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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 168425, 13 pages
http://dx.doi.org/10.1155/2014/168425
Research Article

On Asymptotic Behavior of Solutions of Generalized Emden-Fowler Differential Equations with Delay Argument

1Department of Mathematics and Computer Sciences, Ariel University, 40700 Ariel, Israel
2Department of Mathematics, Tbilisi State University, University Street 2, 0143 Tbilisi, Georgia

Received 4 July 2013; Accepted 8 November 2013; Published 22 January 2014

Academic Editor: Josef Diblik

Copyright © 2014 Alexander Domoshnitsky and Roman Koplatadze. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The following differential equation is considered. Here ,   ,   ,   , and . We say that the equation is almost linear if the condition is fulfilled, while if or , then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new sufficient conditions implying Property A for delay Emden-Fowler equations are obtained.

1. Introduction

This work deals with oscillatory properties of solutions of a functional differential equation of the form where

It will always be assumed that the condition is fulfilled.

Let . A function is said to be a proper solution of (1) if it is locally absolutely continuous together with its derivatives up to order inclusive, and there exists a function such that on and the equality holds for . A proper solution of (1) is said to be oscillatory if it has a sequence of zeros tending to . Otherwise the solution is said to be nonoscillatory.

Definition 1. We say that (1) has Property A if any of its proper solutions is oscillatory when is even and either is oscillatory or satisfies when is odd.

Definition 2. We say that (1) is almost linear if the condition holds, while if   or , then we say that the equation is an essentially nonlinear differential equation.

The Emden-Fowler equation originated from theories concerning gaseous dynamics in astrophysics in the middle of the nineteenth century. In the study of stellar structure at that time it was important to investigate the equilibrium configuration of the mass of spherical clouds of gas. Lord Kelvin in 1862 assumed that the gaseous cloud is under convective equilibrium and then Lane [1] studied the equation

The Emden-Fowler equations were first considered only for second-order equations and written in the form which could be reduced in the case of positive and continuous coefficients to the equation

To avoid difficulties of defining when is negative and is not an integer, the equation was usually considered. The mathematical foundation of the theory of such equations was built by Fowler [2] and the description of the results can be found in Chapter 7 of [3].

We see also the Emden-Fowler equation in gas dynamics and fluid mechanics (see Sansone [4], page 431 and the paper [5]). Nonoscillation of these equations is important in various applications. Note that the zero of such solutions corresponds to an equilibrium state in a fluid with spherical distribution of density and under mutual attraction of its particles. The Emden-Fowler equations can be either oscillatory (i.e., all proper solutions have a sequence of zeros tending to zero) or nonoscillatory, if solutions are eventually positive or negative, or, in contrast with the case of linear differential equations of second order, may possess both oscillating and nonoscillating solutions. For example, for the equation it was proven in [2] that for all solutions oscillate, for —all solutions nonoscillate, and for there are both oscillating and nonoscillating solutions.

The Emden-Fowler equation presents one of the classical objects in the theory of differential equations. Tests for oscillation and nonoscillation of all solutions and existence of oscillating solutions were obtained in the works [68]. In [9] for the case it was obtained that all solutions of the equation oscillate if and only if

The latest research results in this area are presented in the book [8]. Behavior of solutions to th order Emden-Fowler equations can be essentially more complicated. Properties A and B defined by Kiguradze are studied in the abovementioned book.

There are essentially less results on oscillation of delay Emden-Fowler equations. Oscillation properties of nonlinear delay differential equations, where Emden-Fowler equations were also included as a particular case, were studied in [1020]. Results of these papers are discussed in [13, 15], where various examples demonstrating essentialities of conditions are also presented. Note that for delay differential equations there are no results on nonoscillation of all solutions and only existence of nonoscillating solutions is studied. Actually, the results on oscillation of delayed equations are based on the approaches existing for ordinary differential equations with development in the direction of preventing the obstructive influence of delay. In the paper [15] the following equation and its particular case are considered. It was obtained for the last equation under some standard assumptions on the coefficients [15] that in the case , all solutions oscillate. We see that the integral depends on deviation of argument and the power of the equation . For the equation where is the ratio of two positive odd integers, for and as each of the following conditions (a), (b), and (c) ensures oscillation of all solutions:(a) (b) (c)

Most proofs of results on oscillation of all solutions to second order equations utilize the fact that if a nonoscillating solution exists, the signs of the solution and its second derivative are opposite to each other for sufficiently large . Then a growth of nonoscillating solution is estimated and the authors come to contradiction with conditions that proves oscillation of all solutions. Note that delays disturb oscillation. Instead of appears. The principle is clear: for oscillation of all solutions we have to achieve a corresponding smallness of the delay . All this is more complicated if we study th order equations. In this case also the fact that and its th derivative have different signs for sufficiently large is used, but the technique is more complicated.

In the papers [2128] a generalization of Emden-Fowler equations was considered. The powers in these papers can be functions and not constants. In many cases, it leads to essentially new oscillation properties of such equations. Surprisingly, oscillation behavior of equations, with the power and with functional power such that can be quite different. The main purpose of our paper is to study conditions under which the generalized (in this sense) equations preserve the known oscillation properties of Emden-Fowler equations and conditions under which these properties are not preserved. Oscillatory properties of almost linear and essentially nonlinear differential equation with advanced argument have already been studied in [2128]. In this paper we study oscillation properties of th order delay Emden-Fowler equations.

2. Some Auxiliary Lemmas

In the sequel, will denote the set of all functions absolutely continuous on any finite subinternal of along with their derivatives of order up to including .

Lemma 3 (see [28]). Let , ,   for   , and in any neighborhood of . Then there exist and   such that   is odd and

Remark 4. If is odd and   , then it means that in only the second inequalities are fulfilled.

Lemma 5 (see [29]). Let and let be fulfilled for some   with odd. Then
If, moreover, then there exists such that

3. Necessary Conditions for the Existence of a Solution of Type  

The following notation will be used throughout the work:

Clearly , and is nondecreasing and coincides with the inverse of when the latter exists.

Definition 6. Let . By one denotes the set of all proper solutions of (1) satisfying the condition with some .

Lemma 7. Let the conditions (2), (3) be fulfilled, let with   odd, and let be a positive proper solution of (1). If, moreover, , , then for any there exists   such that for any where is given by the first equality of (26) and

Proof. Let ,   with   odd and (see Definition 6) is solution of (1). Since , according to (1), , and , it is clear that condition (22) holds. Thus, by Lemma 5 there exists such that the conditions –(25) with are fulfilled and
Observe that there exists such that for . Thus, by (24), for any we get
According to and , choose such that
By (34) and (35) we have
Let . Since as , without loss of generally we can assume that   for . Then by from (36) we get
It is obvious that where
Thus, according to (37), and (39) from (38) we get
Therefore
Hence, according to (37) and (39) where
Thus, according to (36) and (42) where
Now assume that and . Since for , without loss of generality we can assume that for . Therefore, from (36) we have
Taking into account (46), as above we can find that if , then where
According to (43)–(45) and (47)–(49), it is obvious that for any , , and there exists such that (29)–(31) hold, where is defined by (32). This proves the validity of the lemma.

Analogously we can prove.

Lemma 8. Let conditions (2), (3), be fulfilled, let with odd, , and let be a positive proper solution of (1). Then for any there exists such that for any where is defined by the second equality of (26) and

Remark 9. In Lemma 7, the condition cannot be replaced by the condition . Indeed, let . Consider (1), where is even and
It is obvious that the function is the solution of (1) and it satisfies the condition for . On the other hand, the condition holds, but the condition (22) is not fulfilled.

Theorem 10. Let with be odd, let and the conditions (2), (3), , and let be fulfilled, and for some , . Then for any there exists such that if , and if , then for any and , where is defined by first equality of (26) and is given by (30)–(32).

Proof. Let and such that . By definition, (1) has a proper solution satisfying the condition with some . Due to (1), , and , it is obvious that condition (22) holds. Thus by Lemma 5 there exists such that conditions -(24) with are fulfilled. On the other hand, according to Lemma 7 (and its proof), we see that and there exists such that relation (30) is fulfilled. Without loss of generality we can assume that for . Therefore, by (24), from (58) we have
Assume that . Then by (44) and (59), we have
On the other hand, according to and it is obvious that
Therefore, from (60) we get
Now assume that and . Then according to (47), , and (61), from (59) we have
Thus, we obtain where .
It is obvious that there exist such that
Therefore, from (64) where
From the last inequality we get
Passing to the limit in the latter inequality, we get that is, according to (62) and (69), (56) and (57) hold, which proves the validity of the theorem.

Using Lemma 8 in a similar manner one can prove the following.

Theorem 11. Let with be odd, let (2), (3), , and be fulfilled, and for some , . Then there exists such that if , for any and if , then for any and where is defined by the second equality of (26) and is given by (51)–(53).

4. Sufficient Conditions for Nonexistence of Solutions of the Type

Theorem 12. Let with odd, let the conditions (2), (3), , and be fulfilled, and if , for any large and for some or if , for same and
Then for any one has , where and are defined by (26) and is given by (30)–(32).

Proof. Assume the contrary. Let there exist such that (see Definition 6). Then (1) has a proper solution satisfying the condition . Since the condition of Theorem 10 is fulfilled, there exists such that if (if ), the condition (56) (the condition (57)) holds, which contradicts and . The obtained contradiction proves the validity of the theorem.

Using Theorem 11 analogously we can prove the following.

Theorem 13. Let with odd, let the conditions (2), (3), , and be fulfilled, and if , for any large and for some or if for same and Then for any we have , where is defined by the second equality of (26) and is given by (51)–(53).

Corollary 14. Let with odd, let the conditions (2), (3), and be fulfilled, , , and
Then for any .

Proof. Since it suffices to note that by the conditions and are fulfilled for .

Corollary 15. Let with odd, let the conditions (2) and (3) be fulfilled, , , and If, moreover, for some then for any , .

Proof. Clearly by virtue of conditions and are fulfilled. Let . According to and it is obvious that, for large , . Therefore, by , for ,   holds, which proves the validity of the corollary.

Corollary 16. Let with odd, let the conditions (2), (3), , and be fulfilled, , , and for some
Then for any , , where is defined by the first condition of (3).

Proof. By and (77), for , the condition holds, which proves the validity of the corollary.

Corollary 17. Let with odd and let the conditions (2), (3), and be fulfilled. Moreover, if , , and there exists such that then for any , , where is defined by the first condition of (3).

Proof. By there exist and such that
Let and . Then by and (30), for large ,
Therefore
Thus, by (81) and , it is obvious that holds, which proves the corollary.

Quite similarly one can prove the following.

Corollary 18. Let with odd, let the conditions (2), (3), , and be fulfilled, , and . Moreover, if and for some and then for any one has , where and are defined by the condition of (26).

Corollary 19. Let with be odd, let the conditions (2), (3), and be fulfilled, , and . Moreover, let there exist and such that and let at least one of the conditions or and for some and be fulfilled. Then for any one has , where is defined by (26).

Proof. It suffices to show that the condition is satisfied for some and . Indeed, according to and (88), there exist , , , and such that
By (77), (31), and (91), from (31) we get
Let . Choose such that
Therefore, by (91) from (31) we can find such that
Hence for any there exists such that
Assume that (89) is fulfilled. Choose such that . Then according to (92), (96), and , the condition holds for and . In this case, the validity of the corollary has already been proven.
Assume now that is fulfilled. Let and choose and such that
Then according to (96), (92), and , it is obvious that holds for . The proof of the corollary is complete.

Using Theorem 13, in a manner similar to above we can prove the following.

Corollary 20. Let with odd, let the conditions (2), (3), and be fulfilled, , and
Then for any , , where is given by (26).

Corollary 21. Let with odd, let the conditions (2), (3), and be fulfilled, , and
Moreover, let for some
Then for any one has , where is given by (26).

Corollary 22. Let with odd, let the conditions (2), (3), , and be fulfilled, , and for some
Then for any , .

5. Differential Equation with Property A

Theorem 23. Let the conditions (2), (3) be fulfilled and for any with odd, the conditions and hold. Moreover for any large , if and for some let   be fulfilled or if and , for some , , and , let   be fulfilled. Then, if for odd then (1) has Property , where and are defined by (26) and is given by (30)–(32).

Proof. Let (1) have a proper nonoscillatory solution (the case is similar). Then by (2), (3), and Lemma 3 there exists such that is odd and conditions hold. Since, for any with odd, the conditions of Theorem 12 are fulfilled we have . Now assume that , is odd, and there exists such that for sufficiently large . According to since , from (1) we have where is a sufficiently large number. The last inequality contradicts the condition (102). The obtained contradiction proves that (1) has Property A.

Theorem 24. Let the conditions (2), (3) be fulfilled and for any with odd the conditions and hold. Let moreover, if , for some    hold or if and for some , , and , hold. Then, if for odd (102) is fulfilled, then (1) has Property , where is defined by the second equality of (26) and is given by (51)–(53).

Proof. The proof of the theorem is analogous to that of Theorem 23. We simply use Theorem 13 instead of Theorem 12.

Theorem 25. Let , , let the conditions (2), (3), , and be fulfilled, and If, moreover, for some , , and , holds and for odd n (102) is fulfilled, then (1) has Property , where and are defined by (26) and is given by (30)–(32).

Proof. It suffices to note that by (104) and , for any there exist , , and such that condition is fulfilled.

Theorem 26. Let , and conditions (2), (3), , , and (104) be fulfilled. If, moreover, for some , , and , holds and for odd n (102) is fulfilled, then (1) has Property , where is defined by the second condition of (26) and is given by (51)– .

Proof. The theorem is proved similarly to Theorem 25 if we replace the condition by the condition .

Corollary 27. Let , , and conditions (2), (3), , , and (104) be fulfilled. Then (1) has Property , where and are given by (26).

Proof. By , , and (104), condition (102), and for any    holds. Now assume that (1) has a proper nonoscillatory solution . Then, by (2), (3), and Lemma 3, there exists such that is odd and the condition holds. Since for any with odd the conditions of Corollary 14 are fulfilled, we have . Therefore is odd and . According to (102) and it is obvious that the condition (5) holds. Therefore, (1) has Property .

Using Corollaries 1519, the validity of Corollaries 2832 can be proven similarly to Corollary 27.

Corollary 28. Let , , and conditions (2), (3), , , and (104) be fulfilled. Then (1) has Property , where and are given by (26).

Corollary 29. Let , , conditions (2), (3), (104) be fulfilled, and for some hold and if is odd, condition (102) holds. Then (1) has Property .

Corollary 30. Let , , the conditions (2), (3), , , (104) be fulfilled, and (81) for some holds. Then (1) has Property .

Corollary 31. Let , , the conditions (2), (3), , (104), and and for some and be fulfilled. Then (1) has Property , where and are given by (26).

Corollary 32. Let , , and the conditions (2), (3), , and (104) be fulfilled. Let moreover, there exist and such that and (88) hold. Then either condition (89) or condition is sufficient for (