Int. Journal of Math. Analysis, Vol. 4, 1, no. 36, 1787-1798 A Note on the Positive Nonoscillatory Solutions of the Difference Equation x n+1 = α c ix n i + x n k c ix n i ) Vu Van Khuong 1 and Mai Nam Phong 1 Deartment of Mathematics Hung Yen University of Technology and Education Hung Yen Province, Vietnam vuvankhuong@gmail.com Deartment of Mathematical Analysis University of Transort and Communications Hanoi City, Vietnam mnhong@gmail.com Abstract The aim of this note is to show that the following difference equation x n+1 = α c ix n i + x n k c ix n i where α, >, k N, c i,,,, c i = 1, has ositive nonoscillatory solutions which converge to the ositive equilibrium x = 1+ 1+4α. In the roof of the result we use a method develoed by L. Berg and S. Stević 1-4], 7-15]. ) Mathematics Subject Classification: 39A1 Keywords: Equilibrium, asymtotic, ositive solution, difference equation, nonoscillatory solution
1788 Vu Van Khuong and Mai Nam Phong 1 Introduction Recently, there has been a lot of interest in studying the global attractivity, the boundedness character and the eriodic nature of nonlinear difference equations. For some recent results see, for examle 5-15]. In 5] the authors have studied the behavior of all ositive solutions of the difference equation x n+1 = + x n, x n x n where is a ositive real arameter and the initial conditions x,x 1,x are ositive real numbers. For every the value of the ositive arameter, there exists a unique ositive equilibrium x which satisfies the equation x = x +. In this note we will investigate the behavior of the ositive solution of the difference equation x n+1 = α c ix n i + x n k c ix n i ) 1) where α, >, k N, c i, i =,..., k 1, c i = 1, and the initial conditions x k,x k+1,,x 1,x are arbitrary ositive real numbers. Note that the ositive equilibrium of Eq 1) also satisfies the equation x = x + α We say that a solution x n ) of equation 1) is bounded and ersists if there exists ositive constants P and Q such that P x n Q for n = k, k +1,...,, 1,... A ositive semicycle of a solution x n ) consists of a string of terms {x l,x l+1,..., x m }, all greater than or equal to x, with l k and m + and such that either l = k, or l> k and x l 1 < x, and either m =, or m< and x m+1 < x. A negative semicycle of a solution x n ) consists of a string of terms {x l,x l+1,..., x m }, all less than to x, with l k and m and such that and either l = k, or l> k and x l 1 x, either m =, or m< and x m+1 x. The first semicycle of a solutions starts with the term x k and is ositive if x k x. We now investigate oscillation of ositive solutions of the difference equation 1). We shall rove the following theorem, which is similar to the result from aer 5].
Positive nonoscillatory solutions 1789 Theorem 1.1. Let {x n } + n= k be a ositive solution of Eq1) for which there exists N k such that x N < x and x N+1 x, orx N x and x N+1 < x. Then the solution {x n } + n= k oscillates about the equilibrium x with every semicycle excet ossibly the first) having at most k terms. Proof. Let N k such that x N < x x N+1. The case where x N+1 < x x N is similar and will be omitted. Now suose that the ositive semicycle beginning with the term x N+1 has k terms. Then x N < x x N+i,i=1,,..., k and so α x N+k+1 = c + ix N+k i The roof is comlete. x P N c ix N+k i ) α x + min, x N {x N+k i } ) < α x +1=x. Before we investigate the local stability of the solution of Eq1) we quote the following well known result see 17]). Lemma 1.1. Assume that k i < 1. Then the zero equilibrium of the difference equation k y n+1 + i y n i = is globally asymtotically stable. In this section we study the local stability of the solutions of Eq1). Eq1) has two equilibriums x = 1+ 1+4α, x 1 = 1 1+4α We have the linearized equation for Eq. 1) about the ositive equilibrium x = 1+ 1+4α is αci y n+1 = x y n+1 + αci x + c ] i y n i + x x y n k The characteristic olynomial associated with Eq) is + c ] i y n i x x y n k = ) t k+1 + αci x + c ] i t k i x x = 3)
179 Vu Van Khuong and Mai Nam Phong Since αci < x + c ] i + x x < 1 α + x + x <x = x + α x <x < 1 by Lemma 1. we obtain that the equilibrium x is locally asymtotically stable with << 1. In the case > 1, x is unstable. The linearized equation for Eq1) about the ositive equilibrium x 1 = 1 1+4α < is y n+1 = y n+1 + αci x 1 + c i αci x 1 + c i x 1 The characteristic olynomial associated with Eq4) is t k+1 + x 1 ] y n i + x 1 y n k ] y n i x 1 y n k = 4) αci x + c ] i t k i = 5) 1 x 1 x 1 It is easy to see that αci x + c ] i > 1 1 x 1 x 1 and consequently the equilibrium x 1 is unstable. On the ositive nonoscillatory solutions of the difference equation 1) Our aim in this note is to solve the following roblem. Do there exists nonoscillatory solutions of Eq1)? We will solve this roblem by a method due to L. Berg and S. Stević, see, for examle, ], 7-15].
Positive nonoscillatory solutions 1791 Note that the linearized equation for Eq1) about the ositive equilibrium x can be written in the following equivalent form: x y n+1 + c i α + x)y n i xy n k = 6) The characteristic olynomial associated with Eq6) is gt) =x t k+1 + c i α + x)t k i x = 7) Since g) = x <,g1) = x + c i α + x) x = x + α> and g t) =x k +1)t k + c i k i)α + x)t k i 1 > when t, 1], it follows that for each >,α >, there is unique ositive root t of the olynomial belonging to the interval, 1). As suggested by Stević in 7], this fact motivated us to believe that there are solutions of Eq1) which have the following asymtotics x n = x + at n + otn ) 8) where a R and t is the above mentioned root of the olynomial 7). Asymtotics for solutions of difference equations have been investigated by L. Berg and S. Stević, see, for examle, 1-4], 7-15] and the reference therein. The roblem is solved by constructing two aroriate sequences y n and z n with y n x n z n 9) for sufficiently large n. In 1], ] some methods can be found for the construction of these bounds, see, also 3, 4]. From 5] and results in Berg s aer 3, 4] we exect that for k such solutions have the first three members in their asymtotics in the following form ϕ n = x + at n + bt n 1) The following result lays a crucial art in roving the main result. The roof of the result is similar to that of Theorem 1 in 16], we will give a roof for the benefit of the reader.
179 Vu Van Khuong and Mai Nam Phong Theorem.1. Let f : I k+ I be a continuous and nondecreasing function in each argument on the interval I R, and let y n ) and z n ) be sequences with y n <z n for n n and such that y n k fn, y n k+1,..., y n+1 ), fn, z n k+1,..., z n+1 ) z n k, for n>n + k 1 11) Then there is a solution of the following difference equation with roerty x n k = fn, x n k+1,..., x n+1 ) 1) y n x n z n for n n. 13) Proof. Let N be an arbitrary integer such that N > n + k 1. The solution x n ) of 1) with given initial values x N,x N+1,...,x N+k satisfying condition 13) for n {N,N +1,...,N + k} can be continued by 1) to all n<n. Inequalities 11) and the monotonic character of f imly that 13) holds for all n {n,...,n + k}. Let A N be the set of all k + 1)-tules x n,...,x n +k such that there exist solution x n ) of 1) with these initial values satisfying 13) for all n {n,...,n + k}. It is clear that A N is a closed nonemty set for every N>n + k 1, and that A N+1 A N. It follows that the set A = N=n +k A N is a nonemty subset of R k+1 and that if x n,...,x n +k) A, then the corresonding solution of 1) satisfy 11) for all n n, as desired. 3 The main result In this section, we rove the main result in this note. Theorem 3.1. For each α, > there is a nonoscillatory solution of Eq1) converging to the ositive equilibrium x = 1+ 1+4α, with the asymtotic behavior 1). Proof. First note that Eq1) can be written in the following equivalent form: x n k = x n+1 α ) 1 c ix n i c i x n i
Positive nonoscillatory solutions 1793 since ) x n+1 c i x n i = α + x ) 1 n k c i x n i We have and ) x n+1 c i x n i >α ] 1 x n k = x n+1 c i x n i ) α c i x n i ) 1 1 Let F x n k,x n k+1,...,x n,x n+1 )= ] 1 = x n+1 c i x n i ) α c i x n i ) 1 1 xn k = 14) fu n+1,u n,...,u n k+,u n k+1 )= u n+1 c i u n i ) α defines on the set = u n+1 ] 1 c i u n i ) α c i u n i ) 1 1 c i u n i ) 1 ] 1 A = {u n+1,u n,...,u n k+,u n k+1 ) R k+1 + : u n+1 c i u n i ) >α} We have f u n+1 = 1 u n+1 c iu n i ) α ] 1 c iu n i ) 1 1. c iu n i ) > f u n i = 1 u n+1 c iu n i ) α ] 1 c iu n i ) 1 1 c i u n+1 c iu n i ) 1 α 1)c i ] c iu n i ) = = 1 u n+1 c iu n i ) α ] 1 c iu n i ) 1 1 c i c iu n i ) u n+1 ] c iu n i )+α1 ) >, 1],
1794 Vu Van Khuong and Mai Nam Phong i =, 1,..., k 1 On the other hand, f = 1 u n+1 u n i c i u n i ) α c i u n i ) 1 ] 1 1 ] } c i c i u n i ) { u n+1 c i u n i ) α + α > on the set A, also for >1. Let I =x, ). Since for u n+1,u n,...,u n k+,u n k+1 x, ) u n+1 c iu n i ) > x = x + α>α, we have that x, ) k+1 A, so that f increases in each argument on x, ) and min u n+1,u n,...,u n k+,u n k+1 ) x, ) k+1 fu n+1,u n,...,u n k+,u n k+1 )=fx, x,...,x) =x that is, f : I k+1 I We exect that solutions of Eq1) have the asymtotics aroximation 1). Thus, we can calculate F ϕ n k,ϕ n k+1,...,ϕ n+1,ϕ n+1 ). We have F = x + at n+1 + bt n+ ) c i x + at n i + bt n i ) α c i x + at n i + bt n i ) ] 1 1 x + at n k + bt n k ) ) F = x + at n+1 + bt n+ ) x + a c i t n i + b t n i α x + a c i t n i + b ] 1 ] 1 ] 1 1 t n i x + at n k + bt n k ) F = x + axt n+1 + bxt n+ + ax c i t n i + a c i t n i+1 + ab c i t 3n i+ + + bx c i t n i + ab c i t 3n i+1 + b c i t 4n i+ α x 1 1+ a c it n i + b c it n i x ] 1 ] 1 x + at n k + bt n k )
Positive nonoscillatory solutions 1795 From x = x + α, we have F = x 1+ 1 axt n+1 + bxt n+ + ax c i t n i + a c i t n i+1 + ab c i t 3n i+ + x ) + bx c i t n i + ab c i t 3n i+1 + b c i t 4n i+ + + 1 x axt n+1 + bxt n+ + ax c i t n i + a c i t n i+1 + ab c i t 3n i+ + ) ] + bx c i t n i + ab c i t 3n i+1 + b c i t 4n i+ + 1+ 1 ) a c i t n i + b c i t n i x x + at n k + bt n k ) x t k+1 + F = a c iα + x)t k i x xt k We have + a +1 )x + 1 x a c i t n i + b ] { xt t n k+ + + b c i t i+1 + 1 )x +1 x x t k+1 + c iα + x)t k i x xt k c i t i) + = gt) xt k, 1 )x t ) ] c i t n i + c ] iα + x)t k i x + ]} t n + ot n ) where gt) is the characteristic olynomial 7). We know that there exists the unique root t, 1) such that gt ) =. Let gt ) = xtk+ + c iα + x)t k i x. From this, with t = t we have { F = b gt ) +1 )x + a c i t i+1 + 1 )x +1 x + ot n ), <t <t < 1, gt ) <gt )= Thus, the coefficient of b is negative: We set A = +1 )x gt ) <. c i t i+1 + 1 )x +1 x ]} ) c i t i 1 )x + t t n + c i t i ) + 1 )x t
1796 Vu Van Khuong and Mai Nam Phong Then Set Note that If we obtain F = b gt ) q = a.a gt ) H t b) = gt ) ] + a A t n + otn ). and H t b) =b gt ) < and H t q) =. + a A. ˆϕ n = x + at n + bt n = 1+ 4α +1 + at n + bt n, F ˆϕ n k, ˆϕ n k+1,..., ˆϕ n, ˆϕ n+1 ) b gt ) ] + a A t n = H t b)t n. Since H t b) = gt ) and H t q ) <. With the notations <, we obtain that there are q 1 <qand q >qsuch that H t q 1 ) > y n = x + at n + q 1 t n,z n = x + at n + q t n. We get gt F y n k,y n k+1,...,y n,y n+1 ) q ) ] 1 + a A t n > ] gt F z n k,z n k+1,...,z n,z n+1 ) q ) + a A t n <. These relations show that inequalities 11) are satisfied for sufficiently large n, where f = F + x n k and F is given by 14). Since for all n, y n >, we can aly Theorem.1 with I =x, ) and see that there is an n > and a solution of Eq1) with the asymtotics x n = ˆϕ n + ot n ), for n n, where ˆϕ n is defined by 1) and b = q. In articular, the solution converges monotonically to the ositive equilibrium x = 1+ 1+4α, for n n. Hence, the solution x n+n +k is also such a solution when n k.
Positive nonoscillatory solutions 1797 References 1] L. Berg, Asymtoticsche Darstellungen und Entwicklungen, Dt. Verlag Wiss, Berlin, 1968. ] L. Berg, On the asymtotics of nonlinear difference equations, Z. Anal. Anwendungen 14) ), 161-174. 3] L. Berg, Inclusion theorems for nonlinear difference equations with alications, J. Differ. Equations. Al. 14) 4), 399-48. 4] L. Berg, Corrections to Inclusion theorems for nonlinear difference equations with alications, J. Differ. Equations Al. 11) 5), 181-18. 5] E. Camouzis, R. Devault, and W. Kosmala, On the eriod five trichotomy of all ositive solutions of x n+1 = +x n x n, J. Math. Anal. Al., 91 4), 4-49. 6] R. Devault, C. Kent and W. Kosmala, On the recursive sequence x n+1 = + x n k x n, J. Differ. Equations Al. 9 8) 3), 71-73. 7] S. Stević, Asymtotics of some classes of higher order difference equations, Discrete Dyn. Nat. Soc. Vol. 7, Article ID 56813, 7), ages. 8] S. Stević and K. Berenhaut, A note on ositive nonoscillatory solutions of the difference equation x n+1 = α + x n k, J. Differ. Equations Al. 1 5) 6), 495-499. x n x 9] S. Stević and K. Berenhaut, The difference equation x n+1 = α + P n k has c ix n i solutions converging to zero, J. Math. Anal. Al. 36 7), 1466-1471. 1] S. Stević, Asymtotic behaviour of a nonlinear difference equation, Indian. J. Pure. Al. Math. 341) 3), 1681-1689. 11] S. Stević, On the recursive sequence x n+1 = α + x n 1, J. Al. Math and comuting x n 181-) 5), 9-34. 1] S. Stević, On the recursive sequence x n+1 = 1 x n + A x n 1, Inter. J. Math. Sci. 71) 1). 13] S. Stević, A global convergence results with alications to eriodic solutions, Indian J. Pure Al. Math. 33), 45-53. 14] S. Stević, A note on the difference equations x n+1 = k α i x i, J. Differ. Equations n i Al. 87) ), 641-647.
1798 Vu Van Khuong and Mai Nam Phong 15] S. Stević, Asymtotic behaviour of a nonlinear difference equation, Indian J. Pure Al. Math. 341) 3), 1681-1689. 16] S. Stević, On ositive solutions of a k+1)th order difference equation, Al. Math. Lett., inress. 17] S. Stević, On the recursive sequence x n+1 = α+βxn γ x n k, Bulletin of the Institute of Mathematics academia sinica.31)4), 61-7. Received: March, 1