Lyapunov-Type Inequalities for some Sequential Fractional Boundary Value Problems

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1 Advnces in Dynmicl Systems nd Applictions ISSN , Volume 11, Number 1, pp (2016) Lypunov-Type Inequlities for some Sequentil Frctionl Boundry Vlue Problems Rui A. C. Ferreir Grupo Físic-Mtemátic Universidde de Lisbo Av. Prof. Gm Pinto, 2, Lisbo, Portugl. Abstrct In this work, we consider sequentil frctionl differentil eqution subject to Dirichlet-type boundry conditions nd we perform n nlysis iming to derive Lypunov-type inequlity. We cover the cses of the resercher s most used frctionl differentil opertors, nmely, the Riemnn Liouville nd the Cputo ones. As n ppliction of our results, we present criteri for the nonexistence of rel zeros to generlized sine function. AMS Subject Clssifictions: 34A08, 34A40, 26D10, 34C10. Keywords: Sequentil frctionl derivtive, Lypunov inequlity, specil functions. 1 Introduction The fmous Lypunov inequlity, nmed fter the Russin mthemticin A. M. Lypunov used it to study the stbility of solutions of second order differentil equtions, cn be stted s follows: Theorem 1.1 (See [9]). If the boundry vlue problem y (t) + q(t)y(t) = 0, < t < b, y() = 0 = y(b), hs nontrivil continuous solution, where q is rel nd continuous function, then q(s) ds > 4 b. (1.1) Received June 29, 2016; Accepted August 29, 2016 Communicted by Mrtin Bohner

2 34 Rui A. C. Ferreir Typicl pplictions of this result include bounds for eigenvlues, stbility criteri for periodic differentil equtions, nd estimtes for intervls of disconjugcy. Thus, it is not surprising tht scholrs hve written surveys nd books in which Lypunov-type inequlities ply mjor role [2, 13]. The uthor succeeded to generlize Theorem 1.1 for boundry vlue problems in which the clssicl derivtive y is replced by frctionl derivtive D α (cf. the definitions in Section 2), nmely, the Riemnn Liouville frctionl derivtive or the Cputo frctionl derivtive (cf. [4] nd [5], respectively). In this work we im to obtin Lypunov-type inequlity for different problem from those cited bove. Indeed, we will consider the following sequentil frctionl boundry vlue problem, ( D α D β y)(t) + q(t)y(t) = 0, < t < b, (1.2) y() = 0 = y(b), (1.3) where α, β re some rel numbers to be defined lter nd D α stnds for the Riemnn Liouville or the Cputo frctionl derivtive. Some pplictions of liner ordinry sequentil frctionl differentil equtions my be found in [8, Section 7.6] while initil developments on these (sequentil) opertors my be consulted in the book [10]. At first glnce one might think tht becuse the frctionl boundry vlue problem (1.2) (1.3) hs only minor chnge in its formultion, when compred to the ones studied in [4, 5], then the nlysis will be somehow the sme. However, our experience in deling with this subject tught us tht chnging (only) the frctionl opertor my led to more complex nlysis of the problem (cf. the differences between the Riemnn Liouville nd the Cputo cses in [4] nd [5], respectively). Let us now mke brief description of why this hppens: To obtin Lypunov-type inequlity for frctionl boundry vlue problem we use method tht goes bck t lest to Nehri [11]. The method consists in trnsforming the BVP into n equivlent integrl form nd then find the mximum of the modulus of its Green s function. The dvntge is tht then you will need not hve to use properties of the clssicl clculus tht do not necessrily hold within the frctionl clculus setting (cf. Borg s proof of the Lypunov inequlity in [2, Section 1]); we will only be deling with n integrl eqution nd, in some sense, forget bout frctionl clculus. To mke things more precise, suppose tht given boundry vlue problem is equivlent to the integrl eqution y(t) = G(t, s)q(s)y(s)ds, where G stnds for the Green s function, s known usully in the literture. Then, ssuming y is nontrivil (which implies tht q is not the zero function nd G is not constnt in view tht in our study G(, s) = 0, s [, b]), we get y < mx G(t, s) (t,s) [,b] [,b] q(s) ds y,

3 Lypunov-Type Inequlities 35 or 1 mx (t,s) [,b] [,b] G(t, s) < q(s) ds, (1.4) being (1.4) the desired Lypunov inequlity. The drwbck is tht the Green s function might be very complex to nlyze 1, e.g. so fr we re wre of results for frctionl differentil equtions with order between one nd two nd one of order between three nd four [4 7, 12, 15], in contrst to the clssicl cse (cf. [13, Section ]). The pln of this mnuscript is s follows: in Section 2 we provide to the reder brief introduction to some of the frctionl clculus concepts nd results. In Section 3 we prove our results nd finlly, in Section 4, we present n exmple of ppliction of one of our Lypunov-type inequlities. 2 Frctionl Clculus We will introduce the concepts s well s some results used throughout this work. Definition 2.1. Let α 0 nd f be rel function defined on [, b]. The Riemnn Liouville frctionl integrl of order α is defined by (Iq 0 f)(x) = f(x) nd ( I α f)(t) = 1 Γ(α) t (t s) α 1 f(s)ds, α > 0, t [, b]. Definition 2.2. The Riemnn Liouville frctionl derivtive of order α 0 is defined by ( D 0 f)(t) = f(t) nd ( D α f)(t) = (D m I m α f)(t) for α > 0, where m is the smllest integer greter or equl thn α. Definition 2.3. The Cputo frctionl derivtive of order α 0 is defined by ( C D 0 f)(t) = f(t) nd ( C D α f)(t) = ( I m α D m f)(t) for α > 0, where m is the smllest integer greter or equl thn α. The following sequence of results my be found in the book by Kilbs et l. [8]. Proposition 2.4. Let f be continuous function on some intervl I nd α, β > 0. Then ( I α I β f)(t) = ( I α+β f)(t) = ( I β I α f)(t) on I. Proposition 2.5. Let f be continuous function on some intervl I nd α > 0. Then ( D α I α f)(t) = f(t) on I, with D being the Riemnn Liouville or the Cputo frctionl opertor. 1 We will comment bout the difficulties we hd to obtin the Lypunov inequlity for the Cputo cse when compred to the one studied in [5].

4 36 Rui A. C. Ferreir Proposition 2.6. The generl solution y of the following frctionl differentil eqution ( D α y)(t) = f(t), t >, 0 < α 1, is y(t) = c(t ) α 1 + ( I α f)(t), c R. Proposition 2.7. The generl solution y of the following frctionl differentil eqution ( C D α y)(t) = f(t), t, 0 < α 1, is y(t) = c + ( I α f)(t), c R. 3 Min Results In this section we present nd prove our min ccomplishments. For the reder s benefit we provide the results in two different subsections, the first one contining the results using the Riemnn Liouville frctionl differentil opertor nd the second one using the Cputo frctionl differentil opertor. 3.1 Riemmnn Liouville s Cse Let 0 < α, β 1 nd fix 1 < γ := α + β 2. Moreover, ssume tht q is rel vlued continuous function on [, b]. Deriving Lypunov-type inequlity for the boundry vlue problem ( D α D β y)(t) + q(t)y(t) = 0, < t < b, (3.1) y() = 0 = y(b), (3.2) is ctully esy becuse we cn use [4, Theorem 2.1] in this cse. Indeed, ssuming tht (3.1) (3.2) hs nontrivil solution y C[, b], then it is of the form y(t) = c Γ(α) Γ(γ) (t )γ 1 ( I γ qy)(t), by Proposition 2.6 nd the fct tht I β (t ) α 1 = Γ(α) Γ(α + β) (t )α+β 1 (c R is determined by the condition y(b) = 0). It is cler tht y is integrble on [, b]. Then (see [14, Section ]), ( D α D β y)(t) = ( D γ y)(t). The following result is therefore n immedite consequence of [4, Theorem 2.1].

5 Lypunov-Type Inequlities 37 Theorem 3.1. If the frctionl boundry vlue problem (3.1) (3.2) hs nontrivil continuous solution, then ( ) γ 1 4 q(s) ds > Γ(γ). (3.3) b Remrk 3.2. Note tht if α = β = 1, i.e., γ = 2, then we get the clssicl Lypunov inequlity (1.1). 3.2 Cputo s Cse We consider now the BVP (1.2) (1.3) with D δ = C D δ, for δ {α, β}. Remrk 3.3. Contrrily to the Riemnn Liouville cse, for continuous function y the equlity ( C D αc D β y)(t) = ( C D α+β y)(t) does not hold in generl (see [3, Remrk 3.4. (b)]). Lemm 3.4. Let 0 < α, β 1 be such tht 1 < α + β 2 nd q C[, b] for some < b. Then, y C[, b] is solution of the frctionl boundry vlue problem ( C D αc D β y)(t) + q(t)y(t) = 0, < t < b, (3.4) if, nd only if, y stisfies the integrl eqution where G(t, s) = 1 Γ(α + β) y(t) = y() = 0 = y(b), (3.5) G(t, s)q(s)y(s)ds, (b s) α+β 1 (t ) β (t s) α+β 1, s t b, (b ) β (b s) α+β 1 (t ) β, t s b. (b ) β Proof. The proof is just repeted ppliction of Proposition 2.7 nd the concomitnt use of the boundry conditions. We leve the detils to the reder. The rest of this section is devoted essentilly to determine the mximum of G(t, s) for (t, s) [, b] [, b]. We will not write it s proof since we will comment on the difficulties nd chllenges when tckling this problem, which constitute the min differences to the ones studied before by the uthor [4, 5]. Let us strt by defining function: g 1 (t, s) = (b s)α+β 1 (t ) β (b ) β, t s b.

6 38 Rui A. C. Ferreir g 1 is obviously nonnegtive. Moreover, g 1 (t, s) (b s)α+β 1 (s ) β (b ) β := f(s), s [, b]. So, we re left to find mx f(s). We hve, s [,b] f (s) = 1 (b ) β [ (α + β 1)(b s)α+β 2 (s ) β + (b s) α+β 1 β(s ) β 1 ] = (b s)α+β 2 (s ) β 1 (b ) β [ (α + β 1)(s ) + ], from which follows tht f (α + β 1) + βb (s) = 0 if, nd only if, s = nd f (s) > 0 (α + β 1) + βb for s < nd f (α + β 1) + βb (s) < 0 for s >. We conclude tht, mx g 1(t, s) = f (t,s) [,b] [,b] = Now we define function g 2 by: ( ( ) (α + β 1) + βb b (α+β 1)+βb α+2β 1 ) α+β 1 ( (α+β 1)+βb α+2β 1 (b ) β ) β = (α + β 1)α+β 1 β β () α+2β 1 (b )α+β 1. (3.6) g 2 (t, s) = (b s)α+β 1 (t ) β (b ) β (t s) α+β 1, s t b. This function resembles the corresponding one in [5]. The substntil difference is tht the fctor (t ) δ in [5] is of order δ = 1 while here is of order δ = β. This fct llowed us to determine immeditely in [5] the zero of the derivtive of g 2 with respect to t for n rbitrry but fixed s (cf. [5, Lemm 2] for the detils). However, in this work we cnnot proceed s in [5] in view tht t g 2 (t, s) = (b s)α+β 1 β(t ) β 1 (b ) β (α + β 1)(t s) α+β 2. (3.7) Fortuntely we cn use the Fritz John theorem nd tke dvntge of some kind of symmetry of the prtil derivtives of our problem in order to find the cndidtes to

7 Lypunov-Type Inequlities 39 mxim of the function g 2 (t, s) with < s < t < b. We hve t g 2 (t, s) = (b s)α+β 1 β(t ) β 1 (b ) β (α + β 1)(t s) α+β 2 = 0, (3.8) s g 2 (t, s) = (α + β 1)(b s)α+β 2 (t ) β (b ) β + (α + β 1)(t s) α+β 2 = 0, from which follows tht or (b s) α+β 2 (t ) β 1 (b ) β [ (α + β 1)(t )] = 0, t = + (3.9) α + β 1, (3.10) provided s < t < b. Note tht it is very hrd (if not impossible) to find the solutions in s from inserting (3.10) in (3.8) or (3.9). Therefore, our pproch consists in inserting (3.10) in g 2 nd then perform n nlysis of the resulting function of only the vrible s. Before we strt the mentioned nlysis we need to find out which intervl s belongs. On one hnd we hve, On the other hnd we get, s < t s < + α + β 1 (α + β 1) + βb s <. Let us finlly consider the following function: F (s) : = t < b + α + β 1 < b (3.11) (α + β 1) + b(1 α) s >. β (3.12) (b s) α+β 1 ( + β(b s) α+β 1 ) β (b ) β ( + = (b s)α+2β 1 β β (α + β 1) β (b ) β ( + α + β 1 s ) α+β 1 α + β 1 s, ) α+β 1, (α + β 1) + b(1 α) (α + β 1) + βb with < s <. Fortuntely we needed β not to clculte zeros of the derivtive for this function s it hppens tht it is strictly

8 40 Rui A. C. Ferreir incresing. To prove this lst sttement we strt differentiting F : F (s) = ()(b ( ) s)α+2β 2 β β α+β 2 +(α +2β 1) + (α + β 1) β (b ) β α + β 1 s. By (3.11) nd the fct tht α + β 2 < 0 nd > 0, we get tht F (s) > ()(b s)α+2β 2 β β (α + β 1) β (b ) β + () (b s) α+β 2 = ()(b s) α+β 2 (1 > 0, where the lst inequlity follows from (3.12). Therefore, mx s [ (α+β 1)+b(1 α) β, (α+β 1)+βb α+2β 1 F (s) = F ] ( ) (α + β 1) + b(1 α) in view tht F = 0. β The following result is therefore vlid. (b s) β β β (α + β 1) β (b ) β ( (α + β 1) + βb ), = (α + β 1)α+β 1 β β () α+2β 1 (b )α+β 1, Proposition 3.5. The function G defined in Lemm 3.4 stisfies the following property: ) G(t, s) (b )α+β 1 Γ(α + β) (α + β 1) α+β 1 β β, (t, s) [, b] [, b], () α+2β 1 with equlity if nd only if t = s = (α + β 1) + βb. Theorem 3.6. If the frctionl boundry vlue problem (3.4) (3.5) hs nontrivil continuous solution, then q(s) ds > Γ(α + β) (b ) α+β 1 () α+2β 1 (α + β 1) α+β 1 β β. Remrk 3.7. Observe tht when α = β = 1, then Theorem 3.6 reduces to Theorem 1.1. Agin this shows tht Theorem 3.6 is generliztion of Theorem 1.1.

9 Lypunov-Type Inequlities 41 4 An Exmple of Appliction We will end this work presenting n ppliction of Theorem 3.1. Consider the following sequentil frctionl differentil eqution ( 0 D α 0D α y)(t) + λ 2 y(t) = 0, λ R, t (0, 1), 1 2 < α 1. (4.1) The fundmentl set of solutions to (4.1) is (cf. [1, Exmple 1]) where 2 cosα(λt) = nd sin α (λt) = {cos α (λt), sin α (λt)}, ( 1) j λ 2j t (2j+1)α 1 Γ((2j + 1)α), j=0 ( 1) j λ 2j+1 t (j+1)2α 1 Γ((j + 1)2α). j=0 Therefore the generl solution of (4.1) is y(t) = c cos α (λt) + d sin α (λt). (4.2) Now, the nontrivil solutions of (4.2) for which the boundry conditions y(0) = 0 = y(1) hold, stisfy sin α (λ) = 0, where λ is rel number different from zero (eigenvlue). By Theorem 3.1, the following inequlity then holds λ 2 > Γ(2α)4 2α 1, or in other words: Theorem 4.1. Let 1 2 < α 1. If then sin α (t) hs not rel zeros. t Γ(2α)4 2α 1, t 0, Remrk 4.2. Anlogous results to the ones in this section cn be obtined if we use the Cputo frctionl opertor nd the trigonometric functions s defined in [1, Exmple 2]. 2 The definitions of cos α nd sin α in [1] should be s we re writing them here.

10 42 Rui A. C. Ferreir Acknowledgements The uthor ws supported by the Fundção pr Ciênci e Tecnologi (FCT) through the progrm Investigdor FCT with reference IF/01345/2014. References [1] B. Bonill, M. Rivero nd J. J. Trujillo, Liner differentil equtions of frctionl order, in Advnces in frctionl clculus, 77 91, Springer, Dordrecht. [2] R. C. Brown nd D. B. Hinton, Lypunov inequlities nd their pplictions, in Survey on clssicl inequlities, 1 25, Mth. Appl., 517 Kluwer Acd. Publ., Dordrecht. [3] K. Diethelm, The nlysis of frctionl differentil equtions, Lecture Notes in Mthemtics, 2004, Springer, Berlin, [4] R. A. C. Ferreir, A Lypunov-type inequlity for frctionl boundry vlue problem, Frct. Clc. Appl. Anl. 16 (2013), no. 4, [5] R. A. C. Ferreir, On Lypunov-type inequlity nd the zeros of certin Mittg- Leffler function, J. Mth. Anl. Appl. 412 (2014), no. 2, [6] M. Jleli nd B. Smet, Lypunov-type inequlities for frctionl differentil eqution with mixed boundry conditions, Mth. Inequl. Appl. 18 (2015), no. 2, [7] M. Jleli, L. Rgoub nd B. Smet, A Lypunov-Type Inequlity for Frctionl Differentil Eqution under Robin Boundry Condition, J. Funct. Spces, 2015, Art. ID , 5 pp. [8] A. A. Kilbs, H. M. Srivstv nd J. J. Trujillo, Theory nd pplictions of frctionl differentil equtions, North-Hollnd Mthemtics Studies, 204, Elsevier, Amsterdm, [9] A. M. Lypunov, Probleme générl de l stbilité du mouvement, (French Trnsltion of Russin pper dted 1893), Ann. Fc. Sci. Univ. Toulouse 2 (1907), , Reprinted s Ann. Mth. Studies, No, 17, Princeton, [10] K. S. Miller nd B. Ross, An introduction to the frctionl clculus nd frctionl differentil equtions, A Wiley-Interscience Publiction, Wiley, New York, [11] Z. Nehri, On the zeros of solutions of second-order liner differentil equtions, Amer. J. Mth. 76 (1954),

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