LYAPUNOV-TYPE INEQUALITIES FOR α-th ORDER FRACTIONAL DIFFERENTIAL EQUATIONS WITH 2 < α 3 AND FRACTIONAL BOUNDARY CONDITIONS

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1 Eletroni Journl of ifferentil Eqution, Vol , No. 203, pp ISSN: URL: or LYAPUNOV-TYPE INEQUALITIES FOR α-th ORER FRACTIONAL IFFERENTIAL EQUATIONS WITH 2 < α 3 AN FRACTIONAL BOUNARY CONITIONS SOUGATA HAR, QINGKAI KONG Communited y Jerry Goldtein Atrt. We tudy liner frtionl oundry vlue prolem oniting of n α-th order Riemnn-Liouville frtionl differentil eqution with 2 < α 3 nd ertin frtionl oundry ondition. We derive everl Lypunovtype inequlitie nd pply them to etlih nonexitene, uniquene, nd exitene-uniquene of olution for relted homogeneou nd nonhomogeneou liner frtionl oundry vlue prolem. A peil e, our work extend ome exiting reult for third-order liner oundry vlue prolem. 1. Introdution We onider the α-th order frtionl liner differentil eqution α x t qtx = 0, 2 < α Rell tht for ny γ 0 nd t >, I γ x t := 1 Γγ t t γ1 xd denote the γ-th order left-ided Riemnn-Liouville frtionl integrl of xt t, nd γ x t denote the γ-th order left-ided Riemnn-Liouville frtionl derivtive of xt t defined γ x t := dn dt n I nγ 1 d n t x t = Γn γ dt n t nγ1 xd, 1.2 where n = γ 1 with γ the integer prt of γ nd Γγ = t γ1 e t dt i the 0 Gmm funtion. In the following, we denote with αk α3 x x := lim t := lim t αk I 3α x t for k = 1, 2, 3 x t for 2 < α < Mthemti Sujet Clifition. 34A08, 34A40, 26A33, 34B05. Key word nd phre. Frtionl differentil eqution; frtionl oundry ondition; Lypunov-type inequlitie; oundry vlue prolem; exitene nd uniquene of olution Tex Stte Univerity. Sumitted June 2, Pulihed Septemer 6,

2 2 S. HAR, Q. KONG EJE-2017/203 In thi pper, we derive Lypunov-type inequlitie for the oundry vlue prolem BVP oniting of 1.1 nd one of the following oundry ondition BC: α2 x = α2 x = 0 nd α3 x = 0, ; 1.3 α3 α3 x = α3 x = 0 nd α2 x = 0, < ; 1.4 x = α3 x = 0 nd α1 x = 0, <. 1.5 Lypunov-type inequlitie hve een ued n importnt tool in oilltion, dionjugy, ontrol theory, eigenvlue prolem, nd mny other re of differentil eqution. Beue of their importne, thee inequlitie hve een extended nd generlized in mny diretion y everl uthor. Now we riefly review ome exiting reult on Lypunov-type inequlitie for oth integer-order nd frtionlorder differentil eqution. For the eond-order liner differentil eqution x qtx = 0 on, 1.6 with q C,, R, the following reult i known the Lypunov inequlity, ee 17, 2. Theorem 1.1. Aume 1.6 h olution xt tifying x = x = 0 nd xt 0 for t,. Then qt dt > It w firt noted y Wintner 24 nd lter y everl other uthor tht inequlity 1.7 n e improved y repling qt y q t := mx{qt, 0}, the nonnegtive prt of qt, to eome q tdt > Inequlity 1.8 w generlized to more generl form of eond-order liner differentil eqution y Hrtmn 11, Chpter XI, nd improved y Hrri nd Kong 12 nd Brown nd Hinton 1 lter on. We note tht the numer 4 in the ove inequlitie i the et in the ene tht if it i repled y ny lrger numer, then the inequlitie fil to hold, ee 11, p. 345 nd 16 for exmple. Lypunov-type inequlitie hve een further extended to higher order liner differentil eqution nd hlf-liner differentil eqution y mny uthor. See 5, 7, 19, 20, 26, 27, 25, 28 for the higher order liner e, 3, 4 for the hlf-liner e, nd Pino 21 for n exellent urvey on vriou Lypunov-type inequlitie. Among the ove, hr nd Kong 7 etlihed Lypunov-type inequlitie for odd order liner differentil eqution. Retriting their reult to the third-order eqution x qtx = 0, 1.9 we hve the following reult. Theorem 1.2. Aume 1.9 h nontrivil olution xt tifying x = x = 0 nd x = 0 for,. Then one of the following two ttement hold: i q tdt > 8, 2 ii q tdt > 8, 2

3 EJE-2017/203 FRACTIONAL LYAPUNOV-TYPE INEQUALITIES 3 iii q tdt q tdt > 8 2. A reult, qt dt > 8 2. hr nd Kong 3 lo etlihed Lypunov-type inequlitie for third-order hlf-liner differentil eqution. The following i one of their reult retrited to the liner e. Theorem 1.3. Aume 1.9 h olution xt tifying x = x = x = 0 nd xt 0 for t,. Then q tdt > 4 2. Although Lypunov-type inequlitie hve een developed in mny diretion for the integer-order differentil eqution, there re only few known reult for the frtionl differentil eqution. In 9, Ferreir otined uh inequlitie for Riemnn-Liouville frtionl BVP for the eqution α x t qtx = 0, 1 < α 2, 1.10 where q C,, R. Theorem 1.4. Aume 1.10 h olution xt tifying x = x = 0 nd xt 0 for t,. Then 4 α1. qt dt > Γα 1.11 It w indited in hr nd Kong 6 tht 1.11 n e improved y repling qt y q t to eome 4 α1. q tdt > Γα Moreover, hr nd Kong 6 otined Lypunov-type inequlitie for the frtionl BVP oniting of 1.10 nd the frtionl integrl BC α2 x = α2 x = Theorem 1.5. Aume 1.10 h nontrivil olution xt tifying BC Then { 2α Gt, q } d > 1. mx t, Aume 1.10 h olution xt tifying BC 1.12 nd α2 x t 0 on,. Then Here, mx t, { 2α Gt, q } d > 1. { Gt, := 1 t, t t, t i the Green funtion for the BVP 1.13 u = ht, u = u =

4 4 S. HAR, Q. KONG EJE-2017/203 with h L,, R; 2α Gt, q denote the 2 α-th order right-ided frtionl derivtive of Gt, q with repet to t for fixed t,, i.e., 2α Gt, q := 1 d Γ1 α d τ 2α Gt, τqτdτ; 1.15 nd 2α Gt, q repreent the poitive prt of 2α Gt, q. Reently, Lypunov-type inequlitie for Riemnn-Liouville frtionl differentil equ- tion with 3 < α 4 nd pointwie BC were etlihed y O Regn nd Smet 18. For Cputo frtionl differentil eqution, Lypunov-type inequlitie were derived o fr only for 1 < α 2, ee 8, 13, 14, 15, 22. To the et of our knowledge, no Lypunov-type inequlitie hve een found for 2 < α 3. In thi rtile, we derive Lypunov-type inequlitie for eh of the BVP 1.1, 1.3; 1.1, 1.4 nd 1.1, 1.5 nd utilize them to etlih the exitene nd uniquene for olution of relted homogeneou nd nonhomogeneou liner BVP. Our work over the reult of Theorem 1.2 nd improve tht of Theorem 1.3 for the third-order liner differentil eqution 1.9. Thi rtile i orgnized follow: After thi introdution, we preent the Lypunov-type inequlitie for eh of the BVP 1.1, 1.3; 1.1, 1.4 nd 1.1, 1.5 in Setion 2; nd then pply them to etlih the exitene nd uniquene for olution of ertin relted homogeneou nd nonhomogeneou liner frtionl BVP in Setion Min reult In thi Setion, we let < < < nd ume q L,, R. To prove our reult, we will need the following frtionl integrtion y prt formul, ee 23, 2.64: φ γ ψ d = ψ γ φ d, 0 < γ < 1, 2.1 for ny φ L p,, ψ L r, uh tht p 1 r 1 1 γ, where γ φ := 1 d Γ1 γ d τ γ φτdτ. Similr to the nottion in Setion 1, we denote y 3α Gt, q the 3 α-th order right-ided frtionl derivtive of Gt, q with repet to t for fixed t, nd y 3α Gt, q the poitive nd negtive prt of ± 3α Gt, q, repetively. Firt we preent the Lypunov-type Inequlitie for BVP 1.1, 1.3. Theorem 2.1. Aume 1.1 h nontrivil olution xt tifying BC 1.3. Then 3α Gt, q d dt > Proof. Let yt := α3 x t for < t nd y = α3 x. 2.3 Then yt i ontinuou on,. Note tht xt = 3α yt. We lim tht α x t = y t. 2.4

5 EJE-2017/203 FRACTIONAL LYAPUNOV-TYPE INEQUALITIES 5 In ft, for 2 < α < 3, from 1.2 we hve t α x = d3 dt 3 I 3α x t = d3 dt 3 α3 x t = y t; nd 2.4 hold lerly when α = 3 ine yt = xt. Alo, α2 x t = d α3 dt x t = y t. Then BVP 1.1, 1.3 eome the third-order liner BVP y = qtx, y = y = 0, y = 0 for,. 2.5 We denote zt = y t nd rewrite BVP 2.5 z = qtx, z = z = Uing the Green funtion Gt, defined in 1.13 for BVP 1.14 we hve zt = Gt, qxd = Gt, q 3α yd. For fixed t, nd 2 < α 3, pplying 2.1 with φ = Gt, q, ψ = y, nd γ = 3 α, we otin zt = Gt, q 3α yd = y 3α Gt, qd. 2.7 Repling zt y y t, nd then integrting oth ide from to t nd uing the ft tht y = 0, we ee tht Hene yt = yt = t t Let m := mx t, yt. Then m m from whih it follow tht 1 y 3α Gτ, q d dτ. 2.8 y 3α Gτ, qd dτ y 3α Gτ, q d dτ α Gτ, q d dτ α Gτ, q d dτ. Uing the me rgument given in the proof of 6, Theorem 3.1, we ome to the onluion tht 1 < 3α Gτ, q d dτ. We omit the detil. In the following, we y tht funtion ut doe not hnge ign on n intervl J if ut 0 on J or ut 0 on J. Under the umption tht α3 x t doe not hnge ign on, nd on,, we derive hrper Lypunov-type inequlitie thn 2.2.

6 6 S. HAR, Q. KONG EJE-2017/203 Theorem 2.2. Aume 1.1 h nontrivil olution xt tifying BC 1.3 with,. Furthermore, ume α3 x t doe not hnge ign on, nd on,. Then one of the following hold: 3α Gt, q d dt > 1, 3α Gt, q d dt > 1, 3α Gt, q d dt 3α Gt, q d dt > 1. d 3α Gt, q d dt 3α Gt, q d dt > 1. Proof. Let yt e defined y 2.3. A hown in the proof of Theorem 2.1, 2.8 hold. Sine yt i ontinuou on, nd y = 0, there exit t 1, nd t 2, uh tht yt 1 = mx{ yt : t, } nd yt 2 = mx{ yt : t, }. Without lo of generlity, we my ume yt tifie one of the following e: I yt 0 on,, nd yt 1 yt 2 ; II yt 0 on,, nd yt 1 < yt 2 ; III yt 0 on, nd yt 0 on,, nd yt 1 yt 2 ; IV yt 0 on, nd yt 0 on,, nd yt 1 < yt 2. In the equel, we denote m = mx{ yt 1, yt 2 }. Ce I: In thi e, m = yt 1. Then 2.8 with t = t 1 how tht m = y 3α Gτ, q d dτ m 3α Gτ, q t 1 Similr to the proof of Theorem 2.1, we hve 1 < Gτ, q 3α whih how tht onluion hold. d dτ Ce II: In thi e, m = yt 2. Then 2.8 with t = t 2 how tht t2 m = y 3α Gτ, q d dτ m Gτ, q Agin, thi led to 1 < 3α whih how tht onluion hold. Gτ, q 3α d dτ Ce III: In thi e, m = yt 1. Then 2.8 with t = t 1 how tht m = y 3α Gτ, q d dτ = t 1 m t 1 t 1 t 1 y 3α Gτ, q d dτ y 3α Gτ, q d dτ 3α Gτ, q d dτ m t 1 3α d dτ. d dτ. Gτ, q d dτ.

7 EJE-2017/203 FRACTIONAL LYAPUNOV-TYPE INEQUALITIES 7 One gin, thi how tht onluion hold. Ce IV: The me rgument in Ce III how tht onluion d hold. We omit the detil. A onequene of Theorem 2.2, we hve the following orollry. Corollry 2.3. Aume 1.1 h nontrivil olution xt tifying BC 1.3. Furthermore, ume α3 x t doe not hnge ign on, nd on,. Suppoe 3α Gt, q 0 for, t,,. Then α2 q d > 2Γα Suppoe 3α Gt, q 0 for, t,,. Then Suppoe tht 3α 3α α2 q d > 2Γα Gt, q 0 for, t,, nd tht Gt, q 0 for, t,,. Then α2 q d α2 q d > 2Γα Proof. By umption we hve 3α Gt, q = 3α Gt, q nd 3α Gt, q = 0 for, t,,. It i ey to ee tht in thi e, ll feile inequlitie in -d of Theorem 2.2 led to By the definition of 3α = = = 1 Γα 2 1 Γα 2 1 Γα 2 Hene 2.14 eome 3α Gt, q d dt > Gt, q given in 1.15 we hve 3α Gt, q d dt τ α3 Gt, τqτdτ d dt τ α3 Gt, τqτdτdt Gt, τdt τ α3 qτdτ. Gt, τdt τ α3 qτdτ > Γα 2. Uing the ft tht qt q t, Gt, τ 0 on,, we hve Gt, τdt τ α3 q τdτ > Γα

8 8 S. HAR, Q. KONG EJE-2017/203 Note tht for τ,, Gt, τdt = 1 τ τ. 2 Therefore, 2.15 led to The proof i imilr to e nd hene i omitted. It i ey to ee tht Theorem 2.2, onluion -d led to 3α Gt, q d dt 3α Then the proof i imilr to e nd hene i omitted. Gt, q d dt > 1. Remrk 2.4. Let g := α1 for 2 < α 3. It i ey to ee tht the mximum of g our t d = α 2 /α 1. Hene for,, g gd = α 2α2 α1 α 1 α1. Therefore, eome repetively the following i q d > 2α1α1 Γα2 α2 α2. α1 ii q d > 2α1α1 Γα2 α2 α2. α1 iii q d q d > 2α1α1 Γα2 α2 α2. α1 The following provide upplement to Theorem 2.2. Theorem 2.5. Aume 1.1 h nontrivil olution xt tifying α2 x = α2 x nd α3 x = Then 3α Gt, q d dt > Furthermore, ume α3 x t doe not hnge ign on,. Then 3α Gt, q d dt > Aume 1.1 h nontrivil olution xt tifying α2 x = α2 x nd α3 x = 0. Then 3α Gt, q d dt > 1. Furthermore, ume α3 x t doe not hnge ign on,. Then 3α Gt, q d dt > 1. Proof. A in the proof of Theorem 2.1, we ee tht 2.8 hold with =, i.e., yt = t y 3α Gτ, q d dτ. 2.19

9 EJE-2017/203 FRACTIONAL LYAPUNOV-TYPE INEQUALITIES 9 Hene yt = t y 3α y 3α Gτ, q d dτ Gτ, q d dτ. Let m = mx t, yt. Then hown in the proof of Theorem 2.1, thi led to Furthermore, ume yt = α3 x t doe not hnge ign on,. Without lo of generlity, we my ume yt 0 in,. Then there exit t 2, uh tht m = yt 2 = mx{yt : t, }. Letting t = t 2 in 2.19 we otin m = t2 y 3α Gτ, q d dτ m 3α A hown in the proof of Theorem 2.1, thi led to The proof i imilr to Prt nd hene i omitted. Gτ, q d dτ. Remrk 2.6. Now, we remrk on the peil e of Theorem with α = 3, where BVP 1.1, 1.3 eome the third-order liner BVP x qtx = 0, x = x = 0, x = 0 for, In thi e, onluion 2.2 in Theorem 2.1 eome Note tht Gt, 0 for, t,,. Hene Gt, q d dt = With imple lultion we hve nd hene 2.21 led to Gt, q d dt > Gt, q d dt = Gt, dt = q d > 8 2. Similrly, onluion -d in Theorem 2.2 led to Gt, dt q d. i q tdt > 8, 2 ii q tdt > 8, 2 iii q tdt q tdt > 8. 2 The me remrk pplie to the e when = or given in Theorem 2.5. It i ey to ee tht the ondition xt doe not hnge ign on, nd on, in Theorem 2.2 nd it prllel ondition in Theorem 2.5 re not eentil for the integer-order differentil eqution. Therefore, thee reult gree with Theorem 1.2.

10 10 S. HAR, Q. KONG EJE-2017/203 Next we derive the Lypunov-type inequlitie for BVP 1.1, 1.4. Note from 1.13 tht the Green funtion of BVP i given y u = ht, u = uη = 0 { G η t, := 1 η t, t η, η t η, t η for ny η,. Then the following reult for BVP 1.1, 1.4 re derived from Theorem 2.1 nd 2.5 for BVP 1.1, 1.3. Here, we will ue 3α η G η t, q to denote the 3 α-th order right-ided frtionl derivtive of G η t, q with repet to t η for fixed t,, i.e., 3α η Gt, q := 1 d Γ2 α d η τ 3α G η t, τqτdτ. Theorem 2.7. Aume 1.1 h nontrivil olution xt tifying BC 1.4. Then η η 3α G η t, q d dt > up η, η Aume 1.1 h nontrivil olution xt tifying 1.4 nd α3 x t doe not hnge ign on,. Then η η G η t, q d dt > up η, Proof. Sine α3 η, uh tht α2 3α η x = α3 x = 0, y Rolle Theorem there exit x η = α3 x η = 0. Hene it tifie α2 x = α2 Applying Theorem 2.1 to BVP 1.1, 2.24 we hve η η 3α η x η = 0 nd α3 x = G η t, q d dt > 1. Then 2.22 follow. From the proof of Prt we ee tht there exit η, uh tht α2 x η = 0. Hene xt tifie By the umption, α3 x t doe not hnge ign on, η. Applying the eond prt of Theorem 2.5, Prt to BVP 1.1, 2.24 we hve η η G η t, q d dt > 1. Then 2.23 follow. 3α η With the me rgument in Corollry 2.3, we otin the orollry elow from Theorem 2.7. Corollry 2.8. Aume 1.1 h nontrivil olution xt tifying BC 1.4. Suppoe 3α η G η τ, q 0 on, η, η for every η,. Then q d > 2α 1α1 Γα 2 α 2 α2 α1.

11 EJE-2017/203 FRACTIONAL LYAPUNOV-TYPE INEQUALITIES 11 In the lt prt of thi etion, we derive the Lypunov-type inequlitie for BVP 1.1, 1.5. Theorem 2.9. Aume 1.1 h nontrivil olution xt tifying BC 1.5. Then Gt, q τ dτd > mx t, 3α Aume 1.1 h nontrivil olution xt tifying 1.5 nd α3 x t doe not hnge ign on,. Then Gt, q τ dτd > mx t, 3α Proof. Let yt e defined y 2.3. A hown in the proof of Theorem 2.1, BVP 1.1, 1.5 eome It follow tht y = qtx, y = y = 0 nd y = 0. y = t qτxτdτ = t qτ 3α y τdτ. For ny t,, pplying 2.1 with φτ = qτ, ψτ = 3α y τ, γ = 3 α, nd repled y t, we otin y = t yτ 3α t q τdτ. Uing the Green funtion Gt, given y 1.13 for BVP 1.14 we ee Hene yt = yt = Let m = mx t, yt. Then m m mx t, from whih it follow tht 1 mx t, Gt, yτ 3α q τdτd Gt, yτ 3α q τdτd Gt, yτ 3α q τ dτd. Gt, 3α q τ dτd Gt, 3α q τ dτd. Thi, together with the rgument in the proof of Theorem 2.1, led to From the proof of Prt we ee tht 2.27 hold. Without lo of generlity, ume tht yt 0 on,. Let m = mx t, yt. Then from 2.27 we ee tht m mx t, Gt, yτ 3α q τ dτd. After thi the me rgument in the proof of Theorem 2.1 led u to 2.26.

12 12 S. HAR, Q. KONG EJE-2017/203 With the me rgument in Corollry 2.3, we otin the orollry elow from Theorem 2.9. Corollry Aume 1.1 h nontrivil olution xt tifying BC 1.5. Suppoe 3α q τ 0 for,. Then t α3 q tdt > 8Γα 2 2. Remrk Here we remrk on the peil e of Theorem 2.9 with α = 3 where BVP 1.1,1.5 eome the third-order liner BVP Note tht mx t, x qtx = 0, x = x = 0 nd x = Gt, qτ dτd mx t, mx t, nd y Remrk mx Gt, d =. t, 8 Hene onluion 2.25 in Theorem 2.9 led to qτ dτ > 8 2. Similrly, onluion 2.26 in Theorem 2.9 led to q τdτ > 8 2. Thee inequlitie improve thoe in Theorem 1.3. Gt, qτ dτ d Gt, d qτ dτ 3. Applition to oundry vlue prolem In the lt etion, we pply the reult on the Lypunov-type Inequlitie otined in Setion 2 to tudy the nonexitene, uniquene, nd exitene-uniquene of olution of ertin frtionl order liner BVP. efinition 3.1. A nontrivil olution xt of 1.1 i id to e I-poitive if I nα x t 0 on,, where n = α 1. The following reult i on the nonexitene of ertin olution of BVP 1.1, 1.3. Theorem 3.2. Aume 3α Gt, q d dt Then BVP 1.1, 1.3 h no nontrivil olution. Aume Gt, q d dt 1, 3.2 3α

13 EJE-2017/203 FRACTIONAL LYAPUNOV-TYPE INEQUALITIES 13 3α Gt, q Then BVP 1.1, 1.3 h no I-poitive olution. d dt Proof. Aume the ontrry, i.e., BVP 1.1, 1.3 h nontrivil olution xt. Then y Theorem 2.1, 2.2 hold. Thi ontrdit umption 3.1. Aume the ontrry, i.e., BVP 1.1, 1.3 h n I-poitive olution xt. Then from the proof of Theorem 2.2, we ee tht only Ce I nd II in the proof re feile. Hene either or 3α 3α Thi ontrdit the umption. Gt, q Gt, q d dt > 1 d dt > 1. Next we pply the reult of Theorem 3.2 to tudy the nonhomogeneou liner BVP oniting of the eqution α x t qtx = wt, on, 3.4 nd the BC α2 x = k 1, α2 x = k 2, α3 x = k 3, 3.5 where q, w L,, R, 2 < α 3, nd k 1, k 2, k 3 R. Bed on Theorem 3.2, we otin riterion for BVP 3.4, 3.5 to hve unique olution nd revel reltion mong the olution if the prolem h more thn one olution. Theorem 3.3. Aume 3α Gt, q d dt Then BVP 3.4, 3.5 h unique olution on, for ny k 1, k 2, k 3 R. Aume 3α Gt, q d dt 1 < 3α Gt, q d dt. ± If BVP 3.4, 3.5 h two olution x 1 t nd x 2 t, then there exit d, uh tht I 3α x 1 d = I 3α x 2 d. Proof. By Theorem 3.2, Prt, BVP 1.1, 1.3 h only the zero olution. Then y the Fredholm lterntive theorem 10, we onlude tht BVP 3.4, 3.5 h unique olution. The onluion i lerly true when x 1 t x 2 t on,. Aume x 1 t x 2 t on, nd let xt = x 1 tx 2 t. Then xt i nontrivil olution of BVP 1.1, 1.3. By Theorem 3.2, Prt, xt i not I-poitive on,. With the me reon, xt i not n I-poitive olution on, either. Then there exit d, uh tht I 3α x d = 0, i.e., I 3α x 1 d = I 3α x 2 d. The reult in thi etion n e eily extended to the homogeneou liner BVP 1.1, 1.4 nd 1.1, 1.5, nd their orreponding nonhomogeneou liner BVP. We left the detil to the intereted reder.

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15 EJE-2017/203 FRACTIONAL LYAPUNOV-TYPE INEQUALITIES 15 Sougt hr eprtment of Mthemti nd Sttiti, Univerity of Mine, Orono, ME 04469, USA E-mil ddre: Qingki Kong eprtment of Mthemti, Northern Illinoi Univerity, ekl, IL 60115, USA E-mil ddre: