On the Oscillation of Nonlinear Fractional Difference Equations

Transcription

1 Mathematica Aeterna, Vol. 4, 2014, no. 1, On the Oscillation of Nonlinear Fractional Difference Equations M. Reni Sagayaraj, A.George Maria Selvam Sacred Heart College, Tirupattur , S.India M.Paul Loganathan Department of Mathematics, Dravidian University, Kuppam Abstract In this paper, we study oscillatory behavior of the fractional difference equations of the following form t 1+α pt α xt γ +qtf t s 1 α xs = 0,t N t0 +1 α, where α denotes the Riemann-Liouville difference operator of order α, 0 < α 1 and γ > 0 is a quotient of odd positive integers. We establish some oscillation criteria for the above equation by using Riccati transformation technique and some Hardy type inequalities. An example is provided to illustrate our main results. Mathematics Subject Classification: 26A33, 39A11, 39A12 Keywords: difference equations, oscillation, nonlinear, fractional order. 1 Introduction Oscillatory behavior of fractional differential equations have been dealt by several authors, see [2]-[9] and the theory of fractional differential equations is presented in books, see [16]-[18]. But the qualitative properties of fractional difference equations are studied by few authors, see [10]-[15]. Motivated by [3] and [9], we study the oscillatory behavior of the following fractional difference equation of the form t 1+α pt α xt γ +qtf t s 1 α xs = 0,t N t0 +1 α, 1

2 92 M.R.Sagayaraj, A.G.M.Selvam and P.Loganathan for 0 < α 1. Here α denotes the Riemann-Liouville difference operator, γ > 0 is a quotient of odd positive integers. In this paper we assume the following conditions. H 1. pt and qt are positive sequences and f : R R is a continuous function such thatfx/x n k for a certain constant k > 0 and for all x 0. A solution xt of 1 is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation 1 is said to be oscillatory if all its solutions are oscillatory. 2 Preliminaries and Basic Lemmas In this section, we present some preliminary results of discrete fractional calculus, which will be used throughout this paper. Definition 2.1 see [12] Let ν > 0. The ν-th fractional sum f is defined by ν ft = 1 t ν t s 1 ν 1 fs, Γν s=a where f is defined for s a mod 1 and ν f is defined for t a + ν mod 1 and t ν = Γt+1. The fractional sum Γt ν+1 ν f maps functions defined on N a to functions defined on N a+v. Definition 2.2 see [12] Let µ > 0 and m 1 < µ < m, where m denotes a positive integer, m = µ. Set ν = m µ. The µ-th order Riemann-Liouville fractional difference is defined as µ ft = m ν ft = m ν ft. Lemma 2.3 Let xt be a solution of 1 and let Gt = t 1+α t s 1 α xs 2 then Gt = Γ1 α α xt. 3 Proof: Gt = t 1+α t s 1 α xs = t 1 α t s 1 1 α 1 xs =Γ1 α 1 α xt,

3 Oscillation of Nonlinear Fractional Difference Equations 93 which implies Gt = Γ1 α 1 α xt = Γ1 α α xt. In order to discuss our results in Section 3, now we state the following lemma. Lemma 2.4 Hardy et al. see [1] If X and Y are nonnegative, then mxy m 1 X m m 1Y m for m > 1 4 where equality holds if and only if X = Y. 3 Main Results Theorem 3.1 Suppose that H 1 and p 1/γ s = 5 holds. Furthermore, assume that there exists a positive sequence bt such that kbsqs b +s 2 =, 6 4b 2 s+1rs where Rt = btγ1 αγ b 2 t+1pt and b + s = max[ bs,0]. Then every solution of 1 is oscillatory. Proof: Suppose the contrary that xt is a nonoscillatory solution of 1. Without loss of generality, we may assume that xt is an eventually positive solution of 1. Then there exists t 1 > t 0 such that xt > 0 and Gt > 0 for t t 1, 7 where G is defined as in 2. Therefore, it follows from 1 that pt α xt γ = qtf Gt < 0 for t t 1. 8 Thus pt α xt γ is an eventually non increasing sequence. First we show that pt α xt γ is eventually positive. Suppose there is an integer t 1 > t 0 such that pt 1 α xt 1 γ = c < 0 for t t 1, so that pt α xt γ pt 1 α xt 1 γ = c < 0

4 94 M.R.Sagayaraj, A.G.M.Selvam and P.Loganathan which implies that Gt Γ1 α = α xt c 1/γ p 1/γ t for t t 1. Summing both sides of the last inequality from t 1 to t 1, we get Gt Gt 1 +Γ1 αc 1/γ p 1/γ s as t, 9 which contradicts the fact that Gt > 0. Hence pt α xt γ > 0 is eventually positive. Define the function wt by the Riccati substitution wt = bt pt α xt γ G γ t Then we have wt > 0 for t t 1. It follows that wt = bt wt+1 bt+1 for t t bt pt α xt γ G γ t+1 btpt+1 α xt+1 γ G γ t G γ t+1g γ t b + t wt+1 bt+1 btqtfgt G γ t Now using the inequality see [1] we have Using the above inequality x β y β x y β, pt α xt γ pt+1 α xt+1 γ α xt γ pt+1 α xt+1 γ. pt btpt+1 α xt+1 γ G γ t. G 2γ t+1 wt b + t wt+1 bt+1 kbtqt btpt+1 α xt+1 γ Gt γ G 2γ t+1 b + t wt+1 bt+1 kbtqt btpt+1 α xt+1 γ Γ1 α α xt γ b 2 t+1p 2 t+1 α xt+1 2γ wt+1 2 b + t wt+1 bt+1 kbtqt btγ1 αγ pt+1 pt α xt+1 γ b + t wt+1 bt+1 b 2 t+1pt+1 α xt+1 γ wt+1 2 kbtqt btγ1 αγ b 2 t+1pt wt+12 = b + t wt+1 bt+1 kbtqt Rtwt+12 11

5 Oscillation of Nonlinear Fractional Difference Equations 95 where Rt = btγ1 αγ. We now set b 2 t+1pt X = Rtwt+1 and Y = Using Lemma 2.4 and put m = 2, we obtain b + t 2bt+1 Rt. b + t 2 Rtwt+1 From 11, we conclude that 2bt+1 Rt b + t 2 1 2bt+1 Rt = b +t 2 4b 2 t+1rt Rtwt+1 2 wt kbtqt+ b +t 2 4b 2 t+1rt. Summing the above inequality from t 1 to t 1, we have kbsqs b +s 2 4b 2 wt 1 wt wt 1 <, for t t 1. s+1rs Letting t, we get which contradicts 6. The proof is complete. kbsqs b +s 2 4b 2 wt 1 <, s+1rs Theorem 3.2 Suppose that H 1 and p 1/γ s = hold. Furthermore, assume that there exists a positive sequence bt such that Ht,t = 0 for t t 0 Ht,s > 0 t > s t 0 2 Ht,s = Ht,s+1 Ht,s 0 for t > s t 0. If 1 Ht,t 0 bsqsht, s h 2 + t,s =, 12 4kHt, srs where h + t,s = 2 Ht,s+ Ht,s b +s bs+1 solution of 1 is oscillatory. and b + s = max[ bs,0]. Then every

6 96 M.R.Sagayaraj, A.G.M.Selvam and P.Loganathan Proof: Suppose the contrary that xt is a nonoscillatory solution of 1. Without loss of generality, we may assume that xt is an eventually positive solution of 1. We proceed as in the proof of Theorem 3.1 to get 11 hold. Multiplying 11 by Ht,s and summing from t 1 to t 1, we obtain kbsqsht,s Ht,s ws+ Ht,sRsw 2 s+1 Ht,s b + s ws+1 bs+1 13 Using the summation by parts formula, we obtain Ht,s ws = [Ht,sws] t + =Ht,t 1 wt 1 + ws+1 2 Ht,s ws+1 2 Ht,s. 14 Now, we have k bsqsht,s Ht,t 1 wt ws+1 2 Ht,s Ht,s b + s ws+1 bs+1 Ht,t 1 wt 1 + Ht,sRsw 2 s+1 Ht,t 1 wt 1 + Ht,sRsw 2 s+1 2 Ht,s+ Ht,s b +s bs+1 ws+1 h+ t,sws+1 Ht,sRsw 2 s+1 15 where h + t,s = 2 Ht,s+ Ht,s b +s bs+1 is defined as in Theorem 3.2. Set X = h + t,s Ht,sRsws+1 and Y = 2 Ht,sRs

7 Oscillation of Nonlinear Fractional Difference Equations 97 Using the Lemma 2.4 with m = 2, we get h + t,s 2 Ht,sRsws+1 2 Ht,sRs h + t,s Ht,sRs = h2 + t,s 4Ht,sRs Ht,sRsws+1 From equation 15, we have 2 Ht,s 0 for t > s t 0, 0 < Ht,t 1 Ht,t 0 for t > t 1 t 0 bsqsht,s k 1 Ht,t 1 wt 1 +k 1 bsqsht, s 2 h2 + t,s 4kHt, srs h 2 + t,s k 1 Ht,t 1 wt 1 4kHt, srs k 1 Ht,t 0 wt 1. Since 0 < Ht,s Ht,t 0 for t > s t 0, we have 0 < Ht,s Ht,t 0 1 for t > s t 0. Hence it follows from that 1 Ht,t 0 Letting t, we have 1 Ht,t 0 = 1 Ht,t Ht,t 0 bsqsht,s h2 +t,s 4Ht, srs t 1 1 bsqsht,s h2 +t,s 4Ht, srs bsqsht,s h2 +t,s 4Ht, srs 1 t 1 1 bsqsht,s+k 1 wt 1 Ht,t 0 t 1 1 bsqs+k 1 wt 1. bsqsht, s which is a contradiction to 12. The proof is complete. h 2 t +t,s 1 1 bsqs+k 1 wt 1 <, 4kHt, srs

8 98 M.R.Sagayaraj, A.G.M.Selvam and P.Loganathan Example 3.3 Consider the fractional differential equation t γ 1 α xt γ + 1 t 1+α t 2 t s 1 α xs γ = 0, 16 where α = 0.5, γ > 0 is a quotient of odd positive integers and k = 1, pt = t γ 1, qt = 1 t. Since 2 p 1/γ 1 s = =, t1 1/γ we find that H 1 and 5 hold. We will apply Theorem 3.1 and it remains to satisfy condition 6. Taking bs = s, we obtain kbsqs b +s 2 4b 2 s+1rs = limsup 1 s 1 sγ 1 4 π γ = which implies that 6 holds. Therefore, by Theorem 3.1 every solution of 16 is oscillatory. References [1] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge [2] Said R. Grace, Ravi P. Agarwal, Patricia J.Y. Wong, Ağacik Zafer, On the oscillation of fractional differential equations, FCAA, Vol15, No [3] Da-Xue Chen, Oscillation criteria of fractional differential equations, Advances in Difference Equations 2012, 2012:33. [4] Da-Xue Chen, Oscillatory behavior of a class of fractional differential equations with damping, U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 1, [5] Da-Xue Chen, Pei-Xin Qu, Yong-Hong Lan, Forced oscillation of certain fractional differential equations, Advances in Difference Equations 2013, 2013:125. [6] Chunxia Qi, Junmo Cheng, Interval oscillation criteria for a class of fractional differential equations with damping term, Hindawi Publishing Corporation, Mathematical Problems in Engineering, Volume 2013, Article ID , 8 pages. [7] S.Lourdu Marian, M. Reni Sagayaraj, A.George Maria Selvam, M.Paul Loganathan, Oscillation of fractional nonlinear difference equations, Mathematica Aeterna, Vol. 2, 2012, no. 9, [8] S.Lourdu Marian, M. Reni Sagayaraj, A.George Maria Selvam, M.Paul Loganathan, Oscillation of Caputo like Discrete fractional equations, IJPAM, Vol89 No , ,doi:

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