Oscillation of solutions of some nonlinear difference equations

Publicacions Matem`atiques, Vol 40 (1996), 127–133. OSCILLATION OF SOLUTIONS OF SOME NONLINEAR DIFFERENCE EQUATIONS Zdzislaw Szafra´ nski and Bla˙ zej Szmanda Abstract Sufficient conditions for the oscillation of some nonlinear difference equations are established. 1. Introduction In this note we consider the nonlinear difference equation of the form (1) ∆(rn∆xn)+qnf(xn−τn)=0,n=0,1,2,... , where ∆ denotes the forward difference operator: ∆vn=vn+1 −vnfor any sequence (vn) of real numbers; (qn) is a sequence of real numbers, (τn) is a sequence of integers such that lim n→∞(n−τn)=∞, (rn) is a sequence of positive numbers and Rn= n−1  k=0 1 rk →∞,as n→∞. f:R→Ris a continuous with uf(u)>0(u= 0). By a solution of Equation (1) we mean a sequence (xn) which is defined for n≥min i≥0(i−τi) and satisfies Equation (1) for all large n. A nontrivial solution (xn) of (1) is said to be oscillatory if for every n0>0 there exists an n≥n0such xnxn+1 ≤0. Otherwise it is called nonoscillatory. In several recent papers the oscillatory behaviour of solutions of nonlinear difference equations have been discussed e.g. see [1]-[6]. Our purpose in this paper is to give the sufficient conditions for the oscillation of solutions of Equation (1). The results obtained here extend those in [6]. 128 Z. Szafra´ nski, B. Szmanda 2. Main results Theorem 1. Assume that (i) qn≥0and ∞qn=∞, (ii) lim|u|→∞ inf |f(u)|>0. Then every solution of Equation (1) is oscillatory. Proof: Assume, that Equation (1) has nonoscillatory solution (xn), and we assume that (xn) is eventually positive. Then there is a positive integer n0such that (2) xn−τn>0for n≥n0. From the Equation (1) we have ∆(rn∆xn)=−qnf(xn−τn)≤0,n≥n0, and so (rn∆xn) is an eventually nonincreasing sequence. We first show that rn∆xn≥0 for n≥n0. In fact, if there is an n1≥n0such that rn1∆xn1=c<0 and rn∆xn≤c for n≥n1that is ∆xn≤c rn and hence xn≤xn1+c n−1  k=n1 1 rk →−∞as n→∞ which contradicts the fact that xn>0 for n≥n1. Hence rn∆xn≥0 for n≥n0. Therefore we obtain xn−τn>0,∆xn≥0,∆(rn∆xn)≤0 for n≥n0. Let L= lim n→∞ xn. Then L>0 is finite or infinite. Case 1. L>0 is finite. From the continuity of function f(u)wehave lim n→∞ f(xn−τn)=f(L)>0. Oscillations in difference equations 129 Thus, we may choose a positive integer n3(≥n0) such that (3) f(xn−τn)>1 2f(L)n≥n3. By substituting (3) into Equation (1) we obtain (4) ∆(rn∆xn)+1 2f(L)qn≤0,n≥n3. Summing up both sides of (4) from n3to n(≥n3), we obtain rn+1∆xn+1 −rn3∆xn3+1 2f(L) n  i=n3 qi≤0 and so 1 2f(L) n  i=n3 qi≤rn3∆xn3,n≥n3, which contradicts (i). Case 2. L=∞. For this case, from the condition (ii) we have lim n→∞ inf f(xn−τn)>0 and so we may choose a positive constant cand a positive integer n4 sufficiently large such that (5) f(xn−τn)≥cfor n≥n4. Substituting (5) into Equation (1) we have ∆(rn∆xn)+cqn≤0,n≤n4. Using the similar argument as that of Case 1 we may obtain a contradiction to the condition (i). This completes the proof. Theorem 2. Assume, that (iii) qn≥0and ∞Rnqn=∞, 130 Z. Szafra´ nski, B. Szmanda then every bounded solution of (1) is oscillatory. Proof: Proceeding as in the proof of Theorem 1 with assumption that (xn) is a bounded nonoscillatory solution of (1) we get the inequality (4) and so we obtain (6) Rn∆(rn∆xn)+1 2f(L)Rnqn≤0,n≥n3. It is easy to see that (7) Rn∆(rn∆xn)≥∆(Rnrn∆xn)−rn∆xn∆Rn. From inequalities (6) and (7) we deduce n  k=n3 ∆(Rkrk∆xk)− n  k=n3 ∆xk+1 2f(L) n  k=n3 Rqqk≤0n≥n3, which implies 1 2f(L) n  k=n3 Rkqk≤xn+1 +Rn3rn3∆xn3−xn3,n≥n3. Hence there exists a constant csuch that n  k=n3 Rkqk≤cfor all n≥n3, contrary to the assumption of the theorem. Theorem 3. Assume that (iv) (n−τn)is nondecreasing, where τn∈{0,1,2,...}, (v) there is a subsequence of (rn), say (rnk)such that rnk≤1for k=0,1,2,..., (vi) ∞ n=0 qn=∞, (vii) fis nondecreasing and there is a nonnegative constant Msuch that (8) lim u→0sup u f(u)=M. Oscillations in difference equations 131 Then the difference (∆xn)of every solution (xn)of Equation (1) oscillates. Proof: If not, then Equation (1) has a solution (xn) such that its difference (∆xn) is nonoscillatory. Assume first that the sequence (∆xn) is eventually negative. Then there is a positive integer n0such that ∆xn<0n>n 0 and so (xn) is decreasing for n≥n0which implies that (xn) is also nonoscillatory. Set (9) wn=rn∆xn f(xn−τn),n≥n1≥n0. Then (10) ∆wn=rn+1∆xn+1 f(xn+1−τn+1 )−rn∆xn f(xn−τn) =∆(rn∆xn) f(xn−τn)+rn+1∆xn+1 f(xn−τn)−f(xn+1−τn+1 ) f(xn+1−τn+1 )f(xn−τn) ≤∆(rn∆xn) f(xn−τn)=−qn,n≥n1. Summing up both sides of (10) from n1to n,wehave wn+1 −wn1≤− n  i=n1 qi and, by (vi), we get (11) lim n→∞ wn=−∞, which implies that eventually (12) f(xn−τn)>0 and therefore xn−τn>0. By (11), we can choose n2(≥n1) such that wn≤−(M+1),n≥n2. That is (13) rn∆xn+(M+1)f(xn−τn)≤0,n≥n2. 132 Z. Szafra´ nski, B. Szmanda Set lim n→∞ xn=L. Then L≥0. Now we prove that L=0. IfL>0, then we have lim n→∞ f(xn−τn)=f(L)>0, by the continuity of f(u). Choosing an n3sufficiently large, such that (14) f(xn−τn)>1 2f(L),n≥n3 and substituting (14) into (13), we have (15) ∆xn+1 2rn (M+1)f(L)≤0,n≥n3. Summing up both sides of (15) from n3to nwe get xn+1 −xn3+1 2(M+1)f(L) n  i=n3 1 ri ≤0 which implies that lim n→∞ xn=−∞. This contradicts (12). Hence lim n→∞ xn=0. By the assumptions we have lim n→∞ sup xn−τn f(xn−τn)≤M. From this we can choose n4, such that xn−τn f(xn−τn)<M+1,n≥n4. That is xn−τn<(M+1)f(xn−τn),n≥n4, and so from (13) we get (16) rn∆xn+xn−τn<0,n≥n4. In particular, from (16) for a subsequence (rnk) satisfying the condition (v), we have xnk+1 −xnk+xnk−τnk≤rnk(xnk+1 −xnk)+xnk−τnk<0, for ksufficiently large, which implies that 0<x nk+1 +(xnk−τnk−xnk)<0 for all large k. This is a contradiction. The case that (∆xn) is eventually positive can be treated in a similar fashion and so the proof of Theorem 3 is completed. Oscillations in difference equations 133 References 1. J. W. Hooker and W. T. Patula, Second order nonlinear difference equation: oscillation and asymptotic behaviour, J. Math. Anal. Appl. 91 (1983), 9–29. 2. J. Popenda, Oscillation and nonoscillation theorems for second order difference equations, J. Math. Anal. Appl. 123 (1987), 34–38. 3. Z. Szafra´ nski, On some oscillation criteria for difference equations of second order, Fasc. Math. 14 (1985), 91–99. 4. B. Szmanda, Oscillation criteria for second order nonlinear equations, Ann. Polon. Math. 43 (1983), 225–235. 5. B. Szmanda, Oscillatory behaviour of certain difference equations, Fasc. Math. 21 (1990), 65–78. 6. Z. Wang and J. Yu, Oscillation criteria for second order nonlinear difference equations, Funkcialaj Ekvacioj 34 (1991), 313–319. Institute of Mathematics Pozna´n University of Technology 60-965 Pozna´n POLAND Primera versi´o rebuda el 15 de Maig de 1995, darrera versi´o rebuda el 18 d’Octubre de 1995