Oscillation theorems for second order nonlinear forced differential equations

In this paper, a class of second order forced nonlinear differential equation is considered and several new oscillation theorems are obtained. Our results generalize and improve those known ones in the literature.


Introduction
We consider the oscillation behavior of solutions of second order forced nonlinear differential equation where r, q ∈ C([t 0 , ∞), ℝ), and f, ψ, g ∈ C(ℝ, ℝ) and H is a continuous function on [t 0 , ∞) × ℝ 2 , α is a positive real number. Throughout the paper, it is assumed that the following conditions are satisfied: (A 1 ) r(t) > 0, t ≥ 0; (A 2 ) xg(x) > 0, g ∈ C 1 (ℝ) for x ≠ 0; (A 3 ) H t;x;y ð Þ g x ð Þ ≤ p t ð Þ∀t∈ t 0 ; ∞ ½ Þ; x; y∈ℝ and x≠0: We restrict our attention only to the solutions of the differential equations (1.1) and (1.2) that exist on some ray [t 0 , ∞), where t 0 ≥ t, to may depend on the particular solutions. Such a solution is said to be oscillatory if it has arbitrarily large zeros, and otherwise, it is said to be nonoscillatory. Equations (1.1) and (1.2) are called oscillatory if all its solutions are oscillatory.
The problem of finding oscillation criteria for second order nonlinear ordinary differential equations, which involve the average of integral of the alternating coefficient, has received the attention of many authors because in the fact there are many physical systems are modeled by second order nonlinear ordinary differential equations; for example, the so called Emden -Fowler equation arises in the study of gas dynamics and fluid mechanics. This equation appears also in the study of relativistic mechanics, nuclear physics and in the study of chemically reacting systems.
In this paper, we continue in this direction the study of oscillatory properties of equations (1.1) and (1.2). The purpose of this paper is to improve and extend the above mentioned results. Our results are more general than the previous results. The relevance of our results becomes clear due to some carefully selected examples.

Main results
In this section we prove our main results.
Theorem 2.1. Suppose that, conditions (A 1 ) -(A 3 ) hold, and Then all solutions of equation (1.1) are oscillatory. Proof. Let x(t) be a non-oscillatory solution on [T, ∞), T ≥ T 0 of the equation (1.1). We assume that Then differentiating (2.5), (1.1) and take in account as- In view of (2.1) we conclude that By using the extremum of one variable function it can be proved that Now, by applying this inequality we have ð2:8Þ Taking the limit for both sides of (2.9) and using (2.4), we find w(t) → − ∞. Hence, there exists T 1 ≥ T such that f(x ′ (t)) < 0 ⇒ x ′ (t) < 0, ∀t ≥ T 1 .

Condition (2.4) also implies that
and there exists T 2 ≥ T 1 such that Now, integrating by parts, we get From (2.1) and (2.2), we find From (2.3) and 0 < x(t) ≤ x(T 2 ), this implies that Z x t ð Þ x T 2 ð Þ k 2 ψ y ð Þ ð Þ 1 α dy is lower bounded, but the right side of it tends to mines infinity. Then, this is a contradiction. Example 2.2. Consider the following differential equation Evidently, if we take p t ð Þ ¼ 2 t 3 ; ρ t ð Þ ¼ t and α ¼ 2: Then all conditions of Theorem 2.1 are satisfied, hence, all the solutions are oscillatory.
Case 1: x ′ (t) > 0 for T 1 ≥ T for some t ≥ T 1 ; then from (2.10), we obtain From (2.1) and (2.2), we obtain Using (2.13), we obtain which contradicts to the condition (2.13). Case 2: If x ′ (t) is oscillatory, then there exists a sequence {α n } → ∞ on [T, ∞) such that x ′ (α n ) < 0. Let us assume that N is sufficiently large so that Then, from (2.1), (2.2) and (2.7), we have which contradicts to the assume that x ′ (t) oscillates. Case 3: Let x ′ (t) < 0 for t ≥ T 1 . Condition (2.11) implies that for any t 0 ≥ T 1 such that The remaining part of the proof is similar to that of Theorem 2.1 then will be omitted.
Example 2.4. Let us consider the following equation Remark 2.2. when α = 1, ψ(x(t)) = 1 and f(x′(t)) = x′(t), Theorem 2.1 and 2.2 reduce to Theorem 1 and 2 Remili (2010) and Theorems 2.1 and 2.3 are obtained by analogy with Theorems 2.1 and 2.2 from (Temtek and Tiryaki 2013 Proof. Without loss of generality, let assume that there exists a solution x(t) of (1.2) such that x(t) > 0 on [T, ∞) for some T ≥ t 0 . A similar argument holds also for the case when x(t) < 0. Let w(t) be defined by the Riccati Transformation Derivation this equality we have