Département de Mathématiques

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On the Asymptotic Behaviour and Oscillation of the Solutions of Certain Differential Equations and Systems with Delay
(2010) Ahmed Berkane
The abstract of this thesis, which concerns with the oscillatory behavior of the solutions of di erential equations of second order and of partial equations , is as follows: The rst contribution, subject of chapter two, is concerned with a simple generalization of Parhi and Kirane [28]. Our essential contribution in this part of thesis is to consider the coe cients and the delays as functions. The second contribution, subject of chapter three, is to consider the following equation without forcing term r(t)ψ(x(t))x 0 (t) 0 + p(t)x 0 (t) + q(t)f(x(t)) = 0, t ≥ t0 ≥ 0, (1) where t0 ≥ 0, r(t) ∈ C 1 ([t0,∞); (0,∞)), p(t) ∈ C([t0,∞)); R), q(t) ∈ C([t0, ∞)); R), p(t) and q(t) are not identical to zero on [t?,∞[ for some t? ≥ t0, f(x), ψ(x) ∈ C(R, R) and ψ(x) > 0 for x 6= 0. We do not only prove the oscillation of equation (1), but we also localize its zeros thanks to an idea of Nasr [27]. Our result is obtained under the following conditions: • (C1) For some positive constant K, f(x)/x ≥ K > 0 for all x 6= 0. • (C2) For some two positive constants C1, 0 < C ≤ ψ(x) ≤ C1 • (C3) Suppose further there exists a continuous function u(t) such that u(a) = u(b) = 0, u(t) is di erentiable on the open set (a, b), a, b ≥ t?, and Z b a