Comparison Criteria for Nonlinear Functional Dynamic Equations of Higher Order

Abstract

We will consider the higher order functional dynamic equations with mixed nonlinearities of the form , on an above-unbounded time scale , where , ,   with  , and . The function is a rd-continuous function such that for . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.

1. Introduction

In this paper, we consider comparison criteria for higher order nonlinear dynamic equation with mixed nonlinearities of the formon an above-unbounded time scale , where(i) is an integer, and , , with , , and ;(ii) for .Without loss of generality we assume . For and , we denote by the space of right-dense continuous functions from to and by the set of functions in with right-dense continuous -derivatives. Throughout this paper we make the following assumptions:(iii) and , are constants and for , such that(iv) such that , on .(v) rd-continuous function such that , , and we let be a nondecreasing function on .

Recall that the knowledge and understanding of time scales and time scale notation are assumed. For an excellent introduction to the calculus on time scales, see [1–3]. By a solution of (1) we mean a nontrivial real-valued function for some such that , and satisfies (1) on , where is the space of right-dense continuous functions. An extendable solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is said to be nonoscillatory.

In the last few years, there has been an increasing interest in obtaining sufficient conditions for the oscillation/nonoscillation of solutions of different classes of dynamic equations; we refer the reader to [4–13] and the references cited therein. Special cases of (1) have been studied by many authors. When and for , and , Grace et al. [14] established some oscillation criteria for higher order nonlinear dynamic equation of the form where is the ratio of positive odd integers. In paper by Grace [15], some new criteria for the oscillation of the even order dynamic equation where and are the ratios of positive odd integers, were given. Recently, Hassan and Kong [16] obtained asymptotics and oscillation criteria for the th-order half-linear dynamic equation with deviating argument and Grace and Hassan [17] establish oscillation criteria for more general higher order dynamic equation The purpose of this paper is to derive comparison criteria for higher order nonlinear dynamic equation with mixed nonlinearities (1).

2. Preliminaries

We will employ the following lemmas. Consider the inequalitywhere and are positive real-valued, rd-continuous functions on and satisfies condition (2), is a rd-continuous function and as , and and are positive real numbers.

Now, we present the following lemma.

Lemma 1. If inequality (8) has an eventually positive solution, then the equationhas also an eventually positive solution.

Proof. Let be an eventually positive solution of inequality (8). It is easy to see that eventually. Let be sufficiently large so that , , and for . Then, in view of there is such that , for . Inequality (8) becomesIntegrating (11) from to and letting , we have where Now, we define a sequence of successive approximations as follows:It is easy to show that Then, the sequence is nonincreasing and bounded for each . This means that we may define . Since we find that By Lebesgue’s dominated convergence theorem on time scale, one can easily find Thereforewhere Then Equation (19) then gives Hence (9) has a positive solution . This completes the proof.

The second one is cited from [18, 19].

Lemma 2. Assume that (3) holds. Then there exists an -tuple with satisfying

The next lemma is cited from [17] and improves the well-known lemma of Kiguradze.

Lemma 3. Assume that (2) holds. If (1) has an eventually positive solution , then there exists an integer with being odd such thateventually, andeventually.

3. Main Results

In the following main theorem, we will use the following notations: for and for and, for any , define , , and , , by the following recurrence formulas:with and , provided the improper integrals involved are convergent.

Theorem 4. Assume that for sufficiently large the first-order dynamic equationis oscillatory, where for every number with being odd. Then(1)if is even, every solution of (1) is oscillatory,(2)if is odd andthen every solution of (1) either is oscillatory or tends to zero eventually.

Proof. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, and , for and . By Lemma 3, there exists an integer , , with being odd such that (24) and (25) hold for .
(I) When , from (1), we getFrom (23), we have Using the arithmetic-geometric mean inequality (see [20, Page 17]), we have Then for , This together with (30) shows that Integrating above inequality from to and then using the fact that is strictly increasing and is nondecreasing, we get and by (25) we see that . Hence by taking limits as we have which impliesIntegrating above inequality (37) from to and letting , we get Continuing this process times, we findAlso, from (24) and (25), we get It follows that Then for ,Analogously, we haveThen for From (39) and (44), we get Let ; we get or In view of Corollary 2.3.5 in [21], there exists a positive solution of (27) which contradicts the assumption of the theorem.
(II) When (in this case is odd), thereforeSince eventually, then . Assume that . Then for sufficiently large , we have for and . It follows that where . Then from (1), we have Integrating from to , we getAnd by (48) we see that . Hence by taking limits as we havewhich implies Again integrating above inequality from to and then taking , we get which implies Continuing this process () times, we find Integrating above inequality to , we get Hence by (29), we have , which contradicts the fact that eventually. This shows that This completes the proof.

Theorem 5. Assume that for sufficiently large the second-order dynamic equationis oscillatory, wherefor every integer number with being odd. Then(1)if is even, then every solution of (1) is oscillatory,(2)if is odd and (29) holds, then every solution of (1) either is oscillatory or tends to zero eventually.

Proof. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, and on . By Lemma 3, there exists an integer , , with being odd such that (24) and (25) hold for .
(I) When , as seen in the proof of Theorem 4, we obtain, for ,Hence, we haveSince for , we haveIt follows from (62) that we have, for ,Continuing this process, we have Then for From (60) and (67), we getSet ; we have, for , orIn view of Lemma 1, there exists a positive solution of (58) which contradicts the assumption of the theorem.
(II) When , as shown in the proof of Theorem 4, we show that if (29) holds, then . This completes the proof.

Remark 6. The conclusion of Theorems 4 and 5 remains intact if assumption (29) is replaced by one of the following conditions:

Theorem 7. Assume thatAnd for sufficiently large , the first-order dynamic equation is oscillatory, whereThen(1)if is even, every solution of (1) is oscillatory,(2)if is odd, then every solution of (1) either is oscillatory or tends to zero eventually.

Theorem 8. Assume that (72) holds and, for sufficiently large , the second-order dynamic equation is oscillatory, whereThen(i)if is even, every solution of (1) is oscillatory,(ii)if is odd, every solution of (1) either is oscillatory or tends to zero eventually.

Proof of Theorems 7 and 8. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, it is sufficiently large, such that and , for and . By Lemma 3, there exists an integer , , with being odd such that (24) and (25) hold for .
(I) When , we claim that (72) implies that . In fact, if , then for Since on , then for . Then there exists such that for and . It follows that where . Then from (1), we have Integrating from to , we get And by (48) we see that . Hence by taking limits as we have which impliesIntegrating above inequality from to and then taking , we get which implies Again, integrating above inequality from to and noting that eventually, we get Then by (72), we have , which contradicts the fact that on . This shows that if (72) holds, then . The rest of proof of (I) is similar to proof (I) of Theorems 4 and 5, respectively, with and hence can be omitted.
(II) When (in this case is odd), therefore Since eventually, then . Assume that . Then for sufficiently large , we have for and . It follows that where . Then from (1), we have Integrating from to , we get And using (48) we see that . Hence by taking limits as we have which implies Integrating above inequality from to and then taking , we get which impliesAgain, integrating above inequality from to and noting that eventually, we getThen by (72), we have , which contradicts the fact that on . This shows that . This completes the proof.

Remark 9. The conclusion of Theorems 7 and 8 remains intact if assumption (72) is replaced by one of the following conditions:

Theorem 10. Assume thatThen(1)if is even, every solution of (1) is oscillatory,(2)if is odd, then every solution of (1) either is oscillatory or tends to zero eventually.

Proof. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, it is sufficiently large, such that and , for and . By Lemma 3, there exists an integer , , with being odd such that (24) and (25) hold for .
(I) When , this implies that is strictly increasing on . Then for sufficiently large , we have for . It follows that where . Equation (1) becomesReplacing by in (98), integrating from to , we obtain Hence by (96), we have , which contradicts the fact that eventually.
(II) When (in this case is odd), therefore This implies that is strictly decreasing on . Then . Assume that . Then for sufficiently large , we have for . It follows thatwhere . As is case (I), we get a contradiction with the fact that eventually. This shows that . This completes the proof.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.