Research Article Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions

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1 Inernaional Differenial Equaions Volume 26, Aricle ID , 9 pages hp://dx.doi.org/.55/26/ Research Aricle Exisence of he Soluion for Sysem of Coupled Hybrid Differenial Equaions wih Fracional Order and Nonlocal Condiions Khalid Hilal and Ahmed Kajouni Laboraoire de Mahémaiques Appliquées & Calcul Scienifique, Universié Sulan Moulay Slimane, BP 523, 23 Béni Mellal, Morocco Correspondence should be addressed o Ahmed Kajouni; kajjouni@gmail.com Received February 26; Revised 29 April 26; Acceped 3 May 26 Academic Edior: Gason Mandaa N guérékaa Copyrigh 26 K. Hilal and A. Kajouni. his is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. his paper is moivaed by some papers reaing he fracional hybrid differenial equaions wih nonlocal condiions and he sysem of coupled hybrid fracional differenial equaions; an exisence heorem for fracional hybrid differenial equaions involving Capuo differenial operaors of order <α 2is proved under mixed Lipschiz and Carahéodory condiions. he exisence and uniqueness resul is elaboraed for he sysem of coupled hybrid fracional differenial equaions.. Inroducion Our aim in his paper is o sudy he exisence of soluion for he boundary value problems for hybrid differenial equaions wih fracional order <α 2and nonlocal condiion (BVPHDEFNL for shor) of he form c D α x () ( )=g(, x ()) f (, x ()) a.e. J=[, ], <α 2, x () f (, x ()) = L (x), x () f (, x ()) =x, where c D α ishecapuofracionalderivaive. f C(J R, R \{}), g C(J R, R), L : C(J, R) R is a coninuous funcion and x R. And exploiaion of resuls obained o sudy he exisence of soluions for () a sysem of coupled hybrid fracional differenial equaions is as follows: c D α x () ( f (, x (),y()) )=g (, x (),y()) a.e. [, ], <α 2, c D β y () ( f 2 (, x (),y()) )=g 2 (, x (),y()) a.e. [, ], <β 2, x () f (, x (),y()) = L (x, y) ; x () =, y () f 2 (, x (),y()) = L 2 (x, y) ; y () =, where c D α is he Capuo fracional derivaive. f i C([, ] R R, R\{}), g i C([, ] R R, R),and L i : C([, ], R) C([, ], R) R are coninuous funcions (i =, 2). (2)

2 2 Inernaional Differenial Equaions Fracional differenial equaions are a generalizaion of ordinary differenial equaions and inegraion o arbirary nonineger orders. he origin of fracional calculus goes back o Newon and Leibniz in he seveneenh cenury. I is widely and efficienly used o describe many phenomena arising in engineering, physics, economy, and science. here are several conceps of fracional derivaives, some classical, such as Riemann-Liouville or Capuo definiions. For noeworhy papers dealing wih he inegral operaor and he arbirary fracional order differenial operaor, see [ 7]. he quadraic perurbaions of nonlinear differenial equaions have araced much aenion. We call such fracional hybrid differenial equaions. here have been many works on he heory of hybrid differenial equaions, and we refer he readers o he aricles [8 2]. Dhage and Lakshmikanham [] discussed he following firs order hybrid differenial equaion d d [ x () ]=g(, x ()) a.e. J=[, ], f (, x ()) x( )=x R, where f C(J R, R \{})and g C(J R, R). hey esablished he exisence, uniqueness resuls, and some fundamenal differenial inequaliies for hybrid differenial equaions iniiaing he sudy of heory of such sysems and proved, uilizing he heory of inequaliies, he exisence of exremal soluions and comparison resuls. Zhao e al. [3] have discussed he following fracional hybrid differenial equaions involving Riemann-Liouville differenial operaors: D q x () [ ]=g(, x ()) a.e. J=[, ], f (, x ()) (4) x () =, where f C(J R, R \{})and g C(J R, R). he auhors of [3] esablished he exisence heorem for fracional hybrid differenial equaion and some fundamenal differenial inequaliies. hey also esablished he exisence of exremal soluions. Hilal and Kajouni [4] have sudied boundary fracional hybrid differenial equaions involving Capuo differenial operaors of order <α<as follows: D α x () ( )=g(, x ()) f (, x ()) a.e. J=[, ], (5) x () a f (, x ()) +b x () f (, x ()) =c, where f C(J R, R \{}), g C(J R, R) and a, b,and c are real consans wih a+b=. hey proved he exisence resul for boundary fracional hybrid differenial equaions under mixed Lipschiz and Carahéodory condiions. Some fundamenal fracional differenial inequaliies are also esablished which are uilized o prove he exisence of exremal (3) soluions. Necessary ools are considered and he comparison principle is proved which will be useful for furher sudy of qualiaive behavior of soluions. he nonlocal condiion is a condiion aached o he main equaion; i replaces he classic nonlocal condiion in order o model physical phenomena of he fashion neares from realiy. he nonlocal condiion involves he funcion L (x) = p i= c i x ( i ), (6) where c i, i=,2,...,p, are given consans and < < 2 < < p. Le us observe ha Cauchy problems wih nonlocal condiions were iniiaed by Byszewski and Lakshmikanham [2] and, since hen, such problems have also araced several auhors including A. Aizicovici, K. Ezzinbi, Z. Fan, J. Liu, J. Liang, Y. Lin,.-J. Xiao, G. N Guérékaa, E. Hernàndez, and H. Lee (see [2, 5]). 2. Preliminaries In his secion, we inroduce noaions, definiions, and preliminary facs which are used hroughou his paper. By X = C(J,R) we denoe he Banach space of all coninuous funcions from J=[,]ino R wih he norm y = sup { y (), J}. (7) And le C(J R, R) denoe he class of funcions g:j R R such ha (i) he map g(, x) is measurable for each x R, (ii) he map x g(, x) is coninuous for each J. he class C(J R, R) is called he Carahéodory class of funcions on J R which are Lebesgue inegrable when bounded by a Lebesgue inegrable funcion on J. By L (J; R) we denoe he space of Lebesgue inegrable real-valued funcions on J equipped wih he norm L defined by x L = x (s) ds. (8) Definiion. he fracional inegral of he funcion h L ([a, b], R + ) of order α R + is defined by I α a h () = (s) α h (s) ds, (9) a where Γ is he gamma funcion. Definiion 2. For a funcion h given on he inerval [a, b],he Capuo fracional order derivaive of h is defined by c α D a +h () = Γ (nα) (s) nα h (n) (s) ds, () a where n=[α]+and [α] denoes he ineger par of α.

3 Inernaional Differenial Equaions 3 Lemma 3 (see [6]). Le α>. hen he fracional differenial equaion has soluions h () =c +c + +c n n, c D α h () = () c i R, i=,2,...,n, n=[α] +. Lemma 4 (see [6]). Le α>.hen (2) I α D α h () =h() +c +c + +c n n, (3) for some c i R, i=,2,...,n, n=[α]+. Definiion 5. By a soluion of he BVPHDEFNL () we mean afuncionx C(J,R) such ha (i) d 2 u/d 2 L (J, R),whereu: x/f(, x) for each x R, (ii) x saisfies he equaions in (). 3. Exisence Resul In his secion, we prove he exisence resuls for he boundary value problems for hybrid differenial equaions wih fracional order () on he closed and bounded inerval J= [, ] under mixed Lipschiz and Carahéodory condiions on he nonlineariies involved in i. We defined he muliplicaion in X by (xy) () =x() y (), for x, y X. (4) Clearly, X=C(J;R) is a Banach algebra wih respec o above norm and muliplicaion in i. We prove he exisence of soluion for he BVPHDEFNL () by a fixed poin heorem in Banach algebra due o Dhage []. Lemma 6 (see []). Le S be a nonempy, closed convex, and bounded subse of he Banach algebra X and le A:X X and B:X Xbe wo operaors such ha (a) A is Lipschizian wih a Lipschiz consan α, (b) B is compleely coninuous, (c) x=axby x Sfor all y S, (d) αm <,wherem = B(S) = sup{ B(x) : x S}. hen he operaor equaion AxBx = x has a soluion in S. We make he following assumpions: (H ): he funcion x x/f(, x) is increasing in R almos everywhere for J. (H ): here exiss a consan L>such ha f (, x) f(,y) L xy, (5) for all Jand x, y R. (H 2 ): here exiss a funcion h L (J, R) such ha g (, x) h() a.e. J, (6) for all x R. (H 3 ): here exiss a consan M>such ha L(y) M, for each y C(J;R). As a consequence of Lemmas 3 and 4 we have he following resul which is useful in wha follows. Lemma 7. Assume ha hypohesis (H ) holds. hen for any h L (J; R), hefuncionx C(J;R) is a soluion of he BVPHDEFNL: c α x () D ( f (, x ()) )=h() x () f (, x ()) = L (x), x () f (, x ()) =x a.e. J = [, ], <α 2, ifandonlyifx saisfies he hybrid inegral equaion x () =f(, x ()) [ (s) α h (s) ds (s) α h (s) ds ( )L (x) + x ]. (7) (8) Proof. Assume ha x is a soluion of he problem (8). Applying he Capuo fracional operaor of he order α, weobain he firs equaion in (7). Again, subsiuing =and = in (8) we will have he second equaion in (7). Conversely, c D α (x()/f(, x())) = h(),sowege x () f (, x ()) +c +c =I α h (). (9) hen x()/f(, x()) + c =and c =L(x),andeven x () f (, x ()) +c +c = (s) α h (s) ds. (2) hus, c = (L (x) x + (s) α h (s) ds) (2) implies ha x () f (, x ()) = (s) α h (s) ds (s) α h (s) ds ( )L (x) + x. (22)

4 4 Inernaional Differenial Equaions heorem 8. Assume hypoheses (H ) (H 3 ).Furher,if L ( 2α Γ (α+) h L +M+ x ) <, (23) hen he hybrid fracional order differenial equaion () has a soluion defined on J. Proof. We defined a subse S of X by S={x X N}, (24) x where N = F ((2 α /Γ(α + )) h L +M+ x )/( L((2 α /Γ(α + )) h L +M+ x )) and F = sup J f(, ). I is clear ha S saisfies hypohesis of Lemma 6. By an applicaion of Lemma 7, () is equivalen o he nonlinear hybrid inegral equaion x () =f(, x ()) [ (s) α g (s, x (s)) ds (s) α g (s, x (s)) ds ( )L (x) + x ]. Define wo operaors A:X Xand B:S Xby Ax () =f(, x ()), J, Bx () = (s) α g (s, x (s)) ds (s) α g (s, x (s)) ds ( )L (x) + x. (25) (26) hen he hybrid inegral equaion (25) is ransformed ino he operaor equaion as x () =Ax() Bx (), J. (27) We will show ha he operaors A and B saisfy all he condiions of Lemma 6. Claim.Lex, y X. hen by hypohesis (H ), Ax () Ay() = f (, x ()) f(,y()) L x () y() L xy, (28) for all J.akingsupremumover,weobain for all x, y X. Ax Ay L xy, (29) Claim 2 (we show ha B is coninuous in S). Le (x n ) be a sequence in S converging o a poin x S.henbyLebesgue dominaed convergence heorem, lim n (s) α g(s,x n (s))ds = (s) α lim n g(s,x n (s))ds, b lim (s) α g(s,x n (s))ds n = b (s) α lim g(s,x n n (s))ds. (3) And since L is a coninuous funcion lim L (x n n)=l (x), (3) hen lim Bx n n () = lim [ n (s) α g(s,x n (s))ds (s) α g(s,x n (s))ds ( )L (x n)+ x ]= lim n (s) α g(s,x n (s))ds lim n (s) α g(s,x n (s))ds lim ( n ) L (x n )+ x = (s) α g (s, x (s)) ds (s) α g (s, x (s)) ds ( ) L (x) + x =Bx(), (32) for all J.hisshowshaB is a coninuous operaor on S. Claim 3 (B is compac operaor on S). Firs, we show ha B(S) is a uniformly bounded se in X. Le x S. hen by hypohesis (H 2 ),forall J, Bx () (s)α g (s, x (s)) ds + (s)α g (s, x (s)) ds + L (x) + x α α h L + α α h L + L (x) + x 2 α Γ (α+) h L +M+ x. (33) hus, Bx (2 α /Γ(α + )) h L +M+ x,forallx S.

5 Inernaional Differenial Equaions 5 his shows ha B is uniformly bounded on S. Nex, we show ha B(S) is an equiconinuous se on X. We se p() = h(s)ds. Le, 2 J.henforanyx S, Bx ( )Bx( 2 ) = ( s) α g (s, x (s)) ds 2 ( 2 s) α g (s, x (s)) ds ( 2 ) (s)α g (s, x (s)) ds ( 2 ) L (x) + ( 2 )x α 2 g (s, x (s)) ds + α 2 g (s, x (s)) ds + M+ x 2 α +(α p( )p( 2 ) + M+ x ) 2. (34) Since p is coninuous on compac J, i is uniformly coninuous. Hence, 2 <η Bx ( )Bx( 2 ) <ε, ε>, η> (35) for all, 2 Jand for all x X. his shows ha B(S) is an equiconinuous se in X. hen by Arzelá-Ascoli heorem, B is a coninuous and compac operaor on S. Claim 4 (hypohesis (c) of Lemma 6 is saisfied). Le x X and y Sbe arbirary such ha x=axby.hen, x () = Ax () By () f (, x ()) and so, [ f (, x ()) f(, ) + f (, ) ] ( 2α Γ (α+) h L +M+ x ) (L x () +F ) ( 2α Γ (α+) h L +M+ x ), (36) x () L( 2α Γ (α+) h L +M+ x ) x () which implies F ( 2α (α+) h L +M+ x ), (37) x () F ((2 α /Γ (α+)) h L +M+ x ) L((2 α /Γ (α+)) h L +M+ x (38) ). aking supremum over, x F ((2 α /Γ (α+)) h L +M+ x ) L((2 α /Γ (α+)) h L +M+ x ) (39) =N. hen x S,andhypohesis(c) of Lemma 6 is saisfied. Finally, we have M= B (S) = sup { Bx :x S} So, 2 α (4) Γ (α+) h L +M+ x. αm L ( 2α Γ (α+) h L +M+ x ) <. (4) hus, all he condiions of Lemma 6 are saisfied and hence he operaor equaion AxBx = x has a soluion in S. As a resul, BVPHDEFNL () has a soluion defined on J. his complees he proof. 4. An Example In his secion we give an example o illusrae he usefulness of our main resuls. Le us consider he following fracional boundary value problem: c D α ( (2+ln (+)) (x () +x 2 ()) e ) (s) α g (s, x (s)) ds (s) α g (s, x (s)) ds ( )L (x) + x e x2 () = x 2 () + 2 a.e. J=[, ], < α 2, + n x () f (, x ()) = c i x( i ); i= x () f (, x ()) =, (42)

6 6 Inernaional Differenial Equaions where < < 2 < < n <, c i, i =,2,...,n,aregiven posiive consans. And n i= c i <(π)/2m,wherem=max i n L( i ). We se f (, x) = g (, x) = L (x ()) = e (2+ln (+)) (+x), e x2 x , n i= c i x( i ). Le x, y [, + ) and J.Wehave (, x) [, ] [, + ), f (, x) f(,y) = e 2+ln (+) +x +y xy 2 (+x) ( + y) 2 xy. Hence, condiion (H ) holds wih L=/2.Alsowehave g (, x) = e x 2 x h(), (43) (44) (45) coupled hybrid fracional differenial equaions for (). Consider c α x () D ( f (, x (),y()) )=g (, x (),y()) a.e. [, ], < α 2, c β y () D ( f 2 (, x (),y()) )=g 2 (, x (),y()) a.e. [, ], < β 2, x () f (, x (),y()) = L (x, y), x () =, y () f 2 (, x (),y()) = L 2 (x, y), y () =, (49) where c D α is he Capuo fracional derivaive. f i C(J R R, R \{}), g i C([, ] R R, R),and L i : C([, ], R) C([, ], R) R are coninuous funcions (i =, 2). Main Resuls. LeΩ = {ω() \ ω() C ([, ])} denoe a Banach space equipped wih he norm ω = sup{ ω(), [, ]}, whereω = K R. Noicehaheproducspace (K R, (x, y) ) wih he norm (x, y) = x + y, (x, y) K R is also a Banach space. In view of Lemma 7, we define an operaor Φ:K R K R by where h() = /( + 2 ).Wehave h () d = + 2 = π 4. (46) hen condiion (H 2 ) holds. Furhermore, since L C(J, R), henwesem = max i n L( i ) and we have n L ( i) = c i L ( i ) M i= n i= c i. (47) We will check ha condiion (23) is saisfied wih =. Since n i= c i <(π)/2m,henπ+2m n i= c i <. hus, n π Γ (α+) +2M c i <, (48) which is saisfied for each α (, 2]. hen by heorem 8 problem (42) has a soluion on [, ]. 5. Sysem of Coupled Hybrid Fracional Differenial Equaions he aim of his secion is o obain he exisence resuls, by means of Banach s fixed poin heorem, for he problem of i= where Φ(x,y)() =(Φ (x, y) (),Φ 2 (x, y) ()), (5) Φ (x, y) () =f (, x (),y()) [ (s) α g (s, x (s),y(s))ds (s) α g (s, x (s),y(s))ds () L (x, y)], (5) Φ 2 (x, y) () =f 2 (, x (),y()) [ (s) β g 2 (s, x (s),y(s))ds (s) β g 2 (s, x (s),y(s))ds () L 2 (x, y)].

7 Inernaional Differenial Equaions 7 In he sequel, we need he following assumpions: (H ):hefuncionsf i are coninuous and bounded; ha is, here exis posiive numbers L i such ha f i (, u, V) L i for all (, u, V) [,] R R (i =, 2). (H 2 ): here exis real consans ρ,δ >and ρ i,δ i (i =, 2) such ha g (, x, y) ρ +ρ x + ρ 2 y and g 2 (, x, y) δ +δ x + δ 2 y for all x, y R (i =, 2). (H 3 ): here exis real consans M,M 2 > L (x, y) M and L 2 (x, y) M 2 for each x, y C([, ]). (H 4 ): here exis real consans γ,γ,γ,γ 2 such ha L (x,y )L (x 2,y 2 ) γ x x 2 +γ 2 y y 2 and L 2 (x,y )L 2 (x 2,y 2 ) γ x x 2 +γ 2 y y 2. For breviy, le us se μ = μ 2 = 2L Γ (α+), 2L 2 Γ(β+), (52) μ = min {(μ ρ +μ 2 δ ),(μ ρ 2 +μ 2 δ 2 )}. (53) Now we presen our resul for he exisence and uniqueness of soluions for problem (49). his resul is based on Banach s conracion mapping principle. heorem 9. Suppose ha condiions (H ), (H 3 ),and(h 4 ) hold and ha g,g 2 : [, ] R 2 R are coninuous funcions. In addiion, here exis posiive consans η i, ξ i, i =,2, such ha g (, x,y )g (, x 2,y 2 ) =η x x 2 +η 2 y y 2, g 2 (, x,y )g 2 (, x 2,y 2 ) =ξ x x 2 +ξ 2 y y 2, [, ], x,x 2,y,y 2 R. (54) If μ (η +η 2 )+μ 2 (ξ +ξ 2 )+L (γ +γ 2 )+L 2 (γ +γ 2 )<,hen problem (49) has a unique soluion. Proof. Le us se sup [,] g (,, ) = κ < and sup [,] g 2 (,,) = κ 2 < and define a closed ball: B r = {(x, y) K R : (x, y) r},where r μ κ +μ 2 κ 2 +M L +M 2 L 2 μ (η +η 2 )μ 2 (ξ +ξ 2 ). (55) Claim 5 (we show ha ΦB r B r ). Le (x, y) B r.wehave Φ (x, y) () M [ sup { [,] (s) α g (s, x (s),y(s)) ds + (s) α g (s, x (s),y(s)) ds} + L (x, y) ] =M [ sup { [,] (s) α ( g (s, x (s),y(s))g (s,, ) + g (s,, ) )ds + (s) α ( g (s, x (s),y(s))g (s,, ) + g (s,, ) )ds} + L (x, y) ] M 2 [ Γ (α+) (η x +η 2 y +κ ) 2 +L ] M [ Γ (α+) ((η +η 2 )r+κ )+L ] μ [(η +η 2 )r+κ ]+M L. Hence, (56) Φ (x, y) μ [(η +η 2 )r+κ ]+M L, (57) Φ 2 (x, y) μ 2 [(ξ +ξ 2 )r+κ 2 ]+M 2 L 2. From (57), i follows ha Φ(x, y) r. Nex, for (x,y ), (x 2,y 2 ) K R and for any [,], we have Φ (x 2,y 2 ) () Φ (x,y ) () (s) α L [ g (s, x 2 (s),y 2 (s)) g (s, x (s),y (s)) ds (s) α + g (s, x 2 (s),y 2 (s)) g (s, x (s),y (s)) ds + L (x 2,y 2 ) L (x,y ) ] L 2 [ Γ (α+) (η x 2 () x () +η 2 y 2 () y () )+γ x 2 () x () +γ 2 y 2 () y () ] μ (η x 2 x +η 2 y 2 y )+L (γ x 2 x +γ 2 y 2 y ) [μ (η +η 2 )+L (γ +γ 2 )] ( x 2 x + y 2 y ) (58)

8 8 Inernaional Differenial Equaions which yields Φ (x 2,y 2 )Φ (x,y ) [μ (η +η 2 )+L (γ +γ 2 )] ( x 2 x + y 2 y ). Working in a similar manner, one can find ha Φ 2 (x 2,y 2 )Φ 2 (x,y ) [μ 2 (ξ +ξ 2 )+L 2 (γ +γ 2 )] ( x 2 x + y 2 y ). We deduce ha Φ(x 2,y 2 )Φ(x,y ) [μ (η +η 2 ) +μ 2 (ξ +ξ 2 )+L (γ +γ 2 )+L 2 (γ +γ 2 )] (59) (6) (6) hus, for any x, y S,wecange Φ (x, y) () (s) α L [ g (s, x (s),y(s)) ds which yields (s) β + g (s, x (s),y(s)) ds + () L (x, y)] L N 2 Γ (α+) +M (63) Φ (x, y) =N μ +M. (64) In a similar manner, Φ 2 (x, y) =N 2μ 2 +M 2. (65) We deduce ha he operaor Φ is uniformly bounded. Now we show ha he operaor Φ is equiconinuous. We ake τ,τ 2 [, ] wih τ <τ 2 and obain ( x 2 x + y 2 y ). In view of condiion μ (η +η 2 )+μ 2 (ξ +ξ 2 )+L (γ +γ 2 )+ L 2 (γ +γ 2 )<,ifollowshaφ is a conracion. So Φ has a unique fixed poin. his implies ha problem (49) has a unique soluion on [, ].hiscompleesheproof. In our second resul, we discuss he exisence of soluions for problem (49) by means of Leray-Schauder alernaive. Lemma (see [7]). Le F : J J be a compleely coninuous operaor (i.e., a map ha is resriced o any bounded se in G is compac). Le P(F) = {x J : x = λfx for some < λ < }. hen eiher he se P(F) is unbounded or F has a leas one fixed poin. heorem. Assume ha condiions (H ) (H 3 ) hold. Furhermore, i is assumed ha μ ρ +μ 2 δ <and μ ρ 2 +μ 2 δ 2 <,whereμ and μ 2 are given by (52). hen he boundary value problem (49) has a leas one soluion. Proof. We will show ha he operaor Φ:K R K R saisfies all he assumpions of Lemma. In he firs sep, we prove ha he operaor Φ is compleely coninuous. Clearly, i follows by he coninuiy of funcions f, f 2, g,andg 2 ha heoperaorφ is coninuous. Le S K R be bounded. hen we can find posiive consans N and N 2 such ha g (, x (),y()) N g 2 (, x (),y()) N 2, (x,y) S. (62) Φ (x (τ 2 ),y(τ 2 )) Φ (x (τ ),y(τ )) τ 2 L N (τ 2 s) α +L N τ 2 τ τ L N τ 2 τ τ ds (τ s) α ds (s) α ds + M τ 2 τ (τ s) α (τ 2 s) α ds α (τ 2 s) ds +L N τ 2 τ (s) α ds + M τ 2 τ τ τ 2, Φ 2 (x (τ 2 ),y(τ 2 )) Φ 2 (x (τ ),y(τ )) τ 2 L 2 N 2 (τ 2 s) β τ ds (τ s) β ds +L 2 N 2 τ 2 τ (s) β ds + M 2 τ 2 τ τ L 2 N 2 τ 2 τ β (τ 2 s) (τ s) β (τ 2 s) β ds ds +L 2 N 2 τ 2 τ (s) β ds + M 2 τ 2 τ τ τ 2, (66)

9 Inernaional Differenial Equaions 9 which end o independenly of (x, y). his implies ha he operaor Φ(x, y) is equiconinuous. hus, by he above findings, he operaor Φ(x, y) is compleely coninuous. In he nex sep, i will be esablished ha he se P = {(x, y) K R/(x, y) = λφ(x, y), λ } is bounded. Le (x, y) P.henwehave(x, y) = λφ(x, y).hus,for any [, ],wecanwrie hen, x () x () =λφ (x, y) (), y () =λφ 2 (x, y) (). 2L Γ (α+) (ρ +ρ x +ρ 2 y )+M, y () 2L 2 Γ(β+) (δ +δ x +δ 2 y )+M 2, which imply ha hus, x μ (ρ +ρ x +ρ 2 y )+M, y μ 2 (δ +δ x +δ 2 y )+M 2. x + y (μ ρ +μ 2 δ +M +M 2 ) which, in view of (55), gives +(μ ρ +μ 2 δ ) x +(μ ρ 2 +μ 2 δ 2 ) y, (67) (68) (69) (7) (x, y) μ ρ +μ 2 δ +M +M 2 μ. (7) his shows ha he se is bounded. Hence, all he condiions of Lemma are saisfied and consequenly he operaor Φ has a leas one fixed poin, which corresponds o a soluion of problem (49). his complees he proof. Compeing Ineress he auhors declare ha here are no compeing ineress regarding he publicaion of his paper. [4] A. A. Kilbas, H. M. Srivasava, and J. J. rujillo, heory and Applicaions of Fracional Differenial Equaions, vol.24 of Norh-Holland Mahemaics Sudies, Elsevier Science B.V., Amserdam, he Neherlands, 26. [5] V. Lakshmikanham and A. S. Vasala, Basic heory of fracional differenial equaions, Nonlinear Analysis: heory, Mehods & Applicaions,vol.69,no.8,pp ,28. [6] V. Lakshmikanham, heory of fracional funcional differenial equaions, Nonlinear Analysis: heory, Mehods & Applicaions, vol. 69, no., pp , 28. [7] I. Podlubny, Fracional Differenial Equaions, vol. 98ofMahemaics in Sciences and Engineering, Academic Press, San Diego, Calif, USA, 999. [8] B. C. Dhage, On a condensing mappings in Banach algebras, he Mahemaics Suden,vol.63,no. 4,pp.46 52,994. [9] B. C. Dhage, A nonlinear alernaive in Banach algebras wih applicaions o funcional differenial equaions, Nonlinear Funcional Analysis and Applicaions, vol. 8, pp , 24. [] B. C. Dhage, On a fixed poin heorem in Banach algebras wih applicaions, Applied Mahemaics Leers, vol. 8, no. 3, pp , 25. [] B. C. Dhage and V. Lakshmikanham, Basic resuls on hybrid differenial equaions, Nonlinear Analysis: Hybrid Sysems,vol. 4, no. 3, pp , 2. [2] B. C. Dhage, Quadraic perurbaions of periodic boundary value problems of second order ordinary differenial equaions, Differenial Equaions & Applicaions,vol.2,no.4,pp , 2. [3] Y. Zhao, S. Sun, Z. Han, and Q. Li, heory of fracional hybrid differenial equaions, Compuers and Mahemaics wih Applicaions, vol. 62, no. 3, pp , 2. [4] K. Hilal and A. Kajouni, Boundary value problems for hybrid differenial equaions wih fracional order, Advances in Difference Equaions,vol.25,aricle83,25. [5] K. Ezzinbi, X. Fu, and K. Hilal, Exisence and regulariy in he α-norm for some neural parial differenial equaions wih nonlocal condiions, Nonlinear Analysis: heory, Mehods & Applicaions,vol.67,no.5,pp ,27. [6] S. Zhang, Posiive soluions for boundary value problems of nonlinear fracional differenial equaions, Elecronic Differenial Equaions,vol.36,pp. 2,26. [7] A. Granas and J. Dugundji, Fixed Poin heory, Springer, New York, NY, USA, 23. References [] R. P. Agarwal, V. Lakshmikanham, and J. J. Nieo, On he concep of soluion for fracional differenial equaions wih uncerainy, Nonlinear Analysis: heory, Mehods & Applicaions,vol.72,no.6,pp ,2. [2] L. Byszewski and V. Lakshmikanham, heorem abou he exisence and uniqueness of a soluion of a nonlocal absrac CauchyprobleminaBanachspace, Applicable Analysis,vol.4, no., pp. 9, 99. [3] A. M. El-Sayed, Fracional order evoluion equaions, Journal of Fracional Calculus,vol.7,pp.89,995.

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