R.S Dahiya

Oscillation theorems of Nth-order functional differential equations with forcing terms

where (a) p, q, g, h, f, r: [a, co) -+ R are continuous, (b) p(t) > 0, q(t) > 0, g(r), h(t) are nondecreasing, (c) g(t) t and g(t) --+ co as t + co. A function x(t) E CC&,, 00) is called oscillatory if x(t) has arbitrarily large zeros in [to, co), to >O. Otherwise x(t) is called nonoscillatory. We provide sufficient conditions for the above functional differential equations to be almost oscillatory in the sense that every solution x(t) of (l), (2), and (3) is either oscillatory or else it satisfies lim,, m x”‘(t) = 0, 06i

Oscillation criteria of even-order nonlinear delay differential equations

Abstract It is the purpose of this paper to establish oscillations criteria for even-order nonlinear differential equation in a very general form with delay and to extend result due to Staikos and Petsoulas, which are in turn generalizations of others.

Oscillation and asymptotic behavior of bounded solutions of functional differential equations with delay

Abstract Conditions on a ( t ), g ( t ), and f ( t ) have been found under which the bounded nonoscillatory solutions of the equation y ( n ) ( t ) − a ( t ) y ( g ( t )) = f ( t ) approach zero. For the even order equation y (2 n ) ( t ) − a ( t ) y ( g ( t )) = f ( t ) the delay is shown to be causing the oscillatory behavior.

On the oscillation of a second-order delay equation

Abstract The oscillatory nature of two equations ( r ( t ) y ′( t ))′ + p 1 ( t ) y ( t ) = f ( t ), ( r ( t ) y ′( t ))′ + p 2 ( t ) y ( t − τ ( t ))= 0, is compared when positive functions p 1 and p 2 are not “too close” or “too far apart.” Then the main theorem states that if h ( t ) is eventually negative and a twice continuously differentiable function which satisfies ( r ( t ) h ′( t ))′ + p 1 ( t ) h ( t ) ⩾ 0, then this inequality is necessary and sufficient for every bounded solution of ( r ( t ) y ′( t ))′ + p 2 ( t ) y ( t − τ ( t )) = 0 to be nonoscillatory.

Existence of slow oscillations in functional equations

Abstract For the equation L n y(t) + H(t, y(t)) = ƒ(t) (A), where L n is a disconjugate nonlinear operator, sufficient conditions have been found to ensure that all proper solutions of (A) are slowly oscillating. The operator L n has the form L n = 1 p n (t) d dt 1 p n − 1 (t) ··· d dt 1 p 1 (t) d dt p 0 (t) .