On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions

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1 Alsedi et l. Advnces in Difference Equtions :33 DOI /s RESEARCH ARTICLE OpenAccess On Cputo type sequentil frctionl differentil equtions with nonlocl integrl boundry conditions Ahmed Alsedi 1, Sotiris K Ntouys 1,2, Rvi P Agrwl 1,3 nd Bshir Ahmd 1* * Correspondence: bshirhmd_qu@yhoo.com 1 Deprtment of Mthemtics, Fculty of Science, King Abdulziz University, P.O. Box 823, Jeddh, 21589, Sudi Arbi Full list of uthor informtion is vilble t the end of the rticle Abstrct This pper investigtes boundry vlue problem of Cputo type sequentil frctionl differentil equtions supplemented with nonlocl Riemnn-Liouville frctionl integrl boundry conditions. Some existence results for the given problem re obtined vi stndrd tools of fixed point theory nd re well illustrted with the id of exmples. Some specil cses re lso presented. MSC: 34A8; 34B15 Keywor: frctionl differentil equtions; sequentil frctionl derivtive; integrl boundry conditions; fixed point theorems 1 Introduction The subject of frctionl clculus hs received gret ttention in the lst two decdes. Recent work on frctionl differentil equtions supplemented with vriety of initil nd boundry conditions clerly reflects n overwhelming interest in the subject; for instnce, see [1 12] nd the references cited therein. The tools of frctionl clculus hve considerbly improved the mthemticl modeling of mny rel world problems. One of the primry resons for this development is the nonlocl nture of frctionl-order differentil opertors which cn describe the hereditry properties of mny importnt mterils. One cn find pplictions of the subject in numerous fiel of physicl nd technicl sciences such s biomthemtics, blood flow phenomen, ecology, environmentl issues, viscoelsticity, erodynmics, electro-dynmics of complex medium, electricl circuits, electronnlyticl chemistry, control theory,etc. For further detils, see [13 18].Some more recent results concerning frctionl boundry vlue problems cn be found in series of ppers [19 27]. The concept of sequentil frctionl derivtive is given, for exmple, on p.29 of the monogrph [28]. There is close connection between the sequentil frctionl derivtives nd the non-sequentil Riemnn-Liouville derivtives [29, 3].For some recentwork on sequentil frctionl differentil equtions, we refer the reder to the ppers [31 33]. In [34, 35], the uthors studied sequentil frctionl differentil equtions with different kin of boundry conditions. Recently, the existence of solutions for higher-order sequentil frctionl differentil inclusions with nonlocl three-point boundry conditions ws discussed in [36]. However, to the best of our knowledge, the study of sequentil 215 Alsedi et l.; licensee Springer. This is n Open Access rticle distributed under the terms of the Cretive Commons Attribution License which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly credited.

2 Alsedi et l. Advnces in Difference Equtions :33 Pge 2 of 12 frctionl differentil equtions supplemented with nonlocl Riemnn-Liouville type frctionl integrl boundry conditions hs yet to be initited. In this pper, we investigte the existence of solutions of sequentil frctionl differentil eqution of the form: c D α k c D α 1 xt=f t, xt, t [, 1], 2 < α 3, 1.1 supplemented with the boundry conditions η x =, x =, xζ = xs, β >, 1.2 Ɣβ where c D α denotes the Cputo frctionl derivtive of order α, <η < ζ <1,f is given continuous function, nd k, re pproprite positive rel constnts. Here, we emphsize tht the integrl boundry condition 1.2 cn be understood in the sense tht the vlue of the unknown function t n rbitrry position ζ η,1 is proportionl to the Riemnn-Liouville frctionl integrl of the unknown function η η s β 1 xs, whereη, ζ. Further, for β = 1, the integrl boundry condition reduces to the usul form of nonlocl integrl condition xζ = η Ɣβ xs. The contents of the rticle re rrnged s follows. In Section 2, we present bsic result tht lys the foundtion for defining fixed point problem equivlent to the given problem The min results, bsed on Bnch s contrction mpping principle, Krsnoselskii s fixed point theorem nd nonliner lterntive of Lery-Schuder type, re presented in Section 3. The pper concludes with some illustrting exmples. 2 Bsicresult Before presenting n uxiliry lemm, we recll some bsic definitions of frctionl clculus [13, 14]. Definition 2.1 For n 1-times bsolutely continuous function g :[, R, the Cputo derivtive of frctionl order q is defined s c D q gt= 1 Ɣn q t s n q 1 g n s, n 1<q < n, n =[q]1, where [q] denotes the integer prt of the rel number q. Definition 2.2 The Riemnn-Liouville frctionl integrl of order q is defined s I q gt= 1 gs, q >, Ɣq t s 1 q provided the integrl exists. Definition 2.3 The sequentil frctionl derivtive for sufficiently smooth function gt due to Miller-Ross [28]is defined s D δ gt=d δ 1 D δ2 D δ k gt, 2.1 where δ =δ 1,...,δ k is multi-index.

3 Alsedi et l. Advnces in Difference Equtions :33 Pge 3 of 12 In generl, the opertor D δ in 2.1 cn either be Riemnn-Liouville or Cputo or ny other kind of integro-differentil opertor. For instnce, c D q gt=d n q d dt n gt, n 1<q < n, where D n q is frctionl integrl opertor of order n q. Hereweemphsizetht D p f t=i p f t, p = n q; for more detils, see p.87 of [13]. To define the solution for problem , we estblish the following lemm. Lemm 2.4 For h C[, 1], R, the integrl solution of the liner eqution c D α k c D α 1 xt=ht, t [, 1], 2 < α 3, 2.2 supplemented with the boundry conditions 1.2 is given by xt= kt 1e kt e ks m e kζ s e kt s Proof Solving 2.2, we obtin xt=b e kt b 1 k e kt s Ɣβ m τ α 2 hτ dτ hτ dτ hτ dτ 1 e kt b 2 k 2 kt 1e kt hτ dτ dm. 2.3, 2.4 where b, b 1, b 2 re unknown rbitrry constnts. Using the boundry conditions 1.2in 2.4, we find tht b =,b 1 =nd b 2 = k2 e ks m m τ α 2 hτ dτ dm Ɣβ e kζ s hτ dτ, where = kζ 1e kζ kη β1 Ɣβ ββ 1 ηβ β η e ks. 2.5 Substituting the vlues of b, b 1 nd b 2 in 2.4 yiel the solution 2.3. This completes the proof.

4 Alsedi et l. Advnces in Difference Equtions :33 Pge 4 of 12 3 Existence of solutions Let C = C[, 1], R denote the Bnch spce of ll continuous functions from [, 1] R endowed with the sup norm defined by x = sup{ xt, t [, 1]} <. To simplify the proofs in the forthcoming theorems, we estblish the boun for the integrls rising in the sequel. Lemm 3.1 For h C[, 1], R, we hve η i e ks m Ɣβ m τ α 2 hτ dτ dm ii iii ηαβ 1 ηk e kη 1 h, k 2 ƔαƔβ e kζ s hτ dτ ζ α 1 e kζ h, kɣα e kt s hτ dτ 1 1 e k h. kɣα Proof i Obviously, nd m s Hence η m τ α 2 dτ = m τα 1 Ɣα ks m mα 1 sα 1 e dm Ɣα Ɣα s m = mα 1 Ɣα e ks m dm = sα 1 Ɣα e ks m Ɣβ η s α 1 1 e ks h Ɣβ kɣα η αβ 1 η h 1 e ks = h kɣαɣβ 1 e ks. k m τ α 2 hτ dτ dm η αβ 1 k 2 ƔαƔβ ηk e kη 1. The proofs of ii nd iii re similr. nd For the ske of convenience, we set p = sup kt 1e kt = 1 e k k t [,1] { η αβ 1 = p kη e kη 1 ζ α 1 1 e kζ } 1 e k k 2 ƔαƔβ kɣα kɣα. 3.2 In view of Lemm 2.4,wetrnsformproblem s x = Sx, 3.3

5 Alsedi et l. Advnces in Difference Equtions :33 Pge 5 of 12 where S : C C is defined by Sxt= kt 1e kt Ɣβ e ks m m τ α 2 e kζ s e kt s f τ, xτ dτ f τ, xτ dτ f τ, xτ dτ dm. 3.4 Observe tht problem hs solutions if the opertor eqution 3.3 hsfixed points. Theorem 3.2 Let f :[,1] R R be jointly continuous function stisfying the condition H 1 f t, x f t, y L x y, t [, 1], x, y R, where L is the Lipschitz constnt. Then the boundry vlue problem hs unique solution if <1/L, where is given by 3.2. Proof As the first step, we show tht the opertor S given by 3.3mpsC into itself. For tht, we set sup t [,1] f t, = M <.Then,forx C,wehve Sx = sup t [,1] e ks m kt 1e kt [ η Ɣβ m x α 2 f τ, xτ dτ e kζ s s x α 2 e kt s s x α 2 f τ, xτ dτ sup t [,1] kt 1e kt dm f τ, xτ dτ e ks m Ɣβ m τ α 2 f τ, xτ f τ, f τ, dx dm e kζ s f τ, xτ f τ, f τ, dx f τ, xτ f τ, f τ, dx e kt s s x α 2 L x M { p e ks m Ɣβ e kζ s dτ e kt s m τ α 2 dτ ] dτ dm }

6 Alsedi et l. Advnces in Difference Equtions :33 Pge 6 of 12 L x M { [ η αβ 1 p k 2 ƔαƔβ = L x M <. kη e kη 1 ζ α 1 1 e kζ kɣα ] } 1 e k kɣα This shows tht S mps C into itself. Now, for x, y C nd for ech t [, 1], we obtin Sx Sy = sup Sxt Syt t [,1] sup kt 1e kt [ η t [,1] Ɣβ e ks m m τ α 2 f τ, xτ f τ, yτ dτ dm ] e kζ s f τ, xτ f τ, yτ dτ e kt s f τ, xτ f τ, yτ dτ { L x y p Ɣβ e ks m m x α 2 dx dm e kζ s s x α 2 dx e kt s { [ η αβ 1 L p k 2 ƔαƔβ = L x y, s x α 2 dx } kη e kη 1 ζ α 1 1 e kζ kɣα ] } 1 e k x y kɣα where is given by 3.2. As <1/L, therefore, S is contrction. Thus, the conclusion of the theorem follows by the contrction mpping principle. This completes the proof. Now, we stte known result due to Krsnoselskii [37] which is needed to prove the existenceoftlestonesolutionof Theorem 3.3 Let M be closed, convex, bounded nd nonempty subset of Bnch spce X. Let G 1, G 2 be the opertors such tht:ig 1 x G 2 y Mwheneverx, y M; ii G 1 is compct nd continuous; iii G 2 is contrction mpping. Then there exists z Msuch tht z = G 1 z G 2 z. Theorem 3.4 Assume tht f :[,1] R R is jointly continuous function stisfying H 1. In ddition we suppose tht the following ssumption hol: H 2 f t, x μt, t, x [, 1] R with μ C[, 1], R. Then the boundry vlue problem hs t lest one solution on [,1] if [ η αβ 1 p kη e kη 1 ζ α 1 1 e kζ ] < k 2 ƔαƔβ kɣα

7 Alsedi et l. Advnces in Difference Equtions :33 Pge 7 of 12 Proof Letting sup t [,1] μt = μ,wefix r μ, 3.6 where is given by 3.2, nd consider B r = {x C : x r}. Define the opertors S 1 nd S 2 on B r s S 1 xt= e kt s f τ, xτ dτ, S 2 xt= kt [ 1e kt η Ɣβ e ks m m τ α 2 f τ, xτ dτ e kζ s f τ, xτ dτ For x, y B r, it follows from 3.6tht { [ η αβ 1 S 1 x S 2 y p k 2 ƔαƔβ. dm kη e kη 1 ζ α 1 1 e kζ kɣα ] } 1 e k μ r. kɣα Thus, S 1 x S 2 y B r. In view of condition 3.5, it cn esily be shown tht S 2 is contrction mpping. The continuity of f implies tht the opertor S 1 is continuous. Also, S 1 is uniformly bounded on B r s S 1 x 1 e k μ. kɣα Now we prove the compctness of the opertor S 1. Setting =[,1] B r,wedefine sup t,x f t, x = M r, nd consequently we get S 1 xt 1 S 1 xt 2 = 1 e kt 1 s 2 e kt 2 s s u α 2 f u, xu du s u α 2 f u, xu du M r t1 α kɣα tα 2 t1 α e kt 1 t 2 α e kt 2, which is independent of x nd ten to zero s t 2 t 1.Thus,S 1 is reltively compct on B r. Hence, by the Arzelá-Ascoli theorem, S 1 is compct on B r.thusllthessumptionsof Theorem 3.3 re stisfied nd the conclusion of Theorem 3.3 implies tht the boundry vlue problem hs t lest one solution on [, 1]. This completes the proof. Remrk 3.5 In the bove theorem we cn interchnge the roles of the opertors S 1 nd S 2 to obtin the second result replcing 3.5 by the following condition: 1 e k kɣα <1.

8 Alsedi et l. Advnces in Difference Equtions :33 Pge 8 of 12 In the next theorem we prove the existence of solution for the boundry vlue problem vi Lery-Schuder nonliner lterntive. Lemm 3.6 Nonliner lterntive for single-vlued mps [38] Let E be Bnch spce, Cbeclosed, convex subset of E, UbenopensubsetofCnd U. Suppose tht F : U C is continuous, compct tht is, FU is reltively compct subset of C mp. Then either i F hs fixed point in U, or ii there is u U the boundry of U in C nd λ, 1 with u = λfu. Theorem 3.7 Suppose tht f :[,1] R R is jointly continuous function. Further, it is ssumed tht the following conditions hold: H 3 There exist function φ C[, 1], R nd nondecresing function ψ : R R such tht f t, x φtψ x for ll t, x [, 1] R. H 4 There exists constnt M >such tht M ψm φ >1, where is given by 3.2. Then the boundry vlue problem hs t lest one solution on [, 1]. Proof Consider the opertor S : C C, where Sxt= kt 1e kt Ɣβ e ks m m τ α 2 e kζ s e kt s f τ, xτ dτ f τ, xτ dτ f τ, xτ dτ. dm We show tht S mps bounded sets into bounded sets in C[, 1], R. For positive number r,letb r = {x C[, 1], R: x r} be bounded set in C[, 1], R. Then Ɣβ e ks m m τ α 2 Sxt kt 1e kt e kζ s f τ, xτ dτ f τ, xτ dτ e kt s f τ, xτ dτ p Ɣβ dm

9 Alsedi et l. Advnces in Difference Equtions :33 Pge 9 of 12 Consequently, e ks m m τ α 2 e kζ s e kt s φτψ x dτ φτψ x dτ φτψ x dτ ψ x { [ η αβ 1 φ p k 2 ƔαƔβ = ψ x φ. Sx ψr φ. dm kη e kη 1 ζ α 1 1 e kζ kɣα ] } 1 e k kɣα Next we show tht S mps bounded sets into equicontinuous sets of C[, 1], R. Let t 1, t 2 [, 1] with t 1 < t 2 nd x B r,whereb r is bounded set of C[, 1], R. Then we obtin Sxt 2 Sxt 1 1 e kt 2 s e kt 1 s s u α 2 f u, xu du 2 e kt s u 2 s α 2 t 1 f u, xu du kt 2 t 1 e kt 2 e kt [ 1 η e ks m m τ α 2 Ɣβ f τ, xτ dτ ] dm e kζ s f τ, xτ dτ 1 e kt 2 s e kt 1 s s u α 2 ψrφu du 2 e kt s u 2 s α 2 ψrφu du t 1 kt 2 t 1 e kt 2 e kt [ 1 η Ɣβ e ks m m τ α 2 ψrφτ dτ dm e kζ s ψrφτ dτ ]. Obviously, the right-hnd side of the bove inequlity ten to zero independently of x B r s t 2 t 1. As S stisfies the bove ssumptions, therefore it follows by the Arzelá- Ascoli theorem tht S : C C is completely continuous. The result will follow from the Lery-Schuder nonliner lterntive Lemm 3.6once we hve proved the boundedness of the set of ll solutions to equtions x = λsx for λ [, 1].

10 Alsedi et l. Advnces in Difference Equtions :33 Pge 1 of 12 Let x be solution. Then, for t [, 1], nd using the computtions employed in proving tht S is bounded, we hve xt = λsxt ψ x { [ η αβ 1 φ p k 2 ƔαƔβ = ψ x φ. kη e kη 1 ζ α 1 1 e kζ kɣα ] } 1 e k kɣα Consequently, we hve x ψ x φ 1. In view of H 4, there exists M such tht x M.Letusset U = { x C [, 1], R : x < M }. Note tht the opertor S : U C[, 1], R is continuous nd completely continuous. FromthechoiceofU, thereisnox U such tht x = λsx forsomeλ, 1. Consequently, by the nonliner lterntive of Lery-Schuder type Lemm 3.6, we deduce tht S hs fixed point x U which is solution of problem This completes the proof. 4 Exmples The following exmple is concerned with the illustrtion of Theorem 3.2. Exmple 4.1 Consider the problem { c D 3/2 D 2xt= L 2 t 2 1sin t xttn 1 xt, t 1, x =, x =, x1/2 = 1/3 xs. 4.1 Here, α = 5/2, f t, xt = L 2 t 2 1sin t xttn 1 xt, k =2, =1,η = 1/3, ζ = 1/2, β = 1. Clerly f t, x f t, y L x y tn 1 x tn 1 y /2 L x y. With the given vlues, we find tht.34681, p , For L < 1/ , it follows by Theorem 3.2 tht problem 4.1hsuniquesolution. Next, we discuss n illustrtive exmple for Theorem 3.7. Exmple 4.2 Let us consider problem 4.1withft, xt = e t 4 1t 2 x2 1. We check the conditions of Theorem 3.7. Clerly ssumption H 3 holwith φ = 1/4, ψ x =1 x 2 ; nd by ssumption H 4 wefindthtm 1 < M < M 2,whereM nd M Thus the conclusion of Theorem 3.7 pplies, nd hence problem 4.1 with the given vlue of f t, xt hs solution on [, 1].

11 Alsedi et l. Advnces in Difference Equtions :33 Pge 11 of 12 5 Conclusions We hve obtined some existence results for boundry vlue problem of Cputo type sequentil frctionl differentil equtions with nonlocl Riemnn-Liouville frctionl integrl boundry conditions with the id of Bnch s contrction mpping principle, Krsnoselskii s fixed point theorem nd nonliner lterntive of Lery-Schuder type. We observe tht some new specil results follow by fixing the prmeters involved in the given problem. For instnce, if we choose β = 1, then the results of this pper correspond to Cputo type sequentil frctionl differentil eqution with the boundry conditions of the form x =, x =, xζ = η xs.further,inthelimit,weobtinthe results for the boundry conditions x =, x =, xζ =. Competing interests The uthors declre tht they hve no competing interests. Authors contributions Ech of the uthors, AA, SKN, RPA nd BA, contributed to ech prt of this work eqully nd red nd pproved the finl version of the mnuscript. Author detils 1 Deprtment of Mthemtics, Fculty of Science, King Abdulziz University, P.O. Box 823, Jeddh, 21589, Sudi Arbi. 2 Deprtment of Mthemtics, University of Ionnin, Ionnin, 451 1, Greece. 3 Deprtment of Mthemtics, Texs A&M University-Kingsville, Kingsville, TX , USA. Acknowledgements This rticle ws funded by the Denship of Scientific Reserch DSR, King Abdulziz University, Jeddh, Sudi Arbi. The uthors, therefore, cknowledge technicl nd finncil support of KAU. The uthors lso thnk the reviewers for their useful comments. Received: 2 November 214 Accepted: 19 Jnury 215 References 1. Ahmd, B: Existence of solutions for irregulr boundry vlue problems of nonliner frctionl differentil equtions. Appl. Mth. Lett. 23, Ahmd, B, Nieto, JJ: Riemnn-Liouville frctionl integro-differentil equtions with frctionl nonlocl integrl boundry conditions. Bound. Vlue Probl. 211, Ling, S, Zhng, J: Existence of multiple positive solutions for m-point frctionl boundry vlue problems on n infinite intervl. Mth. Comput. Model. 54, Su, X: Solutions to boundry vlue problem of frctionl order on unbounded domins in Bnch spce. Nonliner Anl. 74, Bi, ZB, Sun, W: Existence nd multiplicity of positive solutions for singulr frctionl boundry vlue problems. Comput. Mth. Appl. 63, Agrwl, RP, O Regn, D, Stnek, S: Positive solutions for mixed problems of singulr frctionl differentil equtions. Mth. Nchr. 285, Cbd, A, Wng, G: Positive solutions of nonliner frctionl differentil equtions with integrl boundry vlue conditions. J. Mth. Anl. Appl. 389, Ahmd, B, Ntouys, SK, Alsedi, A: A study of nonliner frctionl differentil equtions of rbitrry order with Riemnn-Liouville type multistrip boundry conditions. Mth. Probl. Eng. 213,Article ID Zhng, L, Wng, G, Ahmd, B, Agrwl, RP: Nonliner frctionl integro-differentil equtions on unbounded domins in Bnch spce. J. Comput. Appl. Mth. 249, Ahmd, B, Ntouys, SK: Existence results for higher order frctionl differentil inclusions with multi-strip frctionl integrl boundry conditions. Electron. J. Qul. Theory Differ. Equ. 213, O Regn, D, Stnek, S: Frctionl boundry vlue problems with singulrities in spce vribles. Nonliner Dyn. 71, Blenu, D, Mustf, OG, O Regn, D: A uniqueness criterion for frctionl differentil equtions with Cputo derivtive. Nonliner Dyn. 71, Podlubny, I: Frctionl Differentil Equtions. Acdemic Press, Sn Diego Kilbs, AA, Srivstv, HM, Trujillo, JJ: Theory nd Applictions of Frctionl Differentil Equtions. North-Hollnd Mthemtics Studies, vol. 24. Elsevier, Amsterdm Sbtier, J, Agrwl, OP, Mchdo, JAT e.: Advnces in Frctionl Clculus: Theoreticl Developments nd Applictions in Physics nd Engineering. Springer, Dordrecht Tomovski, Z, Hilfer, R, Srivstv, HM: Frctionl nd opertionl clculus with generlized frctionl derivtive opertors nd Mittg-Leffler type functions. Integrl Trnsforms Spec. Funct. 21, Konjik, S, Oprnic, L, Zoric, D: Wves in viscoelstic medi described by liner frctionl model. Integrl Trnsforms Spec. Funct. 22, Keyntuo, V, Lizm, C: A chrcteriztion of periodic solutions for time-frctionl differentil equtions in UMD spces nd pplictions. Mth. Nchr. 284,

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