Cheng, Denghua

Yan, Jurang

English

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaDeng Hua Cheng; Ju Rang YanOscillation and global attractivity in a nonlinear delay difference equationMathematica Slovaca, Vol. 50 (2000), No. 3, 335--344Persistent URL: http://dml.cz/dmlcz/132829Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2000Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czíVtathemaiica Slovaca ©2000 Math. Slovaca, 50 (2000), NO. 3, 335-344 Slovák Academy of Sciences OSCILLATION AND GLOBAL ATTRACTIVITY IN A NONLINEAR DELAY DIFFERENCE EQUATION DENGHUA CHENG* — JURANG YAN** (Communicated by Milan Medved') ABSTRACT. We obtain a necessary and sufficient condition for every positive solution of the nonlinear delay difference equation n = 0 , l , . . . (*) n+1~ a + bxp„_„-П — fc CX Q to oscillate about its positive equilibrium. We also obtain conditions under which the positive equilibrium of (*) is globally attractive. 1. Introduct ion There have been many papers considering the oscillation and the nonoscil-lation of nonlinear delay difference equations, see, for example, [l]-[6] and the references cited in [1]. Our aim in this paper is to investigate the oscillation and global attractivity of the nonlinear delay difference equation n = 0 , l , . . . , (1) where "-Ы n 4- hтp — rтq a -r oxn_k cxn_k o G ( 0 , l ) , 6,p, q G (0, oo), cG (-00,00), k £ N, . ufcq\^ fc ^ \ ^ ^ n (2) p>*> a+b{^) - % ) >°-By a solution of (1) we mean a sequence {xn} of real numbers which is defined for n > — k and satisfies (1) for n — 0 , 1 , . . . . It is easy to see under the initial conditions: xn = An > 0 , n = - f c , - k + l , . . - , 0 , (3) 2000 M a t h e m a t i c s Sub jec t C l a s s i f i c a t i o n : Primary 39A10, 39A12. Key words : oscillation, global attractivity, delay difference equation. 335 DENGHUA CHENG — JURANG YAN equation (1) has a unique positive solution satisfying (3). Equation (1) has a unique positive equilibrium x*. In Section 2, we establish a necessary and sufficient condition for every positive solution of (1) to oscillate about x* and in Section 3, we establish a sufficient condition for the global attractivity of x*. When p = 2 and q = 1 V . L . K o c i c and G. L a d as [1; pp. 166, 167] investigated a similar equation. Our results in this paper extend and improve their results. 2. Oscillation of equation (1) In this section, we study the oscillatory behavior of the solution of (1). As usual, a solution {xn}n>_k of (1) is said to be oscillatory about x* if the terms xn of the sequence are neither eventually greater than x* nor eventually less than x*. Otherwise, the solution is called nonoscillatory about x*. Before we present the main result we state two lemmas which will be useful in the sequel. The first one is extracted from [1; pp. 6, 7]. LEMMA 1. ([1]) Consider the delay equation Wn+i " »« + rfiVn-k) = 0, n = 0 , 1 , . . . , (4) where r G (0, oo), k eN and f € C[R, R]. Assume that uf(u) > 0 /or u / 0 and that lim « ! > _ , . u-»0 U Suppose also there exists a positive number 8 such that either f(u) < u for 0 < u < 8, or f(u) > u for — 5 < u < 0. Then every solution of (4) oscillates if and only if (>1 i/fc = 0, i > (/e+l)^1 */ k - 1 ' The proof of the next lemma is straightforward and will be omitted. 336 OSCILLATION AND GLOBAL ATTRACTIVITY IN A DELAY DIFFERENCE EQUATION LEMMA 2. Assume that (2) holds and set F(x) = a + bxp - cxq . Then there is a unique positive number x* such that F(x*) = 1. Furthermore, FMIH *" »<;<;•• (5) { > 1 for x < x < oo. In addition, if c <0, then F(x) is increasing for x > 0, (6) and if c > 0, then is decreasing for 0 < x < (fjj)J F(x){ ' ' ^ ^ " ' ^ ( 7 ) is increasing for ( f | ) p~q < x < oo . The main result in this section is the following: THEOREM 1. Assume that (2) holds. Then every positive solution of (1) os-cillates about x* if and only if ( > 1 if k = 0, pb(xr-qc(x*y\ :_j^_ ifk>1 (8) ^ > (fc+l)*+- v K -^ l ' P r o o f . The change of variable xn = x* eVn transforms (1) to the difference equation Vn+i -yn + Ha + h^YePVn~k ~<x*)qeqyn~k] =0. (9) Clearly every solution of (1) oscillates about x* if and only if every solution of (9) oscillates about zero. Set f(u) = In [a + b(x* eu)p - c(x* eu)q] , g(u) = f(u)-[pb(x*y-qc(x*)q]u. If c < 0, then clearly «/(«) > 0 for u # 0. (10) Next, assume that c > 0. As p > q > 0 and b(x*)p - c(x*)q = 1 - a > 0 , 337 DENGHUA CHENG — JURANG YAN it follows that f(u)>ln[a+(b(x*)p -c(x*)q)equ] > 0 for u > 0 , f(u) < In [a + (b(x*Y - c(x*)q) equ] < 0 for u < 0 . Hence, (10) holds for c G (—oo,oc). Observe that ^ = pb(x*Yepu-qc(x*)qequ _ _ du a + b(x*)PePu-c(x*)qequ L v > H v ; J < TT—l 7—^ [p&teT eptt -gc(x*)? e«" ~ a + b(x*)PePu-c(x*)qeiul v ' 7 V ; - (p&(.t*)p - gc(£*)9) (a + b(x*)p eptt -c(x*)q equ)] = 77-^ " -—- \pb(x*)p epu -qc(x*)q equ -a(pb(x*)p - Qc(x*)q) a + b(x*)P ePu -c(x*)q equ L v v ' VF v > q K ' > - {pb(x*)p - qc(x*)q)b(x*)pepu+{pb(x*)p - qc(x*)q)c(x*)qequ = 77-1 7—1 \{pb(x*)p ~ b(x*)p(pb(x*Y - qc(x*)q)) e»u a + b(x*)P ePu-c(x*Y equ iX v v > v v > Hy > >> - (qc(x*)q - c(x*)q(pb(x*)p ~ qc(x*)q)) equ -a(pb(x*Y - qc(x*)q)] ^a + b(x*)Peplu-c(x*)qe^pb^P " &***>' ~ qc(x*)q)b(x*Y - qc(x*)q + (pb(x*)p - qc(x*)q)c(x*)q) equ -a(pb(x*)p - qc(x*)q)] < 77—1 -—- \pb(x*)p ~ qc(x*q)(l - b(x*)p + c(x*)q) equ ~ a + b(x*)PePu-c(x*)qequi y ' v >K v ' ^ > J * - a(pb(x*)p - qc(x*)q)] and pb(x*)p ~ qc(x*)q > p\b(x*)p - c(x*)q] = p(l - a) > 0. Hence ^r < — r r - - — - — - — \(pb(x*)» ~ qc(x*)q) (l-a- b(x*)p + c(x*)q)] du ~ a + b(x*)PePu-c(x*)qequlK K > v > A v > v > n = 0 for u < 0. This together with #(0) = 0 implies that g(u) > 0 for u < 0, that is /(«) > \pb(x*)p - gc(x*)9]« for u < 0. We also have ffl^=p&(x*)p-oc(x*)9 au 338 OSCILLATION AND GLOBAL ATTRACTIVITY IN A DELAY DIFFERENCE EQUATION and so ,- /(«) , lim -=—- —— = 1 . u^o [pb(x*)P - qc(x*Y\u Hence, by Lemma 1, every solution of (9) oscillates if and only if (8) holds. The proof is complete. • 3. Global At tract ivity of (1) In this section, we investigate the global attractivity of the positive equilib-rium x* of (1). THEOREM 2. Assume that (2) holds. Then every positive solution of (1) nonosdilatory about x* tends to x* as n -+ 00. P r o o f . Assume that xn > x* for n sufficiently large. The proof when xn < x* for n sufficiently large is similar and will be omitted. Set xn = x* eVn . Then yn > 0 for n sufficiently large and yn+i - yn + Ha + b(x*)p ePyn'k -<x*)q eqyn~k] = 0. (11) Thus for n sufficiently large yn+i -yn<- H" + ( K * T - <x*y) e«*-»] < 0, and so lim y = \x G [0, 00), say, exists. n—>-oo We claim that fi = 0. Otherwise, \i > 0. Take p + q Then there exists N0 > 0 such that for n > JV0, V-s<yn_k<i~L + e. (13) First, assume that c < 0. From (11) and (13), it follows that Vn+i ~yn + Ha+ (b(x*)P ~ <X*Y) eq{^£)] < 0 for n > N0 and by summing this inequality from A"0 to 00 we get a contradiction. Next, assume that c > 0. Then (11) and (13) yield 1/n+i - Vn + Ha + b(x*)P ep{fl~£) -c(x*)q eq{^s)] < 0 . (14) In view of (12), we have In [a + b(x*)p ep{»~£) -c(x*)q eq{^£)] > In [a + (b(x*)p - c(x*)q) eq{^e)] and so (14) yields yn+i -yn + Ha+ ib(x*)p - <x*)q) e ' u + £ ) ] < ° for n > N0 . By summing this inequality from N0 to oc we get a contradiction. The proof is complete. • 339 DENGHUA CHENG — JURANG YAN THEOREM 3- Assume that (2) holds. Set (l)k+1 tfc<0, M ° ^ - - + r i / O O . («+fc(̂ r-c(u)') Suppose that ^ ± ^ ln[a + K * * M 0 ) P - c(a;*M0)9] < 1. (15) T/ien every positive solution of (1) oscillatory about x* tends to x* as n -» oo. P r o o f . Assume that {#n}n>_fc is an solution of (1) oscillatory about x*. We will prove that lim xn = x* . n->-oo Let {n J be an increasing sequence of positive integers such that n- —r oo as i —r oo satisfying x n . < x* and xn.+1 > x* for i = 1,2, . . . . and for each i = 1,2,. . . , some of the terms x • with ni < j < ni+1 are greater than x* and some are less than x*. For each i = 1, 2 , . . . , let mi and M- be the integers in the interval [nvni+1] such that xmi+1 = minfy. : n- < j < n- + 1 } , x M i + 1 = max{x j : n{ < j < n - + 1 } . Then for each i = 1,2, . . . , xmi+1 < x* and A x m i < 0 while x M . + 1 > x* and ^XM. > 0 . By (1), we have 0>Ax = xmJ1-(a + ^mi-fe-c^-fe)] which indicates that a + bxm._k - cxqm._k > 1, that is xm._ fc > x* . Therefore, there exists an integer m~ satisfying max{n i ,m i — fc} < rh~ < mi + 1 and #__ > x*, and x- < x* for j = ra~ + 1 , . . . , rni + 1. (16) Similarly, there exists an integer Mi satisfying max{n., Mi — k}< Mt < Mt + 1 and XM:<X*, and xj > x* for j = ~M{ + 1 , . . . , M- + 1. (17) 340 OSCILLATION AND GLOBAL ATTRACTIVITY IN A DELAY DIFFERENCE EQUATION Now we show that {xn} is bounded from above and bounded from below away from zero. In fact, since xn > 0 for n > 0, it follows by (1) that Xn+l _ xn a + bxpn_k - cxqn_k First, assume c < 0. Then for n > 0, we have < - for n > 0. (18) c » + i - -Hence, by multiplying this ineqality from Mi to Mi we have ^±i<(iy •— ~ \aJ and so / i \k+l xMi+i<x*(±L) = * * M 0 , which clearly implies that X~MІ xn < x * M 0 for n > 0. By using this fact in (18), we find that for n > 0 1 ^n+l > xn ~ a + 6 ( X * M 0 ) P - c(a;*M0)s and so g ^ - + 1 > _ = M , xmT (a + 6 ( X * M 0 ) P - c(x*M0Y)k+1 l ' which implies that x n > x * M 1 for n > 0 . Next, assume that c > 0 . Then, in view of Lemma 2, we see from (18) that for n > 0, X n + 1 < *n - a + b(fp)p-c(fp) Hence, we have x я * М—1 < - = мп and so XMІ+I — x M0 , 341 DENGHUA CHENG — JURANG YAN which implies that Similarly, we have xn < x*M0 for n > 0 . xn > x*M1 for n > 0 . Therefore, we have Mxx* <xn< x*M0 for n > 0 . Now set r ±ln[a + b{x*)pepu-c{x*)qequ] for u ^ O , ^ ~ I I>b(x*)p - q/cOz*)« for u = 0 . Observe that the transformation x n = #* eyn transforms (1) into Vn+i ~ 2/n = -9(yn-k)yn-k • (19) Clearly, to show that lim a; = #*, it suffices to show that n-»oo nlim2/„ = 0 . (20) To this end, observe that In M1<yn< In M0 for n > 0. (21) First we show that there is a 5 > 0 such that 5 < g{yn) < g{lnM0) for n > 0 . (22) Observe that u , f(u) = {f f ""J' t 1 for w = 0 is increasing, / > 0, p > q, and pb{x*)p > qc{x*)q. Thus for u < 0, 5(u) - J ln[l + b(or*)p(e^ - 1 ) - c{x*)q{equ - 1 ) ] < ^ * ) P _ ! ! _ z i _ g c ( x *)?____i — v ' pu v ' gw <(p6(ar*)p - ? c ( x * ) « ) / M ( 2 3 ) < p&(*T ~ tfcfV)9 = 5(0) and g{u) - 1 In [a + b(z* e*)* - c(x* ett)ff] > 0 . (24) 342 OSCILLATION AND GLOBAL ATTRACTIVITY IN A DELAY DIFFERENCE EQUATION Also, as g is increasing for u > 0, it follows that g(0) < g(u) < g(ln(M0)) for 0 < u < lnM 0 . (25) Therefore, by using (21), (23), (24) and (25) and because g is continuous, we see that (22) holds. Next, define the nonnegative function V(yn) = - 2 n yn- ] L 9(Vi)Vi + Y, \9(yi+k+i)Yl9(yj)yj i=n—k i=n—k J = г for n> N0. Calculating the difference of V along the solutions of (19) and using the fact that 2y{yn+1 < y\ + yn+1, we see that v(yn+1) - V(yn) n+1 -| 2 r n yn+i- J2 9(.VÍ)VÍ - yn- J2 9^i)y-i=n—k+l iz=n—k n+1 , n+1 \ n / n + JI {9(yi+k+i)Y.9(yj)yj)- J2 l9(yi+k+1)J29(yj)y2j i—n-k+l ^ j=i ' i=n~k j=i = -g(yn+1)yn+1 n+l 2Уn+i+9(yn+i)yn+1-2 J2 9(УІ)У L i=n—к+l n+1 n + 9(Уn+i)УІ+i j^ 9(yi+к+i) - g(yn+i) Xľ g(Уi)УÌ i=n—k+l i=n n+1 - ЫУn+i)У2n+i + 29(Уn+i)Уn+i £ 9(vJy> i n—к i=n—fc+1 n + 1 - g(Уn+i) 5Z 9{УІ)VÌ - g2(vn+1)yl+i i=n—к+l n+1 + g(yn+1)y2n+1 J2 9(yi+к+1) + g2(yn+i)У2n+i - g(Уn+i)g(Уn-к)Уn-i=n—к+l n+1 < - g(yn+1)y2n+1 n + l 2 " 5 ľ 9{УÙ- S 9{Уi+к+v i=n—k+l i—n—k+l This, in view of (22), yields V(yn+1) - V(yn) < -2[1 - g(lnM0)(k + l)]g(yn+1)y2n+1 343 DENGHUA CHENG — JURANG YAN By summing both side of this inequality wre see that for n > LV0, n-fl y(»n+1) + 2[l-ff(lnM0)(fc + l)] Yl 9(yi)y2i<v(yNQ). i=N0+l Hence, oo Yl^yn)y2n<°°^ n = l which, in view of (22), implies that CO n = l Clearly, this fact implies that (20) holds. The proof is complete. • Acknowledgement The authors are grateful to the referee for some valuable suggestions for the improvement of this paper. R E F E R E N C E S [1] KOCIC, V. L.—LAD AS, G. : Global Behavior of Nonlinear Difference Equations of Higher Order with Application, Kluwer Academic Publishers, Dordrech t-Bos ton-London, 1993. [2] CHEN, MING-PO—YU, J. S.: Oscillation and global attractivity in a delay logistic dif-ference equation, J Differ. Equa t ions Appl. 1 (1995), 227-237. [3] SO, J. W-H—YU, J. S.: On the stability and uniform persistence of a discrete model of Nicholson's blowflies, J. Ma th. Anal. Appl 193 (1995), 233-244. [4] LIU, BING—YAN, J U R A N G : Oscillatory and asymptotic behavior of second order non-linear difference equations, Proc. Edinburgh Ma th. Soc. (2) 39 (1996), 523-533. [5] ZHOU, XIAOLIANG—YAN, J U R A N G : Oscillatory and asymptotic properties of higher order nonlinear difference equations, Nonlinear Anal. 31 (1998), 493-502. [6] HE, X U E Z H O N G : Oscillatory and asymptotic behaviour of second order nonlinear differ-ence equations, J. Ma th. Anal. Appl. 175 (1993), 482-498. Received November 27, 1996 Taiyuan TV University Revised May 25, 1998 Taiyuan, Shanxi 030001 P. R. CHINA ** Department of Mathematics Shanxi University Taiyuan, Shanxi 030006 P. R. CHINA E-mail: jryan@mail.sxu.edu.cn 344 

Oscillation and global attractivity in a nonlinear delay difference equation

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Oscillation and global attractivity in a nonlinear delay difference equation

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Institute of Mathematics AS CR, v. v. i.