Aydin Tiryaki

Oscillation theorems for nonlinear second-order differential equations

In this paper, some new oscillation criteria are given for the second-order nonlinear differential equation [r(t)Φ(x(t))ϕ(x′(t))]′ + c(t)ϕ(x(t)) = 0, t ≥ t0, where r ϵ C([t0, ∞); [0, ∞)), c ϵ C([t0, ∞);R), Φ ϵ C(R;R), and ϕ : R → R is defined by ϕ(s) = |s|p−2s with p > 1 a fixed real number. These criteria involve the use of averaging functions.

Asymptotic behaviour of a class of nonlinear functional differential equations of third order

Abstract This paper deals with nonoscillatory behaviour of solutions of third-order nonlinear functional differential equations of the form y ‴ + p ( t ) y ′ + q ( t ) F ( y ( g ( t ))) = 0. It has been shown that under certain conditions on coefficient functions, the nonoscillatory solutions of this equation tends to either zero or ∓∞ as t → ∞.

Oscillation of second-order nonlinear differential equations with nonlinear damping

This paper is concerned with the oscillation of a class of general type second-order differential equations with nonlinear damping terms. Several new oscillation criteria are established for such a class of differential equations under quite general assumptions. Examples are also given to illustrate the results.

BOUNDEDNESS AND STABILITY PROPERTIES OF SOLUTIONS OF CERTAIN FOURTH ORDER DIFFERENTIAL EQUATIONS VIA THE INTRINSIC METHOD

Received: May 25, 1994; r ev i sed : December 23, 1995 Abstract :In this paper, firstly a Lyapunov function is derived for (1.2) by applying the intrinsic method introduced by Chin[6], Secondly equation (1.2) is examined in two cases : (i) Ρ s 0, (ii) Ρ * 0 . For case (i) the asymptotic stability in the large of the trivial solution χ = 0 is investigated and for case (ii) the boundedness results are obtained for solutions of equation (1.2). These results improve and include several well-known results. AMS Classification numbers: 34D20, 34D99.

On the boundedness and the stability results for the solution of certain fourth order differential equations via the intrinsic method

In this paper, we first present constructing a Lyapunov function for (1.1) and then we show the asymptotic stability in the large of the trivial solution x=0 for case p≡0, and the boundedness result of the solutions of (1.1) for case p≠0. These results improve several well-known results.

Oscillation theorems for certain nonlinear differential equations of second order

Abstract We present new oscillation criteria for the second-order nonlinear differential equations. These criteria involve the use of averaging functions. Our theorems are stated in general form; they complement and extend related results known in the literature. Also, the relevance of our results is to show that some of Manojlovics results contain the superfluous condition.

Nonoscillation and asymptotic behaviour for third order nonlinear differential equations

In this paper we consider the equation y‴+q(t)y′α + p(t)h(y)=0, where p, q are real valued continuous functions on [0, ∞) such that q(t) ≥ 0, p(t) ≥ 0 and h(y) is continuous in (−∞, ∞) such that h(y)y > 0 for y ≠ 0. We obtain sufficient conditions for solutions of the considered equation to be nonoscillatory. Furthermore, the asymptotic behaviour of these nonoscillatory solutions is studied.

Oscillation criteria for nth-order forced functional differential equations

In this paper, by using the method of general means, some new oscillation criteria for forced functional differential equations of the form x(n)(t)+∑i=1n−1aix(i)(t)+q(t)fxg(t)=e(t),t0⩾0, are established, where ai are real constants, q(t), f(t), e(t), and g(t) are real continuous functions, xf(x)>0 whenever x≠0, and limt→∞g(t)=∞.

On the nonoscillatory behavior of solutions of third order differential equations

We are interested in nonoscillatory behavior of solutions of differential equations of the form where p(t) ≥ 0, q(t) ≤ 0 are real valued continuous functions on [0,∞), ∝ > 0 is the ratio of odd integers and h(y) is continuous on (-∞,∞) such that h(y)y>0 for y ≠ 0. We obtain sufficient contitions so that all solutions of the considered equation are nonoscillatory.

More on the explicit solutions for a second-order nonlinear boundary value problem

The explicit solutions to the boundary value problem x″(t)=λ(t)eμ(t)x(t) x(0)=x(1)=0, where λ and μ are continuous functions, are discussed.