- Email: [email protected]
Oscillation theorems for second-order nonlinear partial difference equations
Journal of Computational and Applied Mathematics 132 (2001) 479–482 www.elsevier.nl/locate/cam
Letter to the Editor
Oscillation theorems for second-order nonlinear partial di(erence equations a
Shu Tang Liua; b;∗ , Yong Qing Liua
Department of Automatic Control Engineering, South China University of Technology, Guangzhou, 510641, People’s Republic of China b Department of Mathematics, Binzhou Normal College, Shandong, Binzhou, 256604, People’s Republic of China Received 30 October 2000; received in revised form 15 January 2001
Abstract The oscillation of the second-order nonlinear partial di(erence equation s T (1 ; 2 )[cmn 1 (ymn )] + ai (m; n)fi (ym+1; n ; 1 (ymn )) = 0 i=1
c 2001 is investigated. Some su9cient conditions for the oscillation of all solutions of the above equation are obtained. Elsevier Science B.V. All rights reserved.
1. Introduction Partial di(erence equations are di(erence equations that involve functions of two or more independent integer variables. Such equations arise from considerations of random walk problems, the study of molecular orbits [9], mathematical physics problems [1] and the numerical di(erence approximation problems [2] and from [9], we also have obtained that the asymptotic behavior of molecular orbits has a close a9nity to oscillatory behavior of solution of partial di(erence equation. In this paper, we consider an oscillatory behavior of solutions of the second-order nonlinear partial
The research was supported partially by a NNSF of China (no. 69934034) and a foundation (no. Y98A02005) from Shandong Province of China. ∗ Correspondence address: Department of Automatic Control Engineering South China University of Technology, Guangzhou 510641, People’s Republic of China. E-mail address: [email protected] (S. Tang Liu). c 2001 Elsevier Science B.V. All rights reserved. 0377-0427/01/$ - see front matter PII: S 0 3 7 7 - 0 4 2 7 ( 0 1 ) 0 0 3 7 3 - 9
480
S.T. Liu / Journal of Computational and Applied Mathematics 132 (2001) 479–482
di(erence equation T (1 ; 2 )[cmn 1 (ymn )] +
s
ai (m; n)fi (ym+1; n ; 1 (ymn )) = 0;
(1)
i=1
where T (1 ; 2 ) = 1 + 2 + I; 1 ymn = ym+1; n − ymn ; 2 ymn = ym; n+1 − ymn and I (ymn ) = ymn . Let N0 ={0; 1; 2; :::; }, {ai (m; n)}(m; n)∈N02 are real double sequences, i=1; 2; :::; s; s is a positive integer, the double sequence {cmn }(m; n)∈N02 is assumed to be positive. The following conditions will be assumed: (c1 ) ai (m; n) ¿ 0 for m ¿ m0 ¿ 0; n ¿ n0 ¿ 0; i = 1; 2; : : : ; s; (c2 ) fi : R2 → R; ufi (u; v) ¿ 0 for all u = 0; i = 1; 2; : : : ; s: By a solution of (1) we mean a real double sequence {ymn } satisfying Eq. (1) for m; n ∈ N0 . We consider only such solutions which are nontrivial for all large m; n: A solution {ymn } of (1) is called nonoscillatory if it is eventually positive or eventually negative. Otherwise it is called oscillatory. The problem of determining nonoscillation criteria for nonlinear partial di(erence equation has been the subject of investigation in [3,5,6]. Among the papers dealing with this subject we refer to [4,7,8,10 –12] in wich oscillation criteria for linear di(erence equations have been established. However, for second-order nonlinear partial di(erence equation, there does not seem to be any oscillation criteria up to now. The purpose of this note is to give some criteria (su9cient conditions) for oscillation of all solutions of (1). 2. Main results An elementary identity for double sequences will be used later. Lemma (Zhang and Liu, 1997). m n
(Ai+1; j + Ai; j+1 − Aij )
(2)
i=m−k j=n−l
=
m+1
n
Aij +
i=m+1−k j=n+1−l
m
Ai; n+1 − Am−k; n−l + Am+1; n−l :
i=m−k
Theorem. Suppose there exists an index j such that is continuous on R2 , (i) f j (u; v) ∞ ∞ (ii) i=m3 j=n3 ak (i; j) = ∞: Then every bounded solution of (1) is oscillatory. Proof. Suppose there exists a bounded nonoscillatory {ymn } of (1). Assume that ymn ¿ 0 for m ¿ m1 ¿ 0; n ¿ n1 ¿ 0: Then, by conditions (c1 ) and (c2 ); we see that T (1 ; 2 )[cmn 1 (ymn )] ¡ 0; i.e., {cmn 1 (ymn )} is nonincreasing sequence for m ¿ m1 ; n ¿ n1 : Now, we can derive that 1 (ymn ) ¿ 0
for m ¿ m1 ; n ¿ n1 :
(3)
S.T. Liu / Journal of Computational and Applied Mathematics 132 (2001) 479–482
481
In fact, if there would exist m2 ¿ m1 and n2 ¿ n1 such that 1 (ym2 n2 ) = c ¡ 0; then 1 (ymn ) 6 c for m ¿ m2 ; n ¿ n2 and hence n m− 1 n m− 1 n (ymj − ym2 j ) = 1 (yij ) 6 c = c(m − m2 )(n − n2 ): i=m2 j=n2 +1
j=n2 +1
i=m2 j=n2 +1
Note that the boundedness character of {ymn } yields n n ymj 6 ym2 j + c(m − m2 )(n − n2 ) → −∞ as m; n → ∞; ymn 6 j=n2 +1
j=n2 +1
which contradicts the fact that ymn ¿ 0 for m ¿ m1 ; n ¿ n1 : Thus (3) holds, that is ym+1; n ¿ ymn for m ¿ m1 ;
n ¿ n1 :
(4)
Using a similar argument as before, we also have 2 (ymn ) ¿ 0 or ym; n+1 ¿ ymn
for m ¿ m1 ; n ¿ n1 :
(5)
From (4) and (5), that is, sequence {ymn } is monotone increasing in m and n for m ¿ m1 ; n ¿ n1 : Moreover, since {ymn } is bounded, we must have 1 (ymn ) → 0, ymn → , (0 ¡ ¡ ∞) as m; n → ∞: Consider the sequence {pmn = cmn 1 (ymn )} for m ¿ m1 ; n ¿ n1 : Using (1), (c1 ) and (c2 ) we have T (1 ; 2 )(pmn ) =cm+1n 1 (ym+1; n ) + cm; n+1 1 (ym; n+1 ) − cmn 1 (ymn ) =T (1 ; 2 )[cmn 1 (ymn )] s ai (m; n)fi (ymn ; 1 (ymn )) =− i=1
6 − aj (m; n)fj (ym+1; n ; 1 (ymn )):
(6)
By the continuity of fj , fj (ym+1; n ; 1 (ymn )) → fj (; 0) ¿ 0 as m; n → ∞ and hence there exists m3 ¿ m1 ; n3 ¿ n1 such that fj (ym+1; n ; 1 (ymn )) ¿ 12 fj (; 0) for m ¿ m3 ; n ¿ n3 . Therefore from (6) we deduce T (1 ; 2 )(pmn ) 6 − 12 ak (m; n)fk (; 0)
(7)
and summing up both sides of (7) from m3 to m − 1 and n3 to n − 1; we Ond m− 1 n− 1 m− 1 n− 1 1 1 ak (i; j): T (1 ; 2 )(pij ) 6 − fk (; 0) 2 2 i=m j=n i=m j=n 3
3
3
(8)
3
Moreover, applying Lemma, we Ond pmn − pm3 n3
6 pmn +
m− 1
n
pij +
i=m3 +1 j=n3 +1
=
m− 1 n− 1 i=m3 j=n3
T (1 ; 2 )(pij ):
n− 1
pmj + · · · + − pm3 n3
j=n3 +1
(9)
482
S.T. Liu / Journal of Computational and Applied Mathematics 132 (2001) 479–482
Combining (8) and (9), we obtain m− 1 n− 1 1 ak (i; j): pmn 6 pm3 n3 − fk (; 0) 2 i=m j=n 3
(10)
3
Using (ii), leads to a contradiction for it implies lim [cmn 1 (ymn )] = −∞
m; n→∞
which means that {1 (ymn )} is eventually negative. A similar argument can be used to show bounded and eventually negative solution does not exist. The proof is complete. Example. Consider the partial di(erence equation
e2mn m + n 2 [ 1 (ymn )]2 1 (ymn ) + = 0; T (1 ; 2 ) 3 + 2 m+n 5 1 + ym+1; n
(11)
where f1 (u; v) = (v2 =1 + u2 ), cmn = 3 + (e2mn =m + n) and a1 (m; n) = (m + n=5)2 : It is easy to see that all assumptions of Therome hold. Therefore every solution of (13) oscillates. In fact, {ymn }={(−1)n =en } is such a solution. Remark. We can obtain many su9cient conditions for the oscillation of Eq. (1). Due to limited space, their statements are omitted here. References [1] R. Courant, K. Friedrichs, H. Lewy, On partial di(erence equations of mathematical physics, IBM J. 11 (1967) 215–234. [2] W.G. Kelley, A.C. Peterson, Di(erence Equations; An Introduction with Applications, Academic Press, New York, 1991. [3] S.T. Liu, S.S. Cheng, Nonexistence of positive solution of a nonlinear partial di(erence equation, Tamkang, J. Math. 28 (1) (1997) 51–58. [4] S.T. Liu, S.S. Cheng, Existence of positive solution for partial di(erence equations, Far East, J. Math. Sci. 5 (3) (1997) 387–392. [5] S.T. Liu, X.P. Guan et al., Nonexistence of positive solution of a class of nonlinear delay partial di(erence equation, J. Math. Anal. Appl. 234 (1999) 361–367. [6] S.T. Liu, Y.Q. Liu et al., Existence of monotone positive solution for partial di(erence equation, J. Math. Anal. Appl. 247 (2000) 384–396. [7] S.T. Liu, H. Wang, Necessary and su9cient conditions for oscillations of a class of delay partial di(erence equations, Dynamic Systems Appl. 7 (1998) 495–500. [8] S.T. Liu, B.G. Zhang, Oscillation of a class of partial di(erence equations, PanAmer. Math. J. 8 (1998) 93–100. [9] X.-P. Li, Partial di(erence equations used in the study of molecular orbits, Acta Chim. Sinica 40 (8) (1982) 688-698, in Chinese. [10] B.G. Zhang, S.T. Liu, On the oscillation of two partial di(erence equations, J. Math. Anal. Appl. 206 (1997) 480–492. [11] B.G. Zhang, S.T. Liu, Oscillation of partial di(erence equations with variable coe9cients, Comput. Math. Appl. 36 –38 (1998) 235–242. [12] B.G. Zhang, S.T. Liu, S.S. Cheng, Oscillation of class of delay partial di(erence equations and its applications, J. Di(erential Equations Appl. 1 (1995) 215–226.
Letter to the Editor
Oscillation theorems for second-order nonlinear partial di(erence equations a
Shu Tang Liua; b;∗ , Yong Qing Liua
Department of Automatic Control Engineering, South China University of Technology, Guangzhou, 510641, People’s Republic of China b Department of Mathematics, Binzhou Normal College, Shandong, Binzhou, 256604, People’s Republic of China Received 30 October 2000; received in revised form 15 January 2001
Abstract The oscillation of the second-order nonlinear partial di(erence equation s T (1 ; 2 )[cmn 1 (ymn )] + ai (m; n)fi (ym+1; n ; 1 (ymn )) = 0 i=1
c 2001 is investigated. Some su9cient conditions for the oscillation of all solutions of the above equation are obtained. Elsevier Science B.V. All rights reserved.
1. Introduction Partial di(erence equations are di(erence equations that involve functions of two or more independent integer variables. Such equations arise from considerations of random walk problems, the study of molecular orbits [9], mathematical physics problems [1] and the numerical di(erence approximation problems [2] and from [9], we also have obtained that the asymptotic behavior of molecular orbits has a close a9nity to oscillatory behavior of solution of partial di(erence equation. In this paper, we consider an oscillatory behavior of solutions of the second-order nonlinear partial
The research was supported partially by a NNSF of China (no. 69934034) and a foundation (no. Y98A02005) from Shandong Province of China. ∗ Correspondence address: Department of Automatic Control Engineering South China University of Technology, Guangzhou 510641, People’s Republic of China. E-mail address: [email protected] (S. Tang Liu). c 2001 Elsevier Science B.V. All rights reserved. 0377-0427/01/$ - see front matter PII: S 0 3 7 7 - 0 4 2 7 ( 0 1 ) 0 0 3 7 3 - 9
480
S.T. Liu / Journal of Computational and Applied Mathematics 132 (2001) 479–482
di(erence equation T (1 ; 2 )[cmn 1 (ymn )] +
s
ai (m; n)fi (ym+1; n ; 1 (ymn )) = 0;
(1)
i=1
where T (1 ; 2 ) = 1 + 2 + I; 1 ymn = ym+1; n − ymn ; 2 ymn = ym; n+1 − ymn and I (ymn ) = ymn . Let N0 ={0; 1; 2; :::; }, {ai (m; n)}(m; n)∈N02 are real double sequences, i=1; 2; :::; s; s is a positive integer, the double sequence {cmn }(m; n)∈N02 is assumed to be positive. The following conditions will be assumed: (c1 ) ai (m; n) ¿ 0 for m ¿ m0 ¿ 0; n ¿ n0 ¿ 0; i = 1; 2; : : : ; s; (c2 ) fi : R2 → R; ufi (u; v) ¿ 0 for all u = 0; i = 1; 2; : : : ; s: By a solution of (1) we mean a real double sequence {ymn } satisfying Eq. (1) for m; n ∈ N0 . We consider only such solutions which are nontrivial for all large m; n: A solution {ymn } of (1) is called nonoscillatory if it is eventually positive or eventually negative. Otherwise it is called oscillatory. The problem of determining nonoscillation criteria for nonlinear partial di(erence equation has been the subject of investigation in [3,5,6]. Among the papers dealing with this subject we refer to [4,7,8,10 –12] in wich oscillation criteria for linear di(erence equations have been established. However, for second-order nonlinear partial di(erence equation, there does not seem to be any oscillation criteria up to now. The purpose of this note is to give some criteria (su9cient conditions) for oscillation of all solutions of (1). 2. Main results An elementary identity for double sequences will be used later. Lemma (Zhang and Liu, 1997). m n
(Ai+1; j + Ai; j+1 − Aij )
(2)
i=m−k j=n−l
=
m+1
n
Aij +
i=m+1−k j=n+1−l
m
Ai; n+1 − Am−k; n−l + Am+1; n−l :
i=m−k
Theorem. Suppose there exists an index j such that is continuous on R2 , (i) f j (u; v) ∞ ∞ (ii) i=m3 j=n3 ak (i; j) = ∞: Then every bounded solution of (1) is oscillatory. Proof. Suppose there exists a bounded nonoscillatory {ymn } of (1). Assume that ymn ¿ 0 for m ¿ m1 ¿ 0; n ¿ n1 ¿ 0: Then, by conditions (c1 ) and (c2 ); we see that T (1 ; 2 )[cmn 1 (ymn )] ¡ 0; i.e., {cmn 1 (ymn )} is nonincreasing sequence for m ¿ m1 ; n ¿ n1 : Now, we can derive that 1 (ymn ) ¿ 0
for m ¿ m1 ; n ¿ n1 :
(3)
S.T. Liu / Journal of Computational and Applied Mathematics 132 (2001) 479–482
481
In fact, if there would exist m2 ¿ m1 and n2 ¿ n1 such that 1 (ym2 n2 ) = c ¡ 0; then 1 (ymn ) 6 c for m ¿ m2 ; n ¿ n2 and hence n m− 1 n m− 1 n (ymj − ym2 j ) = 1 (yij ) 6 c = c(m − m2 )(n − n2 ): i=m2 j=n2 +1
j=n2 +1
i=m2 j=n2 +1
Note that the boundedness character of {ymn } yields n n ymj 6 ym2 j + c(m − m2 )(n − n2 ) → −∞ as m; n → ∞; ymn 6 j=n2 +1
j=n2 +1
which contradicts the fact that ymn ¿ 0 for m ¿ m1 ; n ¿ n1 : Thus (3) holds, that is ym+1; n ¿ ymn for m ¿ m1 ;
n ¿ n1 :
(4)
Using a similar argument as before, we also have 2 (ymn ) ¿ 0 or ym; n+1 ¿ ymn
for m ¿ m1 ; n ¿ n1 :
(5)
From (4) and (5), that is, sequence {ymn } is monotone increasing in m and n for m ¿ m1 ; n ¿ n1 : Moreover, since {ymn } is bounded, we must have 1 (ymn ) → 0, ymn → , (0 ¡ ¡ ∞) as m; n → ∞: Consider the sequence {pmn = cmn 1 (ymn )} for m ¿ m1 ; n ¿ n1 : Using (1), (c1 ) and (c2 ) we have T (1 ; 2 )(pmn ) =cm+1n 1 (ym+1; n ) + cm; n+1 1 (ym; n+1 ) − cmn 1 (ymn ) =T (1 ; 2 )[cmn 1 (ymn )] s ai (m; n)fi (ymn ; 1 (ymn )) =− i=1
6 − aj (m; n)fj (ym+1; n ; 1 (ymn )):
(6)
By the continuity of fj , fj (ym+1; n ; 1 (ymn )) → fj (; 0) ¿ 0 as m; n → ∞ and hence there exists m3 ¿ m1 ; n3 ¿ n1 such that fj (ym+1; n ; 1 (ymn )) ¿ 12 fj (; 0) for m ¿ m3 ; n ¿ n3 . Therefore from (6) we deduce T (1 ; 2 )(pmn ) 6 − 12 ak (m; n)fk (; 0)
(7)
and summing up both sides of (7) from m3 to m − 1 and n3 to n − 1; we Ond m− 1 n− 1 m− 1 n− 1 1 1 ak (i; j): T (1 ; 2 )(pij ) 6 − fk (; 0) 2 2 i=m j=n i=m j=n 3
3
3
(8)
3
Moreover, applying Lemma, we Ond pmn − pm3 n3
6 pmn +
m− 1
n
pij +
i=m3 +1 j=n3 +1
=
m− 1 n− 1 i=m3 j=n3
T (1 ; 2 )(pij ):
n− 1
pmj + · · · + − pm3 n3
j=n3 +1
(9)
482
S.T. Liu / Journal of Computational and Applied Mathematics 132 (2001) 479–482
Combining (8) and (9), we obtain m− 1 n− 1 1 ak (i; j): pmn 6 pm3 n3 − fk (; 0) 2 i=m j=n 3
(10)
3
Using (ii), leads to a contradiction for it implies lim [cmn 1 (ymn )] = −∞
m; n→∞
which means that {1 (ymn )} is eventually negative. A similar argument can be used to show bounded and eventually negative solution does not exist. The proof is complete. Example. Consider the partial di(erence equation
e2mn m + n 2 [ 1 (ymn )]2 1 (ymn ) + = 0; T (1 ; 2 ) 3 + 2 m+n 5 1 + ym+1; n
(11)
where f1 (u; v) = (v2 =1 + u2 ), cmn = 3 + (e2mn =m + n) and a1 (m; n) = (m + n=5)2 : It is easy to see that all assumptions of Therome hold. Therefore every solution of (13) oscillates. In fact, {ymn }={(−1)n =en } is such a solution. Remark. We can obtain many su9cient conditions for the oscillation of Eq. (1). Due to limited space, their statements are omitted here. References [1] R. Courant, K. Friedrichs, H. Lewy, On partial di(erence equations of mathematical physics, IBM J. 11 (1967) 215–234. [2] W.G. Kelley, A.C. Peterson, Di(erence Equations; An Introduction with Applications, Academic Press, New York, 1991. [3] S.T. Liu, S.S. Cheng, Nonexistence of positive solution of a nonlinear partial di(erence equation, Tamkang, J. Math. 28 (1) (1997) 51–58. [4] S.T. Liu, S.S. Cheng, Existence of positive solution for partial di(erence equations, Far East, J. Math. Sci. 5 (3) (1997) 387–392. [5] S.T. Liu, X.P. Guan et al., Nonexistence of positive solution of a class of nonlinear delay partial di(erence equation, J. Math. Anal. Appl. 234 (1999) 361–367. [6] S.T. Liu, Y.Q. Liu et al., Existence of monotone positive solution for partial di(erence equation, J. Math. Anal. Appl. 247 (2000) 384–396. [7] S.T. Liu, H. Wang, Necessary and su9cient conditions for oscillations of a class of delay partial di(erence equations, Dynamic Systems Appl. 7 (1998) 495–500. [8] S.T. Liu, B.G. Zhang, Oscillation of a class of partial di(erence equations, PanAmer. Math. J. 8 (1998) 93–100. [9] X.-P. Li, Partial di(erence equations used in the study of molecular orbits, Acta Chim. Sinica 40 (8) (1982) 688-698, in Chinese. [10] B.G. Zhang, S.T. Liu, On the oscillation of two partial di(erence equations, J. Math. Anal. Appl. 206 (1997) 480–492. [11] B.G. Zhang, S.T. Liu, Oscillation of partial di(erence equations with variable coe9cients, Comput. Math. Appl. 36 –38 (1998) 235–242. [12] B.G. Zhang, S.T. Liu, S.S. Cheng, Oscillation of class of delay partial di(erence equations and its applications, J. Di(erential Equations Appl. 1 (1995) 215–226.