Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 R E S E A R C H Open Access Nonlnear fractonal Caputo-Langevn equaton wth nonlocal Remann-Louvlle fractonal ntegral condtons Weera Yukunthorn 1, Sotrs K Ntouyas 2,3 and Jessada Tarboon 1* * Correspondence: essadat@kmutnb.ac.th 1 Nonlnear Dynamc Analyss Research Center, Department of Mathematcs, Faculty of Appled Scence, Kng Mongkut s Unversty of Technology North Bangkok, Bangkok, 18, Thaland Full lst of author nformaton s avalable at the end of the artcle Abstract In ths paper, we study the exstence and unqueness of soluton for a problem consstng of a sequental nonlnear fractonal Caputo-Langevn equaton wth nonlocal Remann-Louvlle fractonal ntegral condtons. A varety of fxed pont theorems, such as Banach s fxed pont theorem, Krasnoselsk s fxed pont theorem, Leray-Schauder s nonlnear alternatve and Leray-Schauder degree theory, are used. Examples llustratng the obtaned results are also presented. MSC: 26A33; 34A8; 34B1 Keywords: fractonal dfferental equatons; nonlocal boundary condtons; fxed pont theorems 1 Introducton In ths paper, we concentrate on the study of exstence and unqueness of soluton for the followng nonlnear fractonal Caputo-Langevn equaton wth nonlocal Remann- Louvlle fractonal ntegral condtons: D p D q λ xt=f t, xt, t [, T], μ I α xη =σ 1, 1.1 ν I β xξ =σ 2, where < p, q 1, 1 < p q 2, D q and D p are the Caputo fractonal dervatves of order q and p, respectvely,i φ s the Remann-Louvlle fractonal ntegral of order φ, where φ = α, β >,η, ξ, Taregvenponts,μ, ν, λ, σ 1, σ 2 R, =1,2,...,m, =1,2,...,n, and f :[,T] R R s a contnuous functon. The sgnfcance of studyng problem 1.1 s that the nonlocal condtons are very general and nclude many condtons as specal cases. In partcular, f α = β = 1, for all =1,2,...,m, =1,2,...,n, then the nonlocal condton of 1.1reducesto { μ 1 η1 xs ds μ 2 ξ1 ν 1 xs ds ν 2 η2 xs ds μ m ξ2 xs ds ν n ηm xs ds = σ 1, ξn xs ds = σ 2, 214 Yukunthorn et al.; lcensee Sprnger. Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense http://creatvecommons.org/lcenses/by/2., whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. 1.2
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 2 of 18 and f σ 1 = σ 2 =,m = n =2,μ 2, ν 2,then1.2 s reduced to η1 ε 1 xs ds = η2 xs ds, ξ1 ε 2 xs ds = ξ2 xs ds, 1.3 where ε 1 = μ 1 /μ 2 andε 2 = ν 1 /ν 2. Note that the nonlocal condtons 1.2and1.3do not contan values of an unknown functon x on the left-hand sde and the rght-hand sde of boundary ponts t =andt = T,respectvely. Fractonal dfferental equatons have been shown to be very useful n the study of models of many phenomena n varous felds of scence and engneerng, such as physcs, chemstry, bology, sgnal and mage processng, bophyscs, blood flow phenomena, control theory, economcs, aerodynamcs and fttng of expermental data. For examples and recentdevelopment of thetopc,see [1 13] and the references cted theren. The Langevn equaton frst formulated by Langevn n 198 s found to be an effectve tool to descrbe the evoluton of physcal phenomena n fluctuatng envronments [14]. For some new developments on the fractonal Langevn equaton, see, for example, [15 24]. In the present paper several new exstence and unqueness results are proved by usng a varety of fxed pont theorems such as Banach s contracton prncple, Krasnoselsk s fxed pont theorem, Leray-Schauder s nonlnear alternatve and Leray-Schauder s degree theory. The rest of the paper s organzed as follows. In Secton 2 we recall some prelmnary facts that we need n the sequel. In Secton 3 we present our exstence and unqueness results. Examples llustratng the obtaned results are presented n Secton 4. 2 Prelmnares In ths secton, we ntroduce some notatons and defntons of fractonal calculus [2, 3] and present prelmnary results needed n our proofs later. Defnton 2.1 Foranatleastn-tmes dfferentable functon g :[, R,theCaputo dervatve of fractonal order q s defned as c D q gt= 1 Ɣn q t t s n q 1 g n s ds, n 1<q < n, n =[q]1, where [q] denotes the nteger part of the real number q. Defnton 2.2 The Remann-Louvlle fractonal ntegral of order q s defned as I q gt= 1 t gs ds, q >, Ɣq t s 1 q provded the ntegral exsts. Lemma 2.1 For q >,the general soluton of the fractonal dfferental equaton c D q ut= s gven by ut=c c 1 t c n 1 t n 1, where c R, =1,2,...,n 1n =[q]1.
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 3 of 18 In vew of Lemma 2.1, t follows that I qc D q ut=utc c 1 t c n 1 t n 1 for some c R, =1,2,...,n 1n =[q]1. In the followng, for the sake of convenence, we set constants 1 = 2 = μ ν η qα Ɣα q 1, 1 = ξ qβ Ɣβ q 1, 2 = μ ν η α Ɣα 1, ξ β Ɣβ 1, and = 1 2 2 1. Lemma 2.2 Let,<p, q 1, 1 < pq 2, α, β >,μ, ν, λ, σ 1, σ 2 R, η, ξ, T, = 1,2,...,m, = 1,2,...,n, and y C[, T], R. Then the nonlnear fractonal Caputo- Langevn equaton D p D q λ xt=yt, 2.1 subect to the nonlocal Remann-Louvlle fractonal ntegral condtons μ I α xη =σ 1, ν I β xξ =σ 2, 2.2 has a unque soluton gven by xt=i qp yt λi q xt 2t q 2 Ɣq 1 σ 1 μ I αqp yη λ μ I αq xη Ɣq 1 1t q 1 Ɣq 1 σ 2 ν I βqp yξ λ ν I βq xξ. 2.3 Ɣq 1 Proof The general soluton of equaton 2.1 s expressed as the followng ntegral equaton: t q xt=i qp yt λi q xtc Ɣq 1 c 1, 2.4 where c and c 1 are arbtrary constants. By takng the Remann-Louvlle fractonal ntegral of order α >for2.4, we get I α xt=i α qp yt λi α q xtc t α q Ɣα q 1 t α c 1. Ɣα 1
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 4 of 18 In partcular, for t = η,wehave I α xη =I α qp yη λi α q xη c η α q Ɣα q 1 α η c 1. Ɣα 1 Repeatng the above process for the Remann-Louvlle fractonal ntegral of order β >, substtutng t = ξ and applyng the nonlocal condton 2.2, we obtan the followng system of lnear equatons: c 1 c 1 1 = σ 1 μ I αqp yη λ μ I αq xη, c 2 c 1 2 = σ 2 ν I βqp yξ λ ν I βq xξ. 2.5 Solvng the lnear system of equatons n 2.5forconstantsc, c 1,wehave c = 2 1 c 1 = 1 σ 1 μ I αqp yη λ μ I αq xη 2 σ 2 σ 2 ν I βqp yξ λ ν I βq xξ, ν I βqp yξ λ ν I βq xξ σ 1 μ I αqp yη λ μ I αq xη. Substtutng c and c 1 nto 2.4, we obtan soluton 2.3. 3 Man results Throughout ths paper, for convenence, the expresson I x φymeans I x φy= 1 Ɣx y y s x 1 φs ds for y [, T]. Let C = C[, T], R denote the Banach space of all contnuous functons from [, T] to R endowed wth the norm defned by u = sup t [,T] ut. As n Lemma 2.2, wedefne an operator K : C C by Kxt=I qp f s, xs t λi q xt 2t q 2 Ɣq 1 σ 1 μ I αqp f s, xs η λ μ I αq xη Ɣq 1 1t q 1 Ɣq 1 σ 2 ν I βqp f s, xs ξ λ ν I βq xξ.3.1 Ɣq 1 It should be notced that problem 1.1 has solutons f and only f the operator K has fxed ponts.
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 5 of 18 In the followng subsectons, we prove exstence, as well as exstence and unqueness results, for problem 1.1 by usng a varety of fxedpont theorems. 3.1 Exstence and unqueness result va Banach s fxed pont theorem Theorem 3.1 Let f :[,T] R R be a contnuous functon. Assume that H 1 there exsts a constant L >such that f t, x f t, y L x y for each t [, T] and x, y R. If L 1 2 <1, 3.2 where constants 1, 2 are defned by 1 := T qp Ɣq p 1 2 T q 2 Ɣq 1 Ɣq 1 m μ η α qp Ɣα q p 1 1 T q 1 Ɣq 1 ν ξ β qp Ɣq 1 Ɣβ q p 1, 3.3 T q 2 := λ Ɣq 1 2 T q 2 Ɣq 1 m μ η α q Ɣq 1 Ɣα q 1 1 T q 1 Ɣq 1 ν ξ β q, 3.4 Ɣq 1 Ɣβ q 1 then problem 1.1 has a unque soluton on [, T]. Proof Problem 1.1 s equvalent to a fxed pont problem by defnng the operator K as n 3.1, whch yelds x = Kx. Usng the Banach contracton mappng prncple, we wll show that problem 1.1 has a unque soluton. Settng sup t [,T] f t, = M <,wedefneaset B r = {x C : x r}, where r M 1 1 L 1 2, 2 T q 2 Ɣq 1 := σ 1 σ 2 Ɣq 1 For any x B r,wehave 1 T q 1 Ɣq 1 Ɣq 1 Kxt I qp f s, xs t λ I q xs t 2 t q 2 Ɣq 1 σ 1 μ I α qp f s, xs η Ɣq 1 λ μ I α q xs η. 3.5
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 6 of 18 1 t q 1 Ɣq 1 σ 2 Ɣq 1 λ ν I β q xs ξ ν I β qp f s, xs ξ I qp f s, xs f s, f s, t λ I q xs t 2 t q 2 Ɣq 1 σ 1 μ I α qp f s, xs f s, Ɣq 1 f s, η λ μ I α q xs η 1 t q 1 Ɣq 1 Ɣq 1 f s, ξ λ σ 2 ν I β qp f s, xs f s, ν I β qxs ξ [ T qp Lr M Ɣq p 1 2 T q 2 Ɣq 1 m μ η α qp Ɣq 1 Ɣα q p 1 1 T q 1 Ɣq 1 ν ξ β qp ] Ɣq 1 Ɣβ q p 1 [ T q r λ Ɣq 1 2 T q 2 Ɣq 1 m μ η α q Ɣq 1 Ɣα q 1 1 T q 1 Ɣq 1 ν ξ β q ] Ɣq 1 Ɣβ q 1 2 T q 2 Ɣq 1 1 T q 1 Ɣq 1 σ 1 σ 2 Ɣq 1 Ɣq 1 =Lr M 1 r 2 r, whch mples that KB r B r. Next, we need to show that K s a contracton mappng. Let x, y C.Then,fort [, T], we have Kxt Kyt I qp f s, xs f s, ys t λ I q xs ys t 2 t q 2 Ɣq 1 m μ I α qpf s, xs f s, ys η Ɣq 1 λ μ I α q xs ys η 1 t q 1 Ɣq 1 Ɣq 1 ν I β qpf s, xs f s, ys ξ
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 7 of 18 λ ν I β q xs ys ξ [ LT pq x y Ɣ1 p q λ T q Ɣ1 q 2 T q 2 Ɣq 1 m Ɣq 1 1 T q 1 Ɣq 1 Ɣq 1 L 1 2 x y, L μ η α qp m Ɣα p q 1 λ L ν ξ β qp Ɣβ q p 1 λ μ η α q Ɣα q 1 ν ξ β q ] Ɣβ q 1 whch leads to Kx Ky L 1 2 x y. SnceL 1 2 <1,K s a contracton mappng. Therefore K hasonlyonefxedpont, whchmplesthatproblem1.1 hasa unque soluton. 3.2 Exstence and unqueness result va Banach s fxed pont theorem and Hölder s nequalty Now we gve another exstence and unqueness result for problem 1.1 by usng Banach s fxed pont theorem and Hölder s nequalty. For σ,1,we set 3 := [ 1 σ 1 σ T qp σ q p σ Ɣq p 2 T q 2 Ɣq 1 Ɣq 1 1 T q 1 Ɣq 1 Ɣq 1 m 1 σ α q p σ 1 σ β q p σ 1 σ μ η α qp Ɣα q p 1 σ ν ξ β qp ]. 3.6 Ɣβ q p Theorem 3.2 Let f :[,T] R R be a contnuous functon. In addton we assume that H 2 f t, x f t, y δt x y for each t [, T], x, y R, where δ L σ [, T], R, σ, 1. Denote δ = T δ 1 σ s ds σ. If 3 δ 2 <1, 3.7 where 2 and 3 are defned by 3.4 and 3.6, respectvely, then problem 1.1 has a unque soluton. Proof For x, y C and each t [, T], by Hölder s nequalty, we have Kxt Kyt I qp δs xs ys t λ I q xs ys t
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 8 of 18 2 t q 2 Ɣq 1 m μ I αqp δs xs ys η Ɣq 1 λ μ I α q xs ys η 1 t q 1 Ɣq 1 Ɣq 1 λ ν I βqp δs xs ys ξ ν I β q xs ys ξ [ t x y t s qp 1 t q δs ds λ Ɣq 1 2 t q 2 Ɣq 1 m η μ η s αqp 1 δs ds Ɣq 1 μ η α q λ Ɣα q 1 1 t q 1 Ɣq 1 ξ ν ξ s βqp 1 δs ds λ x y ν ξ β q ] Ɣβ q 1 [ t 1 σ t t s qp 1 1 σ ds t q σ δs σ ds 1 λ Ɣq 1 2 t q 2 Ɣq 1 m η 1 σ μ η s α qp 1 1 σ ds Ɣq 1 η σ δs σ ds 1 λ 1 t q 1 Ɣq 1 Ɣq 1 ξ σ δs σ ds 1 λ 3 δ 2 x y. μ η α q γ α q 1 ξ ν ξ s β qp 1 1 σ 1 σ ds ν ξ β q ] Ɣβ q 1 Therefore, Kx Ky 3 δ 2 x y. Hence, from 3.7, K s a contracton mappng.banach s fxedpont theoremmples that K has a unque fxed pont, whch s the unque soluton of problem 1.1. Ths completes the proof.
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 9 of 18 3.3 Exstence result va Krasnoselsk s fxed pont theorem Lemma 3.1 Krasnoselsk s fxed pont theorem [25] Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be operators such that a Ax Bx M whenever x, y M;bA s compact and contnuous;cb s a contracton mappng. Then there exsts z Msuchthatz= Az Bz. Theorem 3.3 Let f :[,T] R R be a contnuous functon. Moreover, we assume that H 3 f t, x φt, t, x [, T] R and φ C[, T], R. Then problem 1.1 has at least one soluton on [, T] f 2 <1, 3.8 where 2 s defned by 3.4. Proof We defne the operators A and B on B r by Axt=I qp f s, xs t 2t q 2 Ɣq 1 σ 1 μ I αqp f s, xs η Ɣq 1 1t q 1 Ɣq 1 σ 2 ν I βqp f s, xs ξ, Ɣq 1 Bxt= λi qp xst 2t q 2 Ɣq 1 λ μ I αq xsη Ɣq 1 1t q 1 Ɣq 1 λ ν I βq xsξ, Ɣq 1 where the ball B r s defned by B r = {x C, x r} for some sutable r such that r 1 φ 1 2, wth φ = sup t [,T] φt and 1, 2 and are defned by 3.3, 3.4 and3.5, respectvely. To show that Ax By B r,weletx, y B r.thenwehave AxtByt I qp f s, xs t 2 t q 2 Ɣq 1 Ɣq 1 σ 1 μ I α qp f s, xs η 1 t q 1 Ɣq 1 Ɣq 1 σ 2 ν I β qpf s, xs ξ λ I qp ys t 2 t q 2 Ɣq 1 Ɣq 1 λ μ I α q ys η
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 1 of 18 1 t q 1 Ɣq 1 λ Ɣq 1 1 φ r 2 r. ν I β q ys ξ It follows that Ax By B r, and thus condton a of Lemma 3.1 s satsfed. For x, y C, we have Bx By 2 x y. Snce 2 < 1, the operator B s a contracton mappng. Therefore, condton c of Lemma 3.1 s satsfed. The contnuty of f mples that the operator A s contnuous. For x B r,weobtan Ax 1 φ. Ths means that the operator A s unformly bounded on B r. Next we show that A s equcontnuous. We set sup t [,T] f t, xt = f, and consequently we get Axt 2 Axt 1 1 t1 [ Ɣq p 1 t2 s qp 1 t 1 s qp 1] f s, xs ds t2 t 2 s qp 1 f s, xs ds t 1 2 t q 2 tq 1 σ 1 μ I α qp f s, xs η Ɣq 1 1 t q 2 tq 1 σ 2 ν I βqp f s, xs ξ Ɣq 1 f t qp 2 t qp 1 f 2 t q 2 tq 1 μ η α qp σ 1 Ɣq p 1 Ɣα q p 1 f 1 t q 2 tq 1 σ 2 Ɣq 1 ν ξ β qp, Ɣβ q p 1 whch s ndependent of x and tends to zero as t 1 t 2.ThenA s equcontnuous. So A s relatvely compact on B r, and by the Arzelá-Ascol theorem, A s compact on B r. Thus condton b of Lemma 3.1 s satsfed. Hence the operators A and B satsfy the hypotheses of Krasnoselsk s fxed pont theorem; and consequently, problem 1.1hasat least one soluton on [, T]. 3.4 Exstence result va Leray-Schauder s nonlnear alternatve Theorem 3.4 Nonlnear alternatve for sngle-valued maps [26] Let E be a Banach space, Cbeaclosed, convex subset of E, UbeanopensubsetofCand U. Suppose that A : U C s a contnuous, compact that s, FU s a relatvely compact subset of C map. Then ether A has a fxed pont n U, or there s x U the boundary of U n C and λ, 1 wth x = λax. Theorem 3.5 Let f :[,T] R R be a contnuous functon. Assume that
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 11 of 18 H 4 there exsts a contnuous nondecreasng functon denoted by ψ :[,, and a functon g C[, T], R such that f t, u gtψ x for each t, x [, T] R; H 5 there exsts a constant M >such that M ψm g 1 M 2 >1, where 1, 2 and are defned by 3.3, 3.4 and 3.5, respectvely. Then problem 1.1 has at least one soluton on [, T]. Proof Let the operator K be defned by 3.1. Frstly, we shall show that K maps bounded sets balls nto bounded sets n C. Foranumberr >,letb r = {x C : x r} be a bounded ball n C.Then,fort [, T], we have Kxt I qp f s, xs t λ I q xs t 2 t q 2 Ɣq 1 σ 1 μ I α qp f s, xs η Ɣq 1 λ μ I α q xs η 1 t q 1 Ɣq 1 Ɣq 1 λ ν I β q xs ξ ψr g 1 r 2, σ 2 ν I β qp f s, xs ξ and consequently, Kx ψr g 1 r 2. Next, we wll show that K maps bounded sets nto equcontnuous sets of C. Lett 1, t 2 [, T]wtht 1 < t 2 and x B r.thenwehave Kxt 2 Kxt 1 1 t1 [ Ɣq p 1 t2 s qp 1 t 1 s qp 1] f s, xs ds t2 t 2 s qp 1 f s, xs ds 2 t q 2 tq 1 σ 1 μ I α qp f s, xs η t 1 Ɣq 1 λ μ I α qxs η 1 t q 2 tq 1 σ 2 ν I β qpf s, xs ξ Ɣq 1
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 12 of 18 λ ν I β q xs ξ ψr g qp t 2 t qp 1 Ɣ1 q p 2 t q 2 tq 1 σ 1 Ɣq 1 1 t q 2 tq 1 σ 2 Ɣq 1 ψr g μ η α qp Ɣα q p 1 ψr g ν ξ β qp Ɣβ q p 1 r μ η α q Ɣα q 1 r ν x β q. Ɣβ q 1 As t 2 t 1, the rght-hand sde of the above nequalty tends to zero ndependently of x B r. Therefore, by the Arzelá-Ascol theorem, the operator K : C C s completely contnuous. Let x be a soluton. Then, for t [, T], and followng smlar computatons as n the frst step, we have xt ψ x g 1 x 2, whch leads to x ψ x g 1 x 2 1. By H 5 theresm such that x M.Letusset U = { x C : x < M }. We see that the operator K : U C s contnuous and completely contnuous. From the choce of U,theresnox U such that x = νkx for some ν, 1. Consequently, by the nonlnear alternatve of Leray-Schauder type, we deduce that K has a fxed pont x U whch s a soluton of problem 1.1. Ths completes the proof. 3.5 Exstence result va Leray-Schauder s degree theory Theorem 3.6 Let f :[,T] R R be a contnuous functon. Suppose that H 6 there exst constants γ <1 2 1 1 and M >such that f t, x γ x M for all t, x [, T] R, where 1, 2 are defned by 3.3 and 3.4, respectvely. Then problem 1.1 has at least one soluton on [, T]. Proof We defne an operator K : C C as n 3.1 and consder the fxed pont equaton x = Kx. We shall prove that there exsts a fxed pont x C satsfyng 1.1.
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 13 of 18 Set a ball B r C as { B r = x C : sup xt < r }, t [,T] where a constant radus r >. Hence, we show that K : B r C satsfes the condton x θkx, x B r, θ [, 1]. 3.9 We defne Hθ, x=θkx, x C, θ [, 1]. AsshownnTheorem3.5, the operator K s contnuous, unformly bounded and equcontnuous. Then, by the Arzelá-Ascol theorem, a contnuous map h θ defned by h θ x = x Hθ, x=x θkx s completely contnuous. If 3.9 holds, then the followng Leray- Schauder degrees are well defned, and by the homotopy nvarance of topologcal degree, t follows that degh θ, B r,=degi θk, B r,=degh 1, B r, = degh, B r,=degi, B r,=1, B r, where I denotes the unt operator. By the nonzero property of Leray-Schauder degree, h 1 x=x Kx =foratleastonex B r.letusassumethatx = θkx for some θ [, 1] and for all t [, T]sothat xt = θkxt I qp f s, xs t λ I q xs t 2 t q 2 Ɣq 1 σ 1 μ I α qp f s, xs η Ɣq 1 λ μ I α q xs η 1 t q 1 Ɣq 1 Ɣq 1 λ ν I β qxs ξ σ 2 ν I β qpf s, xs ξ γ xt t qp M Ɣq p 1 2 t q 2 Ɣq 1 m μ η α qp Ɣq 1 Ɣα q p 1 1 t q 1 Ɣq 1 ν ξ β qp Ɣq 1 Ɣβ q p 1 λ xt t q Ɣq 1 2 t q 2 Ɣq 1 m μ η α q Ɣq 1 Ɣα q 1
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 14 of 18 1 t q 1 Ɣq 1 ν ξ β q Ɣq 1 Ɣβ q 1 2 t q 2 Ɣq 1 σ 1 σ 2 Ɣq 1 Takng norm sup t [,T] xt = x,weget 1 t q 1 Ɣq 1 Ɣq 1. x γ x M 1 x 2. Solvng the above nequalty for x yelds x M 1 1 γ 1 2. If r = M 1 1 γ 1 2 1,thennequalty3.9 holds. Ths completes the proof. 4 Examples Example 4.1 Consder the followng fractonal Caputo-Langevn equaton wth Remann- Louvlle fractonal ntegral condtons: D 1 7 D 2 5 1 1 e t x xt= 1 4t1 2 2 x 3 2, t, 1, 3 1 2 I 1 9 x 1 1 3 1 I 4 5 x 3 2 1 5 I 1 7 x 1 5 =5, 4.1 2 5 I 1 3 x 4 5 2 5 I 5 1 x 17 2 1 5 I 1 1 x 9 1 =2. Here p =7/1,q = 2/5, λ = 1/1, T =1,m =3,n =3,μ 1 = 1/2, α 1 =9/1,η 1 = 1/1, μ 2 = 3/1, α 2 = 4/5, η 2 =3/2,μ 3 = 1/5, α 3 =7/1,η 3 = 1/5, σ 1 =5,ν 1 = 2/5, β 1 =3/1,ξ 1 = 4/5, ν 2 = 2/5, β 2 = 1/5, ξ 2 =17/2,ν 3 = 1/5, β 3 = 1/1, ξ 3 =9/1,σ 2 =2andf t, x = 1 e t x /4t 1 2 2 x 32/3.Snce f t, x f t, y 1/4 x y,thenh 1 ssatsfed wth L = 1/4. We can fnd that 1 = T qp Ɣq p 1 2 T q 2 Ɣq 1 Ɣq 1 m μ η α qp Ɣα q p 1 1 T q 1 Ɣq 1 ν ξ β qp Ɣq 1 Ɣβ q p 1 2.84924, T q 2 = λ Ɣq 1 2 T q 2 Ɣq 1 m μ η α q Ɣq 1 Ɣα q 1 1 T q 1 Ɣq 1 ν ξ β q Ɣq 1 Ɣβ q 1 Therefore, we have.524. L 1 2.764296 < 1. Hence, by Theorem 3.1,problem4.1 has a unque soluton on [, 1].
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 15 of 18 Example 4.2 Consder the followng fractonal Caputo-Langevn equaton wth Remann- Louvlle fractonal ntegral condtons: D 3 5 D 3 7 1 1 5 xt= xe 1t sn 2 t/2 x 4 t 7 2 I 2 5 x 3π 5 1 5 I 14 11 x 4π 15 3 7 I 12 17 x 3π 8 2 7 I 4 3 x 2π 9 5 3 I 1 3 x 3π 11 5 2 I 1 19 x π 5 6 5 I 13 18 x π 3 1 4t x 1 5, t, π, 11 15 I 4 5 x π 5 = 3 7, 17 13 I 13 7 x π 4 3 17 I 6 5 x π 16 = 5 6. 4.2 Here p = 3/5, q = 3/7, λ = 1/5, T = π, m =5,n =5,μ 1 = 7/2, α 1 = 5/2, η 1 =3π/5, μ 2 = 1/5, α 2 =14/11,η 2 =4π/15, μ 3 = 3/7, α 3 =17/12,η 3 =3π/8, μ 4 = 2/7,α 4 = 4/3, η 4 = 2π/9, μ 5 = 11/15, α 5 = 4/5, η 5 = π/5, σ 1 = 3/7, ν 1 = 5/3, β 1 =1/3,ξ 1 =3π/11, ν 2 = 5/2, β 2 =19/1,ξ 2 = π/5, ν 3 = 6/5, β 3 =13/18,ξ 3 = π/3, ν 4 = 17/13, β 4 =13/7,ξ 4 = π/4, ν 5 = 3/17, β 5 = 6/5, ξ 5 = π/16, σ 2 = 5/6 and f t, x =xe 1t sn 1 2 t/2 x/4 t 1/4 t x 15.Snce f t, x f t, y e 1t sn 1 2 t/2 x y, thenh2 ssatsfedwth δt=e 1t sn 1 2 t/2 such that δ L 1/2 [, π], R. We can fnd that T q 2 = λ Ɣq 1 2 T q 2 Ɣq 1 m μ η α q Ɣq 1 Ɣα q 1 3 = 1 T q 1 Ɣq 1 ν ξ β q Ɣq 1 Ɣβ q 1 [ 1 σ 1 σ T qp σ q p σ Ɣq p 2 T q 2 Ɣq 1 Ɣq 1 1 T q 1 Ɣq 1 Ɣq 1 17.417544, m 1 σ α q p σ 1 σ β q p σ.35711, 1 σ μ η α qp Ɣα q p 1 σ ν ξ β qp ] Ɣβ q p and δ.35344. Therefore, we have 3 δ 2 966321 < 1. Hence, by Theorem 3.2, problem4.2 hasa unque soluton on[,π]. Example 4.3 Consder the followng fractonal Caputo-Langevn equaton wth Remann- Louvlle fractonal ntegral condtons: D 2 3 D 4 3 3 35 xt=t2 1 x t x lnt 1, t, e 1, t x 1 3 11 I 6 1 x 1 1 1 5 I 5 1 x 1e 11 3 1 7 I 3 8 x 2e 9 4 3 I 7 1 x 2e 5 = 15, 2 5 I 2 1 x 3 8 1 5 I 25 6 x 3e 2 4 1 1 I 4 1 x 5e 6 5 = 5. 4.3 Here p = 2/3, q = 3/4, λ = 3/35, T = e 1,m =4,n =3,μ 1 =3/11,α 1 = 1/6, η 1 = 1/1, μ 2 = 1/5, α 2 = 1/5, η 2 =1e 11/1, μ 3 = 3/7, α 3 = 3/8, η 3 =2e/9, μ 4 = 4/3, α 4 = 1/7, η 4 =
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 16 of 18 2e/5, σ1 =15,ν 1 = 2/5, β 1 = 1/2, ξ 1 = 3/8, ν 2 = 1/5, β 2 =6/25,ξ 2 =3e 2/4,ν 3 = 1/1, β 3 = 1/4, ξ 3 =5e 6/5,σ 2 = 5andf t, x=t 2 1 x t x /t x 1lnt 1. Snce f t, x lnt 1e,thenH 3 s satsfed. We can fnd that T q 2 = λ Ɣq 1 2 T q 2 Ɣq 1 m μ η α q Ɣq 1 Ɣα q 1 1 T q 1 Ɣq 1 ν ξ β q Ɣq 1 Ɣβ q 1.9852. Ths means that 2 <1.ByTheorem3.3 problem 4.3 has as least one soluton on [, e 1]. Example 4.4 Consder the followng fractonal Caputo-Langevn equaton wth Remann- Louvlle fractonal ntegral condtons: D 6 5 D 3 2 1 5 xt= e t x 2 4 sn t sn2t/3, t, π 4 x 6t t1 2 2, 2 2 I 3 8 x π 18 2I 1 1 x 4π 9 3I 4 1 x1 I 2 3 x 3 4 =1, 2 3 I 2 7 x π 6 3I 1 3 x π 3 2 2 7 I 2 11 x 5 4 2 5 I 9 7 x 1 2 =. 4.4 Here p = 5/6, q = 2/3, λ = 1/5, m =4,n =4,μ 1 = 2/2, α 1 = 3/8, η 1 = π/18, μ 2 = 2, α 2 = 1/1, η 2 =4π/9, μ 3 = 3, α 3 = 1/4, η 3 =1,μ 4 = 1,α 4 = 2/3, η 4 = 3/4, σ 1 =1,ν 1 = 2/3, β1 = 2/7, ξ 1 = π/6, ν 2 = 3,β 3 =3/1,ξ 2 = π/3, ν 3 =2 2/7, β 3 = 11/2, ξ 3 = 5/4, ν 4 = 2/5, β 4 = 7/9, ξ 4 = 1/2, σ 2 =,andf t, x=e t x 2 4sn t/4 x 6tsn2t/3/t 1 2. Then we can fnd that 1 = Clearly, T qp Ɣq p 1 2 T q 2 Ɣq 1 Ɣq 1 m μ η α qp Ɣα q p 1 1 T q 1 Ɣq 1 ν ξ β qp Ɣq 1 Ɣβ q p 1 18.47383, T q 2 = λ Ɣq 1 2 T q 2 Ɣq 1 m μ η α q Ɣq 1 Ɣα q 1 1 T q 1 Ɣq 1 ν ξ β q Ɣq 1 Ɣβ q 1 2 T q 2 Ɣq 1 = σ 1 σ 2 Ɣq 1.531365, 1 T q 1 Ɣq 1 Ɣq 1 f t, x e t x 2 t4sn t sn2t/3 = 1 x 1 sn2t/3. 4 x 6t t 1 2 4 By choosng ψ x = x 1 and gt= sn2t/3 /4, we canshow that M ψm g 1 Mλ 2 >1, 3.51534.
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 17 of 18 whch mples M >57.1974.ByTheorem3.5,problem4.4 has at least one soluton on [, π/2]. Example 4.5 Consder the followng fractonal Caputo-Langevn equaton wth Remann- Louvlle fractonal ntegral condtons: D 4 5 D 1 3 1 8 xt= t2 e 2t2 snx π, t, 2π, 4π 3 3 2 I 2 5 x 8 5 1 3 I 1 3 x 4 3 2 11 I 15 2 x 9 2 1 7 I 3 1 x 3 2 =1, 5 2 I 6 1 x6 1 2 I 2 1 x 5 2 4 5 I 9 2 x 3 2 4 3 I 2 7 x 1 2 2 17 I 3 5 x 1 16 = 1. 4.5 Here p = 4/5, q =3/1,λ = 1/8, m =4,n =5,μ 1 = 3/2, α 1 = 2/5, η 1 = 8/5, μ 2 = 1/3, α 2 = 3/1, η 2 = 4/3, μ 3 = 2/11,α 3 = 2/15, η 3 = 9/2, μ 4 = 1/7, α 4 = 1/3, η 4 = 3/2, σ 1 =1,ν 1 = 5/2, β 1 = 1/6, ξ 1 =6,ν 2 = 1/2, β 2 = 1/2, ξ 2 = 5/2, ν 3 = 4/5,β 3 = 2/9, ξ 3 =3/2,ν 4 = 4/3,β 4 = 2/7, ξ 4 = 1/2, ν 5 = 2/17,β 5 = 3/5, ξ 5 = 1/16, σ 2 = 1,andf t, x=t 2 e 2t2 snx π/3/4π. By a drect computaton, we have 1 = T qp Ɣq p 1 2 T q 2 Ɣq 1 Ɣq 1 m μ η α qp Ɣα q p 1 1 T q 1 Ɣq 1 ν ξ β qp Ɣq 1 Ɣβ q p 1 39.248431, T q 2 = λ Ɣq 1 2 T q 2 Ɣq 1 m μ η α q Ɣq 1 Ɣα q 1 1 T q 1 Ɣq 1 ν ξ β q Ɣq 1 Ɣβ q 1.21434. Choosng γ =.2<1 2 1 1.225 and M =.16, we can show that f t, xt t 2 e 2t2 4π x sn π 3 t 2 e 2t2 t 2 e 2t2 x 4π 12.14637 x.15328 γ x M, whch satsfes H 6. By Theorem 3.6,problem4.5 has at least one soluton on [, 2π]. Competng nterests The authors declare that they have no competng nterests. Authors contrbutons All authors contrbuted equally n ths artcle. They read and approved the fnal manuscrpt.
Yukunthorn et al. Advances n Dfference Equatons 214, 214:315 Page 18 of 18 Author detals 1 Nonlnear Dynamc Analyss Research Center, Department of Mathematcs, Faculty of Appled Scence, Kng Mongkut s Unversty of Technology North Bangkok, Bangkok, 18, Thaland. 2 Department of Mathematcs, Unversty of Ioannna, Ioannna, 451 1, Greece. 3 Nonlnear Analyss and Appled Mathematcs NAAM Research Group, Department of Mathematcs, Faculty of Scence, Kng Abdulazz Unversty, P.O. Box 823, Jeddah, 21589, Saud Araba. Acknowledgements The research of JT s supported by Kng Mongkut s Unversty of Technology North Bangkok, Thaland. Receved: 1 September 214 Accepted: 26 November 214 Publshed: 12 Dec 214 References 1. Samko, SG, Klbas, AA, Marchev, OI: Fractonal Integrals and Dervatves: Theory and Applcatons. Gordon & Breach, Yverdon 1993 2. Podlubny, I: Fractonal Dfferental Equatons. Academc Press, San Dego 1999 3. Klbas, AA, Srvastava, HM, Trullo, JJ: Theory and Applcatons of Fractonal Dfferental Equatons. North-Holland Mathematcs Studes, vol. 24. Elsever, Amsterdam 26 4. Baleanu, D, Dethelm, K, Scalas, E, Trullo, JJ: Fractonal Calculus Models and Numercal Methods. Seres on Complexty, Nonlnearty and Chaos. World Scentfc, Boston 212 5. Agarwal, RP, Zhou, Y, He, Y: Exstence of fractonal neutral functonal dfferental equatons. Comput. Math. Appl. 59, 195-11 21 6. Baleanu, D, Mustafa, OG, Agarwal, RP: On L p -solutons for a class of sequental fractonal dfferental equatons. Appl. Math. Comput. 218, 274-281 211 7. Ahmad, B, Neto, JJ: Remann-Louvlle fractonal ntegro-dfferental equatons wth fractonal nonlocal ntegral boundary condtons. Bound. Value Probl. 211, 36 211 8. Ahmad, B, Ntouyas, SK, Alsaed, A: New exstence results for nonlnear fractonal dfferental equatons wth three-pont ntegral boundary condtons. Adv. Dffer. Equ. 211, ArtcleID17384 211 9. O Regan, D, Stanek, S: Fractonal boundary value problems wth sngulartes n space varables. Nonlnear Dyn. 71, 641-652 213 1. Ahmad, B, Ntouyas, SK, Alsaed, A: A study of nonlnear fractonal dfferental equatons of arbtrary order wth Remann-Louvlle type multstrp boundary condtons. Math. Probl. Eng. 213,Artcle ID32415 213 11. Ahmad, B, Neto, JJ: Boundary value problems for a class of sequental ntegrodfferental equatons of fractonal order. J. Funct. Spaces Appl. 213, Artcle ID 149659 213 12. Zhang, L, Ahmad, B, Wang, G, Agarwal, RP: Nonlnear fractonal ntegro-dfferental equatons on unbounded domans n a Banach space. J. Comput. Appl. Math. 249, 51-56 213 13. Lu, X, Ja, M, Ge, W: Multple solutons of a p-laplacan model nvolvng a fractonal dervatve. Adv. Dffer. Equ. 213, 126 213 14. Coffey, WT, Kalmykov, YP, Waldron, JT: The Langevn Equaton, 2nd edn. World Scentfc, Sngapore 24 15. Lm, SC, L, M, Teo, LP: Langevn equaton wth two fractonal orders. Phys. Lett. A 372, 639-632 28 16. Lm, SC, Teo, LP: The fractonal oscllator process wth two ndces. J. Phys. A, Math. Theor. 42, Artcle ID 6528 29 17. Uranagase, M, Munakata, T: Generalzed Langevn equaton revsted: mechancal random force and self-consstent structure.j.phys.a,math.theor.43, Artcle ID 4553 21 18. Densov, SI, Kantz, H, Hängg, P: Langevn equaton wth super-heavy-taled nose. J. Phys. A, Math. Theor. 43, Artcle ID 2854 21 19. Loznsk, A, Owens, RG, Phllps, TN: The Langevn and Fokker-Planck equatons n polymer rheology. In: Handbook of Numercal Analyss, vol. 16, pp. 211-33 211 2. Lzana, L, Ambörnsson, T, Talon, A, Barka, E, Lomholt, MA: Foundaton of fractonal Langevn equaton: harmonzaton of a many-body problem. Phys. Rev. E 81, Artcle ID 51118 21 21. Gambo, YY, Jarad, F, Baleanu, D, Abdelawad, T: On Caputo modfcaton of the Hadamard fractonal dervatve. Adv. Dffer. Equ. 214, 1 214 22. Ahmad, B, Eloe, PW: A nonlocal boundary value problem for a nonlnear fractonal dfferental equaton wth two ndces. Commun. Appl. Nonlnear Anal. 17, 69-8 21 23. Ahmad, B, Neto, JJ, Alsaed, A, El-Shahed, M: A study of nonlnear Langevn equaton nvolvng two fractonal orders n dfferent ntervals. Nonlnear Anal., Real World Appl. 13,599-66 212 24. Sudsutad, W, Tarboon, J: Nonlnear fractonal ntegro-dfferental Langevn equaton nvolvng two fractonal orders wth three-pont mult-term fractonal ntegral boundary condtons. J. Appl. Math. Comput. 43, 57-522 213 25. Krasnoselsk, MA: Two remarks on the method of successve approxmatons. Usp. Mat. Nauk 1, 123-127 1955 26. Granas, A, Dugund, J: Fxed Pont Theory. Sprnger, New York 23 1.1186/1687-1847-214-315 Cte ths artcle as: Yukunthorn et al.: Nonlnear fractonal Caputo-Langevn equaton wth nonlocal Remann-Louvlle fractonal ntegral condtons. Advances n Dfference Equatons 214, 214:315