OSCILLATION OF SOLUTIONS TO NON-LINEAR DIFFERENCE EQUATIONS WITH SEVERAL ADVANCED ARGUMENTS. Sandra Pinelas and Julio G. Dix

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1 Opuscula Mat. 37, no. 6 (2017), ttp://dx.doi.org/ /opmat Opuscula Matematica OSCILLATION OF SOLUTIONS TO NON-LINEAR DIFFERENCE EQUATIONS WITH SEVERAL ADVANCED ARGUMENTS Sandra Pinelas and Julio G. Dix Communicated by Stevo Stević Abstract. Tis work concerns te oscillation and asymptotic properties of solutions to te non-linear difference equation wit advanced arguments x n+1 x n = f i,n(x n+i,n ). We establis sufficient conditions for te existence of positive, and negative solutions. Ten we obtain conditions for solutions to be bounded, convergent to positive infinity and to negative infinity and to zero. Also we obtain conditions for all solutions to be oscillatory. Keywords: advanced difference equation, non-oscillatory solution. Matematics Subject Classification: 39A INTRODUCTION In recent years tere as been a lot of researc concerning te oscillation of solutions to difference and differential equations wit advanced arguments. Tese equations appear in matematical models in wic te present state depends on future states [1, 4, 7, 19]. Te strong interest in tese equations arises from aving applications suc as population dynamics were a difference equation wit constant advanced arguments can serve as a matematical model tat includes a k-t generation [3]. Nowadays tere exists an extensive literature on te oscillation teory of advanced type differential and difference equations. See, for example, te references in tis article, and te references terein. In tis article we study te oscillation and asymptotic properties of solutions to te advanced difference equation x n+1 x n = f i,n (x n+i,n ), (1.1) were {f i,n } n=1 are sequences of real-valued functions and { i,n } n=1 are sequences of positive integers. c Wydawnictwa AGH, Krakow

2 888 Sandra Pinelas and Julio G. Dix In Section 2, we present some conditions for te existence of positive, and of negative solutions to a linear version of (1.1). In Section 3, by extending a result in [5], we obtain conditions for all oscillations to be bounded, and to tend to zero. In Section 4, we obtain conditions for every solution to be oscillatory. Also we compare our conditions wit tose obtained in [13] for constant advances. Also we illustrate our results wit examples. To study te oscillation of solutions, we assume tat solutions exist and are defined for all n large enoug. Quite frequently solutions are obtained as fixed points of contraction mappings, wic is te case in Teorem 3.4 below. In general it is not clear ow to formulate initial-value problems for advanced difference and differential equations. However, in special cases we can obtain a unique of solution. Consider te difference equation x n+1 x n = f n (x n, x n+1,..., x n+m ), (1.2) wit m > 1. For eac fixed set of values n, a 1,..., a m, we assume tat te function f n (a 1, a 2,..., a m, ) is one-to-one and onto from R to R. Ten using te initial data x 1, x 2,..., x m we solve for x m+1 in te equation x 2 x 1 = f 1 (x 1, x 2,..., x m+1 ). Ten using te data x 2,..., x m+1 we solve for x m+2 in te equation x 3 x 2 = f 1 (x 2, x 3,..., x m+2 ). Ten solve for x m+3 in x 4 x 3 = f 1 (x 3, x 4,..., x m+3 ), etc. Tis way we construct te solution by defining one entry at te time. Tis is known as te metod of steps. 2. EXISTENCE OF NON-OSCILLATORY SOLUTIONS By a solution {x n } we mean a sequence of real numbers tat satisfies (1.1) for all n large enoug. A solution is said to be oscillatory if for every positive integer n 0 tere exist n 1, n 2 n 0 suc tat x n1 x n2 0. A non-oscillatory solution is eiter eventually positive or eventually negative. In tis section we restrict our attention to te particular case of (1.1), wen f i,n (x) = x; tus we ave te linear difference equation x n+1 x n = x n+i,n. (2.1) Wen p < n, we use te conventions p i=n x i = 1 and p i=n x i = 0. Teorem 2.1. Assume tat 0 and tat tere exists a sequence {u n } of non-negative terms satisfying u n i,n Ten (2.1) as positive solutions, and negative solutions. u n+j, n n 0. (2.2)

3 Oscillation of solutions to non-linear difference equations Proof. Let {u n } be a solution of (2.2), and let v 1,n := u n. Ten for n n 0 and l 0, we define a double indexed sequence {v l,n } recursively by Ten, by (2.2), v l+1,n+1 = 1 + i,n v l,n+j 0 v 2,n+1 = 1 + i,n v 1,n+j v 1,n+1, n n 0. By induction, we can sow tat 0 v l+1,n+1 v l,n+1 v 1,n+1. Ten, for eac fixed n + 1, te limit lim l + v l,n+1 =: v n+1 exists. Tis limit satisfies Ten for any x n0, te sequence v n+1 = 1 + x n = x n0 i,n n v i v n+j. is a solution of (2.1). Wen x n0 > 0 tis is a positive solution and wen x n0 < 0 tis is a negative solution. Now we consider te difference equation wose coefficients and advanced arguments are smaller tan tose of (2.1), y n+1 y n = b i,n y n+gi,n. (2.3) Teorem 2.2. Assume te conditions of Teorem 2.1 old, and tat 1 g i,n i,n, 0 b i,n for n 1. Ten (2.3) as positive solutions and negative solutions. Proof. Let {u n } be a non-negative solution of (2.2). Ten u n 1 and u n i,n u n+j 1 + i,n b i,n u n+j 1 + b i,n g i,n u n+j. Using tis inequality in Teorem 2.1, we ave te existence of positive solutions and negative solutions. Corollary 2.3. Assume tat te sequences { } n and { i,n } n are bounded as follows: 0 a i and 1 i,n i for 1 i m and n n 0. Also assume tat te inequality λ 1 + a i λ i (2.4) as a non-negative solution λ. Ten (2.1) as positive solutions and negative solutions.

4 890 Sandra Pinelas and Julio G. Dix Proof. We consider te equation wit constant coefficients and constant advances y n+1 y n = a i y n+i. (2.5) Note tat using (2.4), we can sow tat te constant sequence u n = λ satisfies (2.2) wit = a i. Ten by Teorem 2.1, we ave positive solutions and negative solutions to (2.5). Since a i and i,n i, by applying Teorem 2.2, we ave positive solutions and negative solutions to (2.1). Now we consider an equation wit positive and negative coefficients, x n+1 x n = ( x n+i,n b i,n x n+gi,n ). (2.6) Teorem 2.4. Suppose tat 0 b i,n, and 1 g i,n i,n for 1 i m and n n 0. If inequality (2.2) as a nonnegative solution, ten (2.6) as positive solutions and negative solutions. Proof. Let {u n } be a nonnegative solution of (2.2) and let v 1,n = u n. We define te double indexed sequence {v l,n } recursively by v l+1,n+1 = 1 + By (2.2), we ave v 1,n i,n ( i,n g i,n v l,n+j b i,n v l,n+j ), for n n 0, l 1. i,n ( v 1,n+j g i,n ) v 1,n+j b i,n v 1,n+j = v 2,n+1. Since 0 b i,n and 1 g i,n i,n and 1 v 1,n, it follows tat 1 v 2,n. Next, we use induction: Assuming tat 1 v l 1,n, from 0 b i,n it follows tat 1 v l,n. Now assuming tat v l,n v l 1,n, we wis to sow tat v l+1,n v l,n wic by definition as te form i,n ( i,n ( g i,n ) v l,n+j b i,n v l,n+j g i,n v l 1,n+j b i,n v l 1,n+j ).

5 Oscillation of solutions to non-linear difference equations Te above inequality is equivalent to [ gi,n ( v l,n+j [ gi,n ( v l 1,n+j i,n j=g i,n+1 v l,n+j b i,n )] i,n j=g i,n+1 v l 1,n+j b i,n )]. Tis inequality follows from te assumptions 1 v l,n v l 1,n and 0 b i,n. Terefore, 1 v l+1,n v l,n. Consequently, for eac fixed n te limit lim l v l,n := v n exists and is non-negative. Te sequence defined wit tese limits satisfies 0 v n+1 = 1 + Ten for any x n0, te sequence i,n ( x n = x n0 g i,n v j+n b i,n v j+n ), n n 0. n v i is a solution of (2.1). Wen x n0 > 0 tis is a positive solution and wen x n0 < 0 tis is a negative solution. 3. ASYMPTOTIC BEHAVIOR In tis section we study te beavior of solutions to (1.1), as n. Some of our results are analog to tose in [15] for continuous variables. Our first result uses te assumption (H1) Tere exists constants 0 suc tat for i = 1,..., m and n 1: { x if x 0, f i,n (x) x if x < 0. An example of function satisfying te above condition is f i,n (x) = x(2 + sin(x)), wit = 1. Teorem 3.1. Let {x n } be a solution of (1.1). Assume (H1) and + a j,i = +. (3.1) If {x n } is eventually positive ten lim n x n = +. If {x n } is eventually negative ten lim n x n =.

6 892 Sandra Pinelas and Julio G. Dix Proof. Let x n > 0 for n n 0, ten by (1.1) and (H1), {x n } is non-decreasing and m x n+1 x n x n+i,n x n0. Summing bot sides of tis inequality from n 0 to n, we obtain n x n+1 x n0 (1 + a i,j ). Wen n, by (3.1), we obtain lim n + x n = +. Te negative case as a similar proof. For te next result we use te assumption (H2) Tere exists constants 0 suc tat for i = 1,..., m and n 1: { x if x 0, f i,n (x) x if x < 0. Teorem 3.2. Let {x n } be a solution of (1.1). Assume (H2) and + If {x n } is non-oscillatory ten lim n x n = 0. a j,i =. (3.2) Proof. Assume tat {x n } is a positive solution of (1.1). Ten by (1.1) and (H2), {x n } is non-increasing. So, {x n } as a nonnegative limit, α = lim n + x n. If α > 0, ten by (H2), x n+1 x n x n+i,n α. Summing bot sides of tis inequality from n 0 to n and using (3.2), we get lim n x n =. Tis contradicts {x n } begin eventually positive; terefore α = 0. Te eventually negative case as a similar proof. We use te forward difference operator x n = x n+1 x n in te following lemma. In te proof of te lemma we essentially follow te lines of a metod for solving a difference equation, an area of considerable recent interest (see, for example [16 18] and te references terein, were closely related metods were used). Lemma 3.3. Assume tat A n := 1. Ten a sequence {x n } is a solution of (2.1) if and only if it is a solution of x n = x n0 (1 + A i ) + ( m l+ i,l 1 a i,l ) x j i=l+1 (1 + A i ). (3.3)

7 Oscillation of solutions to non-linear difference equations Proof. Using te telescoping property x p x n = p 1 i=n x i, and (2.1), we ave tat {x n } is a solution of (2.1) if and only if Hence, n+ i, x n = x n+i,n = (x n + x j ). x n x n A n = n+ i, j=n j=n x j. Multiplying by n (1 + A i ) 1 bot sides of te last equality, we obtain ) (x n (1 + A i ) 1 = ( m n+ i, j=n x j ) n (1 + A i ) 1. Canging te variable n by l and summing bot sides form l = n 0 to l = n 1, we ave x n (1 + A i ) 1 x n0 = ( m l+ i,l 1 a i,l x j ) l (1 + A i ) 1. Multiplying it by (1+A i ), we obtain (3.3). Now starting from (3.3) and retracing te steps above we obtain equation (2.1). Tis completes te proof. Teorem 3.4. Assume te conditions of Lemma 3.3 are satisfied and tat tere are constants α and n 0 suc tat ( m Also assume tat l+ i,l 1 a i,l k=1 ) a k,j i=l A i α < 1 for n n 0. (3.4) 1 + A i remains bounded as n. (3.5) Ten for eac initial value x n0, tere is a unique solution to (2.1); furtermore, tis solution is bounded. Proof. Let B be te collection of bounded sequences tat ave a common value at n = n 0. Ten B is closed subset of a complete metric space under te supremum norm. Based on Lemma 3.3, we define te operator T : B B by T x n = x n0 (1 + A i ) + ( m l+ i,l 1 a i,l ) x j i=l+1 (1 + A i ). (3.6)

8 894 Sandra Pinelas and Julio G. Dix Note tat wen n = n 0, tere are no terms in te product and no terms in te summation; terefore T x n0 = x n0. To estimate T x n, we note tat by (2.1), Ten by (3.6), we ave T x n x n0 x j k=1 1 + A i + sup n 0 p a k,j x j+k,j sup j p x p ( m x p a k,j. (3.7) k=1 l+ i,l 1 a i,l i=k ) a k,j i=l A i. (3.8) Since {x n } is bounded, by (3.4) and (3.5), te sequence {T x n } is also bounded. Now we sow tat T is a contraction. Let {x n } and {y n } be sequences in B. Te same process as for (3.8) yields T x n T y n sup x p y p p n 0 ( m l+ i,l 1 a i,l Using te norm x = sup n n0 x n and condition (3.4), we ave T x T y α x y. i=k ) a k,j i=l A i. Terefore T is a contraction on B and as a unique fixed point, wic by Lemma 3.3 is te unique solution of (2.1) wit te given initial value x n0. Note tat under te assumptions of Teorem 3.4, all solutions must be bounded. For any solution we use its value x n0 as te initial value in Teorem 3.4. Now we provide an example were te assumptions of Teorem 3.4 are satisfied. Example 3.5. Let m = 5, f i,n (x) = x wit = ( 1) i+n /(i + n) 2. Note tat A n = (n + 1) 2 (3.9) wic satisfies te assumption in Lemma 3.3. Ten 1 + A i ( 1 + To estimate p we use tat ln(1 + x) x for x > 1. ln(p) = ln n n 0 ( 1 + ) 5 (i + 1) 2 =: p. ) 5 (i + 1) 2 5 (i + 1) 2 ( 5 1 x 2 dx 5 1 ) 5. n 0 n n 0

9 Oscillation of solutions to non-linear difference equations Terefore, ( A i exp n 0 ), (3.10) wic indicates tat (3.5) is satisfied. Using (3.9) and tat i,n 4, we ave By (3.10), 4(5)2 (l + 1) 4 exp 5 l+ i,l 1 a i,l k=1 a k,j 4(5)2 (l + 1) 4. ( ) ( ) 5 4(5)2 5 ( ) 1 exp l 5 n 0 (l + 1) 4 4(5)2 5 1 exp 5 n 0 3n 3. 0 In te above inequality we used integration from n 0 to n, as for (3.10). Finally for n 0 = 5 te above expression is less tan 1, and (3.4) is satisfied. Ten by Teorem 3.4, for eac initial value x n0, tere is a unique solution. 4. OSCILLATION OF ALL SOLUTIONS TO (1.1) Li [13] used elaborate estimates to obtain te oscillation of solutions to x n x = x n+i. Here, we use simple estimates to obtain conditions for te oscillation of all solutions to (1.1). Ten we compare our conditions wit tose in [13]. First we extend a result in [5] from an equation wit single and constant advance to multiple and variable advances. Teorem 4.1. Assume (H1) and 2 min i,n =: 0, 1 i m 1 n (4.1) lim inf > 1 (1 1 ) 0 1. n 0 0 (4.2) Ten every solution of (1.1) is oscillatory. Proof. We assume tat tere is an eventually positive solution {x n } of (1.1) and obtain a contradiction. Te proof for eventually negative solutions is similar and is omitted. Let x n > 0 for all n n 0. From (H1) we ave x n+1 x n x n+i,n. (4.3)

10 896 Sandra Pinelas and Julio G. Dix By (H1), {x n } is a non-decreasing sequence; tus x n+0 x n+i,n. Let r n = x n /x n+1. Ten 0 < r n 1 for all n n 0. From (4.3) we ave 1 r n 1 r n+1 r n+2 r +0, for n n 0. Let c be te average between te two sides of inequality (4.2). Ten 1 1 r n > c. r n+1 r n+2 r +0 Let γ = sup n n0 r n. Ten 0 < γ 1, and taking te infimum over n, in bot sides, we ave 1 γ c/γ 0 1, wic implies (1 γ)γ 0 1 c. Te left-and side attains its maximum wen γ = ( 0 1)/ 0. Terefore 1 ( 1 1 ) 0 1 (1 γ)γ 0 1 c > 1 (1 1 ) tis contradiction competes te proof. Teorem 4.2. Assume (H1), (4.1), and lim sup n > 1. (4.4) Ten every solution of (1.1) is oscillatory. Proof. We assume tat tere is an eventually positive solution {x n } of (1.1) and obtain a contradiction. Te proof for eventually negative solutions is similar and is omitted. As in te proof of Teorem 4.1, we divide (4.1) by x n+1 and let r n = x n /x n+1. Since 0 < r n 1 we ave 1 > 1 r n 1 r n+1 r n+2 r +0 Tis contradicts (4.4) and completes te proof. for n n 0. We remark tat condition (7) in [13] and condition (4.4) ere are independent of eac oter. Our values and i,n correspond to p i (n) and k i respectively in [13]. Let f i,n (x) = x, m = 1, i,n = 2 and p i (n) = = { 3/2 if n is a multiple of 3, 0 oterwise.

11 Oscillation of solutions to non-linear difference equations Wen p i (n) = 0, te summand in (7) is zero. Wen p i (n) 0, te terms q 1, q 2,... are zero in (7); tus te summands in (7) are zero. In bot cases (7) is not satisfied wile (4.4) is satisfied. Acknowledgements Te publication was supported by te Ministry of Education and Science of te Russian Federation (te Agreement number No: 02.a ). REFERENCES [1] R. Agarwal, L. Berezansky, E. Braverman, A. Domosnitsky, Nonoscillation Teory of Functional Differential Equations wit Applications, Springer, New York, Dordrect, Heidelberg, London, [2] L. Berezansky, E. Braverman, S. Pinelas, On nonoscillation of mixed advanced-delay differential equations wit positive and negative coefficients, Comput. Mat. Appl. 58 (2009) 4, [3] F.M. Dannan, S.N. Elaydi, Asymptotic stability of linear difference equations of advanced type, J. Comput. Anal. Appl. 6 (2004) 2, [4] L.E. El sgol c, Introduction to te Teory of Differential Equations wit Deviating Arguments, Holden-Day, Inc., San Francisco, [5] L.H. Erbe, B.G. Zang, Ocillation of discrete analogues of delay equations, Differential and Integral Equations 2 (1989) 3, [6] N. Fukagai, T. Kusano, Oscillation teory of first order functional-differential equations wit deviating arguments, Ann. Mat. Pura Appl. 136 (1984) 1, [7] I. Györi, G. Ladas, Oscillation Teory of Delay Differential Equations, Oxford Mat. Monogr., Te Clarendon Press, Oxford University Press, New York, [8] R.G. Koplatadze, T.A. Canturija, Oscillating and monotone solutions of first-order differential equations wit deviating argument, Differ. Uravn. 18 (1982) 2, [in Russian]. [9] M.R. Kulenović, M.K. Grammatikopoulos, Some comparison and oscillation results for first-order differential equations and inequalities wit a deviating argument, J. Mat. Anal. Appl. 131 (1988) 1, [10] T. Kusano, On even-order functional-differential equations wit advanced and retarded arguments, J. Differential Equations 45 (1982) 1, [11] G. Ladas, I.P. Stavroulakis, Oscillations caused by several retarded and advanced arguments, J. Differential Equations 44 (1982) 1, [12] X. Li, D. Zu, Oscillation and nonoscillation of advanced differential equations wit variable coefficients, J. Mat. Anal. Appl. 269 (2002) 2, [13] X. Li, D. Zu, Oscillation of advanced difference equations wit variable coefficients, Ann. Differential Equations 18 (2002) 2,

12 898 Sandra Pinelas and Julio G. Dix [14] H. Onose, Oscillatory properties of te first-order differential inequalities wit deviating argument, Funkcial. Ekvac. 26 (1983) 2, [15] S. Pinelas, Asymptotic beavior of solutions to mixed type differential equations, Electron. J. Differential Equations 2014 (2014) 210, 1 9. [16] S. Stević, On some solvable difference equations and systems of difference equations, Abstr. Appl. Anal (2012), 11 pp. [17] S. Stević, On a solvable system of difference equations of kt order, Appl. Mat. Comput. 219 (2013), [18] S. Stević, M.A. Algamdi, A. Alotaibi, N. Sazad, On a iger-order system of difference equations, Electron. J. Qual. Teory Differ. Equ. Art. (2013), Article ID 47, [19] B.G. Zang, Oscillation of te solutions of te first-order advanced type differential equations, Sci. Exploration 2 (1982) 3, Sandra Pinelas sandra.pinelas@gmail.com RUDN University 6 Mikluko-Maklay St. Moscow, , Russia Julio G. Dix jd01@txstate.edu Department of Matematics Texas State University 601 University Drive San Marcos, TX 78666, USA Received: February 18, Revised: Marc 11, Accepted: Marc 12, 2017.