Unsaturated Solutions of A Nonlinear Delay Partial Difference. Equation with Variable Coefficients

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1 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN Usaturated Solutios of A Noliear Delay Partial Differece Euatio with Variable Coefficiets Xiagyu Zhu Yuahog Tao* Departmet of Mathematics College of Sciece Yabia Uiversity Yaji Jili 1 Abstract: By employig the cocept ad the properties of freuecy measures of ifiite double seuece ad the cocept of usaturated solutio of partial differece euatios the usaturated solutios for the oliear partial differece euatio with variable coefficiets is discussed. Oly usig the cocept of "freuecy measure" of the level sets of the ivolved parameter seueces i euatio the sufficiet coditios of the solutios to be usaturated are preseted thus how freuet the solutios oscillate is well described. Keywords: partial differece euatio; freuecy measure; usaturated solutio 1. Itroductio I recet years may results have bee foud about the oscillatory ad No-oscillatory solutios of differece euatios. However the classical cocept of oscillatio still ca ot describe the oscillatio of seueces. Therefore Chuaju Tia [1] first itroduced the cocept of freuecy measure of seuece ad described freuet oscillatio of seuece. I order to further improve the freuet oscillatio of seuece Zhiiag Zhu et al. [] also defied the cocept of freuetly positive oscillatio ad freuetly egative oscillatio of seuece. At preset there are some results about the freuet oscillatio of solutios of differece euatios see [-9]. Let Z be the set of itegers Z[k l]={i Z i=k k+1 L l} ad Z[k )={i Z i=kk+1 L }. I this paper we discuss the usaturated solutios of the followig oliear partial differece euatio with coefficiets of variable sig u u u p u sg u u sg u (1.1) m m1 m 1 m mk l mk l m mk l mk l where m Z[ ) [1) (1 ); k1 k l1 ad l are all oegative itegers k1 k l1 l. Moreover ad p p m Z [ ) are real double seueces satisfyig the coditio (*) : if p p m Z [ ) has egative item the *This work is supported by Natural Sciece Foudatio of Chia uder umber Correspodece Author: Yua-hog Tao taoyuahog1@16.com Progressive Academic Publishig UK Page 58

2 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN the deomiator of 1 is a positive odd umber; similarly if the the deomiator of 1 is also a positive odd umber. I the seuel p p ad ( ) [ 1 ) [ 1 ) the double seuece euatio (1.1) if it keeps the euatio (1.1) holds. has egative item may have egative items. Geerally for u ( ) [ 1 ) [ 1 ) is called the solutio of. Prelimiary Let Z Z Z we call a elemet of Z to be a lattice. Deote the uio itersectio ad differece of two sets A ad B to be A B AB ad A\ B respectively. If Z the we deote the potetial of to be ; ad deotig ( ) ( s t) s m t. For ay Z ad itegers set Y s t Z s t m X ( ) ( s m t) Z ( s t) ; ( ) ( ) ( ). ad s t X Y s t X Y where ad are all itegers satisfyig the ( s t) Z \ X Y ( s k t l) Z \ k l. (.1) Defiitio.1 [1] Let Z if the upper limit lim sup ( ) exists the we call the limit to be the upper freuet measure of deotig ( ). Similarly if the lower limit lim if ( ) exists the we call the limit to be the lower freuet measure of deotig ( ) *. If ( ) ( ) the the correspodig limit is called the freuet measure of deotig ( ). Defiitio. [1] Let u u be a arbitrary real double seuece if 1 ( u ) the we say that u is freuetly positive if ( u ) the we say that u Progressive Academic Publishig UK Page 59

3 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN is freuetly egative. If u is either freuetly positive or freuetly egative the we say that u is freuetly oscillatory. Lemma.1 [1] ( ) ( ) 1. If A is a arbitrary subset of the ( A) ( A) 1. Particularly if B is a fiite subset of the ( B). Lemma. [1] Let A ad B be subsets of the ( A B) ( A) ( B). If AI B the ( A) ( B) ( A B) ( A) ( B) ( A B) ( A) ( B). Hece ( A) ( \ A) 1. Lemma. [1] Let A ad B be subsets of the ( A) ( B) ( A\ B) ( A) ( B); ( A) ( B) ( A\ B) ( A) ( B). From Lemma. we ca easily get that Lemma.4 [1] Let A ad B be subsets of the ( A) ( B) ( A B) ( A B) ( A) ( B) ( A B); ( A) ( B) ( A B) ( A B) ( A) ( B) ( A B). Lemma.5 [] For ay subset A of we have ( X Y A) ( 1)( 1) ( A); ( X Y A) ( 1)( 1) ( A). where ad are all itegers satisfyig. Lemma.6 [1] Let A1 A L A be subsets of the Ai Ai Ai i1 i1 i1 ( ) ( ) ( 1) ( ) Ai A1 Ai Ai i1 i i1 ( ) ( ) ( ) ( 1) ( ). Lemma.7 [1] Let A ad B be subsets of if ( A) ( B) 1 the AB is a ifiite set. Progressive Academic Publishig UK Page 6

4 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN Usaturated Solutios Let u be a solutio of euatio (1.1) if there exists M Z such that u for ay m M the we call u to be a evetually positive solutio of euatio (1.1). Similarly we ca defie evetually egative solutio evetually o-positive solutio evetually o-egative solutio of euatio (1.1). If the solutio of euatio (1.1) is either evetually positive or evetually egative the we call that the solutio of euatio (1.1) is oscillatory. Obviously the solutio of euatio (1.1) is oscillatory if ad oly if for ay subset ( m ) N : m 1 of there exists ( m1 1) ( m ) such that Defiitio.1 [4] Assume that u u ( ) u m1 u 1 m. is a arbitrary real double seuece if ( u ) (1) the we call that u has usaturated upper positive part. If ( u ) (1) the we call that u has usaturated lower positive part. If ( u ) ( u ) (1) the we call that u has usaturated positive part. Similarly we ca defie that u has usaturated egative part. Obviously if u u evetually positive or evetually egative the ( u ) 1 or ( u ). So if the seuece u u has usaturated upper positive part the it must ever be evetually positive or evetually o-positive that is to say u is oscillatory. We the discuss whether the solutio of euatio (1.1) has usaturated upper positive part. For ay double seuece i j ( i j) vi j c as v c. Similarly we ca defie ( v c) ( v c) v defiig o defie a level set ad v c. For coveiece we assume Z[ 1 ) Z[ 1 ). For ay real double seuece is u ( ) defie the followig two partial differeces: V 1 um um 1 um ; V um um 1 u m. Lemma.1 [6] Let xy ad p if 1 the p p x y x y. p Lemma. Assume that there exists m 1 ad l1 such that p where ( m ) Z[ m 1 m k1] Z[ l1 1]. If u is ay solutio of Progressive Academic Publishig UK Page 61

5 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN euatio (1.1) the V Vu 1um 1 for u where ( m ) Z[ m m k1] Z[ l1 ]; ad 1um 1 V Vu for u where ( m ) Z[ m m k1] Z[ l1 ]. Proof. If u ( m ) Z[ m 1 m k1] Z[ l1 1] the from euatio (1.1) we have So u u u p u u u u. m m1 m 1 m mk1 l 1 m mk l m1 m um 1 ( m ) Z[ m m k ] Z[ l ]. where 1 1 V Vu Similarly if u ( m ) Z[ m 1 m k1] Z[ l1 1] we have V Vu 1um 1 ( m ) Z[ m m k ] Z[ l ]. where 1 1 Assume that mi sice [1) (1 ) the 1. Set p p m m m Z [ 1 ) the uder the assumptio () coditio () ca be deleted. p makes sese. If p the the Theorem.1 Assume that there exists 1 ad (1) such that ( p ) ( ) [( p ) ( )] 1 ( p 1) 4( k)( l)( 1 ). the ay solutio of euatio (1.1) has usaturated upper positive part. Progressive Academic Publishig UK Page 6

6 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN Proof. Let u u be ay solutio of euatio (1.1) the ( u ) ( 1). Otherwise ( u ) or ( u ) 1. If ( u ) the from Lemma. to Lemma.6 we have 1 l1 1 1 l X ky1 p u X ky1 p u 1 \ [( ) ( ) ( )] [( ) ( ) ( )] \ X Y [( p ) ( ) ( u )] 1 l1 k1 1 1 l1 k1 1 4( k 1)( l 1) [( p ) ( )] ( u ) \ X Y [( p ) ( ) ( u )] 4( k 1)( l 1) ( p ) ( ) ( u ) [( p ) ( )] \ X Y [( p ) ( ) ( u )] 4( k 1)( l 1)( ) 1 l1 k1 1 l1 X k Y p u p \ [( ) ( ) ( )] ( 1). the from Lemma.7 we kow that the itersectio 1 1 k1 1 { \ X Y [( p ) ( ) ( u )]} ( p 1) is a ifiite subset of. So accordig to Lemma. there exists m 1 l1 such that p m m 1 (.) p u ( m ) Z[ m 1 m k1] Z[ l1 1]. (.) From (.) ad Lemma. agai we have V u 1um 1 V m Z m m k1 Z l1 ( ) [ ] [ ]. Therefore u u u m k1 l 1 m l m. It follows from euatio (1.1) ad Lemma. that Progressive Academic Publishig UK Page 6

7 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN u u u p u sg u m 1 m 1 m m m k l m k l u sg u m m k l m k l u u u p u u m 1 m 1 m m m k l m m k l 1 m 1 m 1 m m u u u p 1 m um k l m 1 m 1 m m m m u u u p u m m m m u p u m m m [ 1 p ] u. Sice u the m p m m 1 which cotradicts (.). Next assume ( u ) 1 accordig to Lemma. we have ( u ). From Lemma. to Lemma.6 we have 1 l1 1 1 l X ky1 p u X ky1 p u 1 \ [( ) ( ) ( )] [( ) ( ) ( )] \ X Y [( p ) ( ) ( u )] 1 l1 k1 1 4( k 1)( l 1) [( p ) ( ) ( u )] \ X Y [( p ) ( ) ( u )] 1 l1 k1 1 4( k 1)( l 1) ( p ) ( ) ( u ) [( p ) ( )] \ X Y [( p ) ( ) ( u )] 4( k 1)( l 1)( ) 1 l1 k1 1 l1 X k Y p u p \ [( ) ( ) ( )] ( 1). So accordig to Lemma.7 we kow that the itersectio 1 1 k1 1 { \ X Y [( p ) ( ) ( u )]} ( p 1) is a ifiite subset of. Similar to the above discussio ( u ) 1does ot hold so the coclusio is correct. Progressive Academic Publishig UK Page 64

8 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN Theorem. Assume that there exist costats 1 ad (1) such that ( p ) 1 ( ) ( p 1) 1 1 [( p ) ( ) ( p 1)] 8( k11)( l11) the ay solutio of euatio (1.1) has usaturated upper positive part. Proof. To prove ( u ) ( 1). Similar to the proof of Theorem.1 we oly eed to prove ( u ) ad ( u ) 1. Firstly assume ( u ) the form Lemma. to Lemma.6 we have 1 l1 k1 1 { \ X Y [( p ) ( ) ( p 1) ( u )]} 1 l1 k1 1 1 { X Y [( p ) ( ) ( p 1) ( u )]} 1 4( k 1)( l 1){ [( p ) ( ) ( p 1)] ( u )} 1 4( k 1)( l 1){ ( p ) ( ) ( p 1) ( u ) [( p ) ( ) ( p 1)]} 1 1 From Lemma.7 we kow that \ l X Y [( p ) ( ) ( p 1) ( u )] k1 1 is a ifiite set. So accordig to Lemma. there exist m 1 l1 such that p m m 1 p u ( m ) Z[ m 1 m k1] Z[ l1 1]. Similar to Theorem.1 we get the cotradictio. So ( u ) does ot hold the ( u ) 1 i.e. ( u ) ( 1). Hece the theorem is proved. Theorem. Assume that there exist costats 1 4 ad (1) such that ( p ) 1 ( ) ( p 1) [( p ) ( ) ( p 1)] 4 Progressive Academic Publishig UK Page 65

9 Europea Joural of Mathematics ad Computer Sciece Vol. 5 No ISSN Satisfyig 4( k)( l)( 1 4) 1 the ay solutio of euatio (1.1) has usaturated upper positive part. Proof. Similar to the proof of Theorem Example. I euatio (1.1) set p 1 k1 k l1 l 1 mi{ } the the euatio should be u u u u sg u u sg u. m m1 m 1 m m m 1 m 1 Obviously ( p 1) 1 ( p ) ( ) (( p ) ( )) ( p 1) 1. Moreover (( p ) ( ) ( p 1)). The from Theorems. ad. ay solutio of euatio (1.1) has usaturated upper positive part. REFERENCES [1] Chuaju TIAN Shegli XIE Suisheg CHENG. Measures for Oscillatory Seueces[J]. Comput. Math. Applic. 6 (1998): [] Zhiiag ZHU Suisheg CHENG. Freuetly Oscillatory Solutios of Neutral Differece Euatios[J]. Southeast Asia Bulleti of Mathematics. 9 (5): [] Suisheg CHENG. Partial Differece Euatios[M] Taylor ad Fracis Lodo ad New York : [4] Chuaju TIAN Suisheg CHENG Shegli XIE. Freuet Oscillatio Criteria for a Delay Differece Euatio[J]. Fukcialaj Ekvacioj. 46 (): [5] Zhiiag ZHU Suisheg CHENG. Freuetly oscillatory solutios for multi-level partial differece euatios[j]. Iterat. Math. Forum. 1 (6): [6] Ju YANG Yujig ZHANG. Freuet oscillatory solutios of a oliear partial differece euatio[j]. J. Comput. Appl. Math. 4(9): [7] Dogmei LI Yuahog TAO. Usaturated solutios of a class of oliear delay deviatio partial differece euatios[j]. J.Yabia Uiversity(Natural Sciece Editio). 1 6(): [8] Yuahog TAO Xiudog LI. Freuetly Oscillatory Solutios for Noliear Delay Partial Differece Euatios[J]. J. Natural Sciece of Hei Logjiag Uiversity. 1 7(5) : [9] Yuahog TAO Dogmei WU. Freuecy oscillatio of a class of eutral differece euatios[j]. J. Yabia Uiversity(Natural Sciece Editio). 11 7(1): Progressive Academic Publishig UK Page 66