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This dissertation discusses the retarded respectively, with a pulse delay and neutral differential system, using different research methods exist to obtain several classes of anti-periodic solutions of the system sufficient conditions. Thesis consists of four parts. The first chapter is an introduction , a brief introduction Equation to research and development and anti-periodic solutions of the basic situation and relevant background second chapter use Leray-Schauder degree theory to study the second Lienard equation: x \)) x '(t) f2 (x (t)) (x' (t)) 2 prove that under certain conditions, the system has unique anti-periodic solutions to promote the literature [22] and [33] results. following the introduction of (t, x) │ ≤ A, f2 '(x) ≤ 0. (H2.3) │ g1 (t, υ)-g1 (t, ν) │ ≤ L1 │ υ-ν │, │ g2 (υ) -g2 (ν) │ ≤ L2 │ υ-ν │, │ h (t) │ ≤ H. (H2.5) │ f2 (x) │ ≤ B, │ f1 (t, υ)-f1 (t, ν ) │ ≤ F1 │ υ-ν │, │ f2 (υ)-f2 (ν) │ ≤ F2 │ υ-ν │. CHAPTER main conclusions: Theorem 2.3.1 Suppose condition (H2.1), (H2. 2), (H2.3), (H2.5), (H2.6) holds, then the system (2.1.2) existence and uniqueness of anti-periodic solutions chapter by constructing a suitable Lyapunov function to study the pulse delay cell neural network system: the system obtained (3.1.1) Anti-Periodic Solutions and Exponential Stability sufficient conditions of the existing literature [24,42,43,44] the methods and conclusions have extended to both delay pulse situation. introduce conditions: (H3.1) there exist positive Fj, Lj such that: square fj (0) = 0, │ fj (υ) │ ≤ Fj, │ fj (υ)-fj (ν) │ ≤ Lj │ υ-ν │. (H3.2) dik is a real number sequence, and dik> 0, i = 1,2, ..., n, k = 1,2, ...; (H3.3) Π0 lt; tk lt; t (1 dik), i = 1,2, ..., n, is a T-periodic periodic function. (H3.4) there exist positive constants m, M, m lt; M such that (H3.5) There exists a constant δi gt; 0, η gt; 0 and λ gt; 0, i = 1,2, ..., n, and so that the third chapter of the main conclusions: Theorem 3.3.1 Suppose condition (H3.1) - (H3.5 ) hold, then system (3.1.1) has a T-anti-periodic solution z * (t) = {zi * (t)}, and z * (t) = {zi * (t)} is globally exponentially stable. Chapter Exponential dichotomy and the fixed point theory, a class of high-dimensional with infinite delay neutral functional differential equations: existence of anti-periodic solutions. introduce conditions: (H4.1) there exists a positive differentiable function d1 (t), d2 (t), ..., dn (t) (C1 ≤ di (t) ≤ C2, C1, C2 is a positive constant) and the periodic function of consecutive T α (t), such that: (H4 .2) there is a positive differentiable function d1 (t), d2 (t), ..., dn (t) (C1 ≤ di (t) ≤ C2, C1, C2 is a positive constant) and the consecutive T periodic function α ( t), such that: (H4.3) q1 = ∫ - ∞ 0 │ G (s) │ ds <1, q2 = ∫ - ∞ 0 │ Q (s) │ ds is bounded. (H4.4) there exists a positive L1, L2, L3, such that: L1 = sup0 ≤ t ≤ T │ A (t) │, │ g (υ)-g (ν) │ ≤ L2 │ υ-ν │, │ f (t, υ)-f (t, ν) │ ≤ L3 │ υ-ν │. (H4.5) If one k1 = exp (∫ 0Tα (λ) dλ) <1, M = (H4.6) If one of the main conclusions of Chapter IV : Theorem 4.3.1 Suppose condition (H4.1), (H4.3), (H4.4), (H4.5) holds, then the system (4.1.1) exists T-anti-periodic solution. Theorem 4.3.2 assumptions (H4.2), (H4.3), (H4.4), (H4.6) holds, then the system (4.1.1) exists T-anti-periodic solution.
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