Y. Guo. A. Liu, T. Liu, Q. Ma UDC

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1 UDC OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS WITH CONTINUOUS VARIABLES* ОСЦИЛЯЦIЯ КЛАСУ НЕЛIНIЙНИХ ЧАСТКОВО РIЗНИЦЕВИХ РIВНЯНЬ З НЕПЕРЕРВНИМИ ЗМIННИМИ Y. Guo Graduate School Chna Acad. Eng. Phys. P. O. Box 2101, Bejng, , P. R. Chna Guangx Unv. Technology Luzhou, 55006, P. R. Chna A. Lu, T. Lu, Q. Ma School Math. and Phys. Chna Unv. Geosc. Wuhan, 3007, P. R. Chna e-mal:wh_aplu@sna.com Ths paper s concerned wth a class of nonlnear partal dfference equatons wth contnuous varables. Some oscllaton crtera are obtaned usng an ntegral transformaton and nequaltes. Розглянуто клас нелнйних частково рзницевих рвнянь з неперервними змнними. Отримано деяк критерї осциляцї з використанням нтегральних перетворень та нервностей. 1. Introducton. Partal dfference equatons are dfference equatons whch nvolve functons wth two or more ndependent varables. Such equatons arse n nvestgaton of random walk problems, molecular structure problems [1], and numercal dfference approxmaton problems [2], etc. Recently, oscllaton problems for partal dfference equatons wth nvarable coeffcents and dscrete varables have been nvestgated n [3 8]. We can further nvestgate oscllaton propertes of nonlnear equatons wth varable coeffcents and contnuous varables and obtan some oscllaton crtera. In ths paper, we consder a class of nonlnear partal dfference equatons wth contnuous varables, p 1 (x, y)a(x a, y b) p 2 (x, y)a(x a, y) p 3 (x, y)a(x, y b) p (x, y)a(x, y) h (x, y, A(x σ, y τ )) = 0, (1) Ths work was supported by Natural Scence Foundaton of Chna (Grant Nos ), Guangx Natural Scence Foundaton Grant No and Research Foundaton for Outstandng Young Teachers, Chna Unversty of Geoscences (Wuhan) (No.CUGQNL081). c Y. Guo, A. Lu, T. Lu, Q. Ma, 2010 ISSN Нелнйн коливання, 2010, т. 13, N 3 305

2 306 Y. GUO, A. LIU, T. LIU, Q. MA where p 1 (x, y) C(R R, [0, )); p 2 (x, y), p 3 (x, y), p (x, y) C(R R, (0, )); a, b, σ, τ are negatve and h (x, y, u) C(R R R, R), = 1,..., m. Let σ = max,...,m {σ }, τ = max,...,m {τ }. A soluton of (1) s defned to be a contnuous functon A(x, y), for all x σ, y τ, whch satsfes (1) on R R. A soluton A(x, y) of (1) s sad to be oscllatory f t s nether eventually postve nor eventually negatve. Some oscllaton crtera for a soluton of (1) are obtaned usng ntegral transformaton and nequaltes. Our results extend some oscllaton propertes of nonlnear equatons wth nvarable coeffcents and dscrete varables to nonlnear equatons wth varable coeffcents and contnuous varables. 2. Man lemmas. We assume that the followng condtons are satsfed throughout ths paper: (I) p 1 (x, y) p 1 0, p 2 (x, y) p 2 > 0, p 3 (x, y) p 3 > 0, 0 < p (x, y) p, and p, = 1, 2, 3,, are constants and also satsfy p 2, p 3 p ; (II) τ = k a θ, σ = l b ξ, = 1,..., m, where k, l are nonnegatve ntegers and θ (a, 0], ξ (b, 0]. Lemma 1. Assume that () h C(R R R, R), uh (x, y, u) > 0 for u 0, and h (x, y, u), = 1,..., m, s a nondecreasng functon n u; () h (x, y, u), = 1,..., m, s convex n u for u 0. Let A(x, y) be an eventually postve soluton of (1), then there exsts a postve functon x y = 1 A(u, v) du dv eventually satsfyng the followng results: ab xa yb (1) f mn,...,m {k } = 0 and mn,...,m {l } = 0, then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x σ, y τ )) 0; (2) (2) f mn,...,m {k } = 0 and mn,...,m {l } = 0, then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x k a, y l b)) 0; (3) (3) f mn,...,m {k } = 0 and mn,...,m {l } = 0, then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x σ, y l b)) 0; () ISSN Нелнйн коливання, 2010, т. 13, N 3

3 OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS () f mn,...,m {k } = 0 and mn,...,m {l } = 0, then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x k a, y τ )) 0. (5) Proof. From (I), we have the followng nequalty: eventually. Snce p (A(x a, y) A(x, y b) A(x, y)) p 1 A(x a, y b) p 2 A(x a, y) p 3 A(x, y b) p A(x, y) p 1 (x, y)a(x a, y b) p 2 (x, y)a(x a, y) p 3 (x, y)a(x, y b) p (x, y)a(x, y) < 0 = 1 ab x y xa yb A(u, v) du dv, (6) we have x y = 1 (A(x, v) A(x a, v))dv > 0 (7) ab yb and y x = 1 (A(u, y) A(u, y b))du > 0. (8) ab xa From the above, we have s nondecreasng n x and y eventually. Integratng (1), from (I) we have p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p 1 ab xa x yb y h (u, v, A(u σ, v τ )) dv du 0. By (), (), and Jensen s nequalty, we obtan the followng nequalty: p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x σ, y τ )) 0. ISSN Нелнйн коливання, 2010, т. 13, N 3

4 308 Y. GUO, A. LIU, T. LIU, Q. MA Thus (2) holds. Snce a, b, τ, σ are negatve real numbers, there exst nonnegatve ntegers k and l satsfyng σ = k a θ, τ = l b ξ, where a < θ 0, b < ξ 0, = 1, 2,..., m. From (7) and (8), we obtan s nondecreasng eventually. So f mn {k } = 0,,...,m mn {l } = 0,,...,m we have Z(x σ, y τ ) Z(x k a, y l b), = 1, 2,..., m. Snce h (x, y, u), = 1, 2,..., m, s nondecreasng n u, we have p 1 Z(xa, y b)p 2 Z(xa, y)p 3 Z(x, y b) p h (x, y, Z(x k a, y l b)) 0. Hence, (3) holds. Smlarly f mn,...,m {k } = 0, mn,...,m {l } = 0, s nondecreasng n x and y eventually, we have Z(x σ, y τ ) Z(x σ, y l b). Snce h (x, y, u), = 1, 2,..., m, s nondecreasng n u eventually, we have the followng nequalty: p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x σ, y l b)) 0, mplyng (). Smlarly f mn,...,m {k } = 0, mn,...,m {l } = 0, = 1, 2,..., m, s nondecreasng n x and y eventually, we have Z(x σ, y τ ) Z(x k a, y τ ). Snce h (x, y, u) s nondecresng n u, we have the followng nequalty: p 1 Z(xa, y b)p 2 Z(xa, y)p 3 Z(x, y b) p h (x, y, Z(x k a, y τ )) 0. Hence, (5) holds. The proof s completed. By a smlar method, we can obtan propertes of an eventually negatve soluton of (1). Lemma 2. Assume that () h C(R R R, R), uh (x, y, u) > 0 for u 0 and h (x, y, u), = 1,..., m, s a nondecreasng functon n u; () h (x, y, u), = 1,..., m, s concave n u for u 0. Let A(x, y) be an eventually negatve soluton of (1), then there exsts a negatve functon x y = 1 A(u, v)dudv eventually satsfyng the followng results: ab xa yb (1) f mn,...,m {k } = 0 and mn,...,m {l } = 0, then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x σ, y τ )) 0; (9) ISSN Нелнйн коливання, 2010, т. 13, N 3

5 OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS (2) f mn,...,m {k } = 0 and mn,...,m {l } = 0, then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x k a, y l b)) 0; (10) (3) f mn,...,m {k } = 0 and mn,...,m {l } = 0, then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x σ, y l b)) 0; (11) () f mn,...,m {k } = 0 and mn,...,m {l } = 0, then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x k a, y τ )) 0. (12) 3. Man results. In the followng, we nvestgate oscllatory propertes of a soluton of (1) and obtan the man results of ths paper. Theorem 1. Assume that () h (x, y, u) C(R R R, R) s nondecreasng n u and uh (x, y, u) > 0, = = 1, 2,..., m, for all u 0, () lm nf h (x, y, u)/u = S 0, m S > 0, = 1, 2,..., m, x,y,u 0 () h (x, y, u) s convex n u for u > 0, h (x, y, u), = 1,..., m, s concave n u for u < 0, (v) one of the followng condtons holds: S (η 1) η 1 η η (p 1 p 2 p 3 ) η p (η 1) > 1, η = mn{k, l } > 0, = 1,..., m, (13) S k k (k 1) k 1 p k 1 2 p k > 1, mn,...,m {k } > 0, mn {l } = 0, (1),...,m S l l (l 1) l 1 p l 1 3 p l > 1, mn {k } = 0,,...,m mn {l } > 0, (15),...,m 1 p S > 1, Then every soluton of (1) s oscllatory. mn {k } = mn {l } = 0. (16),...,m,...,m ISSN Нелнйн коливання, 2010, т. 13, N 3

6 310 Y. GUO, A. LIU, T. LIU, Q. MA Proof. Assume the contrary. Let A(x, y) be an eventually postve soluton of (1), and be defned by (6). Then by Lemma 1, we obtan lm x,y = ζ 0. In the followng, we clam that ζ = 0. Otherwse, let ζ > 0. By Lemma 1, we know that (2) holds. From (2) and condton (I), we have So p 1 Z(x a, y b) p (Z(x a, y) Z(x, y b) ) p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p 0. Z(x a, y) Z(x, y b) 0. (17) Takng the lmt on both sde of (17), we have ζ 0. Consder ζ 0. Then we have ζ = 0. If mn,...,m {k } > 0, mn,...,m {l } > 0, n the vew of (2), we have (p 1 p 2 p 3 )Z(x a, y b) p p 1Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, Z(x σ, y τ )). (18) Snce s nondecreasng eventually, from (18) for all large x and y we have (p 1 p 2 p 3 )Z(x a, y b) p h (x, y, Z(x η a, y η b)) = = h (x, y, Z(x η a, y η b)) Z(x η a, y η b) η j=1 Z(x ja, y jb) Z(x (j 1)a, y (j 1)b), (19) where η = mn{k, l }, = 1,..., m. Let α(x, y) = /Z(x a, y b). Then α(x, y) > 1 for all large x and y. From (19), we have.e., p 1 p 2 p 3 α(x, y) (p 1 p 2 p 3 ) h (x, y, Z(x η a, y η b)) Z(x η a, y η b) h (x, y, Z(x η a, y η b)) Z(x η a, y η b) η η α(x ja, y jb) p, j=1 α(x ja, y jb)α(x, y) p α(x, y). By (), (20) mples that α(x, y) s bounded. Let lm nf α(x, y) = β. Takng the lmt nferor on both sdes of (20), we obtan x,y (p 1 p 2 p 3 ) j=1 S β η1 p β, (20) ISSN Нелнйн коливання, 2010, т. 13, N 3

7 OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS e., p 1 p 2 p 3 β p S β η < p. (21) Hence, β > p 1 p 2 p 3. Snce m p S β η1 /(p β (p 1 p 2 p 3 )) 1, computng the mnmum of the functon f(x) = x η1 /(p x (p 1 p 2 p 3 )) as x > p 1 p 2 p 3 we obtan p mn β> p 1 p 2 p 3 p So we have m S (η 1) η 1 βη1 p β (p 1 p 2 p 3 ) = (η 1) η1 η η η η (p 1 p 2 p 3 ) η p η 1 (p 1 p 2 p 3 ) η. (13) holds we can obtan that every soluton of (1) s oscllatory. If mn,...,m {k } > 0, mn,...,m {l } = 0, by Lemma 1, we obtan p 1 Z(xa, y b)p 2 Z(xa, y)p 3 Z(x, y b) p p η 1 1 whch s contrary to (13). Therefore f h (x, y, Z(x k a, y τ )) 0. Then we have p 2 Z(x a, y) p h (x, y, Z(x k a, y τ ) = = h (x, y, Z(x k a, y τ )) Z(x a, y τ ) Z(x k a, y τ ) k j=2 Z(x ja, y τ ) Z(x (j 1)a, y τ ). (22) Snce s nondecreasng n x, y eventually, we have Z(x a, y τ )/ > 1 for all large x and y. From (22), we have p 2 Z(x a, y) h (x, y, Z(x k a, y τ )) Z(x k a, y τ ) k j=2 Let α(x, y) = /Z(x a, y) > 1. From (23), we have p m 2 α(x, y) h (x, y, Z(x k a, y τ )) Z(x k a, y τ ) Z(x ja, y τ ) Z(x (j 1)a, y τ ) p. (23) k j=2 α(x ja, y τ ) p. (2) By condton () the above nequalty mples that α(x, y) s bounded. Let lm nf α(x, y) = x,y = β. From (22), we can obtan p 2 h (x, y, Z(x k a, y τ )) Z(x k a, y τ ) k j=1 α(x ja, y τ )α(x, y) p α(x, y). (25) ISSN Нелнйн коливання, 2010, т. 13, N 3

8 312 Y. GUO, A. LIU, T. LIU, Q. MA Takng the lmt nferor on both sdes of (25), we have p 2 m S β k p β. Hence we have p 2 β m S β k 1 p,.e., p 2 β p m S β k 1 < p. Then we obtan β > p 2 and m p S β k 1 /(p β p 2 ) 1. β k Snce mn p β> 2 = pk 1 2 k k p p β p 2 p k (k 1) k 1, we have m S k k (k 1) k 1, 1 whch contradcfs (1). So f (1) holds we can obtan that every soluton of (1) s oscllatory. Smlarly, we can prove that f (15) holds then we can also obtan every soluton of (1) s oscllatory. If mn,...,m {k } = mn,...,m {l } = 0, from Lemma 1, we know that (3) holds. Hence we have Then p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p p k 1 2 p k h (x, y, ) p 1 Z(x a, y b) p 2 Z(x a, y) p 3 Z(x, y b) p h (x, y, ) h (x, y, Z(x σ, y τ )) 0. p 0. (26) Takng the lmt nferor on both sdes of (26), we have m S p, whch s contrary to (16). So f (16) holds we can obtan that every soluton of (1) s oscllatory. If A(x, y) s the eventually negatve soluton of (1), we can obtan a contradcton by assumng that A(x, y) s an eventually negatve soluton of equaton (1). Therefore we know the result s correct. The proof s over. The results ndcate that there are some crtera of oscllatory propertes of solutons of some partal dfference equatons wth forward front dfference. In some sense, the results play some roles n nvestgatng propertes of solutons of advanced partal dfferental equatons. 1. L X. P. Partal dfference equatons used n the study of molecular orbts (n Chnese) // Acta Chm. SINICA P Zhang B. G., Lu S. T., Cheng S. S. Oscllaton of a class of delay partal dfference equatons // J. Dfference Equat. and Appl P Kelley W. G., Peterson A. C. Dfference equatons. New York: Acad. Press, Zhang B. G., Lu S. T. On the oscllaton of two partal dfference equatons // J. Math. Anal. and Appl P Zhang B. G., Lu B. M. Oscllaton crtera of certan nonlnear partal dfference equatons // Comput. Math. Appl P Agarwal R. P., Yong Zhou. Oscllaton of partal dfference equatons wth contnuous varables // Math. and Comput. Modellng P ISSN Нелнйн коливання, 2010, т. 13, N 3

9 OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS Zhang B. G., Tan C. J. Oscllaton crtera of a class of partal dfference equatons wth delays // Comput. Math. Appl P Cu B. T., Lu Y. Q. Oscllaton for partal dfference equaton wth contnuous varables // J. Comput. and Appl. Math P Lu A. P., Guo Y. F. Oscllaton of the solutons of nonlnear delay hyperbolc partal dfferental equatons // Chn. Quart. J. Math ,. P Anpng Lu, Qngxa Ma, Mengxng He. Oscllaton of nonlnear mpulsve parabolc equatons of neutral type // Rocky Mountan J. Math , 3. P Guo Y. F., Lu A. P. Oscllaton of nonlnear mpulsve parabolc dfferental equaton wth several delays // Ann. Dfferent. Equat , 3. P Receved , after revson ISSN Нелнйн коливання, 2010, т. 13, N 3