Local existence for an impulsive fractional neutral integro-differential system with Riemann Liouville fractional derivatives in a Banach space

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1 Klmni et l. Advnces in Difference Equtions 28) 28:6 R E S E A R C H Open Access Locl existence for n impulsive frctionl neutrl integro-differentil system with RiemnnLiouville frctionl derivtives in Bnch spce Plniyppn Klmni *,DumitruBlenu 2,3 nd Mni Mllik Arjunn * Correspondence: klmn@gmil.com Deprtment of Mthemtics, C. B. M. College, Coimbtore, Indi Full list of uthor informtion is vilble t the end of the rticle Equl contributors Abstrct In this mnuscript, we investigte sort of frctionl neutrl integro-differentil equtions with impulsive outcomes nd extend the formul of generl solutions for the impulsive frctionl neutrl integro-differentil system in Bnch spce. By using the nlysis of the limit cse nd the opertor generting compct semigroup, we derive the min results. Finlly, n exmple is discussed to illustrte the efficiency of the results. MSC: Primry 3A8; 35R2; secondry 5J5; 3A3 Keywor: Frctionl differentil equtions; RiemnnLiouville frctionl derivtives; Impulsive Introduction Frctionl clculus is field of mthemtics study tht grows out of trditionl definitions of clculus integrl nd derivtive opertors in much the sme wy frctionl exponents re n outgrowth of exponents with integer vlue. The concept of frctionl frctionl derivtives nd integrls) is populrly believed to hve stemmed from question rised in the yer 695 by Mrquis de L Hopitl 66) to Gottfried Wilhelm Leibniz 666), which sought the mening of Leibniz s currently populr) nottion dn y dx n for the derivtive of order n N when n ; tht is, Wht if n is frctionl?. In his reply, 2 dted 3 September 695, Leibniz wrote to L Hopitl s follows: This is n pprent prdox from which, one dy, useful consequences will be drwn. Tht is, d 2 x re going to be dequte x dx : x. It is typiclly cknowledged tht integer-order derivtives nd integrls hve cler physicl nd geometric interprettions. However, just in cse of frctionl-order integrtion nd differentition, tht represent n pce growing field ech in theory nd in pplictions to plnet issues, it is not thus. Since the looks of the thought on differentition nd integrtion of rbitrry not necessry integer) order, there ws not ny cceptble geometric nd physicl interprettion of those opertions for some three hundred yers. In 22], it is shown tht geometric interprettion of frctionl integrtion is Shdows on the The Authors) 28. This rticle is distributed under the terms of the Cretive Commons Attribution. Interntionl License which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthors) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde.

2 Klmni etl. Advnces in Difference Equtions 28) 28:6 Pge 2 of 26 wlls nd its physicl interprettion is Shdows of the pst. Frctionl differentil equtions bbrevited, FDEs) nd integro-differentil equtions hve gined wide importnce s result of their pplictions in numerous fiel like physics, mechnics, control theory, nd engineering, one will crete relevnce to the books 2,, 22, 29] nd lso the ppers 5,, 2, 25, 26, 332]. The definitions of RiemnnLiouville bbrevited RL) FDEs or integrl initil conditions ply very importnt role in some frctionl problems within the world. Heymns nd Podlubny 9] verified tht it hd been ttinble to ttribute physicl desiring to initil conditions expressed in terms of RL frctionl derivtives or integrls on the sector of the viscoelsticity. For more detils, one cn see, 5, 6, 9, 2]. In ddition, the specultion of impulsive differentil equtions seems to be nturl description of mny rel processes subject to sure perturbtions whose length is negligible s compred with the overll length of the method, such chnges re going to be firly well pproximted s being quick chnges of stte, or inside the design of impulses. This methodology ten to be extr fittingly sculptured by impulsive differentil equtions, which llow for discontinuities inside the evolution of the stte, considered in such fiel s drugs, biology, engineering science, chemicl technology, etc. Therefore, it ppers fscinting to check the frctionl impulsive differentil nd integro-differentil equtions. Furthermore, impulsive frctionl evolution systems with the Cputo frctionl derivtive with completely different conditions were studied by severl uthors, one cn see,, 8, 2, 23, 2]. However, bundnt less is thought regrding the impulsive frctionl evolution systems with RL frctionl derivtive, see, 8, 28]. In specific, the following frctionl order integro-differentil eqution in Bnch spce using the Cputo frctionl derivtive: C D α t ut)aut)f t, ut)) qt s)gt, us)), t, t t k, u ttk I k ut k )), k,...,m, u) u X, ws mentioned by Gou nd Li 8], nd they estblished the locl nd globl existence of mild solution to n impulsive frctionl semiliner integro-differentil eqution with noncompct semigroup. In 8],Liu nd Bin gve the pproximte controllbilityof n impulsive RL frctionl system with the help of the Bnch contrction principle in Bnch spce. Liu et l. ] estblished the pproximte controllbility of impulsive frctionl neutrl evolution equtions with RL frctionl derivtives by using the Bnch contrction principle. Lter, Zhng et l. 28] nlyzed the generl solution of impulsive systems with RL frctionl derivtives by using limit cse s impulse ten to zero). However, locl existence for impulsive frctionl neutrl integro-differentil equtions with RL hs not been fully investigted in the literture. Inspired by the bove-mentioned works, we investigte the following impulsive frctionl integro-differentil eqution with RL frctionl derivtive of the form L D γ t wt)d t, wt) )] A wt)l t, wt) ) qt s)p t, ws) ), I t, T], t t k, k,...,m,.)

3 Klmni etl. Advnces in Difference Equtions 28) 28:6 Pge 3 of 26 I γ w tt k I γ w t k ) γ I w t k ) )) Ik w t k,.2) )] wt)d t, wt) t w H,.3) I γ where L D γ t < γ < ) represents the RL frctionl derivtive of order γ nd I γ denotes the RL integrl of order γ. Throughout this pper, we tke I,T] is n opertionl intervl. Let A : DA ) H H be the infinitesiml genertor of c semigroup T t)} t in Bnch spce H. There exists constnt M suchtht T t) M, D, L, P : I H H, q : I H nd I k : H H re pposite continuous functions. t < t <,...,<t m T.HereI γ wt k )lim ɛ Iγ wt k ɛ)ndi γ wt k )lim ɛ Iγ wt k ɛ) denote the right nd left limits of I γ wt)tt t k,respectively. For impulsive system.).3), we hve L D γ t wt)dt, wt))] lim I,...,I m }.).3) I γ A wt)l t, wt)) qt s)pt, ws)), t, T], wt)dt, wt))] t w H..) As result, it implies tht there exists hidden condition } lim the solution of impulsive system.).3) I,...,I m the solution of system.) }..5) Consequently, the definition of solution for impulsive frmework.).3) isgivenbelow. Definition. Let wt):,t] H be the solution of the frctionl structure.).3) if I γ wt)dt, wt))] t w,theproblem L D γ t wt)dt, wt))] A wt)l t, wt)) qt s)pt, ws)) for ech t, T] is proved, the impulsive conditions Iγ w tt k I k wt k )) here k,...,m) re fulfilled, the confinement of wt) totheintervlt k, t k ] here k,...,m) is continuous, nd therefore condition.5) hol. The rest of this pper is composed s follows. In Sect. 2, we present some preliminries which will be used to prove our necessry nd sufficient conditions. In Sect. 3, the existence of solutions for problem.).3) is nlyzed under pproprite fixed point techniques. In Sect., s finlpoint, exmplesregiven toillustrteourresults. 2 Preliminries In this preliminry, we tend to recll some bsic definitions, lemms, nd theorem which will be used throughout this pper. The norm of Bnch spce H is denoted by H. Let CI, H) represent the Bnch spce of ll H-vlued continuous functions from I to H with the norm w C sup t I wt) H.Sostooutlinethemilolutionofproblem

4 Klmni etl. Advnces in Difference Equtions 28) 28:6 Pge of 26.).3), we tend to dditionlly tke the Bnch spce C γ I, H) w CI, H) : t γ wt) CI, H)} with the norm w Cγ sup t γ wt) H, t I }. Obviously, the spce C γ is Bnch spce. In order to outline the mild solutions of system.).3), we lso consider the Bnch spce PC γ I, H) w :t t k ) γ wt) PC γ I, H) is continuous from left nd hs right limits t t t, t 2,...,t m }} w PCγ mx sup t t k,t k ] t t k ) } γ wt) H : k,,...,m. Definition 2. 3]) Let A be the infinitesiml genertor of n nlytic semigroup T t)} t of uniformly bounded liner opertors on H.If ρa ), where ρa )isthe resolvent set of A,thenfor<η, it is possible to define the frctionl power A η s closed liner opertor on its domin DA η ). For n nlytic semigroup T t)} t,the following properties will be used.. There is M such tht M : sup t, ) T t) <. 2. For ny η, ],thereexistsm η >ensuring tht A η T t) M η, <t T. tη For dditionl detils regrding the semigroup theory nd frctionl powers of opertors, we dvise the reder to refer to 2]. Currently, we offer few bsic definitions nd results of the frctionl clculus theory tht hppen to be used in ddition s chunk of this mnuscript. Definition 2.2 ]) The frctionl integrl of order γ with the lower limit for function f is determined s I γ t f t) f s), t >,γ >, Γ γ ) t s) γ given the right prt is point-wise described on, ), where Γ ) is the gmm function. Definition 2.3 ]) The RL derivtive of order γ with the lower limit for function f L I, H)ischrcterize d n L D γ t f t) Γ n γ ) dt n f s), t >,n <γ < n. t s) nγ Consider the initil vlue problem D γ wt)a wt)f t, wt)), I γ w)w, w C γ C nd Rγ ), ), t, T],

5 Klmni etl. Advnces in Difference Equtions 28) 28:6 Pge 5 of 26 is equivlent to the following nonliner Volterr integrl eqution of the second kind: wt) w Γ γ ) t )γ Γ γ ) t s) γ A ws)f s, ws) )]. The piecewise function for.).3)isgivenby wt) Γ γ ) Iγ w t k ) t tk ) γ D t, wt) ) Γ γ ) t k t s) γ A ws)λ s) ], where Λ s)l s, ws)) qs τ)pτ, wτ)) dτ, t t k, t k ], k,,...,m,with I γ w t k ) γ I w t k ) )) k w t k. By Definition 2.3,wehve D γ wt)d t, wt) )] ) D γ Γ γ ) Iγ w t k ) t tk ) γ D γ Γ γ ) t s) γ A ws)λ s) ] ) t k d t η) γ I γ Γ γ )Γ γ ) dt w t ) k η tk ) γ dη t k Γ γ )Γ γ ) ) d η t η) γ η s) γ A ws)λ s) ] ) dη dt t k d t γ η t η) η s) γ A ws)λ s) ] ) dη Γ γ ) dt t k Γ γ ) t k D γ η η s) γ A ws)λ s) ] ) Γ γ ) t k D γ I γ A wt)l t, wt) ) A wt)l t, wt) ) qt s)p s, ws) ) ]) qt s)p s, ws) ), where ndt t k, t k ]. So, wt) fulfills the condition of frctionl differentil frmework.).3),nd it does not fulfill condition.5). In this wy, we ccept tht wt) is n pproximte solution for the exct solution of impulsive frmework.).3).

6 Klmni etl. Advnces in Difference Equtions 28) 28:6 Pge 6 of 26 Theorem 2. Suppose tht ξ is constnt nd. wt) is generl solution of model.).3) if nd only if wt) fulfills the frctionl integrl eqution Γ γ ) t )γ wt) k i w Γ γ ) t )γ Γ γ ) w t s)γ )A ws)λ s)],, t ], Γ γ ) t s)γ A ws)λ s)] i wt i )) t t Γ γ ) i ) γ k i ξ i wt i )) Γ γ ) )w t ) γ t s)γ A ws)λ s)] w i A ws)λ s)] )t t i ) γ t i t s) γ )A ws)λ s)] }, where nd t t k, t k ], given tht the integrl in 2.) exists. 2.) Proof Necessity : First we re ble to simply verify tht eqution 2.)fulfills the shrouded condition.5). Next, tking the RL frctionl derivtive to eqution 2.) foreveryt t k, t k ], k,,...,m,weget D γ )] wt)d t, wt) D γ w t )γ Γ γ ) k D γ i ) w t ) γ ) D γ Γ γ ) ) i wt i k )) t t i ) γ D γ Γ γ ) i Λ s) ] ) t t i ) γ t s) γ A ws)λ s) ] ) ξ i wt i )) Γ γ ) t s) γ A ws)λ s) ] w t s) γ A ws)λ s) ] } ) t i i A ws) A wt)λ t) ) k t ξ )) i w t i A wt)λ t) ) t A wt) Λ t) ) t t i ]t t k,t k ] A wt)l t, wt) ) i qt s)p s, ws) ) ). t t k,t k ] Therefore, eqution 2.) fulfills the RL frctionl derivtive of model.).3). Using 2.)foreveryt k, k,2,...,m,wehve I γ w t k ) γ I Γ γ ) w t k ) Γ γ ) } t η) γ wη) dη t t k } t η) γ wη) dη k w t k )) ξ k w t k )) w tt k A ws)λ s) )

7 Klmni etl. Advnces in Difference Equtions 28) 28:6 Pge of 26 w k k w t k )). A ws)λ s) ] ) A ws)λ s) ) } t k t t k Therefore, eqution 2.) fulfills the impulsive conditions of model.).3). Then eqution 2.) stisfies the conditions of system.).3)with. Sufficiency : We demonstrte tht the solutions of frmework.).3) fulfillcondition 2.) by scientific induction. By Definition 2.2,the solution ofmodel.).3) fulfills wt) By 2.2), we hve w Γ γ ) t )γ Γ γ ) I γ w t ) γ I w t ) )) w t )) t w w t A ws)λ s) ], nd the pproximte solution wt), t t, t 2 ]isdefinedby wt) Γ γ ) Iγ w t ) t t ) γ D t, wt) ) Γ γ ) Γ γ ) w t t s) γ A ws)λ s) ] D t, wt) ) Γ γ ) t s) γ A ws)λ s) ],, t ]. 2.2) A ws)λ s) ] w t )) ) t t ) γ with e t)wt)wt), t t, t 2 ]. By lim wt) lim wt )) wt )) Γ γ ) t t s) γ A ws)λ s) ], t t, t 2 ], w Γ γ ) t )γ D t, wt) ) t s) γ A ws)λ s) ] w t )) } w Γ γ ) t )γ D t, wt) ) Γ γ ) Λ s) ], t t, t 2 ]. t s) γ A ws) We get lim e } t) lim wt)wt) wt )) wt )) w Γ γ ) t )γ Γ γ ) t s) γ A ws)λ s) ]

8 Klmni etl. Advnces in Difference Equtions 28) 28:6 Pge 8 of 26 Then we ssume Γ γ ) t t ) γ w Γ γ ) A ws)λ s) ] ) t t s) γ A ws)λ s) ]. e t)σ ))) w t lim e t) wt )) σ wt ))) w t ) γ t s) γ A ws)λ s) ] Γ γ ) w A ws)λ s) ] ) t t ) γ t s) γ A ws) t Λ s) ] }, where the function σ ) is n undetermined function with σ ). Thus, wt)wt)e t) σ ))) w t Γ γ ) w t ) γ t s) γ A ws) Λ s) ] ) )) w t t t ) γ σ )))] w t w ) Using 2.3), we get A ws)λ s) ] ) t t ) γ σ )))] w t t s) γ A ws)λ s) ] } D t, wt) ), t t, t 2 ]. 2.3) t I γ w t 2 ) γ I w t 2 ) )) 2 w t 2 )) )) t2 w w t 2 w t 2 A ws)λ s) ], nd the pproximte solution wt), t t 2, t 3 ]isgivenby wt) Γ γ ) Iγ w t 2 ) t t2 ) γ D t, wt) ) Γ γ ) Γ γ ) w t 2 t s) γ A ws)λ s) ] 2 A ws)λ s) ] w t )) 2 w t 2 )) ) )t t 2 ) γ D t, wt) ) Γ γ ) t 2 t s) γ A ws)λ s) ]

9 Klmni etl. Advnces in Difference Equtions 28) 28:6 Pge 9 of 26 for t t 2, t 3 ]. Similrly, we get wt)wt)e 2 t) σ ))) ))) ] w t σ 2 w t Γ γ ) 2 w t ) γ t s) γ A ws)λ s) ] ) )) w t t t ) γ 2 w t 2 )) t t2 ) γ σ w t )))] w A ws)λ s) ] ) t t ) γ σ )))] w t t s) γ A ws)λ s) ] t σ )))] 2 2 w t 2 w A ws)λ s) ] ) t t 2 ) γ σ )))] 2 w t 2 t s) γ A ws)λ s) ] } t 2 D t, wt) ), t t 2, t 3 ]. 2.) Moreover, letting t 2 t,wehve D γ wt)dt, wt))] A wt)l t, wt)) lim t 2 t qt s)pt, ws)), γ, ), t, t 3], t t nd t t 2, I γ w tt k I γ wt k )Iγ wt k ) kwt k )), k,2, I γ wt)dt, wt))] t w H D γ wt)dt, wt))] A wt)l t, wt)) qt s)pt, ws)), γ, ), t, t 3]ndt t, I γ w tt I γ wt )Iγ wt )Iγ wt 2 )Iγ wt 2 ) wt )) 2wt )), I γ wt)dt, wt))] t w H. Using 2.3)nd2.), we hve σ 2 )σ )σ 2 ). Tking ρz)σ z), we hve ρz ω)ρz)ρω)for z, ω C.So,ρz)ξz.Thus, wt) w Γ γ ) t )γ Γ γ ) wt )) Γ γ ) t t ) γ ξ wt )) Γ γ ) t s) γ A ws)λ s) ] t s) γ A ws)λ s) ] w t ) γ

10 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge of 26 w A ws)λ s) ] ) t t ) γ t s) γ A ws)λ s) ] } D t, wt) ), t t, t 2 ]. 2.5) t Continuing in this wy, we obtin, for t t k, t k ], wt) w Γ γ ) t )γ Γ γ ) k i i wt i )) t t i ) γ Γ γ ) t s) γ A ws)λ s) ] k i ξ i wt i )) w t ) γ Γ γ ) i w A ws) t s) γ A ws)λ s) ] Λ s) ] ) t t i ) γ t s) γ A ws)λ s) ] } t i D t, wt) ), t t k, t k ]. 2.6) So, the solution of system.).3) stisfies eqution 2.) with. Therefore, the impulsive system.).3) is equivlent to the integrl eqution 2.)with. Lemm 2. If nd ξ in 2.6) re given by w Γ γ ) tγ Dt, wt)) t Γ γ ) wt) t s)γ A ws)λ s)], t, t ], w Γ γ ) tγ t Γ γ ) t s)γ A ws)λ s)] k i wt i )) i t t Γ γ ) i ) γ, t t k, t k ], 2.) where Λ s)ls, ws)) qs τ)pτ, wτ)) dτ. Then, for t >,we hve where t γ T γ t)w Dt, wt)) t s)γ A T γ t s) )Ds, ws)) t s)γ T γ t s)l s, ws)) qs τ)pτ, wτ)) dτ], t, t ], wt) t γ T γ t)w Dt, wt)) t s)γ A T γ t s) )Ds, ws)) t s)γ T γ t s)l s, ws)) qs τ)pτ, wτ)) dτ] k i T γ t t i )t t i ) γ I i wt i )), t t k, t k ], T γ t)γ θξ γ θ)t t γ θ ) dθ, ξ γ θ) γ θ γ φ γ θ γ ),

11 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge of 26 φ γ θ) π n )n γ n Γ nγ ) θ sinnπγ), θ, ). Here ξ n! γ is the probbility density function in, ), tht is, ξ γ θ) nd ξ γ θ) dθ. Proof The proof is very similr to the proof of 3, 32, Lemm 3., Lemm 3.3], hence here we omit it. Lemm 2.2 3, Lemm 3.2]) If t, T γ t) is liner nd bounded opertor. Tht is, for ny w H, T γ t)w M Γ γ ) w. Lemm 2.3 3, Lemm 3.5]) For ny β, ), η, ], nd for ll w DA ), there exists positive constnt M η in wy tht A T γ t)w A β T γ t)a β w, t T, nd A η T γ t) γ M ηγ 2 η), <t T. t γη Γ γ η)) Lemm 2. 2, Lemm 2.]) Assume ξ, η R, η >nd n N, nd then when t >, we hve I ξ s η ) t ξη Γ ξη) t), ξ η n, Γ η ), ξ η n. Next, we hve tendency to recll some properties of the mesure of noncompctness which will be employed in the proof of our min results. We tend to denote by ω ) the Kurtowski mesure of noncompctness on both the finite sets of H nd CI, H). For more points of interest of the mesure of noncompctness, see 3, 6]. For ny D CI, H) nd t I,letDt)ut) u D} H. IfD CI, H) isbounded,thendt) isbounded in H nd ωdt)) ωd). 3 Existence results In this re, we disply nd demonstrte the existence results for problem.).3). In view of Lemm 2., first, we define the mild solution for model.).3) with the help of probbility density function nd the Lplce trnsform. Definition 3. 2, Definition 3.]) A function w PC γ I, H) issidtobemilolution of model.).3) if the following hold: i) I γ wt)dt, wt))] t w H;

12 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 2 of 26 ii) t γ T γ t)w t s)γ A T γ t s)ds, ws)) Dt, wt)) t s)γ T γ t s)l s, ws)) qs τ)pτ, wτ)) dτ], t, t ], wt) t γ T γ t)w t s)γ A T γ t s)ds, ws)) Dt, wt)) t s)γ T γ t s)l s, ws)) qs τ)pτ, wτ)) dτ] k i T γ t t i )t t i ) γ I i wt i )), t t k, t k ]. Now, we re in position to introduce the hypotheses on frmework.).3)sfollows. H) T t), t >is strongly continuous semigroup nd continuous in the uniform opertor topology. H2) L : I H H is continuous, nd we cn discover constnts N L >in wy tht L t, w)l t, v) H N L w v H, t I, w, v H. H3) P : I H H is continuous, nd we cn discover constnts N P >such tht Pt, w)pt, v) H N P w v H, t I, w, v H. H) The function D : I H H, we cn discover constnts β, ) nd N D >in wys tht D DA β ),ndfornyw, v H, t I,thefunctionA β D, w) is strongly mesurble nd A β Dt, w) stisfies A β Dt, w)a β Dt, v) H N D w v H. H5) There exist constnts <d k < Γ γ )/M k i t i t i ) γ ], k,...,m, ensuring tht I i w)i i v) H d i w v H nd I i w) N I, w, v) H 2. H6) For ny R >nd >,thereexistsl i R, )>, i,2,3, ensuring tht for ny equicontinuous nd countble set D B R w H : w R}, ω L t, D) ) L 2 ωd), ω Pt, D) ) L 3 ωd), ω Dt, D) ) L ωd), t, ]. Remrk 3. Throughout this pper, we define few nottions: A β M nd Kγ, β) M βγ β)τ γβ. βγ γβ)

13 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 3 of 26 Theorem 3. Assume tht hypotheses H)H6) re stisfied. Then, for every w PC γ I, H), there exists τ τ w ), < τ < T, ensuring tht model.).3) hs solution w PC γ, τ ], H). Proof Since we re interested here just in locl solutions, we my ssume tht T <. By using our conditions H)H5), t >ndr >resuchthtb r w )w:t t k ) γ w w r} nd t t k ) γ L t, wt)) N L,t t k ) γ Pt, wt)) N P, t t k ) γ A β Dt, wt)) N D for t t nd w B r w )nelect τ min t Γ γ ), T, MN L q N P ) ] ) γ M Kγ, β)l D, Γ γ ) M Kγ, β) ) N MN L q D u )] } γ, 3.) N P ) where q t M sup t T qt s), u Γ γ ) w k i t i t i ) γ ]. Set Ω w PC γ, τ ], H) :t γ wt) r, t, τ ]}, thenω is closed bll in PC γ, τ ], H) with center θ nd rdius r. Consider the opertor Υ : Ω PC γ, τ ], H)definedby Υ w)t)t γ T γ t)w D t, wt) ) t s) γ A T γ t s)d s, ws) ) t s) γ T γ t s) L s, ws) ) T γ t t k )t t k ) γ )) I k w t k. <t k <t qs τ)p τ, wτ) ) ] dτ For ny w Ω nd t, τ ], by Lemm 2.2 nd Lemm 2.3,wehve t t k ) γ Υ w)t) t t k ) γ t γ Tγ t)w t tk ) γ D t, wt) ) t t k ) γ t s) γ A T γ t s)d s, ws) ) t t k ) γ t s) γ T γ t s) L s, ws) ) qs τ)p τ, wτ) ) dτ] t t k ) γ T γ t t k )t t k ) γ )) I k w t k <t k <t J j, 3.2) j

14 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge of 26 where J t t k ) γ t γ Tγ t)w t tk ) γ D t, wt) ) Tγ t)w A β t tk ) γ A β D t, wt) ) M w M N D, Γ γ ) J 2 t t k ) γ t s) γ A T γ t s)d s, ws) ) t t k ) γ t s) γ A β T γ t s) A β D s, ws) ) t t k ) γ γ M βγ β) t s) γβ A β D s, ws) ) Γ γβ) Kγ, β)n D, J 3 t t k ) γ t s) γ T γ t s) L s, ws) ) t t k ) γ M t γ Γ γ ) γ qs τ)p τ, wτ) ) dτ] M Γ γ ) NL q N P ] t s) γ L s, ws) ) q P s, ws) ) ] M NL q ] N P τ γ, Γ γ ) J t t k ) γ T γ t t k )t t k ) γ )) I k w t k t t k ) γ M Γ γ ) M Γ γ ) <t k <t M Γ γ ) k t t i ) γ )) Ii w t i i k t i t i ) γ t i t i ) γ )) Ii w t i i k t i t i ) γ N I. i Using J J in eqution 3.2), we get t t k ) γ Υ w)t) M w Γ γ ) R. ] k t i t i ) γ N I i N D M Kγ, β) ) M NL q ] N P τ γ Γ γ )

15 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 5 of 26 Therefore, Υ w Ω. Now we show tht Υ is continuous from Ω into Ω. To show this, we first observe tht since L, D, ndp re continuous in I H, for ny ɛ >nd for fixed w B R w ), there exists δ w), δ 2 w) > ensuring tht for ny v B R w )ndlet δw)minδ w), δ 2 w)}.then,fornyv Ω,t t k ) γ wt)vt) < δw)ndchoose A β Kγ, β) ) ND < ɛ δw). Then we hve t t k ) γ Υ w)t)υ v)t) M Γ γ ) τ γ ] N L q ] M N P Γ γ ) md kτ γ t t k ) γ D t, wt) ) D t, vt) ) t t k ) γ t s) γ A T γ t s) D s, ws) ) D s, vs) )] t t k ) γ t s) γ T γ t s) L s, ws) ) L s, vs) )] t t k ) γ t s) γ T γ t s) ) P τ, wτ) ) P τ, vτ) )] dτ qs τ) t t k ) γ T γ t t k )t t k ) γ )) ))] I k w t k Ik v t k <t k <t A β Kγ, β) ) N D ) w v Cγ ɛ. M Γ γ ) τ γ ] N L q ] M N P Γ γ ) md kτ γ Thus, we hve tht Υ : Ω Ω is continuous opertor. Next, we demonstrte tht the opertor Υ : Ω Ω is equicontinuous. For ny w Ω nd t < t 2 τ,wegettht t 2 t k ) γ Υ w)t 2 )t t k ) γ Υ w)t ) t2 t k ) γ t γ 2 T γ t 2 )w t t k ) γ t γ T γ t )w t2 t k ) γ D t 2, wt 2 ) ) t t k ) γ D t, wt ) ) 2 t 2 t k ) γ t 2 s) γ A T γ t 2 s)d s, ws) ) t t k ) γ t s) γ A T γ t s)d s, ws) ) 2 t 2 t k ) γ t 2 s) γ T γ t 2 s) L s, ws) ) qs τ)p τ, wτ) ) ] dτ

16 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 6 of 26 t t k ) γ t s) γ T γ t s) L s, ws) ) qs τ)p τ, wτ) ) ] dτ k t 2 t k ) γ T γ t 2 t i )t 2 t i ) γ )) I i w t i t t k ) γ i k T γ t t i )t t i ) γ )) I i w t i i 9 J j, 3.3) j5 where J 5 t 2 t k ) γ t γ 2 T γ t 2 )w t t k ) γ t γ T γ t )w t 2 t k ) γ t γ 2 t γ ] T γ t 2 )w t γ t 2 t k ) γ T γ t 2 )t t k ) γ T γ t ) w M Γ γ ) t 2 t k ) γ t γ 2 t γ ] w t γ t2 t k ) γ T γ t 2 )t t k ) γ T γ t ) w, J 6 t2 t k ) γ D t 2, wt 2 ) ) t t k ) γ D t, wt ) ) t t k ) γ A β A β D t 2, wt 2 ) ) A β D t, wt ) ) t 2 t k ) γ t t k ) γ ] A β A β D t 2, wt 2 ) ) M t t k ) γ A β D t 2, wt 2 ) ) A β D t, wt ) ) M t2 t k ) γ t t k ) γ ] A β D t 2, wt 2 ) ), 2 J t 2 t k ) γ t 2 s) γ A T γ t 2 s)d s, ws) ) t t k ) γ t s) γ A T γ t s)d s, ws) ) t 2 t k ) γ t2 s) γ t s) γ ] ) A β T γ t 2 s) A β D s, ws) ) t s) γ t 2 s) γ A β T γ t 2 s) t s) γ A β T γ t s) A β D s, ws) ) 2 t 2 t k ) γ t 2 s) γ A β T γ t 2 s) A β D s, ws) ) t γ M βγ β)n D Γ γβ) t s) γ t 2 s) γ A β T γ t 2 s) t 2 s) γβγ t 2 s) γ t s) γ ] t s) γ A β T γ t s) A β D s, ws) )

17 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge of 26 γ M βγ β)n t2 D t 2 t k ) γ t 2 s) γβ, Γ γβ) t 2 J 8 t 2 t k ) γ t 2 s) γ T γ t 2 s) ) L s, ws) ) qs τ)p τ, wτ) ) ] dτ t t k ) γ t s) γ T γ t s) ) L s, ws) ) qs τ)p τ, wτ) ) ] dτ M NL q ) N P t2 s) γ t s) γ ] Γ γ ) t s) γ t2 s) γ T γ t 2 s)t s) γ T γ t s) ) L s, ws) ) qs τ)p τ, wτ) ) dτ M NL q )t 2 t ) γ N P, Γ γ ) γ k J 9 t 2 t k ) γ T γ t 2 t i )t 2 t i ) γ )) I i w t i t t k ) γ M Γ γ ) i k T γ t t i )t t i ) γ )) I i w t i i k N I t2 t i ) γ t t i ) γ ] i k t t i ) γ ) t 2 t i ) γ T γ t 2 t i )t t i ) γ T γ t t i ) ] Ii w t i )). From J 5 )J 9 )ineqution3.3), we hve t 2 t k ) γ Υ w)t 2 )t t k ) γ Υ w)t ) M Γ γ ) t 2 t k ) γ t γ 2 t γ ] w t γ t 2 t k ) γ T γ t 2 )t t k ) γ T γ t ) w M t t k ) γ A β D t 2, wt 2 ) ) A β D t, wt ) ) M t2 t k ) γ t t k ) γ ] A β D t 2, wt 2 ) ) γ M βγ β)n D Γ γβ) i t s) γ t2 s) γ A β T γ t 2 s) t 2 s) γβγ t 2 s) γ t s) γ ] t s) γ A β T γ t s) A β D s, ws) ) γ M βγ β)n D Γ γβ) 2 t 2 t k ) γ t 2 s) γβ t

18 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 8 of 26 M NL q ) N P t2 s) γ t s) γ ] Γ γ ) t s) γ t2 s) γ T γ t 2 s)t s) γ T γ t s) ) L s, ws) ) qs τ)p τ, wτ) ) dτ M NL q )t 2 t ) γ N P Γ γ ) γ k N I t2 t i ) γ t t i ) γ ]] i k t t i ) γ t 2 t i ) γ T γ t 2 t i )t t i ) γ T γ t t i ) ] )) Ii w t i i st 2 t, which mens tht Υ : Ω Ω is equicontinuous. Since t 2 s) γ T γ t 2 s)t s) γ T γ t s) st 2 t becuse T γ )isstrongly continuous. Let B coυ ω). At tht point it is nything but difficult to confirm tht Υ mps B into itself nd B PC γ I, H) is equicontinuous. Now, we prove tht Υ : B B is condensing opertor. For ny E B,by8, Lemm 2.2], there exists countble set E w n } E such tht ω Υ E) ) 2ω Υ E ) ). 3.) By the equicontinuity of B,weknowthtE B is lso equicontinuous. Therefore, by 8, Lemm 2.], ssumption H6), we hve ω Υ E )t) ) ω t t k ) γ t γ T γ t)w D t, w n t) ) t s) γ A T γ t s)d s, w n s) ) t s) γ T γ t s) L s, w n s) ) T γ t t k )t t k ) γ )) )} I k w t k <t k <t qs τ)p τ, w n τ) ) ] dτ 2 A β ω t t k ) γ A β D t, w n t) )) 2ω t t k ) γ t s) γ A T γ t s)d s, w n s) ) ) 2M t Γ γ ) ω t k ) γ t s) γ T γ t s) L s, w n s) ) qs τ)p τ, w n τ) ) ] ) dτ 2 A β L ω t t k ) γ w n t) ) γ M βγ β)t γβ L ω t t k ) γ γβγ γβ)

19 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 9 of 26 )w n t) ) 2M Γ γ ) 2 M L Kγ, β)l t γ γ ω t t k ) γ L t, w n t) ) q P t, w n t) )]) M L2 q ] L 3 τ γ Γ γ ) ) ωe ). 3.5) Since Υ E ) B is bounded nd equicontinuous, we know from 8, Lemm 2.3] tht ω Υ E ) ) mx t I ω Υ E )t) ). 3.6) Therefore, from 3.), 3.)3.6), we know tht ω Υ E) ) M Kγ, β) ) ) M L L2 q ] L 3 τ γ ωe) Γ γ ) ωe). 3.) Thus, Υ : B B is condensing opertor. It follows from 8, Lemm 2.5] tht Υ hs t lest one fixed point wt ) B, which is the mild solution of model.).3) ontheintervl, τ ]. Applictions Consider the subsequent initil-boundry vlue problem of impulsive frctionl integrodifferentil model with RL frctionl derivtives: L D 3 t ] ut, x) e 2s us, x) us, x) 2 ut, x) e ts 2et ut, x) ut, x) e s, t, 2] nd t,.) 5 us, x) I u, x ) sin u, x ) ),.2) u x)i t ] ut, x) e 2s us, x) us, x) t,.3) ut,)ut, π), t, T], x, π],.) where L D 3 is the RiemnnLiouville frctionl derivtives of order 3,< 3, I is the RL integrl of order, u x) H. To study this problem, consider H L 2, π], R). Let the opertor A by A y y, with the domin DA ) y ) H : y, y re bsolutely continuous, y H, yt,)yt, π) }. Then A genertes c semigroup Tt)} t which is compct, nlytic. Besides, A cn be composed s A y n 2 y, e n e n, y DA ), n

20 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 2 of 26 2 where e n x) sin nx, x π, n,2,...,isnorthonormlbsisofh.wehve π Tt)y e n2t y, e n e n, y H, nd Tt) e t M, t. n For ech y H, A 2 y n n y, e n e n. In specific, A 2. The opertor A 2 is given by A 2 y n y, e n e n n on the spce D A ) 2 y ) H, } n y, e n e n H. n From Theorem 2., the generl solution of impulsive model.).)isobtinefol- lows: ut, x) u Γ 3 )t Γ 3 ) eτ esτ t, ], u Γ 3 )t t s) 2 us,x) 2es us,x) us,x) 5 uτ,x) dτ] Γ 3 ) e2s us,x) us,x), t s) 2 us,x) 2es us,x) us,x) eτ esτ 5 uτ,x) dτ] sin Γ 3 ) u, x) )t ) ξ sin Γ 3 ) u, x) )u t ξ Γ 3 ) sin u, x) ) ) 2 us,x) 2es us,x) us,x) Γ 3 ) e2s us,x) us,x) t s) eτ esτ 5 uτ,x) dτ] u 2 us,x) 2es us,x) us,x) eτ esτ 5 uτ,x) dτ] )t ) t s) 2 us,x) 2es us,x) us,x) eτ esτ dτ] }, t, 2]. 5 uτ,x).5) Next, it is verified tht eqution.5) stisfies the condition of system.).). Tking the RiemnnLiouville frctionl derivtive to both sides of eqution.5), we hve:

21 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 2 of 26 i) for t, ], D 3 ut, x) D 3 u e 2s Γ 3 )t Γ 3 ) e sτ ) us, x) us, x) e τ 5 uτ, x) dτ d Γ 3 )Γ 3 ) dt 2es us, x) us, x) 2 us, x) e sτ 2 ut, x) ii) for t, 2], D 3 ut, x) D 3 u e sτ t s) 2 us, x) ] ) t η) 3 u η 2es us, x) us, x) e τ e τ 5 uτ, x) dτ 2et ut, x) ut, x) e 2s Γ 3 )t Γ 3 ) e sτ 5 uτ, x) dτ ] } e ts ) us, x) us, x) e τ 5 uτ, x) dτ D 3 ξ sin u, x) ) Γ 3 ) ] t,] } t,] e s 2es us, x) us, x) η ] ) 5 us, x). η s) 2 us, x) } dη t,] t s) 2 us, x) 2es us, x) us, x) ] ) D 3 sin u t e τ Γ 3 ) 2es us, x) us, x) e sτ 5 uτ, x) dτ 2 us, x) u 2es us, x) us, x) t s) 2 us, x) 2es us, x) us, x) e sτ e τ ] }) 5 uτ, x) dτ d Γ 3 )Γ 3 ) dt 2es us, x) us, x) d Γ 3 )Γ ) dt e sτ u, x ) ) ) t ) Γ 3 ) t s) 2 us, x) ] e sτ t η) 3 u η e τ 5 uτ, x) dτ t η) 3 η ) sin η ] ) e τ ] ) 5 uτ, x) dτ t ) η s) 2 us, x) } dη t,2] ) u, x ) dη } t,2]

22 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 22 of 26 d Γ 3 )Γ ) dt η e sτ e sτ η s) e τ 5 uτ, x) dτ t η) 3 η ) 2 us, x) 2es us, x) us, x) ] u e τ ξ sin u, x) ) ] ) η ) Γ 3 ) u η 2 us, x) 2es us, x) us, x) t s) 2 us, x) 5 uτ, x) dτ 2es us, x) s us, x) e sτ e τ ] } } 5 uτ, x) dτ dη t,2] 2 us, x) 2es us, x) s us, x) e sτ e τ 5 uτ, x) dτ ξ sin u, x) ) t 2 us, x) Γ 3 ) 2es us, x) us, x) ξ sin u, x) ) t Γ 3 ) e sτ e τ } dτ 5 uτ, x) ξ sin u, x) ) d t Γ 3 ) Γ 3 )Γ ) t η) 3 u dt e sτ 2es us, x) us, x) 5 uτ, x) dτ t s) 2 us, x) 2es us, x) η us, x) e sτ e τ ] } } 5 uτ, x) dτ dη 2 ut, x) t e τ } t,2] ] ) η ) ] } ] } t,2] t,2] 2et ut, x) ut, x) e ts e s ] 5 us, x) t ξ sin u, x) ) 2 ut, x) Γ 3 ) 2et ut, x) ut, x) e ts e s ] 2 5 us, x) ut, x) 2et ut, x) t ut, x) e ts e s ] } 5 us, x) t t,2] 2 ut, x) 2et ut, x) t ut, x) e ts e s ] 5 us, x). t,2] t,2] 2 us, x) So, eqution.5) stisfies the RL frctionl derivtive condition of system.).). By Definition 2.2,weobtin I 3 u, x ) I 3 u, x ) t } Γ ) t η) 3 uη) dη t Γ ) t η) 3 uη) dη } t

23 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 23 of 26 sin u, x) ) t } Γ 3 )Γ 3 ) t η) 3 dη t ξ sin u, x ) ) Γ 3 )Γ 3 ) η η s) 3 2 us, x) 2es us, x) us, x) 2es us, x) us, x) e sτ e sτ e sτ e sτ e τ 5 uτ, x) dτ e τ 5 uτ, x) dτ t η) 3 u η ] u 2es us, x) us, x) 5 uτ, x) dτ η η s) 2 us, x) 2es us, x) s us, x) sin u, x ) ) ξ sin u, x ) ) u e τ ] ] ) η ) e sτ 2 us, x) e τ ] } 5 uτ, x) dτ dη} t 2 us, x) u 2es us, x) us, x) e τ ] ) 5 uτ, x) dτ 2 us, x) 2es us, x) s us, x) e sτ e τ 5 uτ, x) dτ sin u, x ) ). 2 us, x) ] } t Tht is, eqution.5) stisfies impulsive condition.2). Therefore, clerly eqution.5) fulfills the following limit cse: lim sin u,x) ) lim sin u,x) ) D 3 ut, x) 2 ut,x) e2s us,x) us,x) ] 2et ut,x) ut,x) t, 2] nd t, I ut, x) es ets 5 us,x), I u) t I u, x)i u, x) sin u, x) ) H, I ut, x) us,x) e2s us,x) ] t u x) H, D 3 ut, x) us,x) e2s us,x) ] 2 ut,x) 2et ut,x) ut,x) es ets 5 us,x), t, 2], e2s us,x) us,x) ] t u x) H. So, eqution.5) is the generl solution of system.).3). Chrcterize the dministrtors L, D, P : I H H nd q : I H by L t, u) 2et ut, x) ut, x),

24 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 2 of 26 e t Pt, u) 5 us, x), Dt, u) e 2s us, x) us, x), qt s)e ts nd I k u, x )) sin u, x ) ). Then the impulsive frctionl differentil system.).) cnbeconvertedintothe bstrct form problem.).3). Next, we shll show tht hypotheses H2)H5) re stisfied. For this, u, v PC γ, 2], H). i) L t, u)l t, v) 2e t u u 2et v v 2e t u u v v 2e t u v u) v) 2 u v. ii) Hypothesis H2) hol if N L 2. Pt, u)pt, v) e t 5u et 5v e t u v 5 u)5 v) e u v. 25 If N L e, condition H3) is stisfied. 25 iii) Choose β 2,wehve t A ) 2 Dt, u)a ) 2 Dt, v) e 2s u u v v e2t u v 2 u) v) e u v. 2 Here, condition H) hol with N P e 2.

25 Klmni etl.advnces in Difference Equtions 28) 28:6 Pge 25 of 26 iv) Finlly, Ik u )) I k u )) sin u, x ) ) sin v, x ) ) ) u) sin u sin u v. And Ik u) sin u, x ) ) u. From iv), we notice tht hypothesis H5) hol with d i N I. Let t I,thenwehve q sup t I t qt s) e ts e t e. Therefore, ll the conditions of Theorem 3. re verified. Hence, problem.).)hs solution in, 2]. 5 Conclusion In this mnuscript, we hve studied the locl existence for n impulsive frctionl neutrl integro-differentil system with RiemnnLiouville frctionl derivtives in Bnch spce. More precisely, by utilizing the semigroup theory, frctionl powers of opertors, nd condensing fixed point theorem, we investigte the impulsive frctionl neutrl integro-differentil system in Bnch spce. To vlidte the obtined theoreticl results, n exmple is nlyzed. The frctionl differentil equtions re very efficient to describe rel life phenomen; thus, it is essentil to extend the present study to estblish the other qulittive nd quntittive properties such s stbility nd pproximte controllbility. There re two direct issues which require further study. First, we will investigte the globl existence of mild solution to impulsive frctionl semiliner integro-differentil equtions with noncompct semigroup. Secondly, we will be devoted to studying the pproximte controllbility of impulsive frctionl neutrl integro-differentil systems with RiemnnLiouville frctionl derivtives in Bnch spce both in the cse of noncompct opertor nd norml topologicl spce. Acknowledgements The uthors re grteful to the nonymous referees for their constructive comments nd helpful suggestions. Funding Not pplicble. Avilbility of dt nd mterils Not pplicble. Competing interests The uthors declre tht they hve no competing interests.

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