( ), ( ( )) 2 ( ) ( ) ( ) ( ) ( ( )) u t h t u t u t t a t c t u u t u t t ξ μ ξ + − − − 2 ( ) 2 ( ) ( ) ( ( )) 2 ( ) ( ( )) ( , ) ( ( )) t t t b t c t u t t c t u t t k t s f u s ds ξ ξ ξ − − − + − ∫ 2 2 2 ( ) ( ( )) ( , ( ), ( ( )) ( ) (1 ( )) ( ( )) c t u t t h t u t u t t u t t u t t ξ ξ ξ ξ ′ + − − + − − − 2 2 ( ) ( ) ( ( )) (1 ( )) ( ( )) t t t t f u t t f u v dv ξ ξ ξ − ′ + − − ∫ 2 ( ) 2 ( ) ( ) 2 ( ) ( ) ( ( )) 2 ( ) ( ( )) t t t a t u t b t u t u t t u t f u s ds ξ ξ β − ≤ − − − + ∫ 2 1 2 2 ( ) 2 ( ) ( ( )) 2 ( ) ( ) ( ) ( ( )) q u t q u t u t t a t c t u t u t t ξ ξ + + − − − MJEN MANAS Journal of Engineering, Volume 9 (Special Issue 1) © 2021 www.journals.manas.edu.kg Y. Altun / MANAS Journal of Engineering 9 (2021) 49-57 52 2 ( ) 2 ( ) ( ) ( ( )) 2 ( ) ( ( )) ( ( )) t t t b t c t u t t c t u t t f u s ds ξ ξ β ξ − − − + − ∫ 2 2 1 2 2 ( ) ( ) ( ( )) 2 ( ) ( ( )) ( ) q c t u t u t t q c t u t t u t ξ ξ + − + − + 2 2 2 ( ) (1 ( )) ( ( )) ( ) ( ( )) (1 ( )) ( ( )) t t t t u t t t f u t t f u v dv ξ ξ ξ ξ ξ − ′ ′ − − − + − − ∫ ( ) ( ) 2 1 0 2 ( ) 2 1 ( ) 2 ( ) ( ) ( ) ( ( )) a t q u t c a t b t u t u t t ξ ≤ − + + − + − ( ) ( ) 2 2 2 2 2 ( ) ( ) ( ( )) ( ) ( ( )) t t t u t f u s ds q u t u t t ξ β ξ − + + + + − ∫ ( ) 2 2 0 0 2 2 ( ) (1 ( )) ( ( )) q c c b t t u t t ξ ξ ′ + − − − − ( ) ( ) 2 2 2 2 0 1 0 ( ) ( ( )) ( ( )) ( ) ( ( )) t t t c u t t f u s ds q c u t u t t ξ β ξ ξ − + − + + + − ∫ 2 2 ( ) ( ) ( ( )) (1 ( )) ( ( )) . t t t t f u t t f u v dv ξ ξ ξ − ′ + − − ∫ From (2), we obtain ( ) ( ) 2 1 0 0 2 2 ( ) 2 ( 1) 1 ( ) dW a t q c q u t dt β ξ ≤ − + + + + + + ( ) 0 2 ( ) ( ) ( ) ( ( )) c a t b t u t u t t ξ − + − ( ) ( ) 2 2 0 0 1 2 2 ( ) 2 (1 ) ( ( )) q c b t q q u t t βξ δ ξ + + − + + + − − − 2 0 ( ) [ (1 )] ( ( )) . t t t c f u s ds ξ β β δ − + + − − ∫ The last estimate implies that ( ) ( ), T dW t t dt ψ ≤ ΛΨ where 2 1 2 ( ) ( ) [ ( ) ( ( ) ( ( ( )) ) ] t T t t t u t u t t f u s ds ξ ψ ξ − = − ∫ and Λ is defined in (5). Therefore if the matrix Λ is negative definite, dW dt is negative. (5) implied that there exists a constant sufficiently small 0 > η such that t dW u dt η ≤ − . Thus, equation (4) is asymptotically stable according to [6, Theorem 8.1, pp. 292–293]. This completes the proof of Theorem 1. Example 1: As a special case of the equation (4), we consider the following the first order NVIDE MJEN MANAS Journal of Engineering, Volume 9 (Special Issue 1) © 2021 www.journals.manas.edu.kg Y. Altun / MANAS Journal of Engineering 9 (2021) 49-57 53 0.3 [ ( ) 0.24 ( 0.3)] 1.8 ( ) 0.5 ( 0.3) 0.6 ( ) , 0. t t d u t u t u t u t u s ds t dt − + − = − − − − ≥ ∫ (6)