Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

This is a part of the Handbook of Differential Equations (A.
Canada, P. Drabek, A. Fonda, editors), where recent results in
the qualitative theory of half-linear differential equations
are presented.

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Half-linear differential equations

This book summarizes the qualitative theory of differential equations with or without delays, collecting recent oscillation studies important to applications and further developments in mathematics, physics, engineering, and biology. The authors address oscillatory and nonoscillatory properties of first-order delay and neutral delay differential equations, second-order delay and ordinary differential equations, higher-order delay differential equations, and systems of nonlinear differential equations. The final chapter explores key aspects of the oscillation of dynamic equations on time scales-a new and innovative theory that accomodates differential and difference equations simultaneously.

Nonoscillation and Oscillation Theory for Functional Differential Equations

On solution of

Oscillation Theory for Functional Differential Equations (L. H. Erbe, Q. K. Kong, and B. G. Zhang)

Oscillations caused by several retarded and advanced arguments

Oscillation criteria for second-order delay differential equations

In this paper we obtain sufficient conditions under which every solution of the retarded differential equation (1) x'(t) + p (t)x(t -T) = O, t :,2 to where T is a nonnegative constant, and p(t) > 0, is a continuous function, tends to zero as t o0. Also, under milder conditions, we prove that every oscillatory solution of (1) tends to zero as t o0. More precisely the following theorems have been established. THEOREM 1. Assume that J0pp(t) dt= +oo and lim,I OC f-fTp(s) ds < T/2 or lim sup,t 0tt-TP(S) ds < 1. Then every solution of (1) tends to zero as t oo0. THEOREM 2. Assume that lim sup,t J0 tt-TP(S) ds < 1. Then every oscillatory solution of (1) tends to zero as t o0.

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Asymptotic behavior of solutions of retarded differential equations

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form x′(t) + p(t)x(τ(t)) = 0, t ≥ t0, (1) where p, τ ∈ C([t0,∞),R),R = [0,∞), τ(t) is non-decreasing, τ(t) < t for t ≥ t0 and limt→∞ τ(t) =∞. Let the numbers k and L be defined by

/pdf/oscillation-criteria-for-delay-equations-2lyprrnnfb.pdf

Oscillation criteria for delay equations

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form (1) x′(t) + p(t)x(τ(t)) = 0, t ≥ T, where p, τ ∈ C([T,∞),R+), R+ = [0,∞), τ(t) is nondecreasing, τ(t) < t for t ≥ T and limt→∞ τ(t) = ∞. Let the numbers k and L be defined by

/pdf/oscillation-tests-for-delay-equations-1at12xvmeu.pdf

Ioannis P. Stavroulakis

Papers

Oscillations caused by several retarded and advanced arguments

Oscillation criteria for second-order delay differential equations

Asymptotic behavior of solutions of retarded differential equations

Oscillation criteria for delay equations

Oscillation Tests for Delay Equations