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

and Applied Analysis 3 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 x(t) and its second derivative x(t) are opposite to each other for sufficiently large t. 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 t,σ(t) appears. The principle is clear: for oscillation of all solutions we have to achieve a corresponding smallness of the delay t −σ(t). All this is more complicated if we study nth order equations. In this case also the fact that x(t) and its nth derivative x(t) have different signs for sufficiently large t is used, but the technique is more complicated. In the papers [21–28] 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 λ and with functional power μ(t) such that limt→∞μ(t) = λ, 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 [21–28]. In this paper we study oscillation properties of nth order delay Emden-Fowler equations. 2. Some Auxiliary Lemmas In the sequel, Cloc([t0; +∞)) will denote the set of all functions u : 0 +∞) → R absolutely continuous on any finite subinternal of 0 +∞) along with their derivatives of order up to including n − 1. Lemma 3 (see [28]). Let u ∈ C loc 0 +∞)), u(t) > 0, u (n) (t) ≤ 0 for t ≥ t0, and u(t) ̸ ≡ 0 in any neighborhood of +∞. Then there exist 1 ≥ 0 and l ∈ {0, . . . , n − 1} such that l + n is odd and u (i) (t) > 0 for t ≥ 1 (i = 0, . . . , l − 1) , (−1) i+l u (i) (t) > 0 for t ≥ 1 (i = l, . . . , n − 1) . l Remark 4. If n is odd and l = 0, then it means that in 0 only the second inequalities are fulfilled. Lemma 5 (see [29]). Let u ∈ C loc 0 +∞)) and let l be fulfilled for some l ∈ {0, . . . , n − 1} with l + n odd. Then ∫ +∞ t0 t n−l−1 󵄨 󵄨 󵄨 󵄨 u (n) (t) 󵄨 󵄨 󵄨 󵄨 󵄨 dt < +∞. (21) If, moreover,

A proper solution  : [ 0 : +∞) →  of ( 1) is said to be oscillatory if it has a sequence of zeros tending to +∞.
Otherwise the solution  is said to be nonoscillatory.
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 The Emden-Fowler equations were first considered only for second-order equations and written in the form   ( ()   ) +  ()   = 0,  ≥ 0, 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   () +  () | ()|  sign  () = 0,  ≥ 0, 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  ≥ −2 > −( + 3)/2 all solutions oscillate, for  < −( + 3)/2-all solutions nonoscillate, and for −( + 3)/2 ≤  < −2 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 [6][7][8].In [9] for the case 0 <  < 1, 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 nth 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 [10][11][12][13][14][15][16][17][18][19][20].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 0 <  < 1, 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  = 1, . . ., , and () → ∞ as  → ∞, each of the following conditions (a), (b), and (c) ensures oscillation of all solutions: (a) 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 [21][22][23][24][25][26][27][28] 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 lim  → ∞ () = , 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 [21][22][23][24][25][26][27][28].In this paper we study oscillation properties of nth order delay Emden-Fowler equations.

Necessary Conditions for the Existence of a Solution of Type (20 ℓ )
The following notation will be used throughout the work: Clearly  (−1) () ≥ , and  (−1) is nondecreasing and coincides with the inverse of  when the latter exists.Definition 6.Let  0 ∈  + .By U ℓ, 0 one denotes the set of all proper solutions  : [ 0 , +∞) →  of (1) satisfying the condition (20 ℓ ) with some  1 ≥  0 .
Analogously we can prove.
Consider (1), where  is even and It is obvious that the function () =  − (1/) is the solution of (1) and it satisfies the condition (20 1 ) for  ≥ (2/).On the other hand, the condition (28 1 ) holds, but the condition ( 22) is not fulfilled.
Using Lemma 8 in a similar manner one can prove the following.
Using Theorem 11 analogously we can prove the following.
Using Theorem 13, in a manner similar to above we can prove the following.Then for any  0 ∈  + , U ℓ, 0 = 0, where  is given by (26).