Nonlocal cauchy problem for delay fractional integrodifferential equations of neutral type

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1 Li Advances in Difference Euations 22, 22:47 RESEARCH Open Access Nonlocal cauchy problem for delay fractional integrodifferential euations of neutral type Fang Li Correspondence: com School of Mathematics, Yunnan Normal University, Kunming 6592, People s Republic of China Abstract This article concerns the existence of mild solutions for the fractional integrodifferential euations of neutral type with finite delay and nonlocal conditions in a Banach space X. The existence of mild solutions is proved by means of measure of noncompactness. As an application, the existence of mild solutions for some integrodifferential euation is obtained. Mathematics Subject Classification: 34K5; 34A2. Keywords: fractional integrodifferential euation, mild solution, finite delay, nonlocal condition, neutral type, measure of noncompactness Introduction The fractional differential euations have received increasing attention during recent years and have been studied extensively see, e.g., [- and references therein since they can be used to describe many phenomena arising in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. Moreover, the Cauchy problem for various delay euations in Banach spaces has been receiving more and more attention during the past decades see, e.g., [2,3,6,7,-2. As in [5,8,3-5 and the related references given there, we pay attention to the nonlocal condition because in many cases a nonlocal condition v +gv =v is more realistic than the classical condition v =v in treating physical problems. To the author s knowledge, few articles can be found in the literature for the solvability of the fractional order delay integrodifferential euations of neutral type with nonlocal conditions. In this article, we concern with the following nonlocal neutral delay fractional integrodifferential euations c D v t h t, v t = Av t + f t, v t + v t = g vt + φ t, t [ r,, a t, s, v s ds, t [, T, : where T>, <<, <r<. The fractional derivative is understood here in the Caputo sense. X is a separable Banach space. A is a closed operator. Here h: [, T C [-r,,x X, f: [,T C[-r,, X X, a: Δ C[-r,,X XΔ ={t, s Î [, 22 Li; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Li Advances in Difference Euations 22, 22:47 Page 2 of 23 T [,T: t s}, g: C[-r,,X C[-r,,X, j Î C[-r,,X, where C[a, b, X denotes the space of all continuous functions from [a, b to X. For any continuous function v defined on [-r, T and any t Î [, T, we denote by v t the element of C[-r,, X defined by v t θ =vt +θ, θ Î [-r,. This article is organized as follows. In Section 2, we recall some basic definitions and preliminary results. In Section 3, we give the existence theorem of mild solution of. and its proof. In the last section, an example is given to show an application of the abstract result. 2 Preliminaries Throughout this article, we denote by X a separable Banach space with norm, by LX the Banach space of all linear and bounded operators on X, and by C[a, b, X the space of all X-valued continuous functions on [a, b with the remum norm as follows: x [a,b = x C[a,b,X = { x t : t [ a, b }, for any x C [ a, b, X. Moreover, we abbreviate u L p [,T,R + with u L p, for any u Î L p [, T, R +. In this article, A is the infinitesimal generator of an uniformly bounded analytic semigroup of linear operators {St} t in X. We will assume that Î ra ra is the resolvent set of X and that S t M, for all t [, T. Under these conditions it is possible to define the fractional power -A a,< a <, as closed linear operator on its domain D-A. Recall the knowledge in [6, we have there exists a constant M > such that A α M. 2 for any a Î,, there exists a positive constant C a such that A α S t C α t α. Let us recall the following known definitions. For more details see [9. Definition 2.. [9 The fractional integral of order with the lower limit zero for a function f Î AC[, is defined as I f t = t Ɣ t s f s ds, t >, < <, provided the right side is point-wise defined on [,, where Γ is the gamma function. Definition 2.2. [9 Riemann-Liouville derivative of order with the lower limit zero for a function f Î AC[, can be written as L D f t = Ɣ d t s f s ds, dt t >, < <.

3 Li Advances in Difference Euations 22, 22:47 Page 3 of 23 Definition 2.3. [9 The Caputo derivative of order for a function f Î AC[, can be written as c D f t = L D f t f, t >, < <. Remark 2.4. If ft Î C [,, then c D f t = t Ɣ t s f s ds = I f t, t >, < <. 2 The Caputo derivative of a constant is eual to zero. We will need the following facts from the theory of measures of noncompactness and condensing maps see, e.g., [7,8. Definition 2.5. Let E be a Banach space and A, a partially ordered set. A function β : P E Ais called a measure of noncompactness MNC in E if β co = β for every P E, where PE denotes the class of all nonempty subsets of E. A MNC b is called: i monotone, if Ω, Ω Î PE, Ω Ω implies bω b Ω ; ii nonsingular, if b{a } Ω =bω for every a Î E, Ω Î PE; iii invariant with respect to union with compact sets, if b {D} Ω =bω for every relatively compact set D E, Ω Î PE. If Ais a cone in a normed space, we say that the MNC b is iv algebraically semiadditive, if bω + Ω bω +bω for each Ω, Ω Î P E; v regular, if bω =is euivalent to the relative compactness of Ω; vi real, if Ais [, + with the natural order. As an example of the MNC possessing all these properties, we may consider the Hausdorff MNC X =inf { ε>: has a finite ε -net }. Now, let G : [, h P E be a multifunction. It is called: i integrable, if it admits a Bochner integrable selection [ g :, h [ E, gt G t for a.e. t, h ; ii integrably bounded, if there exists a function ϑ L [, h, E such that G t := { g : g G t} ϑ t a.e t [, h. We present the following assertion about c-estimates for a multivalued integral [[8, Theorem Proposition 2.6. For an integrable, integrably bounded multifunction [ G :, h P X where X is a separable Banach space, let

4 Li Advances in Difference Euations 22, 22:47 Page 4 of 23 χ G t t, for a.e. t [, h, where L + [, h. Then χ G s ds sds for all t [, h. Let E be a Banach space, b a monotone nonsingular MNC in E. Definition 2.7. A continuous map F : Y E E is called condensing with respect to a MNC b or b-condensing if for every bounded set Ω Y which is not relatively compact, we have β F β. The following fixed point principle see, e.g., [7,8 will be used later. Theorem 2.8. Let Wbe a bounded convex closed subset of E and F : W W a b- condensing map. Then FixF = {x : x = F x} is nonempty. Theorem 2.9. Let V E be a bounded open neighborhood of zero and F : V E a b-condensing map satisfying the boundary condition x = λf x for all x Î V and < λ. Then FixF is a nonempty compact set. We state a generalization of Gronwall s lemma for singular kernels [[9, Lemma 7... Lemma 2.. Let x, y: [, T [, + be continuous functions. If y is nondecreasing and there are constants a > and < ξ < such that x t y t + a t s ξ x s ds, then there exists a constant = ξ such that x t y t + κa t s ξ y s ds, for each t [, T. According to Definitions , we can rewrite the nonlocal Cauchy problem. in the euivalent integral euation v t = g v + φ + h t, v t h, φ + g v t + Ɣ t s [Av s + f s, v s +k vs ds, t [, T, v t = g vt + φ t, t [ r, 2: provided that the integral in 2. exists, where k vt = a t, s, v s ds and h, j + gv: = h, jθ +gvθ, θ Î [-r,.

5 Li Advances in Difference Euations 22, 22:47 Page 5 of 23 Set v λ = e λt v t dt, ĥ λ = f λ = e λt f t, v t dt, w λ = e λt h t, v t dt, e λt k vt dt. Using the similar method in [, formally applying the Laplace transform to 2., we have then thus v λ = g v + φ h, φ + g v + h λ + λ λ A v λ + [ f λ + λ w λ, λ A v λ = λ g v + φ h, φ + g v [ + λ h λ + f λ + w λ, ˆv λ =λ λ A g v + φ h, φ + g v + λ λ A ĥ λ + λ A [ˆf λ + ŵ λ =λ + ĥ λ + e λs S s g v + φ h, φ + g v ds e λs AS s ĥ λ ds + e λs S s [ˆf λ + ŵ λ ds, 2:2 provided that the integral in 2.2 exists. We consider the one-sided stable probability density in [4 as follows: ϖ σ = n σ n Ɣ n + sin nπ, σ,, π n! n= whose Laplace transform is given by e λσ ϖ σ dσ = e λ,,. 2:3 Then, using 2.3, we have λ = = λ = = e λs S s g v + φ h, φ + g v ds λt e λt S t g v + φ h, φ + g v dt e λt d dt e λt S t g v + φ h, φ + g v dt e λtσ σϖ σ S t g v + φ h, φ + g v dσ dt t g ϖ σ S v + φ h, φ + g v dσ dt 2:4

6 Li Advances in Difference Euations 22, 22:47 Page 6 of 23 and = = = e λs S s ˆf λ ds e λτ τ S τ e λττ ϖ σ S τ λθ θ e = = = t τ e λt e λt f t, v t dt dτ e λt f t, v t dt dσ dτ θ ϖ σ S e λt f t, v t dt dσ dθ λτ τ t e ϖ σ S λτ τ t e ϖ σ S Similarly, we have = e λs S s ŵ λ ds e λt τ t f t, v t dτdt dσ τ t f t, v t dtdτ dσ t t t s ϖ σ S f s, v s dσ ds dt. t s t s ϖ σ S k vs dσ ds dt 2:5 2:6 and = e λs AS s ĥ λ ds e λt t s t s ϖ σ AS h s, v s dσ ds dt. Thus, from , we obtain ˆv λ = e λt t g ϖ σ S v + φ h, φ + g v dσ + h t, vt t s t s ϖ σ AS t s t s ϖ σ S h s, v s dσ ds f s, v s dσ ds t s t s ϖ σ S k vs dσ ds dt. 2:7

7 Li Advances in Difference Euations 22, 22:47 Page 7 of 23 We invert the last Laplace transform to obtain v t = = t g ϖ σ S v + φ h, φ + g v dσ + h t, vt t s t s ϖ σ AS h s, v s dσ ds t s t s ϖ σ S f s, v s dσ ds t s t s ϖ σ S k vs dσ ds ξ σ S t σ g v + φ h, φ + g v dσ + h t, v t σ t s ξ σ AS t s σ h s, v s dσ ds σ t s ξ σ S t s σ f s, v s dσ ds σ t s ξ σ S t s σ k vs dσ ds, where ξ is a probability density function defined on, such that ξ σ = σ ϖ σ. For any z Î X, we define operators {Qt} t and {Rt} t by Q t z = R t z = ξ σ S t σ zdσ, σ t ξ σ S t σ zdσ. Then from above induction, we can give the following definition of the mild solution of.. Definition 2.. A continuous function v: [-r, T X is a mild solution of. if the function v satisfies the euation v t = Q t g v + φ h, φ + g v + h t, v t + AR t s h s, v s ds + R t s [ f s, v s + k vs ds, t [, T, v t = g vt + φ t, t [ r,.

8 Li Advances in Difference Euations 22, 22:47 Page 8 of 23 Remark 2.2. i Noting that ξ σ dσ =, we obtain Q t M. 2:8 ii According to [4, direct calculation gives that σ ν ξ σ dσ = then, we can obtain R t ν ϖ σ dσ = Ɣ +ν Ɣ, ν,, +ν M Ɣ + t, t >. 2:9 iii For any a Î,, we have A α R t t α C α Ɣ 2 α Ɣ + α, t >. 2: Indeed, for any z Î X we can see that A α R t z = σ t ξ σ A α S t σ zdσ t σξ σ C α t α dσ z σ = t α C α Ɣ 2 α Ɣ + α z. 3 Main results We will reuire the following assumptions. Hf f: [,T C[-r,,X X satisfies f, w: [, T X is measurable for all w Î C[-r,, X and ft, : C[-r,, X X is continuous for a.e. t Î [, T, and there exists a positive function μ L p [, T,R + p > α > such that f t, w μ t w [ r, for almost all t Î [, T. 2 There exists a nondecreasing function h Î L p [, T, R + such that for any bounded set D C[-r,, X, χ f t, D η t θ [ r, χ D θ, a.e. t [, T. Hh h: [, T C[-r,, X X is continuous and there exists constants a Î, and M, L h > such that h Î D-A a and for any Î C[-r,, X the function -A a h, is strongly measurable and satisfies A α h t, ϕ M ϕ [ r, +, 3:

9 Li Advances in Difference Euations 22, 22:47 Page 9 of 23 and for t, t 2 [, T, ϕ, ϕ C [ r,, X, A α h t, ϕ A α h t 2, ϕ L h t t 2 + ϕ ϕ [ r,. 3:2 2 There exists M 2 > such that for any bounded set D C[-r,, X, χ A α h t, D M 2 θ [ r, χ D θ, a.e. t [, T. Ha a: Δ C[-r,, X X and at, s, : C[-r,, X X is continuous for a.e. t, s Î Δ, and for each w Î C[-r,,X, the function a,, w: Δ X is measurable. Moreover, there exists a function m: Δ R + with t t [,T m t, s ds := m < such that a t, s, w m t, s w [ r,, for almost all t, s Î Δ. 2 For any bounded set D C[-r,,X and s t T, there exists a function ζ: Δ R + such that χ a t, s, D ζ t, s θ [ r, χ D θ, where ζ t, s ds := ζ <. t [,T Hg There exists a continuous function L g :[-r, R + such that g v t g v 2 t L g t v t v 2 t, t [ r,. 2 The function gv : [-r, C[-r,,X is euicontinuous and uniformly bounded, that is, there exists a constant N> such that g v [ r, N for all v C [ r,, X. Theorem 3.. Assume that Hf, Hh, Ha and Hg are satisfied, if M M max {M, L h } + Ɣ + l p p,t μ L p <, 2 L g +MM M 2 + M + M M 2 <, p where p lp, := p and L g = L g t t [ r,. Then the mild solutions set of αp problem. is a nonempty compact subset of the space C[-r, T, X. Proof. Define the operator F : C [ r, T, X C [ r, T, X in the following way: g vt + φ t, t [ r,, Q t g v + φ h, φ + g v + h t, v t Fvt = t + AR t s h s, v s ds + R t s [ f s, v s + k vs ds, t [, T. It is clear that the operator F is well defined. The operator F canbewrittenintheformf = 2 i= F i, where the operators F i, i =,2 are defined as follows:

10 Li Advances in Difference Euations 22, 22:47 Page of 23 { g vt + φ t, t [ r,, F vt = Q t g v + φ h, φ + g v + h t, v t, t [, T,, t [ r,, F 2 vt = AR t s h s, v s ds + R t s [ f s, v s + k vs ds, t [, T. Let {v n } nîn beaseuencesuchthatv n v in C[-r, T, X asn. By the continuity of g and h, we can see that F is continuous. Moreover, noting 2., we have AR t s h t s, v n s ds AR t s h s, v s ds A α R t s A α h s, v n s A α h s, v s ds C αɣ +α Ɣ +α L h v n v [ r,t t C αɣ +α αɣ +α L ht α v n v [ r,t, t s α ds as n. Since f satisfies Hf and a satisfies Ha, for almost every t Î [, T and t, s Î Δ, we get f t, v n t f t, vt, as n, a t, s, v n s a t, s, vs, as n. Noting that v n v in C[-r, T, X, we can see that there exists ε > such that v n - v [-r, T ε for n sufficiently large. Therefore, we have f t, v n t f t, vt μ t v n t [ r, + μ t v t [ r, μ t v n t v t [ r, +2μ t v t [ r, μ t ε +2 v [ r,t, and similarly, a t, s, v n s a t, s, vs m t, s ε +2 v [ r,t, a t t, s, v n s ds a t, s, v s ds m ε +2 v [ r,t. It follows from the Lebesgue s dominated convergence theorem that a t t, s, v n s ds a t, s, v s ds, as n,

11 Li Advances in Difference Euations 22, 22:47 Page of 23 and s s Rt s f s, v s n + as, τ, v n τ dτ f s, v s + as, τ, v τ dτ ds M t s f s, v s n s f s, v s + as, τ, v n τ Ɣ + as, τ, v τ dτ ds, as n. Therefore, we obtain that lim n F 2v n F 2 v [ r,t =. 3:3 Now, from 3.3, we can see that F is continuous. Let c be a Hausdorff MNC in X, we consider the measure of noncompactness b in the space C[-r, T, X with values in the cone R 2 + of the following way: for every bounded subset Ω C[-r, T, X, β =, mod c, where mod c Ω is the module of euicontinuity of Ω given by: mod c = lim max vt vt 2, δ v t t 2 δ = t [ r, χ t + t [,T e Lt χ s s [,t, where L> is a constant chosen so that C α Ɣ + α Ɣ + α M Ɣ + t [,T M Ɣ + max{m, M 2 } t [,T t [,T t s α e Lt s ds = L <, 3:4 t s ηs+ζ e Lt s ds = L 2 <, 3:5 t s μs+m e Lt s ds = L 3 <. 3:6 Noting that for any g Î L [, T, X, we have lim L + t [,T e Lt s γ sds =, so, we can take the appropriate L to satisfy Next, we show that the operator F is b-condensing on every bounded subset of C [-r, T, X. Let Ω C[-r, T, X be a nonempty, bounded set such that βf β. 3:7

12 Li Advances in Difference Euations 22, 22:47 Page 2 of 23 Firstly, we estimate ΨΩ. For t Î [-r,, v, v 2 Î Ω, we have gv t gv 2 t L g v t v 2 t, then χg t L g χ t, t [ r, t [ r, therefore, χf t = t [ r, χg t L g t [ r, χ t. t [ r, 3:8 For t Î [, T, one gets Qt gv gv 2 ML g v v 2, thus, we have χqtg ML g χ ML g. 3:9 Moreover, we see that χ Qth, φ + g M χ A α A α h, φ + g MM M 2 χg s s [ r, MM M 2 L g s [ r, χ s MM M 2 L g. 3: For t Î [, T, noting that χ t + θ = χ s θ [ r, s [t r,t χ s + χ s s [ r, s [,t e Lt e Lt, χ s + e Lt χ s s [ r, s [,t 3: then we have χht, t = χ A α A α ht, t M M 2 χ t + θ θ [ r, M M 2 e Lt, where Ω t ={v t : v Î Ω}. Therefore t [,T e Lt χhs, s s [,t M M 2. 3:2

13 Li Advances in Difference Euations 22, 22:47 Page 3 of 23 Now, from , and 3.2, we can see F = χf t + t [ r, t [,T e Lt χf s s [,t L g t [ r, χ t+mm M 2 L g + ML g + M M 2 [L g + MM M 2 + M+M M 2. 3:3 For any t Î [, T, we set t = ARt shs, v s ds : v. We consider the multifunction s Î [, t Gs, Gs = {ARt shs, v s : v }. Obviously, G is integrable, and from Hh it follows that it is integrably bounded. Moreover, noting that Hh2 and 3., we have the following estimate for a.e. s Î [, t: χgs = χ { A α Rt s A α hs, v s :v } = χ A α Rt s A α hs, s C αɣ + α t s α M 2 χ s + θ Ɣ + α θ [ r, C αɣ + α t s α M 2 e Ls. Ɣ+α Applying Proposition 2.6, we have χ t = χ Gsds C αɣ + α Ɣ + α M 2 t s α e Ls ds. Therefore, t [,T C αɣ + α Ɣ + α =L. e Lt χ s s [,t M 2 t [,T t s α e Lt s ds 3:4 Similar to above induction, if we set 2 t = Rt s[f s, v s +kvsds : v, then we can see that the multifunction s [, t Gs, Gs ={Rt s[f s, v s +kvs : v }

14 Li Advances in Difference Euations 22, 22:47 Page 4 of 23 is integrable, and from Hf, Ha it follows that it is integrably bounded. Moreover, noting that Hf2, Ha2, Proposition 2.6 and 3., we have the following estimate for a.e. s Î [, t: χ Gs M Ɣ + t s [ηs+ζ e Ls, furthermore, t [,T e Lt χ 2 s s [,t M Ɣ + = L 2. t [,T t s [ηs+ζ e Lt s ds 3:5 Now, it follows from 3.4 and 3.5 that F 2 2 L i. Then, noting i= 3.3 and choose L i i =, 2 such that [ 2 F L g + MM M 2 + M+M M 2 + L i L, 3:6 where < L <. From 3.7, we have ΨΩ =. Next, we will prove mod c Ω =. Let δ >,t, t 2 Î [, T such that < t -t 2 δ and v Î Ω, we obtain ht, v t ht, v t2 A α A α ht, v t ht 2, v t2 M L h t t 2 + v t v [ r, t2 i= = M L h t t 2 + vt + θ vt 2 + θ θ [ r,, t t 2 δ M L h t t 2 + vs vs 2. s,s 2 [ r,t, s s 2 δ Moreover, noting that Hg2 and the continuity of St intheuniformoperator topology for t> we have mod c F M L h mod c. For <t 2 <t T and v Î Ω, we have 2 ARt shs, v s ds ARt 2 shs, v s ds 2 ARt s Rt 2 shs, v s ds t 2 A α Rt s A α hs, v s ds σ [t s t 2 s ξ σ ASt s σ hs, v s dσ ds σ t 2 s ξ σ ASt s σ St 2 s σ hs, v s dσ ds + M v [ r,t + t t 2 α C αɣ + α αɣ + α =I + I 2 + I 3. Noting that s ASt -s s h s, v s is integrable on [, t, then I, as t 2 t. In view of the fact that {St} t is an analytic semigroup, then for s Î [, t, s Î,, the operator function s ASt -s s is continuous in the uniform operator

15 Li Advances in Difference Euations 22, 22:47 Page 5 of 23 topology in [, t and noting that s hs, v s is integrable on [, t, I 2, as t 2 t. Obviously I 3, as t 2 t. Moreover, nothing that f s, v s + kvs μs+m v [ r,t, we have 2 Rt s[f s, v s +kvsds [Rt s Rt 2 s[f s, v s +kvs ds + v [ r,t M v [ r,t Ɣ + 2 Rt 2 s[f s, v s +kvsds t 2 [Rt s [f s, v s +kvs ds σ [t s t 2 s ξ σ St s σ μs+m dσ ds σ t 2 s ξ σ St s σ St 2 s σ μs+m dσ ds t 2 t s μs+m ds. 3:7 Clearly, the first term on the right-hand side of 3.7 tends to as t 2 t. The second term on the right-hand side of 3.7 tends to as t 2 t as a conseuence of the continuity of St in the uniform operator topology for t>. It is easy to see that the third term on the right-hand side of 3.7 tends to as t 2 t. Thus, the set { F 2 v :v } is euicontinuous, then mod c F 2 =. Since mod c F 2 mod c F i M L h mod c, i= then mod c Ω =, which yields from 3.7. Hence β =,. The regularity property of b implies the relative compactness of Ω. Now, it follows from Definition 2.7 that F is b-condensing. Choose ρ> Ñ + MN + φ + M M Ñ + + M M + C αɣ + α αɣ + α M T α where Ñ := N + φ [ r,. Consider the set B ρ = { v C[ r, T, X : v [ r,t ρ }. M M,

16 Li Advances in Difference Euations 22, 22:47 Page 6 of 23 Let us introduce in the space C[-r, T, X the euivalent norm defined as v c = vt + e Lt vs. t [ r, t [,T s [,t Noting A α ht, v t M v [ r, + vs + s [,t M e v Lt [ r, + e Lt vs s [,t M e Lt v c +, + M 3:8 and ht, v t = A α A α ht, v t M M e Lt v c +, for t Î [, T, wehave Qtgv + φ h, φ + gv + ht, v t then MN + φ + M M Ñ + + M M e Lt v c +, F v c Ñ + t [,T e Lt Qsgv + φ h, φ + gv + hs, vs s [,t Ñ + MN + φ + M M Ñ + + M M v c +. Moreover, for t Î [, T, from 2. and 3.8 we have ARt shs, v s ds A α Rt s A α hs, v s C αɣ + α M Ɣ + α ds t s α e Ls v c +ds, and from 2.9, Hf and Ha we get Rt s[f s, v s +kvsds M t s μs+m v Ɣ + [ r, + vτ ds τ [,s M t s μs+m e Ls v Ɣ + C ds. 3:9 3:2

17 Li Advances in Difference Euations 22, 22:47 Page 7 of 23 Therefore, from 3.9 and 3.2, we obtain F 2 v c = e Lt t [,T s [,t C αɣ + α Ɣ + α + M Ɣ + s M t [,T t [,T C αɣ + α Ɣ + α + s ARs τhτ, v τ dτ+ M v C M Ɣ + v C Let L +, we obtain Hence t [,T t s α e Lt s v c + e Lt ds t s μs+m e Lt s v C ds t [,T Rs τ[f τ, v τ +kvτdτ t s α e Lt s ds+ C αɣ + α αɣ + α M T α t s μs+m e Lt s ds. F 2 v C C αɣ+α αɣ+α M T α. 3:2 Fv C F v C + F 2 v C Ñ + MN + φ + M M Ñ + + M M v C + ρ. + C αɣ + α αɣ + α M T α Now, we show that there exists some r > suchthatfb ρ B ρ. According to Theorem 2.8, problem. has at least one mild solution. Next, for δ,, we consider the following one-parameter family of maps : [, C[ r, T, X C[ r, T, X δ, v δ, v = δfv. We will prove that the fixed point set of the family Π, Fix = {v δ, v for some δ, } is a priori bounded. Noting that the Hölder ineuality, we have t s α μsds l p, t pα p μ L p l p, Tα p μ L p.

18 Li Advances in Difference Euations 22, 22:47 Page 8 of 23 Let v Î FixΠ, for t Î [, T, we have vt Qtgv + φ h, φ + gv + ht, vt + t ARt shs, vs ds + Rt s[f s, v s +kvs ds MN + φ +MM M Ñ ++M M v t [ r, + + M C α Ɣ + α t t s α v s Ɣ + α [ r, +ds M t s + t s μs v s Ɣ + [ r, + ms, τ v τ [ r, dτ ds MN + φ +MM M Ñ ++M M Ñ + vs + s [,t + M C α Ɣ + α t t s α Ñ + vτ + ds Ɣ + α τ [,s + MT α Ɣ + a + a s [,t t s α μs+m vs + a 2 t Ñ + τ [,s t s α vτ ds, τ [,s vτ ds 3:22 where a =MN + φ +M +M M Ñ ++ M C α Ɣ + α T α Ñ + αɣ + α + M Ɣ + l p p,t μ L p Ñ + Mm T αɣ + Ñ, a =M M + a 2 = M C α Ɣ + α Ɣ + α We denote we can see M Ɣ + l p,t p μ L p, + Mm T α. Ɣ + γ t := vs.let t [, t such that γ t = v t. Then, by 3.22, s [,t γ t a + a γ t+a 2 t s α γ sds. By Lemma 2., there exists a constant = a such that γ t a ka a t 2 + a a 2 t s α ds a + ka a 2 T α := ϖ. a 2 α a Hence vt ϖ. t [,T

19 Li Advances in Difference Euations 22, 22:47 Page 9 of 23 Conseuently v [ r,t v [ r, + v [,T Ñ + ϖ. Now we consider a closed ball B R = { v C[ r, T, X : v [ r,t R } C[ r, T, X. We take the radius R> large enough to contain the set FixΠ inside itself. Moreover, from the proof above, F : B R C[ r, T, X is b-condensing and it remains to apply Theorem 2.9. If we assume μ Î C[, T, R + in Hf, then we have Theorem 3.2. Assume that Hf, Hh, Ha and Hg are satisfied, if M max{m, L h } <, 2 L g + MM M 2 + M+M M 2 <, where L g = L g t. Then the mild solutions set of problem. is a nonempty t [ r, compact subset of the space C[-r, T, X. From the proof in Theorem 3., we can see that Theorem 3.2 holds. 4 Application In this section, let X=L 2 [, π, we consider the following integrodifferential problem: [ t u t, x e t π γ x, y ut θ, y dy = 2 u t, x + u t θ, y x2 + k t sin ut θ, x + t s 2 u s θ, x 3 γ 2 θ sin dθds, 4: u t, = u t, π =, π u θ, x = c x, y sin + u θ, y dy + u θ, x, r θ, where t Î [,, r>, x Î [, π and u t θ, x=ut + θ, x. g : [, π [, π R, g 2 :[-r, R and cx, y Î L 2 [, π [, π, R are functions to be specified later. To treat the above problem, we define D A = H 2 [, π H [, π, Au = u. The operator -A is the infinitesimal generator of an analytic semigroup {Tt} t on X. Moreover, A has a discrete spectrum, the eigenvalues are n 2, n Î N, with the corresponding normalized eigenvectors ω n x = 2 π sin nx and the following properties are satisfied: a if w Î DA, then Aω = n= n2 ω, ω n ω n. b For each ω X, T t ω = exp n 2 t ω, ω n ω n and Tt for t. n= c For each ω X, A 2 ω = n= n ω, ω n ω. In particular, n A 2 =. r s 3

20 Li Advances in Difference Euations 22, 22:47 Page 2 of 23 d A 2 is given by A 2 ω = n ω, ω n ω n with domain n= { } D A2 = ω X : n ω, ω n ω n X. n= We assume that the following conditions hold. The function g : [, π [, π R is continuously differential with g, y =g π, y =fory Î [, π and π π γ x, y x 2 2 dydx <. 2 The function g 2 :[-r, R is a continuous function, and r γ 2 θ dθ <. 3 The function cx, y, x, y Î [, π, is measurable and there exists a constant N such that π c x, y dy N. For x Î [, π and Î C[-r,, X, we set v tx = u t, x, φ θx = u θ, x, θ [ r,, π h t, ϕx = e t γ x, y ϕ θ y + ϕ θ y dy, g ϕ θ x = π c x, y sin + ϕ θ y dy, f t, ϕx = k t sin ϕ θx, a t, s, ϕx = t s 2 ϕ θx 3 γ 2 θ sin dθ, r where k t L P [,, R + p > 2. Then the above Euation 4. can be reformulated as the abstract.. For t Î [,, we can see f t, ϕ k t ϕ [ r,. For any ϕ, ϕ C[ r,,x, f t, ϕx f t, ϕx k t ϕ θx ϕ θx. s 3

21 Li Advances in Difference Euations 22, 22:47 Page 2 of 23 Therefore, for any bounded set D C[-r,, X, we have X f t, D k t Noting that h t, ϕ, ω n = π = e t n r θ ω n x π χ D θ, t [,. e t x π γ x, y ϕ θ y + ϕ θ y dy dx γ x, yϕ θ y + ϕ θ y dy, and A 2 h t, ϕ = n h t, ϕ, ω n ω n and n= π π M ϕ [ r, 2 γ x, y 2 dydx x A 2 h t, ϕ A2 h t 2, ϕ = n h t, ϕ h t 2, ϕ, ω n ω n n= π + π π π π γ x, y x γ x, y x dydx t t 2 2 dydx L h t t 2 + ϕ ϕ [ r,, where M = π π γ x, y x 2 π π 2 cos nx π 2dy 2 ϕ θ y 2dy 2 ϕ θ y ϕ θ y 2 dydx, L h = π π π can see that for any bounded set D C[-r,, X and t Î [,, χ A2 h t, D L h χ D θ. θ [ r, γ x, y x 2 2 dydx,we

22 Li Advances in Difference Euations 22, 22:47 Page 22 of 23 Moreover, a t, s, ϕx = t s 2 3 r t s 2 3 s 3 where m t, s := t s 2 3 s 3 and t [, mt, sds = r ϕ θx γ 2 θ sin dθ r γ 2 θ dθ r s 3 γ 2 θ dθ ϕ [ r,, γ 2 θ dθ. Then, we obtain t [, a t, s, ϕx a t, s, ϕx t s 2 3 s 3 Hence, for any bounded set D C[-r,, X, we have χ a t, s, D ζ t, s r θ χ D θ, t s 2 3 s 2 3 ds = B 3, 3 r γ 2 θ dθ r γ 2 θ ϕ θx ϕ θx dθ. where ζ t, s := t s 2 3 s 3 r γ 2 θ dθ, 2 ζ t, s ds = B t [, 3,. r 3 γ 2 θ dθ. For ϕ, ϕ C [ r,, X, θ [ r,, we can get g ϕx g ϕx π π c 2 x, y dydx /2 ϕ ϕ := L g ϕ ϕ. and Moreover, g ϕx π c x, y dy N. Suppose further that there exist constants M, M, such that p 2p 2 Lh + Ɣ p k p < M,, p 2 k + p 2 L g 2+L h + L h < M,, then 4. has a mild solution by Theorem 3.. Acknowledgements This work was ported by the NSF of Yunnan Province 29ZC54M. Competing interests The authors declare that they have no competing interests. Received: 8 December 2 Accepted: 7 April 22 Published: 7 April 22

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