On Asymptotical Behaviors of Solutions of Emden-Fowler Advanced Differential Equation

ON ASYMPTOTICAL BEHAVIORS OF SOLUTIONS OF EMDEN-FOWLER ADVANCED DIFFERENTIAL EQUATION

Roman Koplatadze

Department of Mathematics of TbilisiStateUniversity, Tbilisi,Georgia

Abstract. In the paper the following differential equation

(0.1)

is considered, where , , for and . New sufficient conditions for the equation (0.1) to have Property A or Property B are established.

1. Introduction

Consider the differential equation

(1.1)

where , , and for .

Oscillatory properties of the equation (1.1) in the case where and were first studied by Atkinson [1]. Kiguradze [2] studied the analogous problem for the higher order ordinary differential equation () in the case where is even and . For the ordinary differential equation Ličko and Švec [7] proved necessary and sufficient conditions for both even and odd and and . For the delayed equation () the analogous topics were quite thoroughly investigated in [4] while for the advanced equation (1.1) () attempts were made [3] for obtaining optimal conditions for oscillation of solutions. In the works [5] and [6] results essentially different from those presented in [3] were proved. Moreover, some results therefrom are substantial generalizations of some results of [3].

It will always be assumed that either the condition

(1.2)

or

(1.3)

is fulfilled.

Let . A function is said to be a proper solution of the equation (1.1) if it is locally absolutely continuous along with its derivatives up to the order inclusive, for and it satisfies (1.1) almost everywhere on .

A proper solution of the equation (1.1) is said to be oscillatory if it has a sequence of zeros tending to . Otherwise the solution is said to be nonoscillatory.

Definition 1.1. We say that the equation (1.1) has Property A if any of its proper solutions is oscillatory when is even and either is oscillatory or satisfies

(1.4)

when is odd.

Definition 1.2. We say that the equation (1.1) has Property B if any of its proper solutions either is oscillatory or satisfies (1.4) or

(1.5)

when is even, and either is oscillatory or satisfies (1.5) when is odd.

In the present paper sufficient conditions of new type will be given for the equation (1.1) to have Property A or B.

Let and . By we denote the set of all proper solutions of the equation (1.1) satisfying the condition

(1.6l)

2. Sufficient Conditions of Nonexistence of Monotone Solutions

Theorem 2.1. Let the condition (1.2) ((1.3)) be fulfilled, withodd (even) and

(2.1l)

If, moreover, for someand

(2.2l)

then for anywe have Ø, where

(2.3l)

Remark 2.1. (2.1l) is a necessary condition for the equation (1.1) to have a solution of the type (1.6l).

Corollary 2.1. Let the condition (1.2) ((1.3)) be fulfilled, withodd (even) and (2.1l) hold. If, moreover, for some

then for anywe have Ø, whereisdefined by (2.3l).

Corollary 2.2. Let the condition (1.2) ((1.3)) be fulfilled, with odd (even) and

then for anywe have Ø.

Remark 2.2. Corollary 2.1 is a generalization of Theorem 1.1 [3].

Theorem 2.2. Let the condition (1.2) ((1.3)) be fulfilled, withodd (even) and for someand

whereis defined by (2.3l). Then the condition (2.1l) is necessary and sufficient for Ø for any .

Corollary 2.3. Let the condition (1.2) ((1.3)) be fulfilled, withodd (even) and for some

Then the condition (2.1l) is necessary and sufficient for Ø for any .

3. Differential Equations with Properties A and B

Theorem 3.1. Let the condition (1.2) be fulfilled and for anywith odd the condition (2.1l) hold as well as the condition (2.2l) for someand, where is defined by (2.3l). If, moreover, for add

(3.1)

then the equation (1.1) has Property A.

Theorem 3.2. Let the condition (1.3) be fulfilled and for anywitheven the condition (2.1l) hold as well as the condition (2.2l) for someand , whereis defined by (2.3l). If moreover, for even (3.1) holds, then the equation (1.1) has Property B.

Theorem 3.3. Let (1.2) ((1.3)) be fulfilled, be even (be odd) and

(3.2)

If moreover, for someand

whereis defined by (2.31), then the equation (1.1) has Property A (B).

Corollary 3.1. Let (1.2) and (3.2) ((1.3) and (3.2)) be fulfilled, be even (be odd) andfor some

(3.3)

Then the equation (1.1) has Property A ( B).

Corollary 3.2. Let (1.2) be fulfilled and for some

(3.4)

If moreover, for some

then the equation (1.1) has PropertyA.

Corollary 3.3. Let (1.3) be fulfilled and for some (3.4) holds and

If moreover, for some

then the equation (1.1) has PropertyB.

Corollary 3.4. Let (1.2) ((1.3)) be fulfilled, be even (be odd) and for some

(3.5)

If, moreover,

then the equation (1.1) has PropertyA (B).

Theorem 3.4. Let (1.2), (3.1) and (3.2) ((1.3), (3.1) and (3.2)) be fulfilled, be odd (be even) and for someand

whereis defined by (2.32). Then the equation (1.1) has PropertyA (B).

Corollary 3.5. Let (1.2) and (3.2) ((1.3) and (3.2)) be fulfilled, be odd (be even) and for some

Then the equation (1.1) has Property A ( B).

Corollary 3.6. Let (1.2) and (3.1) ((1.3) and (3.1)) be fulfilled as well as (3.5) for some, , be odd (be even) and

Then the equation (1.1) has Property A ( B)..

Theorem 3.5. Let (1.2) and (3.1) be fulfilled and

(3.6)

If, moreover, for someand

whereis defined by (2.3n-1), then the equation (1.1) has Property A.

Theorem 3.6. Let (1.3), (3.1) and (3.6) be fulfilled and for someand

Then the equation (1.1) has Property B.

References

1. Atkinson F. V.: 'On second order nonlinear oscillations'. Pacific J. Math 1955 5 (1) 643-47.

2. Kiguradze I. T.: 'On oscillatory solutions of some ordinary differential equations'. Soviet Math. Dokl. 1962 144 33-36.

3. Kiguradze I. T. and Stavroulakis I. P.: 'On the oscillation of solutions of higher order Emden-Fowler advanced differential equations'. Appl. Anal. 1998 70 (1-2) 97-112.

4. Koplatadze R.: 'On oscillatory properties of solutions of functional differential equations'. Mem. Differential Equations Math. Phys. 1994 3 1-179.

5. Koplatadze R.: 'On Oscillatory properties of solutions of generalized Emden-Fowler type differential equations'. Proc. A. Razmadze Math. Inst. 2007 145 117-121.

6. Koplatadze R.: 'On asymptotic behaviors of solutions of almost linear and essential nonlinear functional differential equations'. Nonlinear analysis Theory, Methods and Applications (submit).

7. LičkoI. and Švec M.: 'Le caractère oscillatorie des solutions de 'equation ' Crech. Math. J. 1963 13 481-489.

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