Certain New Oscillation criteria for Fourth Order Non – Linear Difference Equations

Certain New Oscillation criteria for Fourth Order Non – Linear Difference Equations

S. Nalini(1) , Dr. S. Mehar Banu(2)

1. Lecturer, Department of Mathematics, Narasus Sarathy Institute of Technology, Salem 636 305 Tamilnadu

2. Asst.Professor,Department of Mathematics Govt.Arts college for women ,Salem 636008,

Tamilnadu.

Abstract

In this paper deals with the oscillatory behaviour of fourth order non linear difference equation of the form

(1.1)

Where is the forward

difference operator defined by .By a solution of (1.1) is consider as exist an N(a) for some . Examples are given to illustrate the importance of the results.

1. Introduction

   Consider the non linear difference equation

   (1.1)

   Where . is the forward

   difference operator defined by .By a solution of (1.1) is consider as exist an N(a) for some . The function f satisfies the following conditions.

   (H1): are real positive sequences, for infinitely many values of n.

   (H2): is continuous and

   for

   all

   (H3):

   (H4):

   Difference equations manifest themselves as mathematical models describing real life situations in probability theory, queuing problems, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, quanta in radiation, genetics in biology, economics, psychology, sociology, etc., unfortunately, these are only considered as the discrete analogs of differential equations. It is an indisputable fact that difference equations appeared much earlier than differential equations and were instrumental in paving the way for the development of the latter. It is only recently that difference equations have started receiving the attention they deserve. Perhaps this is largely due to the advent of computers, where differential equations are solved by using their approximate difference equation formulations. The theory of difference equations has grown at an accelerated pace in the past decade. It now occupies a central position in applicable analysis and will no doubt continue to play an important role in mathematics as a whole.

   In this study we compared to second and higher order difference equations, the study of third and fourth order difference equations has received less attention. Some fourth order difference equations can be found in

   [1] to [10 ]. However, it seems there is very very less known regarding the oscillation of equation (1.1).It is a extended version of [11]. Our aim of in this paper is to present some oscillation criteria for equation (1.1).

2. Main results: Theorem 2.1 :

   If is a oscillatory solution of equation (1.1) for

   .From lemma (2.1) it follows that

   ,

   such that .

   Proof :

   define,

   for .we

   Let be a non oscillatory solution of equation (1.1) without loss of generality we may assume that

   ,for .

   From (1.1) we have

   .We have to prove that

   .

   Assume that contrary for

   . Since is

   decreasing such that

   .Summing the last inequality from to n we

   (2.2)

   =

   get

   (2.3)

   Letting then

   there is an integer such that Summing the last inequality from we get

   Since from the inequality we have From equation (2.3)

   Letting then thus there is

   an integer such that for

   which implies that

   . Summing this inequality from

   we get .

   Let then , this is contradiction.

   Therefore .

   Theorem 2.2 :

   Assume that the difference equation (1.1) holds the condition (H1) to (H4) and there exist a positive sequence such that

   Consider,

   Therefore

   For

   (2.4)

   Then every solution of equation (1.1) is oscillatory.

   Proof :

   Let be a non oscillatory solution of equation (1.1) without loss of generality, we may assume that

   (2.5)

   If

   Then we have

   where

   Summing this inequality from to . We get

   (2.7)

   Then every solution of equation(1.1) is oscillatory

   Proof :

   Let be a non oscillatory solution of equation (1.1),which may assume to be eventually positive.

   From equation (2.2) &(2.3) we define the sequence as follows

   (2.6)

   This is contradiction. Hence it is complete the proof.

   Corollary : Assume that the difference equation (1.1) holds the condition of theorem 2.2 except the condition

   2.1 is replaced by

   Then every solution of(1.1) is Oscillatory.

   Here we apply the double sequence Sn,m in the difference equation (1.1)

   Definition:

   Let Sn,m be a double sequence of real numbers.

   if for every ,there exist such that if n,m then

   if for every ,there exist such that if n,m then

   Theorem 2.3 :

   .

   Assume that (H1) to (H4) holds and let {bn} be a positive sequence and assume that there exist a double sequence such that

   Sn,m >0 for n>m>0

   =0

   This yields after summing by parts

   = 0 for

   Where

   >

   >

   Which clearly contradicts (2.7).

   Remarks :

   By choosing the sequence in appropriate manners, we can derive several oscillation criteria for(1.1).

   Let us consider the double sequence defined by

   Corollary: Assume that all the assumptions of Theorem 2.3 hold except that condition (2.7) is replaced by

   Then every solution of equation (1.1) is oscillatory.

   Example : 1

   Consider the difference equation

   Satisfies all the condition of Theorem 2.2 &2.3 for Sn,m=n+m & . Hence all the solution of equation (1.1) are oscillatory.

3. References

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