We correct a recent result concerning the fractional derivative at extrema points. We then establish new results for the Caputo and Riemann-Liouville fractional derivatives at extrema points

Al-Refai, Mohammed

Mohammed Al-Refai

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Electronic journal of qualitative theory of differential equations

On the fractional derivatives at extrema points

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Electronic Journal of Qualitative Theory of Differential Equations2012, No. 55, 1-5; http://www.math.u-szeged.hu/ejqtde/On the Fractional Derivatives at Extreme PointsMohammed Al-RefaiDepartment of Mathematical SciencesUnited Arab Emirates University,P.O.Box 17551, Al Ain, UAE.m−alrefai@uaeu.ac.aeAbstractWe correct a recent result concerning the fractional derivative at extreme points. Wethen establish new results for the Caputo and Riemann-Liouville fractional derivativesat extreme points.Key words and phrases: Fractional differential equations, Caputo fractional derivative,Riemann-Liouville fractional derivative.1 IntroductionIn recent years several authors have discussed the existence and uniqueness results for wideclasses of fractional differential equations [1, 2, 3, 4, 6, 7, 9]. The techniques implementedare mainly fixed point theorems, maximum principle and the method of lower and uppersolutions. In this paper we correct a result obtained in [9] and obtain new results concerningthe fractional derivatives at extreme points. These results will be of interest for manyresearchers, especially for those who are working in extending the method of lower andupper solutions to fractional boundary value problems [1, 7]. In the following we presentsome definitions and main results concerning the Caputo and Riemann-Liouville fractionalderivatives.Definition 1.1. Let f ∈ C[0, 1], δ ≥ 0, and Γ is the Euler gamma function. The leftRiemann-Liouville fractional integral is defined byIδf(t) ={1Γ(δ)∫t0(t− s)δ−1f(s)ds, δ > 0,f(t), δ = 0.(1.1)Definition 1.2. Let f ∈ Cn[0, 1], the left Caputo fractional derivative is defined byDδCf(t) = In−δ dndtnf(t) ={1Γ(n−δ)∫t0(t− s)n−δ−1f (n)(s)ds, n− 1 < δ < n ∈ Z+,f (n)(t), δ = n ∈ Z+.Definition 1.3. Let f ∈ Cn[0, 1], the left Riemann-Liouville fractional derivative is definedbyDδRf(t) =dndtnIn−δf(t) ={1Γ(n−δ)dndtn∫t0(t− s)n−δ−1f(s)ds, n− 1 < δ < n ∈ Z+,f (n)(t), δ = n ∈ Z+.EJQTDE, 2012 No. 55, p. 1It is well-known that if f(0) = f ′(0) = · · · = f (n−1)(0) = 0, then DδCf(t) = DδRf(t).In general the relation between the Caputo and Riemann-Liouville fractional derivatives isgiven by [5, 8]DδCf(t) = DδR(f(t)−n−1∑k=0tkk!f (k)(0)), (1.2)whereDδRtk =Γ(k + 1)Γ(k − δ + 1)tk−δ. (1.3)2 Main ResultsWe first show that the following result claimed in [9] is not correct. The following is claimedas Theorem 2.2 of [9].• Let a function f ∈ C2(0, 1) ∩ C[0, 1], attain its minimum over the interval [0, 1] at thepoint t0 ∈ (0, 1]. Then DδCf(t0) ≥ 0, for all 1 < δ ≤ 2.As a counter example we consider f(t) = t(t− 12)(t−1), 0 ≤ t ≤ 1. Direct calculations implythat f(t) has absolute minimum value at t0 =3+√36< 1. For 1 < δ < 2, we haveDδCt3 =Γ(4)Γ(4− δ)t3−δ, DδCt2 =Γ(3)Γ(3− δ)t2−δ and DδCt = 0.Thus,D1.1C f(t0) =(3 +√3)1.960.9Γ(2.9)− 30.1(3 +√3)0.920.9Γ(1.9)= −0.4277 · · · < 0,which contradicts the result in Theorem 2.2 of [9]. We correct the above result by imposingmore conditions on f . We haveTheorem 2.1. Let f ∈ C2[0, 1] attain its minimum at t0 ∈ (0, 1), thenDδCf(t0) ≥t−δ0Γ(2− δ)[(δ − 1)(f(0)− f(t0))− t0f ′(0)], for all 1 < δ < 2. (2.1)Proof. We define the auxiliary function h(t) = f(t)− f(t0), t ∈ [0, 1]. Then h(t) satisfies thefollowing in [0, 1]h(t) ≥ 0, h(t0) = h′(t0) = 0, h′′(t0) ≥ 0 and DδCh(t) = DδCf(t).Integration by parts ofDδCh(t0) =1Γ(2− δ)∫t00(t0 − s)1−δh′′(s)ds,yieldsΓ(2− δ)DδCh(t0) = (t0 − s)1−δh′(s)|t00 − (δ − 1)∫t00(t0 − s)−δh′(s)ds. (2.2)Since h′(t0) = 0 and h′′(t0) is bounded, there exists µ1(t) ∈ C[0, 1] such thath′(t) = (t0 − t)µ1(t). We have for 1 < δ < 2limt→t0h′(t)(t0 − t)δ−1 = limt→t0(t0 − t)µ1(t)(t0 − t)δ−1 = limt→t0(t0 − t)2−δµ1(t) = 0.EJQTDE, 2012 No. 55, p. 2HenceΓ(2− δ)DδCh(t0) = −t1−δ0 h′(0)− (δ − 1)∫t00(t0 − s)−δh′(s)ds. (2.3)Since h(t0) = h′(t0) = 0 and h′′(t0) is bounded, there exists µ2(t) ∈ C[0, 1] such thath(t) = (t0 − t)2µ2(t). Thus∫t00(t0 − s)−δ−1h(s)ds =∫t00(t0 − s)−δ+1µ2(s)ds,is bounded andlimt→t0h(t)(t0 − t)δ = limt→t0(t0 − t)2µ2(t)(t0 − t)δ = limt→t0(t0 − t)2−δµ2(t) = 0.Integrating Eq. (2.3) by parts and using the above result together with h(t) ≥ 0 on [0, 1]yieldsΓ(2− δ)DδCh(t0) = −t1−δ0 h′(0)− (δ − 1)[(t0 − s)−δh(s)|t00 − δ∫t00(t0 − s)−δ−1h(s)ds],= −t1−δ0 h′(0)− (δ − 1)[− t−δ0 h(0)− δ∫t00(t0 − s)−δ−1h(s)ds]= −t1−δ0 h′(0) + (δ − 1)t−δ0 h(0) + δ(δ − 1)∫t00(t0 − s)−δ−1h(s)ds≥ −t1−δ0 h′(0) + (δ − 1)t−δ0 h(0) = −t1−δ0 f ′(0) + (δ − 1)t−δ0 (f(0)− f(t0))and the result is obtained.Corollary 2.1. Let f ∈ C2[0, 1] attain its minimum at t0 ∈ (0, 1), and f ′(0) ≤ 0. ThenDδCf(t0) ≥ 0, for all 1 < δ < 2.Proof. By Theorem 2.1 there holds DδCf(t0) ≥ 1Γ(2−δ)[(δ − 1)t−δ0 (f(0)− f(t0))− t1−δ0 f ′(0)].Since f(t0) ≤ f(0), t0 > 0 and f ′(0) ≤ 0, we obtain DδCf(t0) ≥ 0.The following result is obtained as Theorem 1 of [7].• Let a function f ∈ W 1t((0, T )) ∩ C([0, T ]) attain its maximum over the interval [0, T ] atthe point τ = t0, t0 ∈ (0, T ]. ThenDδCf(t0) ≥ 0, 0 < δ < 1,where W 1t ((0, T )) denotes the space of functions f ∈ C1((0, T ]) such that f ′ ∈ L((0, T )) andL((0, T )) being the set of functions Lebesgue integrable on (0, T ).By substituting g = −f , we have the following result.• Let a function g ∈W 1t ((0, T ))∩C([0, T ]) attain its minimum over the interval [0, T ] at thepoint τ = t0, t0 ∈ (0, T ]. Then DδCg(t0) ≤ 0, 0 < δ < 1.The following result is a simple generalization to the above one for t ∈ (0, 1).Theorem 2.2. Let f ∈ C1[0, 1] attain its minimum at t0 ∈ (0, 1), thenDδCf(t0) ≤ t−δ0Γ(1− δ) [f(t0)− f(0)] ≤ 0, for all 0 < δ < 1. (2.4)EJQTDE, 2012 No. 55, p. 3Proof. We define the auxiliary function h(t) = f(t) − f(t0), t ∈ [0, 1]. Then h(t) ≥ 0, on[0, 1], h(t0) = h′(t0) = 0 and h(t) = (t0 − t)µ3(t) for some µ3(t) ∈ C[0, 1]. Integration byparts ofDδCh(t0) =1Γ(1− δ)∫t00(t0 − s)−δh′(s)ds,yieldsΓ(1− δ)DδCh(t0) = (t0 − s)−δh(s)|t00 − δ∫t00(t0 − s)−δ−1h(s)ds. (2.5)For 0 < δ < 1, we have∫t00(t0 − s)−δ−1h(s)ds =∫t00(t0 − s)−δµ3(s)ds is bounded andlimt→t0h(t)(t0 − t)δ = limt→t0(t0 − t)1−δµ3(t) = 0.ThusΓ(1− δ)DδCh(t0) = −t−δ0 h(0)− δ∫t00(t0 − s)−δ−1h(s)ds ≤ −t−δ0 h(0) = −t−δ0 (f(0)− f(t0)),and the result is obtained.In the following we present analogous results concerning the Riemann-Liouville fractionalderivative.Theorem 2.3. Let f ∈ C2[0, 1] attain its minimum at t0 ∈ (0, 1), thenDδRf(t0) ≥ t−δ0Γ(2− δ)(δ − 1)f(t0) for all 1 < δ < 2. (2.6)Moreover, if f(t) ≥ 0 in [0, 1], then DδRf(t0) ≥ 0.Proof. From Eq.’s (1.2)-(1.3) we have for 1 < δ < 2DδRf(t) =t−δΓ(2− δ)[(1− δ)f(0) + tf ′(0)]+DδCf(t).Applying the result in Eq. (2.1) yieldsDδRf(t0) ≥ t−δ0Γ(2− δ)[(1− δ)f(0) + tf ′(0)]+t−δ0Γ(2− δ)[(δ − 1)(f(0)− f(t0))− t0f ′(0)]=t−δ0Γ(2− δ) [(δ − 1)f(t0)].If f(t) ≥ 0 then f(t0) ≥ 0 and finally DδRf(t0) ≥ 0.Theorem 2.4. Let f ∈ C1[0, 1] attain its minimum at t0 ∈ (0, 1), thenDδRf(t0) ≤ t−δ0Γ(1− δ)f(t0), for all 0 < δ < 1. (2.7)Moreover, if f(t0) ≤ 0, then DδRf(t0) ≤ 0.EJQTDE, 2012 No. 55, p. 4Proof. From Eq.’s (1.2)-(1.3) we have for 0 < δ < 1DδRf(t) =t−δΓ(1− δ)f(0) +DδCf(t).Using the result in Eq. (2.4) we obtainDδRf(t0) ≤t−δ0Γ(1− δ)f(0)−t−δ0Γ(1− δ)(f(0)− f(t0)) =t−δ0Γ(1− δ)f(t0),and DδRf(t0) ≤ 0 provided f(t0) ≤ 0.Remark 2.1. Analogous results for the fractional derivatives at absolute maximum pointsare obtained by applying the above results on −f(t).References[1] M. Al-Refai, M. Hajji, Monotone iterative sequences for nonlinear boundary valueproblems of fractional order, Nonlinear Analysis Series A: Theory, Methods and Ap-plications, 74(2011), 3531-3539.[2] S. Abbas and M. Benchohra, Upper and lower solutions method for impulsive partialhyperbolic differential equations with fractional order, Nonlinear Analysis: HybridSystems, 4(2010), 406-413.[3] XiWang Dong, JinRong Wang, Yong Zhou, On nonlocal problems for fractional dif-ferential equations in Banach spaces, Opuscula Mathematica, 3, 31(2011).[4] Rahmat Khan, Existence and approximation of solutions to three-point boundaryvalue problems for fractional differential equations, Electronic Journal of QualitativeTheory of Differential Equations, 58(2011), 1-8.[5] Changpin Li, Weihua Deng, Remarks on fractional derivatives, Applied Mathematicsand Computation, 187(2007), 777-784.[6] Sihua Liang, Jihui Zhang, Positive solutions for boundary value problems of nonlinearfractional differential equations, Nonlinear Analysis, 71(2009), 5545-5550.[7] Yury Luchko, Maximum principle for the generalized time-fractional diffusion equa-tion, Journal of Mathematical Analysis and Applications, 351(2009), 218-223.[8] I. Podlubny, Fractional Differential Equations, Academic Press, New york, (1999).[9] A. Shi and S. Zhang, Upper and lower solutions method and a fractional differen-tial equation boundary value problem, Electronic Journal of Qualitative Theory ofDifferential Equations, 30(2009), 1-13.(Received April 11, 2012)EJQTDE, 2012 No. 55, p. 5

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