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1 Comuers and Mahemaics wih Alicaions 6 (11) Conens liss available a ScienceDirec Comuers and Mahemaics wih Alicaions journal homeage: Nonlocal roblems for fracional inegrodifferenial equaions via fracional oeraors and oimal conrols JinRong Wang a,, Yong Zhou b, Wei Wei a, Honglei Xu a,c a Dearmen of Mahemaics, Guizhou Universiy, Guiyang, Guizhou 555, PR China b Dearmen of Mahemaics, Xiangan Universiy, Xiangan, Hunan 41115, PR China c Dearmen of Mahemaics and Saisics, Curin Universiy of Technology, Perh, WA 6845, Ausralia a r i c l e i n f o a b s r a c Keywor: Nonlocal roblems Fracional inegrodifferenial equaions Mild soluions Exisence Oimal conrols This aer invesigaes nonlocal roblems for a class of fracional inegrodifferenial equaions via fracional oeraors and oimal conrols in Banach saces. By using he fracional calculus, Hölder inequaliy, -mean coninuiy and fixed oin heorems, some exisence resuls of mild soluions are obained under he wo cases of he semigrou T(), he nonlinear erms f and h, and he nonlocal iem g. Then, he exisence condiions of oimal airs of sysems governed by a fracional inegrodifferenial equaion wih nonlocal condiions are resened. Finally, an examle is given o illusrae he effeciveness of he resuls obained. 11 Elsevier Ld. All righs reserved. 1. Inroducion During he as wo decades, fracional differenial equaions have araced much aenion (see for insance [1 4]). This is mosly because hey efficienly describe many racical dynamical henomena arising in engineering, hysics, economy and science. In aricular, we can find numerous alicaions in viscoelasiciy, elecrochemisry, conrol, orous media, elecromagneic (see for insance [5 1]). There has been a significan develomen in nonlocal roblems for fracional differenial equaions or inclusions (see for insance [11 5]). Very recenly, Chang [6] invesigae he fracional order inegrodifferenial equaions wih nonlocal condiions in he Riemann Liouville fracional derivaive sense. Meanwhile, Wang and Zhou [7] considered a class of fracional evoluion equaions and oimal conrols in he Cauo derivaive sense and obained some ineresing resuls. However, o he bes of our knowledge, he nonlocal roblems for fracional inegrodifferenial equaions in he Cauo derivaive sense have no been discussed exensively. Esecially, few auhors discuss he oimal conrol roblems of sysems governed by he fracional inegrodifferenial equaions wih nonlocal condiions in infinie dimensional saces. As we all know, he main difficuly o sudy he fracional evoluion equaions is how o obain he suiable fracional resolven family generaed by he infiniesimal generaor A in Banach sace. In order o solve his roblem, some auhors inroduced an -resolven family under he Riemann Liouville fracional derivaive and some consrains, see, for examle, [8 31], and he ohers inroduced suiable oeraor families wih he Cauo fracional derivaive in erms of some robabiliy densiy funcions and oeraor semigrou [4,5]. For he laer, a ioneering work has been reored by El-Borai [3,33]. This work is suored by Tianyuan Secial Fun of he Naional Naural Science Foundaion of China (1161), Naional Naural Science Foundaion of China ( , 19619), Naional Naural Science Foundaion of Guizhou Province (1, No. 14). Corresonding auhor. Tel.: ; fax: addresses: wjr9668@16.com (J. Wang), yzhou@xu.edu.cn (Y. Zhou), wwei@gzu.edu.cn (W. Wei), h.xu@curin.edu.an (H. Xu) /$ see fron maer 11 Elsevier Ld. All righs reserved. doi:1.116/j.camwa.11..4

2 148 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) Here, moivaed by [1,3 7,34 4], he main urose of his aer is o consider he following fracional nonlinear inegrodifferenial evoluion equaions wih nonlocal iniial condiions: C D q x() = Ax() + n f (, x(), (Hx)()), J, n Z +, q (, 1), (1) x() = g(x) + x X, where C D q denoes Cauo derivaive, A : D(A) X is he infiniesimal generaor of an analyic semigrou {T(), }, he oeraor H is defined by (Hx)() = T h(, s, x(s)), he nonlinear erm f : J X X X(or X ) is a given funcion where J = [, T], X = D(A ) ( < < 1) is a Banach sace wih he norm x = A x for x X. The funcions f, h and g will be secified laer. In he resen aer, using he same idea in our revious resul [4], we will inroduce suiable mild soluions for sysem (1). Then we will rove he exisence resuls of mild soluions for sysem (1). Our resuls will cover he cases for he nonlineariy f aking values in X or X, f, h are Lischiz coninuous or no Lischiz coninuous, and he nonlocal erm g saisfies he Lischiz coninuousness or comleely coninuousness. The main echniques used here are fracional calculaion, Hölder inequaliy, -mean coninuiy via Banach conracion rincile and Schauder s fixed oin heorem for comac mas. Furhermore, we will consider he Bolza roblem of sysems governed by (1) and he exisence resul of oimal conrols will be resened. The res of his aer is organized as follows. In Secion, we give some reliminary resuls on he fracion owers of he generaor of an analyic comac and inroduce he mild soluions for sysem (1). In Secion 3, we sudy he exisence of mild soluions for sysem (1). In Secion 4, we inroduce a class of admissible conrols and rove an exisence resul of oimal conrols for he Bolza roblem (P). A las, an examle is given o demonsrae he alicabiliy of our resuls.. Preliminaries We denoe by X a Banach sace wih he norm and A : D(A) X is he infiniesimal generaor of an analyic comac semigrou of uniformly bounded linear oeraors {T(), }. This means ha here exiss a M > 1 such ha T() M. We assume wihou loss of generaliy ha ρ(a). This allows us o define he fracional ower A for < < 1, as a closed linear oeraor on is domain D(A ) wih inverse A (see [41]). We will inroduce he following basic roeries of A. Theorem.1 ([41], ). (1) X = D(A ) is a Banach sace wih he norm x = A x for x X. () T() : X X for each >. (3) A T()x = T()A x for each x X and. (4) For every >, A T() is bounded on X and here exis M > and ν > such ha A T() M e ν M. (5) A is a bounded linear oeraor in X wih X = Im(A ). (6) If < µ < 1, hen D(A µ ) D(A ). In wha follows, we also use f L (J,R + ) o denoe he L (J, R + ) norm of f whenever f L (J, R + ) for some wih 1 < <. We se (, 1) and denoe by C, he Banach sace C(J, X ) endowed wih sunorm given by x su J x, for x C. Le us recall he following known definiions. For more deails, see [1]. Definiion.. The fracional inegral of order γ wih he lower limi zero for a funcion f is defined as I γ f () = 1 Ɣ(γ ) f (s), >, γ >, ( s) 1 γ rovided he righ side is oin-wise defined on [, ), where Ɣ( ) is he gamma funcion. Definiion.3. The Riemann Liouville derivaive of order γ wih he lower limi zero for a funcion f : [, ) R can be wrien as L D γ 1 f () = Ɣ(n γ ) d n d n f (s), >, n 1 < γ < n. ( s) γ +1 n Definiion.4. The Cauo derivaive of order γ for a funcion f : [, ) R can be wrien as n 1 C D γ f () = L D γ k f () k! f (k) (), >, n 1 < γ < n. k=

3 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) Remark.5. (1) If f () C n [, ), hen C D γ 1 f () = Ɣ(n γ ) f (n) (s) ( s) γ +1 n = In γ f (n) (), >, n 1 < γ < n. () The Cauo derivaive of a consan is equal o zero. (3) If f is an absrac funcion wih values in X, hen inegrals which aear in Definiions. and.3 are aken in Bochner s sense. We need our revious work ([4], Lemma 3.1 and Definiion 3.1) and he following definiion of mild soluions. Definiion.6. By he mild soluions of sysem (1), we mean ha he funcion x : J X which saisfies where x() = T ()[x + g(x)] + T () = ξ q (θ)t( q θ)dθ, ξ q (θ) = 1 q θ 1 1 q ϖq θ 1 q, ϖ q (θ) = 1 π ( s) q 1 s n S ( s)f (s, x(s), (Hx)(s)), J, S () = q ( 1) n 1 qn 1 Ɣ(nq + 1) θ n! n=1 sin(nπq), ξ q is a robabiliy densiy funcion defined on (, ), ha is ξ q (θ), θ (, ) and ξ q (θ)dθ = 1. Remark.7. I is no difficul o verify ha for v [, 1], θ v ξ q (θ)dθ = θ qv ϖ q (θ)dθ = Ɣ(1 + v) Ɣ(1 + qv). θξ q (θ)t( q θ)dθ, θ (, ), The following resuls are very useful and will be used hroughou his aer. Lemma.8 (Lemma.9, [7]). The oeraors T and S have he following roeries: (1) For fixed, T () and S () are linear and bounded oeraors. For any x X, T ()x M x and S ()x x. () {T (), } and {S (), } are srongly coninuous. (3) For every >, T () and S () are also comac oeraors. (4) For any x X, β (, 1) and (, 1), we have AS ()x = A 1 β S ()A β x, J, A S () M qɣ( ) Ɣ(1 + q(1 )) q, < T. (5) For fixed and any x X, we have T ()x M x and S ()x x. (6) T () and S (), > are uniformly coninuous, ha is for each fixed >, and ϵ >, here exiss h > such ha where T ( + ϵ) T () < ε, S ( + ϵ) S () < ε, T () = ξ q (θ)t ( q θ)dθ, for + ϵ and ϵ < h for + ϵ and ϵ < h S () = q θξ q (θ)t ( q θ)dθ. Lemma.9 (-Mean Coninuiy, [4]). For each ψ L (J, X) wih 1 < +, we have lim h T ψ(+h) ψ() d = where ψ(s) = for s does no belong o J.

4 143 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) To end his secion, we recall he following known resuls. Lemma.1 (Bochner s Theorem). A measurable funcion V : J X is Bochner inegral if V is Lebesgue inegrable. Lemma.11 (Schauder Fixed Poin Theorem). If B is a closed bounded and convex subse of a Banach sace X and F : B B is comleely coninuous, hen F has a fixed oin in B. 3. Exisence of mild soluions 3.1. The case ha T() is no comac, f is in X, and f, h and g are Lischiz coninuous We firs make he following assumions. [Hf1]: f : J X X X is coninuous and here exis m 1, m > such ha f (, x 1, x ) f (, y 1, y ) m 1 x 1 y 1 + m x y for all x i, y i X, i = 1, and J. [Hh1]: Le D h = {(, s) R ; s, T}, where h : D h X X. Then, here exis a m h (, s) C(D h, R + ) and such ha H = su J m h (, s) < h(, s, x) h(, s, y) m h (, s) x y for each (, s) D h and x, y X. [Hg1]: g : C X is coninuous and here exiss a consan l g > such ha g(x) g(y) l g x y, for arbirary x, y C. [HΩ]: The funcion Ω n,,q, : J R +, n Z +, 1 < q < 1 wih some 1 < < defined by Ω n,,q, () = Ml g + M qɣ( )(m 1 + H [ ] 1 m ) 1 n+q q, Ɣ(1 + q(1 )) + (q 1) 1 saisfies < Ω n,,q, () ω < 1, for all J. Now we are ready o give our firs resul which are based on he Banach conracion maing rincile. Theorem 3.1. Assume ha [Hf1], [Hh1], [Hg1] and [HΩ] are saisfied. If x X, hen sysem (1) has a unique mild soluion x C. Proof. Define he funcion Ɣ : C C by (Ɣx)() = T ()[x + g(x)] + Noe ha Ɣ is well defined on C. Now, ake J and x, y C. Then we have (Ɣx)() (Ɣy)() T ()[g(x) g(y)] + M g(x) g(y) + ( s) q 1 s n S ( s)f (s, x(s), (Hx)(s)), J. () ( s) q 1 s n S ( s)[f (s, x(s), (Hx)()) f (s, y(s), (Hy)(s))] ( s) q 1 s n A S ( s) f (s, x(s), (Hx)(s)) f (s, y(s), (Hy)(s)), which according o [Hf1], [Hh1], [Hg1], (4) (5) of Lemma.8 and Hölder inequaliy, lea o (Ɣx)() (Ɣy)() Ml g x y + q M qɣ( ) Ɣ(1 + q(1 )) m 1 x(s) y(s) ( s) q 1 s n + q M qɣ( ) Ɣ(1 + q(1 )) m (Hx)(s) (Hy)(s) ( s) q 1 s n Ml g x y + q M qɣ( ) m 1 x y + m Ɣ(1 + q(1 ))

5 Therefore, we can deduce ha Ɣx Ɣy J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) T 1 m h (s, τ) x(τ) y(τ) dτ ( s) (q 1) 1 1 s n Ml g x y + M qɣ( ) Ɣ(1 + q(1 )) q (m 1 + H 1 m ) + (q 1) 1 Ml g + M qɣ( )(m 1 + H m ) Ɣ(1 + q(1 )) Ω n,,q, (T) x y (q 1) (q 1) 1 n+ 1 x y. 1 + n 1 n+q q x y Hence, [HΩ] allows us o conclude in view of he conracion maing rincile, ha Ɣ has a unique fixed oin x C, and x() = T ()[x + g(x)] + is he unique mild soluion of sysem (1). ( s) q 1 s n S ( s)f (s, x(s), (Hx)(s)) 3.. The case ha T() is comac, f is in X, and f, h and g are no Lischiz coninuous We assume he following condiions. [Hf]: f : J X X X is Carahéodory funcion and here exiss a osiive funcion ρ L (J, R + ) for some wih 1 < < such ha f (, x, y) ρ() for all x, y X and J. [Hh]: The funcion h : D h X X is coninuous and here exis L 3, L 4 > such ha h(, s, x) h(, s, y) L 3 x y + L 4 for each (, s) D h and x, y X. [Hg]: g : C X is comleely coninuous and here exis β 1, β > such ha g(x) β 1 x + β. Now we are ready o sae our second exisence resul which are based on he well known Schauder s fixed oin heorem. Theorem 3.. Assume ha he condiions [Hf], [Hh], [Hg] are saisfied. If x X, 1 < +1 1 < <, hen sysem (1) has a leas one mild soluion on J rovided ha Mβ 1 < 1. < q < 1 for some Proof. Define he funcion F : C C by (Fx)() = T ()[x + g(x)] + ( s) q 1 s n S ( s)f (s, x(s), (Hx)(s)), and for n Z +, we choose r such ha 1 M( x + β ) + 1 T n+q 1 ρ L Ɣ(1+q) +(q 1) 1 (J,R + ) r. 1 Mβ 1 Le B r = {x C x r}. Then, we will roceed in hree ses. Se 1. We show ha FB r B r.

6 143 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) Le x B r. Then for J, using (5) of Lemma.8 and he Hölder inequaliy, we have (Fx)() T ()[x + g(x)] + M( x + g(x) ) + ( s) q 1 s n S ( s)f (s, x(s), (Hx)(s)) ( s) q 1 s n f (s, x(s), (Hx)(s)), which, according o [Hf], [Hg] and 1 < q( + (q 1) 1) >, gives (Fx)() M( x + β 1 x + β ) + ( s) q 1 s n ρ(s) M( x + β 1 x + β ) + 1 ( s) (q 1) 1 (s n ρ(s)) M( x + β 1 x + β ) (q 1) 1 s n (ρ(s)) + (q 1) 1 M( x + β 1 x + β ) (q 1) 1 1 n ρ + (q 1) 1 n + 1 L (J,R + ) M( x + β 1 x + β ) (q 1) 1 M( x + β 1 x + β ) (q 1) 1 r 1 1 for J. Hence, we deduce ha Fx r. Se. We rove ha F is coninuous. Le {x m } be a sequence of B r such ha x m x in B r. I comes from [Hh] and ha h(s, τ, x m (τ)) h(s, τ, x(τ)), h(s, τ, x m (τ)) h(s, τ, x(τ)) L 3 x m (τ) x(τ) + L 4 L 3 r + L 4. By means of he Lebesgue dominaed convergence heorem, one can rove ha Then, h(s, τ, x m (τ))dτ h(s, τ, x(τ))dτ, as m. 1 +(q 1) 1 n+ ρ L (J,R + ) n+q 1 ρ L (J,R + ) f (s, x m (s), (Hx m )(s)) f (s, x(s), (Hx)(s)) as m (3) because he funcion f is coninuous on J X X. Now for J, according o [Hf], [Hg], (5) of Lemma.8 and he Hölder inequaliy, we have (Fx m )() (Fx)() T ()[g(x m ) + g(x)] + ( s) q 1 s n S ( s)[f (s, x m (s), (Hx m )(s)) f (s, x(s), (Hx)(s))] M g(x m ) g(x) + ( s) q 1 s n f (s, x m (s), (Hx m )(s)) f (s, x(s), (Hx)(s)) M g(x m ) g(x) + 1 ( s) (q 1) 1 s n f (s, x m (s), (Hx m )(s)) f (s, x(s), (Hx)(s)) 1

7 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) M g(x m ) g(x) (q 1) 1 M g(x m ) g(x) + 1 +(q 1) n s n 1 1 f (s, x m (s), (Hx m )(s)) f (s, x(s), (Hx)(s)) 1 + (q 1) 1 1 +(q 1) 1 n+ 1 f (s, x m (s), (Hx m )(s)) f (s, x(s), (Hx)(s)) M g(x m ) g(x) (q 1) n f (s, x m (s), (Hx m )(s)) f (s, x(s), (Hx)(s)) 1. 1 n T n+q n n Obviously, i is clear ha g(x m ) g(x) as m since g is comleely coninuous on C. Moreover, by virue of (3) and Lebesgue s dominaed convergence heorem, f (s, x m (s), (Hx m )(s)) f (s, x(s), (Hx)(s)), as m. Therefore, i is easy o see ha lim Fx m Fx =, as m. m Tha is, F is coninuous. Se 3. We show ha F is comac. To his end, we use he famous Ascoli Arzela s heorem. We firs rove ha {(Fx)() x B r } is relaively comac in X, for all J. Obviously, {(Fx)() x B r } is comac. Le (, T], and for each h (, ), arbirary δ > and x B r, define he oeraor F h,δ by (F h,δ x)() = T ()[x + g(x)] + T(h q δ) = T ()[x + g(x)] + q h h δ ( s) q 1 s n q δ θξ q (θ)t(( s) q θ h q δ)dθ θ( s) q 1 s n ξ q (θ)t(( s) q θ)f (s, x(s), (Hx)(s)) dθ. f (s, x(s), (Hx)(s)) Then he ses {(F h,δ x)() x B r } are relaively comac in X since by Lemma 3.3 of [1], he oeraors T (h q δ), h q δ > are comac in X. Moreover, using [Hf] and he Hölder inequaliy, we have δ (Fx)() (F h,δ x)() q θ( s) q 1 s n ξ q (θ)t(( s) q θ)f (s, x(s), (Hx)(s)) dθ + q θ( s) q 1 s n ξ q (θ)t(( s) q θ)f (s, x(s), (Hx)(s)) dθ δ h θ( s) q 1 s n ξ q (θ)t(( s) q θ)f (s, x(s), (Hx)(s)) dθ δ δ q + q h δ θ( s) q 1 s n ξ q (θ) T(( s) q θ)f (s, x(s), (Hx)(s)) dθ θ( s) q 1 s n ξ q (θ) T(( s) q θ)f (s, x(s), (Hx)(s)) dθ

8 1434 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) Using he fac ha δ θ( s) q 1 s n ξ q (θ)ρ(s)dθ + h δ θ( s) q 1 s n ξ q (θ)ρ(s)dθ δ ( s) q 1 s n ρ(s) θξ q (θ)dθ + ( s) q 1 s n ρ(s) θξ q (θ)dθ h δ ( s) q 1 s n ρ(s) θξ q (θ)dθ + ( s) q 1 s n ρ(s). h 1 ( s) q 1 s n ρ(s) ( s) (q 1) 1 (s n ρ(s)) 1 + (q 1) (q 1) (q 1) (q 1) 1 +(q 1) 1 ρ L (J,R + ) 1 s n (ρ(s)) n 1 1 +(q 1) n 1 +n ρ L (J,R + ) and h ( s) q 1 s n ρ(s) 1 + (q 1) 1 1 ρ L (J,R + ) h +(q 1) n, we obain 1 1 (Fx)() (F h,δ x)() ρ + (q 1) 1 L (J,R + ) ρ + (q 1) 1 L (J,R + h ) +(q 1) δ +n θξ q (θ)dθ +(q 1) n. Therefore, here are relaively comac ses {(F h,δ x)() x B r } arbirarily close o he se {(Fx)() x B r } for (, T]. Hence, {(Fx)() x B r } is relaively comac in X for all (, T] and since i is comac a = we have he relaively comacness in X for all J. Nex, le us rove ha F(B r ) is equiconinuous. By he comacness of he se g(b r ), we can rove ha he funcions Fx, x B r are equiconinuous a =. For < s < < 1 T, we have Denoe (Fx)( 1 ) (Fx)( ) (T ( 1 ) T ( ))[x + g(x)] + ( 1 s) q 1 s n [S ( 1 s) S ( s)]f (s, x(s), (Hx)(s)) + [( 1 s) q 1 ( s) q 1 ]s n S ( s)f (s, x(s), (Hx)(s)) 1 + ( 1 s) q 1 s n S ( 1 s)f (s, x(s), (Hx)(s)). I 1 = (T ( 1 ) T ( ))[x + g(x)], I = ( 1 s) q 1 s n [S ( 1 s) S ( s)]f (s, x(s), (Hx)(s)), I 3 = [( 1 s) q 1 ( s) q 1 ]s n S ( s)f (s, x(s), (Hx)(s)),

9 I 4 = 1 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) ( 1 s) q 1 s n S ( 1 s)f (s, x(s), (Hx)(s)). Now, we need o rove ha I 1, I, I 3, I 4 end o indeendenly of x B r when 1. In fac, leing x B r, one can deduce ha I 1 (T ( 1 ) T ( ))[x + g(x)] T ( 1 ) T ( ) ( x + su x B r g(x) ), from which we deduce ha lim 1 I 1 = by (6) of Lemma.8. Afer some calculaion, we ge I ( 1 s) q 1 s n (S ( 1 s) S ( s))f (s, x(s), (Hx)(s)) Ɣ() Ɣ() Ɣ() Ɣ() M 1+ M 1+ M 1+ ( 1 s) q 1 s n ρ(s)[( 1 s) q ( s) q ] 1 [s n ρ(s)] 1 1 ( 1 s) (q 1) [( 1 s) q ( s) q ] n ( 1) ( 1 s) (q 1) 1 M 1+ 1 (q 1) n ( 1) ( 1) T ( 1) +n ρ L 1 (J,R + ) [( 1 s) q ( s) q ] ( 1) (q 1) ( 1 ) T ( 1) +n ρ L 1 (J,R + ) (q 1) [( 1 s) q ( s) q ] Using Lagrange mean value heorem, one can obain ( 1 s) q ( s) q as 1, for s J. By Lemma.9, we can deduce [( 1 s) q ( s) q ] as 1. Thus, we deduce ha lim 1 I = by (6) of Lemma.8 again. Also, noe ha I 3 [( 1 s) q 1 ( s) q 1 ]s n f (s, x(s), (Hx)(s)) [( 1 s) q 1 ( s) q 1 ]s n ρ(s) 1 [s n ρ(s)] 1 1 [( 1 s) q 1 ( s) q 1 ] n ( 1) ( 1) T 1. ( 1) T 1 +n ρ [( 1 s) q 1 ( s) q 1 ]. L 1 (J,R + ) By Lagrange mean value heorem and Lemma.9 again, we have lim 1 I 3 =. From he following inequaliy I 4 1 ( 1 s) q 1 s n T( 1 s)f (s, x(s), (Hx)(s))

10 1436 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) ( 1 s) q 1 s n ρ(s) ( 1 s) (q 1) ( 1 s) (q 1) 1 + (q 1) (q 1) s n (ρ(s)) ( 1 ) +(q 1) 1 ρ L (J,R + ) 1 s n (ρ(s)) 1 n + 1 ( 1 ) +(q 1) 1 n + 1 κ n 1 (1 ) n + 1 1, < κ < 1, 1 n n ρ L (J,R + ) one can deduce ha lim 1 I 4 =. In summary, we have roven ha F(B r ) is relaively comac, for J, {Fx x B r } is a family of equiconinuous funcions. Hence by Arzela Ascoli s heorem, F is comac. By Schauder s fixed oin heorem F has a fixed oin x B r. Consequenly, sysem (1) has a leas one mild soluion on J. 4. Exisence of oimal conrols We suose ha Y is anoher searable reflexive Banach sace from which he conrols u ake he values. We denoe a class of nonemy closed and convex subses of Y by W f (Y). The mulifuncion ω : J W f (Y) is measurable and ω( ) E where E is bounded se of Y, he admissible conrol se U ad = Sω = {u L (E) u() ω() a.e.}, 1 < <. Then U ad Ø (Proosiion 1.7 and Lemma 3., [43]). Consider he following conrolled sysem C D q x() = Ax() + n f (, x(), (Hx)()) + C()u(), J, u U ad, n Z +, q (, 1), (4) x() = g(x) + x. Assumion [HC]: C L (J, L(Y, X )). I is easy o see ha Cu L (J, X ) for all u U ad. By Theorem 3., we have he following resul. Theorem 4.1. In addiion o he assumions of Theorem 3., suose he assumion [HC] hol. For every u U ad and q(1 ) > 1, sysem (4) has a mild soluion corresonding o u given by he soluion of he following inegral equaion x u () = T ()[x + g(x)] + ( s) q 1 s n S ( s)f (s, x(s), (Hx)(s)) + ( s) q 1 s n S ( s)c(s)u(s). Proof. Comared wih Theorem 3., he key se is o check he erm conaining conrol olicy. Consider Ϝ() = ( s) q 1 s n S ( s)c(s)u(s). Using he Hölder inequaliy, we have Ϝ() ( s) q 1 s n S ( s)c(s)u(s) ( s) q 1 s n A S ( s) C(s)u(s) M qɣ( ) C T n Ɣ(1 + q(1 )) M qɣ( ) C T n Ɣ(1 + q(1 )) M qɣ( ) C T n Ɣ(1 + q(1 )) ( s) q 1 u(s) ( s) (q 1) (q 1) u(s) Y 1 T +(q 1) 1 u L (J,Y),

11 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) where C is he norm of oeraor C in Banach sace L (J, L(Y, X )). I is easy o see ha ( ) q 1 ( ) n S ( )C( )u( ) is Lebesgue inegrable wih resec o s [, ] for all J from Lemma.11. Hence Ϝ( ) C. Using Theorem 3., one can verify i immediaely. Le x u denoe he mild soluion of sysem (4) corresonding o he conrol u U ad. We consider he Bolza roblem (P): find an oimal air (x, u ) C U ad such ha where J(x, u ) J(x u, u), for all u U ad, J(x u, u) = l(, x u (), u())d + Ψ (x u (T)). We inroduce he following assumion on l and Ψ. Assumion [HL]: [HL1] The funcional l : J X Y R { } is Borel measurable. [HL] l(,, ) is sequenially lower semiconinuous on X Y for almos all J. [HL3] l(, x, ) is convex on Y for each x X and almos all J. [HL4] There exis consans d, e >, ϕ is nonnegaive and ϕ L 1 (J, R) such ha l(, x, u) ϕ() + d x + e u Y. [HL5] The funcional Ψ : X R is coninuous and nonnegaive. In order o obain he exisence of oimal conrols, we need he following imoran lemma. Lemma 4. (Lemma 4., [7]). Oeraor Q : L (J, Y) C for some q(1 ) > 1, given by (Qu)( ) = is srongly coninuous. S ( s)c(s)u(s) Now we can give anoher main resul of his aer, he exisence of oimal conrols for Bolza roblem (P). Theorem 4.3. Le A be he infiniesimal generaor of an analyic comac semigrou {T(), }. If he assumions [Hg1], [HL] and he assumions of Theorem 4.1 hold, hen he Bolza roblem (P) admis a leas one oimal air on C U ad rovided ha Ml g < 1. Proof. If inf{j(x u, u) u U ad } = +, here is nohing o rove. So we assume ha inf{j(x u, u) u U ad } = ϵ < +. Using [HL], we have J(x u, u) ϕ()d + d x u () d + e u() Y d + Ψ (xu (T)) η >, where η > is a consan. Hence, ϵ η >. By definiion of infimum here exiss a minimizing sequence of feasible air {(x m, u m )} A ad where A ad {(x, u) x is a mild soluion of sysem (4) corresonding o u U ad }, such ha J(x m, u m ) ϵ as m +. Since {u m } U ad, {u m } is bounded in L (J, Y), here exiss a subsequence, relabeled as {u m }, and u L (J, Y) such ha u m w u in L (J, Y). Since U ad is closed and convex, by Marzur Lemma, we have u U ad. Suose x m are he mild soluions of sysem (4) corresonding o u m (m =, 1,,...) and x m saisfied he following inegral equaion x m () = T ()[x + g(x m )] + ( s) q 1 s n S ( s)f (s, x m (s), (Hx m )(s)) + ( s) q 1 s n S ( s)c(s)u m (s). Le f m (θ) f (θ, x m (θ), (Hx m )(θ)). Then by [Hf], we obain ha f m is a bounded coninuous oeraor from J ino X. Hence, f m ( ) L (J, X ). Furhermore, {f m ( )} is bounded in L (J, X ), and here exiss a subsequence, relabeled as {f m ( )}, and f ( ) L (J, X ) such ha f m ( ) w f ( ) in L (J, X ). By Lemma 4., we have Qf m s Q f in C. Now, we urn o consider he following conrolled sysem C D q x() = Ax() + n f () + C()u (), J, u U ad, n Z +, x() = g(x) + x. (5)

12 1438 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) By Theorem 4.1, i is clear ha sysem (5) has a mild soluion Define x() = T ()[x + g(x)] + η (1) m () = T ()[g(xm ) g(x)], η () η (3) ( s) q 1 s n S ( s) f (s) + ( s) q 1 s n S ( s)c(s)u (s). () = m ( s) q 1 s n S ( s) f m (s) f (s), () = m ( s) q 1 s n S ( s)c(s) u m (s) u (s). By [Hg1], we can obain η (1) m () = M g(xm ) g(x) Ml g x m x. Using he Hölder inequaliy again, we have η () () m T n ( s) q 1 S ( s) fm (s) f (s) T n Similarly, we also have ( s) (q 1) 1 1 Tn,,q S ( s)(f m (s) f (s)) S ( s)(f m (s) f (s)) 1. 1 η (3) () m Tn,,q C S ( s)(u m (s) u (s)) 1, where Tn,,q = and 1 +(q 1) 1 1 n+ T. +(q 1) 1 By virue of Lemma 4. and Lebesgue s dominaed convergence heorem, T As a resul, S ( s)(f m (s) f (s)) as m, S ( s)(u m (s) u (s)) as m. η (), m η(3) m as m. For each J, x m ( ),x( ) X, we have x m () x() Ml g x m () x() + η () m which imlies ha + η(3) m, x m x η() m + η (3) m, (Ml g < 1 1 Ml g > ). 1 Ml g So, we can infer ha x m s x in C as m. Furhermore, using [Hf], [Hk] and [Hh], we can obain f m ( ) s f (,x( ), (Hx)( )) in C as m. Using he uniqueness of limi, we have f () = f (,x, (Hx)()). (6)

13 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) Thus,x can be given by x() = T ()[x + g(x)] + ( s) q 1 s n S ( s)f (s,x, (Hx)(s)) + which is jus a mild soluion of sysem (4) corresonding o u. Since C L 1 (J, X ), using [HL] and Balder s heorem, we can obain ϵ = lim m l, x m (), u m () d + Ψ (x m (T)) l,x(), u () d + Ψ (x(t)) = J x, u ϵ. This shows ha J aains is minimum a (x, u ) C U ad. 5. An examle Consider he following conrolled sysem 19 x(, y) = x(, y) + e [ ] cos x(, y) + cos(s)x(s, y) + e 19 y e + e 1 + k (y, τ)u(τ, )dτ, y [, 1],, s (, T], u U ad, x(, ) = x(, 1) =, >, σ x(, y) = k 1 (y, τ)x( i, τ)dτ + i= wih he cos funcion J(x, u) = x(, y) dy + σ i= u(, y) dy + k (y, τ) τ x( i, τ)dτ, x(t, y) dy, ( s) q 1 s n S ( s)c(s)u (s), where < T 1, σ N, < < 1 < < σ < T and k : [, 1] [, 1] R is coninuous, k i L ([, 1] [, 1]) for i = 1,. We assume: (i) The funcion k 1 (, τ) is measurable and [ ] 1 c 11 = k (, τ)ddτ 1 <. For each τ [, 1], he funcion k 1(, τ) is measurable, k 1 (, τ) = k 1 (1, τ) =, c 1 = k 1(, τ) ddτ1 <, and here is a nonnegaive funcion G 1 L 1 (, T) such ha k 1(, τ) G 1 () for all (, τ) [, T] [, 1]. (ii) The wo firs-order arial derivaives of he funcion k (, ) are measurable, k (, ) = k (, 1) =, k (, τ) = k (1, τ) = and c 1 = k 1(, τ) ddτ1 <, c = τ k 1(, τ) ddτ1 <. Le X = Y = (L ([, 1]), ). The oeraor A : D(A) X defined by D(A) = {f X f, f X, f () = f (1) = } wih Af = f. Then A generaes a comac, analyic semigrou T( ) of uniformly bounded linear oeraor. Moreover, he eigenvalues of A are n π and he corresonding normalized eigenvecors are e n (u) = sin(nπu), n = 1,,.... We ake he funcions u : Tx([, 1]) R, such ha u L (Tx([, 1])) as he conrols. This claim is ha u(, ) going from [, T] ino Y is measurable. Se U() = {u Y u Y ϑ}, where ϑ L (J, R + ). We resric he admissible conrols U ad o be all he u L (Tx([, 1])) such ha u(, ) ϑ(), a.e. (7)

14 144 J. Wang e al. / Comuers and Mahemaics wih Alicaions 6 (11) Le X 1 = A 1 D(A 1 ), 1, where 1 = A 1 and he oeraor A 1 is given by = z, e n e n n=1 for each z D(A 1 ) = f X z, n=1 e n e n X and A 1 = 1. We denoe by C 1, he Banach sace C J, X 1 cos(s)x(s, y), C()u()(y) = 1 k (y, τ)u(τ, )dτ. Le f : [, T] X 1 ] and g : C 1 e f (, x(), (Hx)()) (y) = cos e + e X 1 by σ g(x)(y) = i= (Kx)( i ) where K : X 1 X 1 is defined by Obviously, (Kφ)(τ) = K(φ φ) (τ) = k 1 (y, τ)φ(τ)dτ + [ x() + (y) for x C 1, k 1 (y, τ)(φ φ)(τ)dτ + equied wih sunorm, x()(y) = x(, y), (Hx)()(y) = T cos(s)x(s) X 1 by (y) + e, k (y, τ)φ (τ)dτ, for all φ X 1. k (y, τ)(φ φ) (τ)dτ, for all φ,φ X 1. Thus sysem (7) can be ransformed ino C D q x() = Ax() + n f (, x(), (Hx)()) + C()u(), J, u U ad, n Z +, q (, 1), x() = g(x) + x, wih he cos funcion J(u) = x() 1 + u() Y d + x(t) 1. I is no difficul o see ha f (, x(), (Hx)()) 1 e e + e + e = ρ(), ρ L 1 (J, R + ). Meanwhile, i comes from he examle in [1] ha g is comleely coninuous oeraor from C 1 X 1 and saisfies g(x) 1 σ (c 1 + c ) x, g(x) g(z) 1 σ (c 1 + c ) x z. Since q(1 ) = > 1 and q = = 95 > 91 = +1 > 1 = 1, sysem (7) has a leas one oimal air while he condiion M 1 σ (c 1 + c ) < 1 hol. References [1] A.A. Kilbas, H.M. Srivasava, J.J. Trujillo, Theory and Alicaions of Fracional Differenial Equaions, in: Norh-Holland Mahemaics Sudies, vol. 4, Elsevier Science B.V., Amserdam, 6. [] V. Lakshmikanham, S. Leela, J.V. Devi, Theory of Fracional Dynamic Sysems, Cambridge Scienific Publishers, 9. [3] K.S. Miller, B. Ross, An Inroducion o he Fracional Calculus and Differenial Equaions, John Wiley, New York, [4] I. Podlubny, Fracional Differenial Equaions, Academic Press, San Diego, [5] K. Diehelm, A.D. Freed, On he soluion of nonlinear fracional order differenial equaions used in he modeling of viscoelasiciy, in: F. Keil, W. Mackens, H. Voss, J. Werher (E.), Scienific Comuing in Chemical Engineering II-Comuaional Fluid Dynamics, Reacion Engineering and Molecular Proeries, Sringer-Verlag, Heidelberg, 1999, [6] L. Gaul, P. Klein, S. Kemfle, Daming descriion involving fracional oeraors, Mech. Sys. Signal Process. 5 (1991) [7] W.G. Glockle, T.F. Nonnenmacher, A fracional calculus aroach of self-similar roein dynamics, Biohys. J. 68 (1995) [8] R. Hilfer, Alicaions of Fracional Calculus in Physics, World Scienific, Singaore,. [9] F. Mainardi, Fracional calculus, Some basic roblems in coninuum and saisical mechanics, in: A. Carineri, F. Mainardi (E.), Fracals and Fracional Calculus in Coninuum Mechanics, Sringer-Verlag, Wien, 1997, [1] F. Mezler, W. Schick, H.G. Kilian, T.F. Nonnenmache, Relaxaion in filled olymers: a fracional calculus aroach, J. Chem. Phys. 13 (1995) [11] R.P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differenial equaions and inclusions involving Riemann Liouville fracional derivaive, Adv. Difference Equ. 9 (9) Aricle ID 98178, 47 ages. (8)

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