Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 901942, 18 pages doi:10.1155/2012/901942 Research Article A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations Fang Li,1 Jin Liang,2 Tzon-Tzer Lu,3 and Huan Zhu4 1 School of Mathematics, Yunnan Normal University, Kunming 650092, China Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China 3 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan 4 Department of Mathematics, University of Science and Technology of China, Hefei 230026, China 2 Correspondence should be addressed to Jin Liang, jliang@ustc.edu.cn Received 12 February 2012; Accepted 3 March 2012 Academic Editor: Yonghong Yao Copyright q 2012 Fang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach space X. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained. 1. Introduction Nonlocal Cauchy problem for equations is an initial problem for the corresponding equations with nonlocal initial data. Such a Cauchy problem has better effects than the normal Cauchy problem with the classical initial data when we deal with many concrete problem coming from engineering and physics cf., e.g., 1–10 and references therein. Therefore, the study of this type of Cauchy problem is important and significant. Actually, as we have seen from the just mentioned literature, there have been many significant developments in this field. On the other hand, fractional differential and integrodifferential equations arise from various real processes and phenomena appeared in physics, chemical technology, materials, earthquake analysis, robots, electric fractal network, statistical mechanics biotechnology, medicine, and economics. They have in recent years been an object of investigations with much increasing interest. For more information on this subject see for instance 9, 11–18 and references therein. 2 Journal of Applied Mathematics Throughout this paper, X is a separable Banach space; LX is the Banach space of all linear bounded operators on X; A is the generator of an analytic and uniformly bounded semigroup {T t}t≥0 on X with T tLX ≤ M for a constant M > 0, and Ca, b, X is the space of all X-valued continuous functions ona, b with the supremum norm as follows: xa,b : max{xt : t ∈ a, b}, for any x ∈ Ca, b, X. 1.1 Let 0 < q < 1, T > 0, Δ {t, s ∈ 0, T × 0, T : t ≥ s}, f : 0, T × C0, T , X → X, and h : Δ × C 0, T , X → X. The nonlocal Cauchy problem for abstract fractional integrodifferential equations, with which we are concerned, is in the following form: c Dq xt Axt ft, xt t kt, sht, s, xsds, t ∈ 0, T , 0 1.2 x0 gx x0 , where k and g are given functions to be specified later and the fractional derivative is understood in the Caputo sense, this means that, the fractional derivative is understood in the following sense: c Dq xt : L Dq xt − x0, t > 0, 0 < q < 1, 1.3 and where L d 1 Dq xt : Γ 1 − q dt t t − s−q xsds, t > 0, 0 < q < 1 1.4 0 is the Riemann-Liouville derivative of order q of xt, where Γ· is the Gamma function. Our main purpose is to establish an existence theorem for the mild solutions to the nonlocal Cauchy problem based on a special measure of noncompactness under weak assumptions on the nonlinearity f and the semigroup {T t}t≥0 generated by A. 2. Existence Result and Proof As usual, we abbreviate uLp 0,T , R with uLp , for any u ∈ Lp 0, T , R . As in 16, 17, we define the fractional integral of order q with the lower limit zero for a function f ∈ AC0, ∞ as 1 I q ft Γ q t t − sq−1 fsds, t > 0, 0 < q < 1, 2.1 0 provided the right side is point-wise defined on 0, ∞. Now we recall some very basic concepts in the theory of measures of noncompactness and condensing maps see, e.g., 19, 20. Journal of Applied Mathematics 3 Definition 2.1. Let E be a Banach space, 2E the family of all nonempty subsets of E, A, ≥ a partially ordered set, and α : 2E → A. If for every Ω ∈ 2E : αcoΩ αΩ, 2.2 then we say that α is a measure of noncompactness in E. Definition 2.2. Let E be a Banach space, and F : Y ⊆ E → E is continuous. Let α be a measure of noncompactness in E such that i for any Ω0 , Ω1 ∈ 2E with Ω0 ⊂ Ω1 , αΩ0 ≤ αΩ1 ; 2.3 α{a0 } ∪ Ω αΩ. 2.4 ii for every a0 ∈ E, Ω ∈ 2E , If for every bounded set Ω ⊆ Y which is not relatively compact, 2.5 αFΩ < αΩ, then we say that F is condensing with respect to the measure of noncompactness α or α-condensing. Definition 2.3. Let ∞ 1 n−1 −qn−1 Γ nq 1 q σ sin nπq , −1 σ π n1 n! σ ∈ 0, ∞ 2.6 be a one-sided stable probability density, and ξq σ 1 −1−1/q −1/q , σ q σ q σ ∈ 0, ∞. 2.7 For any z ∈ X, we define operators {Y t}t≥0 and {Zt}t≥0 by Y tz ∞ ξq σT tq σzdσ, 0 Ztz q 2.8 ∞ σt 0 q−1 ξq σT t σzdσ. q 4 Journal of Applied Mathematics If a continuous function x : 0, T → X satisfies xt Y t gx x0 t Zt − s fs, xs axs ds, t ∈ 0, T , 2.9 0 then the function x is called a mild solution of 1.2. Our main result is as follows. Theorem 2.4. Assume that 1 f·, w and h·, ·, w are measurable for each w ∈ C0, T , X; kt, · is measurable for each t ∈ 0, T ; 2 ft, · is continuous for a.e. t ∈ 0, T ; g is completely continuous; ht, s, · is continuous for a.e. t, s ∈ Δ; the map t → kt : kt, · is continuous from 0, T to L∞ 0, T , R; 3 there exist two positive functions μ·, η· ∈ Lp 0, T, R p > 1/q > 1 and two positive functions m·, · and ζ·, · on Δ with t sup t∈0,T t ∗ mt, sds : m < ∞, 0 sup t∈0,T ζt, sds : ζ∗ < ∞, 2.10 0 such that ft, w ≤ μtw a.e. t ∈ 0, T , ht, s, w ≤ mt, sw a.e. t, s ∈ Δ, 2.11 for all w ∈ C0, T , X, and χ ft, D ≤ ηtχD, χht, s, D ≤ ζt, sχD, for any bounded set D noncompactness: ⊂ a.e. t ∈ 0, T, a.e. t, s ∈ Δ, 2.12 C0, T , X, where χ is the Hausdorff measure of χΩ inf{ε > 0 : Ω has a finite ε-net}. 2.13 4 g· satisfies gx ≤ b, ∀ x ∈ C0, T , X, 2.14 for a positive constant b, and kt : ess sup{|kt, s|, 0 ≤ s ≤ t} 2.15 Journal of Applied Mathematics 5 is bounded on 0, T . Then the mild solutions set of problem 1.2 is a nonempty compact subset of the space C0, T , X, in the case of p − 1 p−1/p q−1/p qM μ p < 1. T L Γ 1 q pq − 1 2.16 Proof. First of all, let us prove our definition of the mild solution to problem 1.2 is well defined and reasonable. Actually, the proof is basic. We present it here for the completeness of the proof as well as the convenience of reading. Write axt t kt, sht, s, xsds, 0 xλ ∞ e−λt xtdt, 0 fλ 2.17 ∞ e −λt ft, xtdt, 0 aλ ∞ e−λt axtdt. 0 Clearly, the nonlocal Cauchy problem 1.2 can be written as the following equivalent integral equation: 1 xt gx x0 Γ q t t − sq−1 Axs fs, xs axs ds, t ∈ 0, T , 2.18 0 provided that the integral in 2.18 exists. Formally taking the Laplace transform to 2.18, we have xλ 1 1 1 q fλ gx x0 q Axλ aλ . λ λ λ Therefore, if the related integrals exist, then we obtain xλ λq−1 λq − A−1 gx x0 λq − A−1 fλ aλ λq−1 q ∞ ∞ 0 0 q e−λ s T s gx x0 ds ∞ 0 q λtq−1 e−λt T tq gx x0 dt q e−λ s T s fλ aλ ds 2.19 6 Journal of Applied Mathematics ∞ ∞ q e−λτ qτ q−1 T τ q e−λt ft, xt axt dt dτ 0 0 ∞ 1 d −λtq e T tq gx x0 dt λ 0 dt ∞ ∞ ∞ −λτσ q−1 q −λt e qτ q σT τ e ft, xt axt dt dσ dτ − 0 0 ∞ ∞ 0 q 0 e−λtσ σq σT tq gx x0 dσdt 0 ∞ ∞ e 0 ∞ θq σT q σq σq −λθ θ 0 q−1 ∞ ∞ e −λt ft, xt axt dt dσ dθ 0 tq gx x0 dσ dt e q σT σq 0 0 ∞ ∞ ∞ q−1 τ − tq −λτ τ − t q e q σT ft, xt axt dτ dt dσ σq σq 0 0 t −λt ∞ e −λt 0 q 0 e ∞ e 0 q σT ∞ ∞ τ 0 ∞ 0 −λt ∞ −λτ 0 tq gx x dσ dt 0 σq τ − tq−1 τ − tq ft, xt axt dt dτ dσ q σT σq σq q σT 0 tq gx x dσ dt 0 σq t ∞ ∞ t − sq−1 t − sq −λt q e q σT fs, xs axs dσ ds dt. σq σq 0 0 0 2.20 Now using the uniqueness of the Laplace transform cf., e.g., 21, Theorem 1.1.6, we deduce that xt ∞ q σT 0 q t ∞ 0 q 0 ∞ 0 0 t ∞ 0 tq gx x0 dσ q σ t − sq−1 t − sq fs, xsdσ ds σT q σq σq t − sq−1 t − sq q σT axsdσ ds σq σq ξq σT tq σ gx x0 dσ Journal of Applied Mathematics q t ∞ 0 q 7 σt − sq−1 ξq σT t − sq σ fs, xsdσ ds 0 t ∞ 0 σt − sq−1 ξq σT t − sq σ axsdσ ds. 0 2.21 Consequently, we see that the mild solution to problem 1.2 given by Definition 2.3 is well defined. Next, we define the operator F : C0, T , X → C0, T , X as follows: Fxt Y t gx x0 t Zt − s fs, xs axs ds, t ∈ 0, T . 2.22 0 It is clear that the operator F is well defined. The operator F can be written in the form F F1 F2 , where the operators Fi , i 1, 2 are defined as follows: F1 xt Y t gx x0 , F2 xt t t ∈ 0, T , Zt − s fs, xs axs ds, t ∈ 0, T . 2.23 0 The following facts will be used in the proof. 1 ∞ ξq σdσ 1, 2.24 Y t ≤ Const; 2.25 0 which implies that 2 ∞ σ ν ξq σdσ 0 ∞ 0 1 Γ1 ν q σdσ , σ qν Γ 1 qν ν ∈ 0, 1, 2.26 which implies that qM q−1 t , Zt ≤ Γ 1q t > 0. 2.27 8 Journal of Applied Mathematics Let {xn }n∈N ⊂ C0, T , X such that lim xn − x0,T 0, n→∞ 2.28 for an x ∈ C0, T , X. Then by the assumptions, we know that for almost every t ∈ 0, T and t, s ∈ Δ: lim ft, xn t ft, xt, n→∞ lim ht, s, xn s ht, s, xs. 2.29 n→∞ Therefore, for sufficiently large n, we have ft, xn t − ft, xt ≤ μt 1 2x 0,T , ht, s, xn s − ht, s, xs ≤ mt, s 1 2x0,T , t t kt, sht, s, xn sds − kt, sht, s, xsds ≤ m∗ k∗ 1 2x0,T , 0 0 2.30 where k∗ : sup kt. t∈0,T 2.31 Hence, t t lim kt, sht, s, xn sds − kt, sht, s, xsds 0. n → ∞ 0 0 2.32 Thus, F2 xn − F2 x0,T −→ 0, as n −→ ∞, since 2.27 implies that s t Zt − s fs, xn s ks, τhs, τ, xn τdτ 0 0 s − fs, xs ks, τhs, τ, xτdτ ds 0 2.33 Journal of Applied Mathematics qM ≤ Γ 1q t 9 t − sq−1 fs, xn s − fs, xs 0 s ds. ks, τhs, τ, x − hs, τ, xτdτ τ n 0 2.34 By 2.33 and our assumptions, we see that F is continuous. Since χ is the Hausdorff measure of noncompactness in X, we know that χ is monotone, nonsingular, invariant with respect to union with compact sets, algebraically semiadditive, and regular. This means that i for any Ω0 , Ω1 ∈ 2E with Ω0 ⊂ Ω1 , χΩ0 ≤ χΩ1 ; 2.35 χ{a0 } ∪ Ω χΩ; 2.36 ii for every a0 ∈ E, Ω ∈ 2E , iii for every relatively compact set D ⊂ E, Ω ∈ 2E , χ{D} ∪ Ω χΩ; 2.37 χΩ0 Ω1 ≤ χΩ0 χΩ1 ; 2.38 iv for each Ω0 , Ω1 ∈ 2E , v χΩ 0 is equivalent to the relative compactness of Ω. Noting that for any ψ ∈ L1 0, T , X, we have t lim sup L → ∞ t∈0,T e−Lt−s ψsds 0. 2.39 0 So, there exists a positive constant L such that qM sup Γ 1 q t∈0,T t t 0 L1 < 0 qMk∗ ζ∗ sup Γ 1 q t∈0,T qM sup Γ 1 q t∈0,T t − sq−1 ηse−Lt−s ds t t − sq−1 e−Lt−s ds L2 < 0 t − sq−1 μs m∗ k∗ e−Lt−s ds 1 , 3 1 , 3 L3 < 2.40 1 . 3 10 Journal of Applied Mathematics For every bounded subset Ω ⊂ C0, T , X, we define mod c Ω : lim sup max vt1 − vt2 , δ → 0 v∈Ω |t1 −t2 |≤δ ΨΩ : sup e−Lt χΩt , 2.41 t∈0,T αΩ : ΨΩ, mod c Ω. Then mod c Ω is the module of equicontinuity of Ω, and α is a measure of noncompactness in the space C0, T , X with values in the cone R2 . Let Ω ⊂ C0, T , X be a nonempty, bounded set such that αFΩ ≥ αΩ. 2.42 By the assumptions and the continuity of T t in the uniform operator topology for t > 0, we get mod c F1 Ω 0. 2.43 Clearly, fs, xs axs ≤ μs m∗ k∗ x 2.44 0,T . Let δ > 0, t1 , t2 ∈ 0, T such that 0 < t1 − t2 ≤ δ and x ∈ Ω. Then t2 t1 Zt2 − s fs, xs axs ds Zt1 − s fs, xs axs ds − 0 0 ≤ x0,T t 2 Zt1 − s − Zt2 − s μs m∗ k∗ ds 0 t1 ∗ ∗ Zt1 − s μs m k ds t2 t ∞ 2 ≤ qx0,T t1 − sq−1 − t2 − sq−1 σξq σ T t1 − sq σ μs m∗ k∗ dσ ds 0 0 t2 ∞ 0 t2 − sq−1 σξq σ T t1 − sq σ − T t2 − sq σ 0 qMx0,T t1 × μs m k dσ ds t1 − sq−1 μs m∗ k∗ ds Γ 1q t2 ∗ ∗ Journal of Applied Mathematics qMx0,T t2 ≤ t1 − sq−1 − t2 − sq−1 μs m∗ k∗ dσ ds Γ 1q 0 t1 q−1 ∗ ∗ μs m k ds t1 − s 11 t2 t2 ∞ 0 t2 − sq−1 σξq σ T t1 − sq σ − T t2 − sq σ μs m∗ k∗ dσ ds. 0 2.45 It is not hard to see that the right-hand side of 2.45 tend to 0 as t2 → t1 . Thus, the set {F2 x· : x ∈ Ω} is equicontinuous, then mod c F2 Ω 0. Combining with 2.43, we have mod c FΩ 0, which implies mod c Ω 0 from 2.42. Next, we show that ΨΩ 0. It is easy to see that ΨF1 Ω 0. 2.46 For any t ∈ 0, T , we define 2 Ωt : F t Zt − sfs, xsds : x ∈ Ω . 2.47 0 We consider the multifunction s ∈ 0, t Gs: Gs Zt − sfs, xs : x ∈ Ω . 2.48 Obviously, G is integrable, that is, G admits a Bochner integrable selection g : 0, h → E, and gt ∈ Gt, for a.e. t ∈ 0, h. 2.49 From 2.27 and our assumptions, it follows that G is integrably bounded, that is, there exists a function ∈ L1 0, h, E such that Gt : sup{g : g ∈ Gt} ≤ t, a.e. t ∈ 0, h. 2.50 12 Journal of Applied Mathematics Moreover, we have the following estimate for a.e. s ∈ 0, t: qM χGs ≤ t − sq−1 χ fs, Ωs Γ 1q qM ≤ t − sq−1 ηsχΩs Γ 1q qM t − sq−1 ηseLs e−Ls χΩs Γ 1q 2.51 qM ≤ t − sq−1 ηseLs ΨΩ. Γ 1q Therefore, since X is a separable Banach space, we know by 20, Theorem 4.2.3 that t t qM 2 Ωt χ χ F Gsds ≤ t − sq−1 ηseLs ds · ΨΩ. Γ 1q 0 0 2.52 So sup e −Lt t∈0,T t qM χ F2 Ωt ≤ sup t − sq−1 ηse−Lt−s ds · ΨΩ Γ 1 q t∈0,T 0 2.53 L1 ΨΩ. Similarly, if we set 2 Ωt F t Zt − saxsds : x ∈ Ω , 2.54 0 then we see that the multifunction s ∈ 0, t Gs, Gs {Zt − saxs : x ∈ Ω} 2.55 is integrable and integrably bounded. Thus, we obtain the following estimate for a.e. s ∈ 0, t: qMk∗ ζ∗ ≤ χ Gs t − sq−1 eLs ΨΩ, Γ 1q sup e t∈0,T −Lt t qMk∗ ζ∗ χ F2 Ωt ≤ sup t − sq−1 e−Lt−s ds · ΨΩ Γ 1 q t∈0,T 0 L2 ΨΩ. 2.56 Journal of Applied Mathematics 13 Now, from 2.53 and 2.56, it follows that ΨFΩ ≤ ΨF1 Ω ΨF2 Ω ≤ L1 L2 ΨΩ LΨΩ, 2.57 < 1. Then by 2.42, we get ΨΩ 0. Hence αΩ 0, 0. Thus, Ω is relatively where 0 < L compact due to the regularity property of α. This means that F is α-condensing. Let us introduce in the space C0, T , X the equivalent norm defined as x∗ sup e−Lt xt . 2.58 Br {x ∈ C0, T , X : x∗ ≤ r}. 2.59 t∈0,T Consider the set Next, we show that there exists some r > 0 such that FBr ⊂ Br . Suppose on the contrary that for each r > 0 there exist xr · ∈ Br , and some t ∈ 0, T such that Fxr t∗ > r. From the assumptions, we have F1 xr t∗ ≤ Mb x0 . 2.60 Moreover, F2 xr t ≤ t Zt − s fs, xr s axr s ds 0 qM ≤ Γ 1q qM Γ 1q t 0 t − sq−1 μsxr s m∗ k∗ eLs xr ∗ ds t t − sq−1 μseLs e−Ls xr sds ∗ ∗ m k t q−1 Ls e ds · xr ∗ t − s 0 qMr ≤ Γ 1q 2.61 0 t t − sq−1 μs m∗ k∗ eLs ds. 0 Therefore, r < sup e−Lt Fxr t t∈0,T qMr ≤ Mb x0 sup Γ 1 q t∈0,T t 0 t − s q−1 ∗ ∗ μs m k e 2.62 −Lt−s ds. 14 Journal of Applied Mathematics Dividing both sides of 2.62 by r, and taking r → ∞, we have qM sup Γ 1 q t∈0,T t t − sq−1 μs m∗ k∗ e−Lt−s ds ≥ 1. 2.63 0 This is a contradiction. Hence for some positive number r, FBr ⊂ Br . According to the following known fact. Let M be a bounded convex closed subset of E and F : M → M a α-condensing map. Then Fix F {x : x Fx} is nonempty. we see that problem 1.2 has at least one mild solution. Next, for c ∈ 0, 1, we consider the following one-parameter family of maps: H : 0, 1 × C0, T , X −→ C0, T , X 2.64 c, x −→ Hc, x cFx. We will demonstrate that the fixed point set of the family H, 2.65 Fix H {x ∈ Hc, x for some c ∈ 0, 1} is a priori bounded. Indeed, let x ∈ Fix H, for t ∈ 0, T , we have xt ≤ M gx x0 t Zt − s fs, xs axs ds 0 qM ≤ Mb x0 Γ 1q t t − sq−1 μsxsds 2.66 0 ∗ ∗ m k t t − s 0 q−1 sup xτds . τ∈0,s Noting that the Hölder inequality, we have t t − sq−1 μsds ≤ 0 ≤ p−1 pq − 1 p−1 pq − 1 p−1/p p−1/p · tpq−1/p · μ Lp · T q−1/p · μ Lp . 2.67 Therefore, from 2.66, we obtain p − 1 p−1/p q−1/p qM μ p sup xs T xt ≤ Mb x0 L pq − 1 Γ 1q s∈0,t qMm∗ k∗ Γ 1q t 0 t − sq−1 sup xτds. τ∈0,s 2.68 Journal of Applied Mathematics 15 We denote yt : sup xs. 2.69 s∈0,t Let t ∈ 0, t such that yt xt. Then, by 2.68, we can see p − 1 p−1/p q−1/p qM μ p yt T yt ≤ Mb x0 L Γ 1 q pq − 1 qMm∗ k∗ t t − sq−1 ysds. Γ 1q 0 2.70 By a generalization of Gronwall’s lemma for singular kernels 22, Lemma 7.1.1, we deduce that there exists a constant κ κq such that yt ≤ ≤ Mb x0 p−1/p q−1/p μ p 1 − qM/Γ 1 q p − 1 / pq − 1 T L t ∗ ∗ κMb x0 qMm k /Γ 1 q q−1 p−1/p q−1/p 2 0 t − s ds μ p 1 − qM/Γ 1 q p − 1 / pq − 1 T L Mb x0 p−1/p q−1/p μ p 1 − qM/Γ 1 q p − 1 / pq − 1 T L q ∗ ∗ κMb x0 qMm k /Γ 1 q T p−1/p q−1/p 2 μ p q 1 − qM/Γ 1 q p − 1 / pq − 1 T L 2.71 : w. Hence, supt∈0,T xt ≤ w. Now we consider a closed ball: BR x ∈ C0, T , X : x0,T ≤ R ⊂ C0, T , X. 2.72 We take the radius R > 0 large enough to contain the set Fix H inside itself. Moreover, from the proof above, F : BR → C0, T , X is α-condensing. Consequently, the following known fact implies our conclusion: Let V ⊂ E be a bounded open neighborhood of zero and F : V → E a α-condensing map satisfying the boundary condition: x / λFx, for all x ∈ ∂V and 0 < λ ≤ 1. Then, Fix F is nonempty compact. 2.73 16 Journal of Applied Mathematics 3. Example In this section, let X L2 0, π, we consider the following nonlocal Cauchy problem for an integrodifferential problem: q ∂t ut, ξ ∂2 1 ut, ξ ut, ξ √ · k 2 ∂ξ k t 1 ut, ξ t √ t − s sin 0 ut, 0 ut, π 0, s · us, ξ ds, t t ∈ 0, 1 t ∈ 0, 1 3.1 j π u ti , y u0, ξ ci ξ, y dy u0 ξ, 1 u ti , y i0 0 q where ∂t is the Caputo fractional partial derivative of order 0 < q < 1; ξ ∈ 0, π; k > 0 is a constant to be specified later; u0 ξ ∈ X; j ∈ N ; 0 < t0 < t1 < · · · < tj < 1; 3.2 ci ·, · i 0, 1, . . . , j are continuous functions and there exists a positive constant b such that j π ci ξ, y dy ≤ b. i0 3.3 0 For t ∈ 0, 1, ξ ∈ 0, π, we set xtξ ut, ξ, j π xti y gxξ ci ξ, y dy, 1 xti y 0 i0 kt, s t − s, √ s · xsξ ht, s, xsξ sin , t 3.4 1 xtξ . ft, xtξ √ · k k t 1 xtξ On the other hand, it is known that the operator A Au u with DA H 2 0, π ∩ H01 0, π generates an analytic semigroup and uniformly bounded semigroup {T t}t≥0 on X with T tLX ≤ 1. Therefore, 3.1 is a special case of 1.2. Moreover, we have 1 for all t ∈ 0, 1, 1 ft, x ≤ √ x : μtx; kkt 3.5 Journal of Applied Mathematics 17 2 for any w, w ∈ X, 1 ft, w − ft, w ≤ √ w − w, kkt 3.6 that is, for any bounded set D ⊂ X, 1 χ ft, D ≤ √ χD, kkt 3.7 √ s · xsξ ≤ mt, sx, sin ht, s, x t 3.8 for a.e. t ∈ 0, 1; 3 for almost all t, s ∈ Δ, where mt, s : √ s/t, and m∗ sup t t∈0,1 t √ mt, sds sup 0 0 t∈0,1 t s ds 2 ; 3 3.9 4 √ ≤ ht, s, w − ht, s, w t s w − w, 3.10 that is, for any bounded set D ⊂ X, χht, s, D ≤ ζt, sχD, where ζt, s : √ 3.11 s/t, and t sup t∈0,1 0 ζt, sds 2 . 3 3.12 Therefore, Theorem 2.4 implies that the problem 3.1 has at least a mild solution when q· p−1/p p − 1 / pq − 1 μ p < 1. L Γ 1q 3.13 18 Journal of Applied Mathematics Acknowledgment The work was supported partly by the Chinese Academy of Sciences, the NSF of Yunnan Province 2009ZC054M and the NSF of China 11171210. References 1 S. Aizicovici and M. 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