Asymptotic Behavior Of Solutions For Several Kinds Of Impulsive Delay Differential Equations

In this dissertation, the asymptotic behavior of solutions for several kinds of impulsive delaydifferential equations is investigated. This paper is organized as follows:In chapter 1, we brieffy outline the development and applications of impulsive delay differentialequations.In chapter 2, by means of definition of asymptotic behavior, the asymptotic behavior of solu-tions for the following equationis investigated. Especially, the results obtained are also applied to some models. Some suffcientconditions of asymptotic behavior solutions are obtained. Introduce the following conditions:(H2.1) There exist two positive constants A > 0 and B > 0 such that(H2.2) For any ff > 0, there exists a positive constant N such that(H2.4) There exists P∈(0, 1) for suffciently large t such that and (H2.5) The inequalities-∞< u≤0≤v < +∞, have a unique solution u = v = 0.The main results of chapter 2 are as follows:Theorem 2.2.1 Assume that conditions (H2.1)-(H2.3) hold. Then every solution of theinitial value problem (2.1.1)-(2.1.2) tends to zero as t→∞.Theorem 2.2.2 Assume that conditions (H2.2),(H2.4) hold. Then every oscillation solutionof the initial value problem (2.1.1)-(2.1.2) is bounded.Theorem 2.2.3 Assume that conditions (H2.1),(H2.2),(H2.4),(H2.5) hold. Then every oscil-lation solution of the initial value problem (2.1.1)-(2.1.2) tends to zero as t→∞.Corollary 2.3.1 Assume that condition (H2.2) hold. If there exists a P > 0 and P(P+12 ) < 1 such thatandfor suffciently large t. Then every solution of (2.1.6) tends to zero as t→∞.Corollary 2.3.2 Assume that condition (H2.2) hold. If there exists a P > 0 and P (P + 21) < 1such thatfor suffciently large t. Then every solution of (2.1.7) tends to zero as t→∞.The results obtained in chapter 2 improve and extend the works in [38,39].In chapter 3, by using Liapunov functional, the asymptotic behavior of solutions of nonlinearimpulsive delay differential equations of the form is discussed. Introduce the hypothesis thatThe main results of chapter 3 are as follows:Theorem 3.2.1 Assume that conditions (H3.1)-(H3.5) hold, if there are finite impulsivepoints in every interval (t .τi, t)(i = 1, 2,···, n), then every solution of the initial value problem(3.1.1)-(3.1.2) tends to a constant as t→∞.Theorem 3.2.2 Assume that conditions (H3.1)-(H3.5) hold, if there are finite impulsivepoints in every interval (t.τi, t)(i = 1, 2,···, n), then every oscillatory solution of the initial valueproblem (3.1.1)-(3.1.2) tends to zero as t→∞.Theorem 3.2.3 Assume that the conditions (H3.1)-(H3.5) hold, if there are finite impulsivepoints in every interval (t .τi, t)(i = 1, 2,···, n), and that for anyα> 0, there is a constantβ> 0such thatThen every solution of the initial value problem (3.1.1)-(3.1.2) tends to zero as t→∞.Corollary 3.2.1 Assume that condition (H3.1),(H3.3),(H3.4) hold, if there are finite impul-sive points in every interval (t .τi, t)(i = 1, 2,···, n), andThen every solution of (3.2.22) tends to a constant as t→∞.Corollary 3.2.2 Assume that condition (H3.4) hold, if there are finite impulsive points inevery interval (t .τi, t)(i = 1, 2,···, n), andThen every solution of (3.2.23) tends to a constant as t→∞.The results obtained in chapter 3 improve and extend the corresponding ones in [27,37]. In chapter 4, by applying Liapunov functional, the asymptotic behavior of solutions of non-linear impulsive neutral differential equations of the followingis studied. Let us introduce the following conditions:(H4.4) There exists a constant M > 0 such that |x|≤|f(x)|≤M|x|, x∈R, xf(x) > 0, x 6= 0;(H4.8) tk .τis not an impulsive point, 0 < bk≤1, and there exists a constantβ> 0 such thattk . tk.1≥βfor all k.The main results of chapter 4 are as follows:Theorem 4.2.1 Assume that (H4.1)-(H4.7) hold, if there are finite impulsive points in everyinterval (t .σ_i, t)(i = 1, 2,···, n), then every solution of the initial value problem (4.1.1)-(4.1.2)tends to a constant as t→∞.Theorem 4.2.2 Assume that (H4.1)-(H4.6),(H4.8) hold, if there are finite impulsive pointsin every interval (t .σ_i, t)(i = 1, 2,···, n), then every solution of the initial value problem (4.1.1)-(4.1.2) tends to a constant as t→∞.Theorem 4.2.3 If the conditions of Theorem 4.2.1 or the conditions of Theorem 4.2.2 hold.Then every oscillation solution of the initial value problem (4.1.1)-(4.1.2) tends to zero as t→∞.Theorem 4.2.4 Assume that (H4.1)-(H4.7) hold, if there are finite impulsive points in everyinterval (t .σ_i, t)(i = 1, 2,···, n), andThen every solution of the initial value problem (4.1.1)-(4.1.2) tends to zero as t→∞ The works obtained in chapter 4 generalize the corresponding results in [26,27,33,45].