COMPARISON THEOREM FOR OSCILLATION OF FOURTH-ORDER NONLINEAR RETARDED DYNAMIC EQUATIONS

This work is concerned with oscillation of a class of fourth-order nonlinear delay dynamic equations on a time scale. A new comparison theorem is established that improves related results reported in the literature.


Introduction
Fourth-order differential equations naturally appear in models concerning physical, biological, and chemical phenomena, for instance, problems of elasticity, deformation of structures, or soil settlement; see [5].In this work, we study oscillation of a fourth-order nonlinear delay dynamic equation (1.1) x ∆ 4 (t) + p(t)x γ (τ (t)) = 0 on an arbitrary time scale T, where γ > 0 is the quotient of odd positive integers, p is a real-valued positive rd-continuous function defined on T, τ ∈ C rd (T, T), τ (t) ≤ t, and τ (t) → ∞ as t → ∞.
Since we are interested in oscillatory behavior, we assume throughout this paper that the given time scale T is unbounded above, i.e., it is a time scale interval of the form [t 0 , ∞) T := [t 0 , ∞) ∩ T with t 0 ∈ T.
By a solution of (1.1) we mean a nontrivial real-valued function x ∈ C 4 rd [T x , ∞) T , T x ∈ [t 0 , ∞) T which satisfies (1.1) on [T x , ∞) T .The solutions vanishing in some neighbourhood of infinity will be excluded from our consideration.A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory.Equation (1.1) is called oscillatory if all its solutions oscillate.
Han et al. [15] studied a second-order Emden-Fowler delay dynamic equation For the oscillation of higher-order dynamic equations on time scales, Erbe et al. [9] considered a third-order dynamic equation Grace et al. [11] studied a fourth-order dynamic equation Monotone and oscillatory behavior of solutions to a fourth-order dynamic equation with the property that were established by Grace et al. [12].Grace et al. [14] The purpose of this paper is to improve those results obtained in [14].This paper is organized as follows: In the next section, we present the basic definitions and the theory of calculus on time scales.In Section 3, we establish some new oscillation results for (1.1).
In what follows, all functional inequalities are assumed to hold eventually, that is, for all sufficiently large t.

Some preliminaries
A time scale T is an arbitrary nonempty closed subset of the real numbers R. Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above and is a time scale interval of the form [t 0 , ∞) T .On any time scale we define the forward and backward jump operators by σ(t) := inf{s ∈ T|s > t} and ρ(t) := sup{s ∈ T|s < t}, where inf ∅ := sup T and sup ∅ := inf T, ∅ denotes the empty set.
A point t ∈ T is said to be left-dense if ρ(t) = t and t > inf T, right-dense if σ(t) = t and t < sup T, left-scattered if ρ(t) < t, and right-scattered if σ(t) > t.The graininess µ of the time scale is defined by µ(t) := σ(t) − t.
A function f : T → R is said to be rd-continuous if it is continuous at each right-dense point and if there exists a finite left limit in all leftdense points.The set of rd-continuous functions f : Fix t ∈ T and let f : T → R. Define f ∆ (t) to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of t (i.e., U = (t − δ, t + δ) ∩ T for some δ > 0) such that In this case, f ∆ (t) is called the (delta) derivative of f at t. f is said to be differentiable if its derivative exists.The set of functions f : T → R that are differentiable and whose derivative is rd-continuous function is denoted by If t is right-dense, then f is differentiable at t iff the limit t − s exists as a finite number.In this case Let f be a real-valued function defined on an interval [a, b] T .We say that f is increasing, decreasing, nondecreasing, and nonincreasing on , and f (t 2 ) ≤ f (t 1 ), respectively.Let f be a differentiable function on [a, b] T .Then f is increasing, decreasing, nondecreasing, and nonincreasing on [a, b] T if f ∆ (t) > 0, f ∆ (t) < 0, f ∆ (t) ≥ 0, and f ∆ (t) ≤ 0 for all t ∈ [a, b) T , respectively.
We will make use of the following product and quotient rules for the derivative of the product f g and the quotient f /g (where g(t)g(σ(t)) = 0) of two differentiable functions f and g For a, b ∈ T and a differentiable function f, the Cauchy integral of f ∆ is defined by

Main results
In this section, we present some sufficient conditions which ensure that every solution of (1.1) is oscillatory.We begin with the following lemma.
Lemma 3.1.Assume there exists T ∈ [t 0 , ∞) T such that Then, there exists a constant Proof.The proof is similar to that of [21, Lemma 1], and hence is omitted.
Lemma 3.3.Assume x is an eventually positive solution of (1.1).Then there are only the following two possible cases for t Proof.The proof is simple, and so is omitted.EJQTDE, 2013 No. 22, p. 5 Theorem 3.4.Assume there exists a positive function m ∈ C 1 rd (T, R) such that for some l ∈ (0, 1).Suppose further that there exists a positive function T , sufficiently large.If both second-order dynamic equations are oscillatory for all sufficiently large t l and for some k ∈ (0, 1), then (1.1) is oscillatory.
We may assume without loss of generality that there exists a t 1 ∈ [t 0 , ∞) T such that x(t) > 0 and x(τ (t)) > 0 for t ∈ [t 1 , ∞) T .It follows from Lemma 3.3 that x satisfies either case (1) or case (2).Assume case (1).Set y := x ∆ .It follows from Lemma 3.2 that for t ∈ [t l , ∞) T and for given l ∈ (0, 1).Since we see that x ∆ /m is nonincreasing.Then, we obtain On the other hand, we find by (3.6) that Hence x/v is nonincreasing.Thus, we get (3.8)x(τ Using (3.7) and (3.8), we obtain for given k ∈ (0, 1).Thus, we get by x ∆ > 0 that Letting z → ∞ in the above inequality, we obtain Letting z → ∞ in the last inequality, from lim z→∞ (−x ∆ 2 (z)) ≥ l 2 ≥ 0, we have It follows from [14, Lemma 2.1] that equation (3.4) has positive solutions, which is a contradiction.The proof is complete.
Remark 3.5.From Theorem 3.4, one can obtain some corollaries for the oscillation of (1.1).For example, if we use some related results in [15], then we get the following results.