Research Article Some Existence Results for Impulsive Nonlinear Fractional Differential Equations with Closed Boundary Conditions

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1 Hndaw Publshng Corporaton Abstract and Appled Analyss Volume 2012, Artcle ID , 15 pages do: /2012/ Research Artcle Some Exstence Results for Impulsve Nonlnear Fractonal Dfferental Equatons wth Closed Boundary Condtons Hlm Ergören 1 and Adem Klçman 2 1 Department of Mathematcs, Faculty of Scences, Yuzuncu Yl Unversty, Van, Turey 2 Department of Mathematcs and Insttute for Mathematcal Research, Unversty Putra Malaysa, Serdang, Malaysa Correspondence should be addressed to Adem Klçman, alcman@putra.upm.edu.my Receved 30 September 2012; Accepted 22 October 2012 Academc Edtor: Beata Rzepa Copyrght q 2012 H. Ergören and A. Klçman. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal wor s properly cted. We nvestgate some exstence results for the solutons to mpulsve fractonal dfferental equatons havng closed boundary condtons. Our results are based on contractng mappng prncple and Burton-Kr fxed pont theorem. 1. Introducton Ths paper consders the exstence and unqueness of the solutons to the closed boundary value problem BVP, for the followng mpulsve fractonal dfferental equaton: C D α x t f t, x t, t J : 0,T, t/, 1 <α 2, ( ( Δx I )), Δx I ( ( )), 1, 2,...,p, x T ax 0 btx 0, Tx T cx 0 dtx 0, 1.1 where C D α s Caputo fractonal dervatve, f C J R, R, I,I C R, R, Δx x ) x ), 1.2

2 2 Abstract and Appled Analyss wth x ) lm h 0 x h, x ) lm x h, h and Δx has a smlar meanng for x t, where 0 <t 1 <t 2 < <t p <t p 1 T, 1.4 a, b, c,andd are real constants wth Δ : c 1 b 1 a 1 d / 0. The boundary value problems for nonlnear fractonal dfferental equatons have been addressed by several researchers durng last decades. That s why, the fractonal dervatves serve an excellent tool for the descrpton of heredtary propertes of varous materals and processes. Actually, fractonal dfferental equatons arse n many engneerng and scentfc dscplnes such as, physcs, chemstry, bology, electrochemstry, electromagnetc, control theory, economcs, sgnal and mage processng, aerodynamcs, and porous meda see 1 7. For some recent development, see, for example, On the other hand, theory of mpulsve dfferental equatons for nteger order has become mportant and found ts extensve applcatons n mathematcal modelng of phenomena and practcal stuatons n both physcal and socal scences n recent years. One can see a notceable development n mpulsve theory. For nstance, for the general theory and applcatons of mpulsve dfferental equatons we refer the readers to Moreover, boundary value problems for mpulsve fractonal dfferental equatons have been studed by some authors see and references theren. However, to the best of our nowledge, there s no study consderng closed boundary value problems for mpulsve fractonal dfferental equatons. Here, we notce that the closed boundary condtons n 1.1 nclude quas-perodc boundary condtons b c 0 and nterpolate between perodc a d 1,b c 0 and antperodc a d 1,b c 0 boundary condtons. Motvated by the mentoned recent wor above, n ths study, we nvestgate the exstence and unqueness of solutons to the closed boundary value problem for mpulsve fractonal dfferental equaton 1.1. Throughout ths paper, n Secton 2, we present some notatons and prelmnary results about fractonal calculus and dfferental equatons to be used n the followng sectons. In Secton 3, we dscuss some exstence and unqueness results for solutons of BVP 1.1, that s, the frst one s based on Banach s fxed pont theorem, the second one s based on the Burton-Kr fxed pont theorem. At the end, we gve an llustratve example for our results. 2. Prelmnares Let us set J 0 0,t 1, J 1 t 1,t 2,...,J 1 1,, J, 1, J : 0,T \{t 1,t 2,...,t p } and ntroduce the set of functons: PC J, R {x : J R : x C, 1,R, 0, 1, 2,...,p and there exst x t and x t, 1, 2,...,pwth x x } and PC 1 J, R {x PC J, R, x C, 1,R, 0, 1, 2,...,p and there exst x t and x t, 1, 2,...,p wth x t x } whch s a Banach space wth the norm x sup t J { x PC, x PC } where x PC : sup{ x t : t J}. The followng defntons and lemmas were gven n 4.

3 Abstract and Appled Analyss 3 Defnton 2.1. The fractonal arbtrary order ntegral of the functon h L 1 J, R of order α R s defned by I α 0 h t 1 0 t s α1 h s ds, 2.1 where Γ s the Euler gamma functon. Defnton 2.2. For a functon h gven on the nterval J, Caputo fractonal dervatve of order α>0 s defned by C D α 0 h t 1 Γ n α 0 t s nα1 h n s ds, n α 1, 2.2 where the functon h t has absolutely contnuous dervatves up to order n 1. Lemma 2.3. Let α>0, then the dfferental equaton C D α h t has solutons h t c 0 c 1 t c 2 t 2 c n1 t n1, c R, 0, 1, 2,...,n 1, n α The followng lemma was gven n 4, 10. Lemma 2.4. Let α>0, then I αc D α h t h t c 0 c 1 t c 2 t 2 c n1 t n1, 2.5 for some c R, 0, 1, 2,...,n 1, n α The followng theorem s nown as Burton-Kr fxed pont theorem and proved n Theorem 2.5. Let X be a Banach space and A, D : X X two operators satsfyng: a A s a contracton, and b D s completely contnuous. Then ether the operator equaton x A x D x has a soluton, or the set ε {x X : x λa x/λ λd x } s unbounded for λ 0, 1. Theorem 2.6 see 22, Banach s fxed pont theorem. Let S be a nonempty closed subset of a Banach space X, then any contracton mappng T of S nto tself has a unque fxed pont.

4 4 Abstract and Appled Analyss Next we prove the followng lemma. Lemma 2.7. Let 1 <α 2 and let h : J R be contnuous. A functon x t s a soluton of the fractonal ntegral equaton: x t 0 t s α1 h s ds Ω 1 t Λ 1 Ω 2 t Λ 2, t J 0 α1 t s h s ds 1 Ω 1 t T Ω 1 t Ω 2 t t 1 t t 1 t 1 t 1 α2 α1 t s h s ds α2 t s Γ α 1 h s ds t s 1 Ω 1 t Γ α 1 h s ds Ω 1 t [ α2 T s Ω 2 t Γ α 1 h s ds 1 Ω ( ( )) 1 t I I 1 )) Ω 1 t 1 t t I ( ( )) 1 Ω2 t I ( ( ( ( )), t J, 1, 2,...,p T t I α1 T s h s ds ( ( )) ] 2.6 f and only f x t s a soluton of the fractonal BVP C D α x t h t, t J, ( ( Δx I )), Δx I ( ( )), x T ax 0 btx 0, Tx T cx 0 dtx 0, 2.7 where Λ 1 : α1 T s h s ds t 1 α1 t s h s ds T t 1 α2 1 t s Γ α 1 h s ds ( ( )) 1 I T t I )) I t t 1 )), α2 t s Γ α 1 h s ds

5 Abstract and Appled Analyss 5 Λ 2 : α2 T s Γ α 1 h s ds t 1 1 d Ω 1 t : Δ ct TΔ, Ω 2 t : 1 b T Δ α2 1 t s Γ α 1 h s ds I ( ( )), 1 a t Δ. 2.8 Proof. Let x be the soluton of 2.7.Ift J 0, then Lemma 2.4 mples that x t I α t s α1 h t c 0 c 1 t h s ds c 0 c 1 t, 0 x t s α2 t Γ α 1 h s ds c 1, for some c 0,c 1 R. If t J 1, then Lemma 2.4 mples that x t t 1 x t α1 t s h s ds d 0 d 1 t t 1, t 1 α2 t s Γ α 1 h s ds d 1, 2.10 for some d 0,d 1 R. Thus we have x ( t ) t1 1 x ( t ) t1 1 α1 t 1 s h s ds c 0 c 1 t 1, x 1) d0, α2 t 1 s Γ α 1 h s ds c 1, x ) 1 d Observng that Δx t 1 x 1 ) ( ) ( ( 1 I1 1)), Δx t 1 x 1) x ) ( ( 1 I 1 1)), 2.12

6 6 Abstract and Appled Analyss then we have d 0 1 d 1 1 α1 t 1 s ( ( h s ds c 0 c 1 t 1 I 1 1)), α2 t 1 s Γ α 1 h s ds c 1 I ( ( )) 1, hence, for t t 1,t 2, x t x t t 1 α1 t s t1 h s ds t t 1 t 1 1 α1 t 1 s h s ds α2 t 1 s Γ α 1 h s ds I ( ( 1 1)) t t1 I ( ( 1 1)) c0 c 1 t, α2 t s t1 Γ α 1 h s ds α2 t 1 s Γ α 1 h s ds ( ( I 1 1)) c If t J 2, then Lemma 2.4 mples that x t t 2 x t α1 t s h s ds e 0 e 1 t t 2, t 2 α2 t s Γ α 1 h s ds e 1, 2.15 for some e 0,e 1 R. Thus we have x ( t ) t2 2 t 1 x 2) e0, α1 t 2 s t1 h s ds t 2 t 1 1 α1 t 1 s h s ds α2 t 1 s Γ α 1 h s ds I ( ( 1 1)) t2 t 1 I ( ( 1 1)) c0 c 1 t 2, 2.16 x ( t ) t2 2 t 1 x 2) e1. α2 t 2 s t1 Γ α 1 h s ds α2 t 1 s Γ α 1 h s ds ( ( I 1 1)) c1, Smlarly we observe that Δx t 2 x ( ) ( ( 2) 2 I2 2)), Δx t 2 x 2) ( x t ) ( ( 2 I 2 2)), 2.17

7 Abstract and Appled Analyss 7 thus we have Hence, for t t 2,t 3, e 0 x ) ( ( 2 I2 e 1 x 2 ) I 2 2 2)), )) x t t 2 α1 t s t1 h s ds t 2 t 1 t t 2 1 [1 α2 t 1 s Γ α 1 h s ds α1 t 1 s t2 Γ α 1 h s ds t 1 ( ( )) ( ( I 1 1 I2 2)) t t1 I 1 α1 t 1 s t2 h s ds t 1 ] α1 t 2 s Γ α 1 h s ds 1 )) I 2 α1 t 2 s h s ds ( t 2)) c0 c 1 t By a smlar process, f t J, then agan from Lemma 2.4 we get x t x t α1 t s h s ds t s t { t 1 ( ( )) 1 I t 1 α1 t s h s ds α2 1 Γ α 1 h s ds t t I α2 t s Γ α 1 h s ds t 1 )) I t t 1 )) c0 c 1 t, α2 1 t s Γ α 1 h s ds I α2 t s Γ α 1 h s ds ( ( )) c Now f we apply the condtons: x T ax 0 btx 0, Tx T cx 0 dtx 0, 2.21 we have Λ 1 1 a c 0 T 1 b c 1, Λ 2 c T c 0 1 d c 1, c 0 1 d Λ 1 Δ 1 b TΛ 2, Δ 2.22 c 1 cλ 1 TΔ 1 a Λ 2. Δ

8 8 Abstract and Appled Analyss In vew of the relatons 2.8, when the values of c 0 and c 1 are replaced n 2.9 and 2.20, the ntegral equaton 2.7 s obtaned. Conversely, assume that x satsfes the mpulsve fractonal ntegral equaton 2.6, then by drect computaton, t can be seen that the soluton gven by 2.6 satsfes 2.7. The proof s complete. 3. Man Results Defnton 3.1. A functon x PC 1 J, R wth ts α-dervatve exstng on J s sad to be a soluton of 1.1, fx satsfes the equaton C D α x t f t, x t on J and satsfes the condtons: ( ( Δx I )), Δx I ( ( )), x T ax 0 btx 0, Tx T cx 0 dtx For the sae of convenence, we defne Ω 1 sup t J Ω 1 t, Ω 2 sup Ω 2 t, Ω 1 sup Ω 1 t, t J t J 2 sup Ω 2 t. Ω t J 3.2 The followngs are man results of ths paper. Theorem 3.2. Assume that A1 the functon f : J R R s contnuous and there exsts a constant L 1 > 0 such that f t, u f t, v L 1 u v, for all t J, and u, v R, A2 I, I : R R are contnuous, and there exst constants L 2 > 0 and L 3 > 0 such that I u I v L 2 u v, I u I v L 3 u v for each u, v R and 1, 2,...,p. Moreover, consder the followng: [ L1 T α Γ α 1 ( )( ) 1 Ω L 1 T α1 ( ) 1 1 p 2pα Ω 2 1 p ( ] 1 Ω )( ) ( 1 pl2 L 3 Ω 1 T Ω 2 T) pl 3 < Then, BVP 1.1 has a unque soluton on J. Proof. Defne an operator F : PC 1 J, R PC 1 J, R by Fx t α1 t s f s, x s ds 1 Ω 1 t T Ω 1 t Ω 2 t t t 1 t 1 α1 t s f s, x s ds α2 t s f s, x s ds Γ α 1

9 Abstract and Appled Analyss 9 1 Ω 1 t Ω 1 t Ω 2 t Ω 1 t 1 1 t t 1 α1 T s f s, x s ds α2 t s f s, x s ds Γ α 1 α2 T s Γ α 1 f s, x s ds 1 Ω 1 t T t I ( ( )) 1 Ω2 t I [ I )) I ( ( )) 1 t t I ( ( )) ] ( ( )). 3.4 Now, for x, y PC J, R and for each t J,weobtan Fx t ( Fy ) t t s α1 ( ) f s, x s f s, y s ds 1 Ω 1 t t 1 t s α1 ( ) f s, x s f s, y s ds T Ω 1 t Ω 2 t t t 1 α2 t s ( ) f s, x s f s, y s ds Γ α 1 1 Ω 1 t Ω 1 t Ω 2 t 1 T s t t 1 α1 α2 t s f s, x s f ( s, y s ) ds Γ α 1 ( ) f s, x s f s, y s ds α2 T s ( ) f s, x s f s, y s ds Γ α 1 [ ( ( )) ( ( )) ( ( 1 Ω 1 t I I y t I Ω 1 t Ω 2 t t t I T t I I )) I )) I )) I ( y ( y )) )), ( ( )) y t )) I ] ( ( )) y t

10 10 Abstract and Appled Analyss [ ( ) Fx t Fy t L1 T α ( )( ) 1 Ω L 1 T α1 ( ) 1 1 p 2pα Γ α 1 Ω 2 1 p ( ] 1 Ω )( ) ( 1 pl2 L 3 Ω x s 1 T Ω 2 T) pl 3 y s. 3.5 Therefore, by 3.3, the operator F s a contracton mappng. In a consequence of Banach s fxed theorem, the BVP 1.1 has a unque soluton. Now, our second result reles on the Burton-Kr fxed pont theorem. Theorem 3.3. Assume that (A1)-(A2) hold, and A3 there exst constants M 1 > 0, M 2 > 0, M 3 > 0 such that f t, u M 1, I u M 2, I u M 3 for each u, v R and 1, 2,...,p. Then the BVP 1.1 has at least one soluton on J. Proof. We defne the operators A, D : PC 1 J, R PC 1 J, R by Ax t 1 Ω 1 t Dx t Ω 1 t 1 [ I T t I )) I ( ( )) ] ( ( )) 1 Ω2 t I α1 t s f s, x s ds 1 Ω 1 t T Ω 1 t Ω 2 t t t 1 ( ( )) 1 t t I t 1 ( ( )), α1 t s f s, x s ds α2 t s f s, x s ds Γ α Ω 1 t 1 t t 1 α2 t s f s, x s ds Γ α 1 Ω 1 t α1 T s f s, x s ds Ω 2 t α2 T s f s, x s ds. Γ α 1 It s obvous that A s contracton mappng for ( )( ) ( 1 Ω 1 pl2 L 3 Ω 1 T Ω 2 T) pl 3 < Now, n order to chec that D s completely contnuous, let us follow the sequence of the followng steps.

11 Abstract and Appled Analyss 11 Step 1 D s contnuous.let{x n } be a sequence such that x n x n PC J, R. Then for t J, we have Dx n t Dx t t s α1 f s, xn s f s, x s ds 1 Ω 1 t t 1 t s α1 f s, x n s f s, x s ds T Ω 1 t Ω 2 t t t 1 α2 t s f s, xn s f s, x s ds Γ α 1 1 Ω 1 t Ω 1 t Ω 2 t 1 T s t t 1 α1 α2 t s f s, xn s f s, x s ds Γ α 1 f s, xn s f s, x s ds α2 T s f s, x n s f s, x s ds. Γ α Snce f s contnuous functon, we get Dx n t Dx t 0 as n. 3.9 Step 2 D maps bounded sets nto bounded sets n PC J, R. Indeed, t s enough to show that for any r>0, there exsts a postve constant l such that for each x B r {x PC J, R : x r}, we have D x l. By A3, we have for each t J, Dx t t s α1 f s, x s ds 1 Ω 1 t t 1 t s α1 T Ω 1 t Ω 2 t t 1 Ω 1 t 1 t t 1 f s, x s ds t 1 α2 t s f s, x s ds Γ α 1 α2 t s f s, x s ds Γ α 1

12 12 Abstract and Appled Analyss Ω 1 t T s α1 f s, x s ds Ω 2 t α2 T s f s, x s ds, Γ α 1 D x M 1T α ( )( ) 1 Ω M 1 T α1 1 1 p 2pα Ω ) Γ α 1 2( 1 p : l Step 3 D maps bounded sets nto equcontnuous sets n PC 1 J, R. Letτ 1,τ 2 J, 0 p wth τ 1 <τ 2 and let B r be a bounded set of PC 1 J, R as n Step 2, and let x B r. Then ( Dy ) τ 2 D τ 1 τ2 ( Dy ) ds s L τ2 τ 1, τ where Dx t α2 t s f s, x s ds Ω 1 Γ α 1 t T t Ω 1 t Ω 2 t 1 Ω 1 t 1 Ω 1 t t t 1 T s α1 t 1 t 1 t s α1 α2 t s f s, x s ds Γ α 1 α2 t s f s, x s ds Γ α 1 f s, x s ds Ω 2 t f s, x s ds α2 T s f s, x s ds, Γ α 1 M 1T α Γ α 1 Ω 1 ( 1 p 2pα ) M 1 T α1 ( 1 Ω 2)( 1 p ) : L Ths mples that A s equcontnuous on all the subntervals J, 0, 1, 2,...,p. Therefore, by the Arzela-Ascol Theorem, the operator D : PC 1 J, R PC 1 J, R s completely contnuous. To conclude the exstence of a fxed pont of the operator A D, t remans to show that the set ε { ) } x X : x λa λd x for some λ 0, 1 λ 3.13 s bounded. Let x ε, for each t J, ) x t λd x t λa t. λ 3.14

13 Abstract and Appled Analyss 13 Hence, from A3, we have x t λ t s α1 f s, x s ds λ 1 Ω 1 t λ T Ω 1 t Ω 2 t t t 1 t 1 t s α1 f s, x s ds α2 t s f s, x s ds Γ α 1 1 t t 1 α2 t s λ 1 Ω 1 t f s, x s ds Γ α 1 α1 T s T α2 λ Ω 1 t f s, x s ds T s λ Ω2 t f s, x s ds Γ α 1 [ ( ( )) t ( ))] λ 1 Ω 1 t I λ I λ ( 1 ( )) λ Ω 1 t T t I λ ( 1 ( )) t 1 λ Ω 2 t I λ λ ( )) t t I λ, 3.15 x t [ M1 T α Γ α 1 ( 1 Ω 1 )( 1 p 2pα ) M 1 T α1 Ω ( ) 2 1 p ( 1 Ω )( ) ( 1 pm2 M 3 Ω 1 T Ω 2 T) pm 3 ]. Consequently, we conclude the result of our theorem based on the Burton-Kr fxed pont theorem. 4. An Example Consder the followng mpulsve fractonal boundary value problem: C D 3/2 x t ( ) 1 Δx 3 sn 2t x t t x t, t 0, 1, t / 1 3, x 1 /3 15 x 1 /3, Δx ( ) 1 3 x 1 /3 10 x 1 /3, 4.1 x 1 5x 0 2x 0, x 1 x 0 4x 0.

14 14 Abstract and Appled Analyss Here, a 5, b 2, c 1, d 4, α 3/2, T 1, p 1. Obvously, L 1 1/25, L 2 1/15, L 3 1/10, Ω 1 4/15, Ω 2 7/15. Further, [ L1 T α Γ α 1 ] ( )( ) 1 Ω L 1 T α1 ( ) ( 1 1 p 1 L2 2pα L 3 Ω 2 1 p Ω 1 T Ω 2 T) pl π < Snce the assumptons of Theorem 3.2 are satsfed, the closed boundary value problem 4.1 has a unque soluton on 0, 1. Moreover, t s easy to chec the concluson of Theorem 3.3. Acnowledgments The authors would le to express ther sncere thans and grattude to the revewer s for ther valuable comments and suggestons for the mprovement of ths paper. The second author also gratefully acnowledges that ths research was partally supported by the Unversty Putra Malaysa under the Research Unversty Grant Scheme RU. References 1 K. Dethelm, The Analyss of Fractonal Dfferental Equatons, Sprnger, N. Heymans and I. Podlubny, Physcal nterpretaton of ntal condtons for fractonal dfferental equatonswth Remann Louvlle fractonal dervatves, Rheologca Acta, vol. 45, no. 5, pp , R. Hlfer, Applcatons of Fractonal Calculus n Physcs, World Scentfc, Sngapore, A. A. Klbas, H. M. Srvastava, and J. J. Trujllo, Theory and Applcatons of Fractonal Dfferental Equatons, vol. 204 of North-Holland Mathematcs Studes, Elsever Scence B.V., Amsterdam, The Netherlands, V. Lashmantham, S. Leela, and J. Vasundhara Dev, Theory of Fractonal Dynamc Systems, Cambrdge Academc Publshers, Cambrdge, UK, J. Sabater, O. P. Agrawal, and J. A. T. Machado, Eds., Advances n Fractonal Calculus: Theoretcal Developmentsand Applcatons n Physcs and Engneerng, Sprnger, Dordrecht, The Netherlands, S. G. Samo, A. A. Klbas, and O. I. Marchev, Fractonal Integrals and Dervatves. Theory and Applcatons, Gordon and Breach Scence, Yverdon, Swtzerland, R. P. Agarwal, M. Benchohra, and S. Haman, Boundary value problems for fractonal dfferental equatons, Georgan Mathematcal Journal, vol. 16, no. 3, pp , R. P. Agarwal, M. Benchohra, and S. Haman, A survey on exstence results for boundary value problems of nonlnear fractonal dfferentalequatons and nclusons, Acta Applcandae Mathematcae, vol. 109, no. 3, pp , Z. Ba and H. Lü, Postve solutons for boundary value problem of nonlnear fractonal dfferental equaton, Journal of Mathematcal Analyss and Applcatons, vol. 311, no. 2, pp , M. Belme, J. J. Neto, and R. Rodríguez-López, Exstence of perodc soluton for a nonlnear fractonal dfferental equaton, Boundary Value Problems, vol. 2009, Artcle ID , 18 pages, M. Benchohra, S. Haman, and S. K. Ntouyas, Boundary value problems for dfferental equatons wth fractonal order and nonlocal condtons, Nonlnear Analyss, vol. 71, no. 7-8, pp , H. A. H. Salem, On the fractonal order m-pont boundary value problem n reflexve Banach spaces and wea topologes, Journal of Computatonal and Appled Mathematcs, vol. 224, no. 2, pp , W. Zhong and W. Ln, Nonlocal and multple-pont boundary value problem for fractonal dfferental equatons, Computers & Mathematcs wth Applcatons, vol. 59, no. 3, pp , 2010.

15 Abstract and Appled Analyss V. Lashmantham, D. D. Baĭnov,andP.S.Smeonov,Theory of Impulsve Dfferental Equatons, World Scentfc, Sngapore, Y. V. Rogovcheno, Impulsve evoluton systems: man results and new trends, Dynamcs of Contnuous, Dscrete and Impulsve Systems, vol. 3, no. 1, pp , A. M. Samoĭleno and N. A. Perestyu, Impulsve Dfferental Equatons, World Scentfc, Sngapore, B. Ahmad and S. Svasundaram, Exstence of solutons for mpulsve ntegral boundary value problems of fractonal order, Nonlnear Analyss, vol. 4, no. 1, pp , M. Benchohra and B. A. Slman, Exstence and unqueness of solutons to mpulsve fractonal dfferental equatons, Electronc Journal of Dfferental Equatons, vol. 10, pp. 1 11, G. Wang, B. Ahmad, and L. Zhang, Some exstence results for mpulsve nonlnear fractonal dfferental equatons wth mxed boundary condtons, Computers & Mathematcs wth Applcatons, vol. 62, no. 3, pp , T. A. Burton and C. Kr, A fxed pont theorem of Krasnosels-Schaefer type, Mathematsche Nachrchten, vol. 189, pp , A. Granas and J. Dugundj, Fxed Pont Theory, Sprnger Monographs n Mathematcs, Sprnger, New Yor, NY, USA, 2003.

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