Oscillation of Nonlinear Delay Partial Difference Equations. LIU Guanghui [a],*

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1 Studies in Mthemtil Sienes Vol. 5, No.,, pp. [9 97] DOI:.3968/j.sms ISSN [Print] ISSN [Online] Osilltion of Nonliner Dely Prtil Differene Equtions LIU Gunghui [],* [] College of Siene, Hunn Institute of Engineering, Chin. * Corresponding uthor. Address: College of Siene, Hunn Institute of Engineering, 88 Est Fuxing Rod, Xingtn, Hunn 44, Chin; E-Mil: gh9@63.om Reeived: September 3, / Aepted: November 3, / Published: November 3, Abstrt: In this pper, we onsider ertin nonliner prtil differene equtions A m+,n A m,n+ A m,n + p i m, n)a m σi,n τ i = where, b,, ), u is positive integer, p i m, n), i =,,, u) re positive rel sequenes. σ i, τ i N = {,, }, i =,,, u. A new omprison theorem for osilltion of the bove eqution is obtined. Key words: Nonliner prtil; Differene equtions; Eventully positive solutions LIU, G. ). Osilltion of Nonliner Dely Prtil Differene Equtions. Studies in Mthemtil Sienes, 5 ), Avilble from sms/rtile/view/j.sms DOI:.3968/j.sms INTRODUCTION In this pper we onsider nonliner prtil differene eqution A m+,n A m,n+ A m,n + p i m, n)a m σi,n τ i =.) where, b,, ), u is positive integer, p i m, n), i =,,, u) re positive rel sequenes. σ i, τ i N = {,, }, i =,,, u. The purpose of this pper is to obtin new omprison theorem for osilltion of ll solutions of.). 9

2 LIU, G./Studies in Mthemtil Sienes, 5 ),. MAIN RESULTS To prove our min result, we need severl preprtory results. Lemm.. Assume tht {A m,n } is positive solution of.). Then i : A m+,n θ A m,n, A m,n+ A m,n,.) nd ii : A m σi,n τ i θ σ σi θ τ τi A m σ,n,.) where θ =, = b, σ = min i u {σ i}, = min i u {τ i}. Proof. Assume tht {A m,n } is eventully positive solutions of.). From.), we hve A m+,n A m,n+ A m,n = p i m, n)a m σi,n τ i, nd so A m+,n A m,n+ A m,n. Hene A m+,n θ A m,n nd A m,n+ A m,n. From the bove inequlity, we n find A m,n θ σ A m σ,n θ σi A m σ i,n, A m σ,n θ τ A m σ,n, nd A m σi,n θ τ A m σ i,n θ τi A m σ i,n τ i. Hene A m,n θ σ θτ A m σ,n θ σi θτi A m σ i,n τi. The proof of Lemm. is ompleted. Lemm.. [] If x, y R + nd x y, then rx r x y) > x r y r > ry r x y), for r >. Theorem.. If the differene inequlity A m+,n A m,n+ A m,n + θ σ σi θ τ τi p i m, n)a m σ,n.3) hs no eventully positive solutions, then every solution of eqution.) osilltes. Proof. Assume tht {A m,n } is positive solution of eqution.). Then, by.) nd Lemm., we obtin A m+,n A m,n+ A m,n + p i m, n)a m σi,n τ i.4) Substituting.) into.4), we hve A m+,n A m,n+ A m,n + This ontrdition ompletes the proof. θ σ σi θ τ τi p i m, n)a m σ,n. 9

3 Osilltion of Nonliner Dely Prtil Differene Equtions Define set E by where Q m,n = u E = {λ > λq m,n >, eventully} θ σ σi θ τ τi p i m, n). Theorem.. Assume tht i) lim Q m,n > ; ii) there exists M m, N n suh tht if σ > >, λ E,M m,n n nd if > σ >, λ E,M m,n n σ j= σ σ j= σ λq m i j,n i ) < ) τ, θ σ λq m i,n i j ) < Then every solution of.) osilltes. θ..5).6) Proof. Suppose, to the ontrry, A m,n is n eventully positive solution. We define subset S of the positive numbers s follows: Sλ) = {λ > A m+,n A m,n+ [ λq m,n ]A m,n, From.3) nd Lemm., we hve A m+,n A m,n+ θ σ θ τ Q m,n )A m,n, eventully}. whih implies θ σ θ τ Sλ). Hene, Sλ) is nonempty. For λ S, we hve eventully tht λq m,n >, whih implies tht S E, Due to ondition i), the set E is bounded, nd hene, Sλ) is bounded. Let u S. Then from Lemm., we hve ) A m+,n+ A m+,n A m,n+ θ uq m,n )A m,n. If σ > >, then A m,n ) τ uq m i,n i )A m τ,n τ θ, nd for j =,,, σ, we hve A m j,n θ j ) τ τ θ uq m i j,n i )A m τ j,n ) τ τ uq m i j,n i )A m σ,n. 9.7)

4 LIU, G./Studies in Mthemtil Sienes, 5 ), Now, from Lemm. nd.7), it follows tht A σ τ m,n i.e., A m,n θ ) τσ ) ) θ ) τσ ) ) Similrly, if > σ >, then A m,n θ ) σ σ ) ) [ σ j= [ σ j= σ j= uq m i j,n i ) A σ τ m σ,n, ] σ uq m i j,n i ) Am σ,n τ..8) ] σ uq m i,n i j ) Am σ,n. Substituting.8) nd.9) into.3), we get respetively, for σ >,.9) A m+,n A m,n+ A m,n +Q m,n ) τ θ nd for > σ, σ j= σ uq m i j,n i ) Am,n, A m+,n A m,n+ A m,n +Q m,n θ Hene, for σ >, σ σ j= σ uq m i,n i j ) Am,n. A m+,n A m,n+ { Q m,n ) τ θ [ σ τ ] τ j= uq } m i j,n i) σ Am,n, m M,n N 93.)

5 Osilltion of Nonliner Dely Prtil Differene Equtions nd for > σ, A m+,n A m,n+ { Q m,n θ [ τ σ ] σ j= uq } m i,n i j) σ Am,n. m M,n N.) From.) nd.), we get ) τ θ nd θ m M,n N m M,n N [ σ j= [ τ σ j= ] τ uq m i j,n i) ] σ uq m i,n i j) σ S for σ >,.) σ S for > σ..3) On the other hnd,.5) implies tht there exists, ) we n hoose the sme) suh tht for σ > λ E,m M,n N σ j= σ λq m i j,n i ) ) τ, θ.4) nd.6) implies tht there exists, ) we n hoose the sme) suh tht for > σ >, λ E,M m,n n σ σ j= σ λq m i,n i j ) In prtiulr,.4) nd.5) led to when λ = u), respetively, ) τ [ σ τ θ λ E,m M,n N j= 94 uq m i j,n i ) ] θ..5) σ u for σ >,.6)

6 LIU, G./Studies in Mthemtil Sienes, 5 ), nd [ σ θ λ E,M m,n n j= σ uq m i,n i j ) ] σ u for > σ..7) Sine u S nd u u implies tht u S, it follows from.) nd.6) for σ >,.3) nd.7) for > σ tht u S. Repeting the bove rguments with u repled by u u, we get S, where, ). Continuing in this wy, u we obtin S, where i, ). This ontrdits the boundedness of S. i The proof is omplete. Corollry.. In ddition to i) of Theorem., ssume tht for σ > >, σ lim inf σ ) j= nd for > σ >, τ lim inf σ σ )σ j= σ Q m i j,n i > Q m i,n i j > Then every solution of.) osilltes. Proof. We note tht We shll use this for mx λ e >λ> λe)τ = τ+ τ e + ) τ+. τ+ τ τ + ) τ+ ) τ, θ σ+ σ σ σ + + ) σ. θ e = σ ) σ j= Q m i j,n i. Clerly, σ τ λ[ j= λ[ σ ) λq m i j,n i )] σ j= λ λ[ σ ) σ j= 95 σ λq m i j,n i )] τ Q m i j,n i )] τ

7 Osilltion of Nonliner Dely Prtil Differene Equtions Similrly, we hve σ σ j= τ+ τ τ e + ) τ+ < θ ) τ. σ λq m i,n i j ) < θ. By Theorem., every solutions of.) osilltes. The proof is omplete. By similr rgument, we hve the following results: Corollry.. If the ondition of Theorem. holds, nd lim inf Q m,n = q > σ+ σ σ σ + + ) σ, θ then every solution of.) osilltes. Theorem.3. Assume tht i) lim Q m,n > ; ii) for σ, >, lim inf Q m,n = q >,.) nd lim Q m,n > θ σ θτ θ q >..) Then every solution of.) osilltes. Proof. Suppose, to the ontrry, A m,n is n eventully positive solution. From.3) nd.), for ny ɛ >, we hve Q m,n > q ɛ for m M, n N. From.3), Lemm. nd bove inequlity, we obtin A m,n A m,n q ɛ) A m σ,n τ q ɛ) θ σ θ τ A m,n, nd A m,n q ɛ) θ σ θ τ A m,n, q ɛ) θ σ θ τ A m,n. Substituting bove inequlities into.3), we get [ ] θ σ θ τ θ σ θ τ q ɛ) + Q m,n θ σ θ τ A m,n <, whih implies lim Q m,n θ σ θτ θ q >. This ontrdits.). The proof is omplete. 96

8 LIU, G./Studies in Mthemtil Sienes, 5 ), Theorem.4. Assume tht i) lim Q m,n > ; ii) σ = =, nd lim Q m,n >..). Then every solution of.) osilltes. Proof. Let u S. Then from.3) nd Lemm., we hve + Q m,n A m,n <, whih implies lim Q m,n. This ontrdits.). The proof is omplete. REFERENCES [] YU, J. S., ZHANG, B. G., & WANG, Z. C. 994). Osilltion of dely differene eqution. Applible Anlysis, 53, 8-4. [] ZHANG, B. G., & ZHOU, Y. ). Osilltion of kind of two-vrible funtion eqution. Computers nd Mthemtis with Applition, 4, [3] Bohner, M., & Cstillo, J. E. ). Mimeti methods on mesure hins. Comput. Mth. Appl., 4, [4] Tnigw, T. 3). Osilltion nd nonosilltion theorems for lss of fourth order qusiliner funtionl differentil equtions. Hiroshim Mth., 33, [5] Bohner, M., & Peterson, A. ). Dynmi equtions on time sles: n introdution with pplitions. Boston: Birkhnser. [6] LIU, G. H., & LIU, L. CH. ). Nonosilltion for system of neutrl dely dynmi eqution on time sles. Studies in Mthemtil Sienes, 3, 6-3. [7] ZHANG, B. G., & YANG, B. 999). Osilltion in higher order nonliner differene eqution. Chznese Ann. Mth.,, 7-8. [8] ZHOU, Y. ). Osilltion of higher-order liner differene equtions. Advne of Diflerene Equtions III: Speil Issue of Computers Mth. Applition, 4, [9] Llli, B. S., & ZHANG, B. G. 99). On existene of positive solutions nd bounded osilltions for neutrl differene equtions. J. Mth. Anl. Appl., 66, [] ZHANG, B. G., & ZHOU, Y. ). Osilltion nd nonosilltion of seond order liner differene eqution. Computers Mth. Applition, 39, -7. [] Erbe, L., & Peterson, A. ). Osilltion riteri for seond order mtrix dynmi equtions on time sle. In R. P. Agrwl, M. Bohner, & D. O Regn Eds.), Speil Issue on Dynmi Equtions on Time Sles. J. Comput. Appl. Mth., 4, [] Erbe, L., & Peterson, A. 4). Boundedness nd osilltion for nonliner dynmi equtions on time sle. Pro. Amer. Mth. So., 3,