International Journal of Difference Equations ISSN 0973-6069, Volume 3, Number, pp. 9 3 208 http://campus.mst.edu/ijde Oscillaty Solutions of Nonlinear Fractional Difference Equations G. E. Chatzarakis School of Pedagogical and Technological Education ASPETE Department of Electrical and Electronic Engineering Educats N. Heraklio, Athens, 42, Greece. geaxatz@otenet.gr and gea.xatz@aspete.gr P. Gokulraj, T. Kalaimani and V. Sadhasivam Thiruvalluvar Government Arts College PG and Research Department of Mathematics Rasipuram, Namakkal - 637 40, Tamil Nadu, India. gokulxlr8@gmail.com, kalaimaths4@gmail.com, ovsadha@gmail.com Abstract In this paper, we study the oscillaty behavi of the fractional difference equation of the fm α xt γ + qtfxt = 0, t N t0 + α, where α denotes the Riemann left fractional difference operat of der α, 0 < α and γ > 0 is a quotient of odd positive integers. We establish some oscillaty criteria f the above equation, using the Riccati transfmation and Hardy type inequalities. Examples are provided to illustrate our theetical results. AMS Subject Classifications: 26A33, 39A2. Keywds: Oscillation, difference equations, fractional sum. Introduction Fractional difference equations have received considerable attention during the recent years. Fractional calculus finds significant application in the fields of viscoelasticity, Received July 4, 207; Accepted December 9, 207 Communicated by Agnieszka Malinowska
20 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam capacit they, electrical circuits, electro-analytical chemistry, tum growth models, neurology, control they, statistics and a review on this direction, see [6, 8, 20, 22, 23, 25, 26]. Significant progress has been made in the study of fractional differential equations, see [6, 7, 0, 2, 4, 27, 3]. In contrast, very little progress has been made in they of fractional difference equations, see [ 5, 8, 9,, 5, 7, 2]. In particular, we observe that the oscillation of fractional difference equations has been studied by many auths in recent researches [3,9,24,28,29]. This is one of reasons to study difference equations with fractional der. Strong interest in the fractional difference equation. is motivated by the fact that it represents a discrete analogue of the following fractional differential equation D α a xt + qtfxt = 0 f 0 α <, t [a, + ], a > 0, where D α a denotes the Riemann Liouville differential operat of der α and the above problem was investigated by Wang et al [30]. The objective is to study the oscillaty behavi of the solutions of fractional difference equations of the fm α xt γ + qtfxt = 0 f 0 < α, t N t0 + α.. Here α denotes the Riemann left fractional difference operat and γ > 0 is a quotient of odd positive integers. In the paper, we assume the conditions H qt is a positive sequence and f : R R is a continuous function such that fx k f a positive constant k, n is a natural number f all x 0 and [ c qt H2 x n ] γ N f t t0 where c < 0 and N > 0. xt α xt + M, xt α xt M, t t 0 f all α xt + 0, α xt 0 and f some positive constants M, M xt 2 xtxt + J, 2 xt T f some positive constants J and T. A solution xt of. is said to be oscillaty if it has no last zero, i.e., if xt = 0, then there exists a t 2 > t such that xt 2 = 0. Equation. itself is said to be oscillaty if every solution of. is oscillaty. A solution xt which is not oscillaty is called nonoscillaty. 2 Preliminaries In this section, we present some preliminary results from discrete fractional calculus. We will make use of these results, throughout the paper.
Oscillaty Solutions of Nonlinear Fractional Difference Equations 2 Definition 2. See [23]. Let ν > 0. The νth fractional sum f is defined by ν ft = Γν t ν t s ν fs, s=a where f is defined f s a mod, ν ft is defined f t a + ν mod and t ν Γt + = Γt ν +. The fractional sum ν f maps functions defined in N a to functions defined in N a+ν. Definition 2.2 See [23]. Let µ > 0 and m < µ < m, where m denotes a positive integer, m = µ. Set ν = m µ. The µth der Riemann left fractional difference is defined as µ ft = m ν ft = m ν ft, where ν ft is νth fractional sum. Lemma 2.3 See [29]. If Gt = t s α xs, t +α then Gt = Γ α α xt. Lemma 2.4 See [7]. If X and Y are nonnegative, then mxy m X m m Y m f m >. 3 Main Results Theem 3.. Suppose that H and H2 hold and q γ s =. Furtherme, assume that there exists a positive sequence rt such that krsqs r +s 2 =, 3. 4rs + where r + s = max{ rs, 0}. Then every solution of. is oscillaty.
22 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam Proof. Suppose to the contrary that xt is a nonoscillaty solution of.. Without loss of generality, we can assume that xt is an eventually positive solution of.. Then there exists t > t 0 such that xt > 0, Gt > 0 and fxt > 0 f t t, where Gt is defined as in Lemma 2.3. From. we have α xt γ = qtfxt < 0 f t t. Thus α xt γ is an eventually non increasing sequence. Next we show that α xt γ is eventually positive. Suppose there exists an integer t > t 0 such that α xt γ = c < 0 f t t, so that α xt γ α xt γ = c < 0, α xt γ c, α xt c γ. Applying Lemma 2.3, we get that Gt Γ α c qt γ γ,, qt γ Thus i.e., Gt Γ α c γ qt γ qt γ. c γ Gt Γ α qt γ, qt Gt Γ α Nqt γ. Summing both sides of the last inequality from t to t, we get Gt Gt + Γ α Nqt γ as t, which contradicts the fact that Gt > 0. Hence α xt γ is eventually positive. Define the function wt by the Riccati substitution wt = rt α xt γ. x γ t
Oscillaty Solutions of Nonlinear Fractional Difference Equations 23 Since rt > 0, xt > 0 and α xt γ > 0, we have wt > 0. Now [ ] rt α xt γ wt = x γ t [ ] α xt γ = rt + α xt + γ rt x γ t x γ t + [ ] x γ t α xt γ α xt γ x γ t = rt x γ tx γ t + r [ ] +t qtfxt wt + + rt rt rt + x γ t + r +t wt + rtqtk rt + [ ] α xt γ x γ t w 2 t + rt + α xt γ α xt + γ r [ +t rt + wt + rtqtk rt + r +t wt + rtqtk M rt + wt + + rt rt + [ α xt γ x γ t x γ tx γ t + xt α xt + ] γ w 2 t + γ rt + w2 t +. 3.2 r + t Let X = wt + and Y = rt + 2. Using Lemma 2.4 and setting rt + M γ m = 2, we obtain M 2 γ r + t wt + rt + 2 rt + M γ rt + w2 t + r +t 2 4rt + which implies that r + t wt + rt + rt + w2 t + r +t 2 4rt + M, γ wt krtqt + r +t 2 4rt +. Summing the above inequality from t to t, we get ws krsqs + r +s 2 4rs + wt wt krsqs + r +s 2 4rs +,, ]
24 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam i.e., krsqs r +s 2 wt 4rs + wt wt < f t t. Letting t, we have krsqs r +s 2 wt 4rs + <, which contradicts 3.. The proof is complete. Theem 3.2. Suppose that H and H2 hold and q γ s =. Furtherme, assume that there exists a positive sequence rt, and a double positive sequence Ht, s such that Ht, t = 0 f t t 0, Ht, s > 0 f t > s t 0, s Ht, s = Ht, s + Ht, s 0 f t > s t 0. If Ht, t 0 rsqsht, s k h2 +t, srs + =, 3.3 4Ht, s where h + t, s = 2 Ht, s + Ht, s r +s rs + and r +s = max{ rs, 0}, then every solution of equation. is oscillaty. Proof. Suppose to the contrary that xt is a non-oscillaty solution of.. Without loss of generality, we can assume that xt is an eventually positive solution of.. Proceeding as in Theem 3., we arrive at equation 3.2. Multiplying 3.2 by Ht, s and summing from t to t, we get Ht, s ws r+ s ws + Ht, s rs + rsqskht, s Ht, s rs + w2 s +,
Oscillaty Solutions of Nonlinear Fractional Difference Equations 25 rsqskht, s Ht, s ws r+ s + ws + Ht, s Ht, s rs + rs + w2 s +. Using the summation by parts fmula, we have that Ht, s ws = [Ht, sws] t + which implies that k rsqsht, s = Ht, t wt + ws + 2 Ht, s ws + 2 Ht, s, Ht, t wt + ws + 2 Ht, s r+ s + ws + Ht, s Ht, s rs + rs + w2 s + Ht, t wt + 2 Ht, s + r +s Ht, s ws + rs + Ht, s rs + w2 s + Ht, t wt + h + t, sws + Ht, s rs + w2 s +, where h + t, s = 2 Ht, s + r +s Ht, s rs + Ht, t wt + h + t, sws + Ht, s rs + w2 s +. 3.4 Set X = Ht, s rs + ws + and Y = h + t, s. 2 Ht, s rs +
26 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam Using Lemma 2.4 with m = 2, we have that 2 Ht, s rs + ws + h + t, s 2 Ht, s rs + Ht, s rs + w2 s + h2 +t, srs +, 4Ht, s h + t, sws + Ht, s rs + w2 s + h2 +t, srs +. 4Ht, s From equation 3.4, we have 2 Ht, s 0 f t > s t 0, 0 < Ht, t Ht, t 0 f t > s t 0, rsqsht, s k Ht, t wt + k h2 +t, srs +, 4Ht, s rsqsht, s k h2 +t, srs + k Ht, t 4Ht, s wt k Ht, t 0 wt. Since 0 < Ht, s Ht, t 0 f t > s t 0 then we have 0 < t > s t 0. Hence it follows that + Ht, t 0 Ht, t 0 Ht, t 0 t rsqsht, s k h2 +t, srs + 4Ht, s rsqsht, s k h2 +t, srs + 4Ht, s rsqsht, s k h2 +t, srs + 4Ht, s t rsqsht, s + k wt Ht, t 0 t Letting t, we have rsqs + k wt. Ht, t 0 rsqsht, s k h2 +t, srs + 4Ht, s Ht, s Ht, t 0 f
Oscillaty Solutions of Nonlinear Fractional Difference Equations 27 t which contradicts 3.3. The proof is complete. rsqs + k wt <, Theem 3.3. Suppose that H and H2 hold. Furtherme assume that there exists a positive sequence rt such that [ [ ] γ J qsk + T ] M M r +s =, 3.5 where r + s = max{ rs, 0}. Then every solution of. is oscillaty. Proof. Suppose to the contrary that xt is a nonoscillaty solution of.. Without loss of generality, we can assume that xt is an eventually positive solution of.. We proceed as in Theem 3. to get that α xt γ is positive. Now define the following function, using Riccati substitution wt = α xt γ x γ t + α xt + rt. Thus [ α xt γ wt = x γ t [ ] α xt γ wt = x γ t ] + α xt + rt, + [ α xt] + rt. Hence, [ ] wt xγ t [ α xt γ ] α xt γ x γ t xt + + rt x γ tx γ t + M = qtfxt [ ] α xt γ x γ t xt x γ t + x γ t x γ t + + + rt M [ ] α γ [ ] γ xt xt qtk + xt xt + M 2 xt + rt [ ] γ [ ] γ xt xt qtk + T M xt xt + M + rt = qtk [ ] xt 2 γ + T M γ xtxt + M + rt qtk J γ M + T γ M + r +t [ ] γ J = qtk + T M M + r +t.
28 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam Summing the above inequality from t to t, we get wt wt [ [ ] γ J qsk + T ] M M + r +s, [ [ ] γ J qsk + T ] M M r +s wt wt wt <. Taking t and sup, we get [ [ ] γ J qsk + T ] M M r +s wt <, which contradicts 3.5. The proof is complete. 4 Examples Example 4.. Consider the nonlinear fractional difference equation 0.5 xt γ + tfxt = 0 f t N t0 +0.5, 4. where α = 0.5, qt = t and γ > 0 is a quotient of odd positive integers. We apply Theem 3.3 with rt = t γ+, qt = t, T =, M = and J =. It is easy to see that H and H2 hold. Then we have [ [ ] γ J qsk + T ] M M r +s = = s = s =, [ s s γ+ ] [s 2 s γ ] [ ] s γ+2 that is condition 3.5 of Theem 3.3 is satisfied. Therefe, all solutions of 4. are oscillaty. Example 4.2. Consider the nonlinear fractional difference equation 0.5 xt γ + tfxt = 0 f t N t0 +0.5, 4.2 s γ
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