Some New Uniqueness Results of Solutions to Nonlinear Fractional Integro-Differential Equations
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1 Annals of Pure and Applied Mahemaics Vol. 6, No. 2, 28, ISSN: X (P), (online) Published on 22 February 28 DOI: hp://dx.doi.org/.22457/apam.v6n2a Annals of Some New Uniqueness Resuls of Soluions o Nonlinear Fracional Inegro-Differenial Equaions Mohammed S. Abdo and Saish K. Panchal 2 Deparmen of Mahemaics Dr. Babasaheb Ambedkar Marahwada Universiy Aurangabad (M.S.), 434 India. msabdo977@gmail.com; 2 drpanchalsk@gmail.com Received 9 February 28; acceped 22 February 28 Absrac. The aim of his paper is o prove he new uniqueness of soluions for nonlinear inegr-odifferenial equaions wih fracional of arbirary order which includes he Capuo fracional operaor derivaive. The echniques employed are a variey of ools such as fracional calculus and is properies, Bihari's inequaliy, and Banach conracion mapping principle. Keywor: Fracional derivaive and inegral, Inegro-differenial equaions, Inegral inequaliies, fixed poin heorem. AMS Mahemaics Subjec Classificaion (2): 26A33, 26D, 47H. Inroducion This paper is concerned wih he uniqueness resuls for fracional inegro differenial equaions of he ype c (.) D y ( ) = a( ) y ( ) f (, y ( )) K (, s, y ( s )), [, b ], wih iniial condiion y () = y, (.2) where <, a C ([, b ], R ), f C ([, b ] RR, ). K C ([, b ] [, b ] RR, ) and saisfying some condiions which will be menioned laer and c D is he Capuo fracional derivaive. The sudy of fracional differenial equaions is relaed o he exensive applicaions of fracional calculus in mechanics, physics, chemical echnology, bioechnology, and so forh. In laes years, here has been an ineres in he look a of he heory fracional differenial equaions, i has seen significan improvemen, see as an insance he monographs of Delboso and Rodino [8], Diehelm [9], Kilbas e al. [], Miller and Ross [] Oldham and Spanier [3] and he references herein. Recenly, uniqueness sandar for he various fracional inegro-differenial equaions have been aken ino consideraion by way some auhors, for greaer info, (see [,2,3,4,5,6,7,2,6]). For example in [2], 345
2 M. S. Abdo and S. K. Panchal Momani e al. sudied he local uniqueness resul by applying Bihari's inequaliy for he following problem ( ) (, ( )) c D y f y K (, s, y ( s )), = y () = y, n n n n where <, f C ([, b ] R, R ), K C ([, b] [, b ] R, R ) and c D is he Capuo fracional operaor. In [6], Balachandran and Trujillo, discussed he exisence and uniqueness of soluions of quasilinear and semilinear of inegro-differenial equaions wih fracional order in Banach space by means of Banach fixed poin heorem. In his paper, we prove he uniqueness resuls of he fracional inegro-differenial equaions (.)-(.2) wih he aid of using Bihari's inequaliy and Banach conracion principle. The res of his paper is spli as follows. In Secion 2, we presen some mahemaical ools and lising he assumpions which have advanage on his paper. Secion 3 is dedicaed o proving he uniqueness of soluions for problem (.)-(.2). Finally, he conclusion and fuure work are given in Secion Some mahemaical ools In his secion, some required noaions, definiions and a lemmas are given. Le J = [, b ] and C ( J, R ) be he Banach space endowed wih he infiniy norm g = sup{ g ( ) : J }, for any g C ( J, R ), we also n C ( J, R ) be space of all real valued coninuous funcion which are coninuously differeniable on J. Definiion 2.. [] The Riemann-Liouville fracional inegral of order > of he funcion g C ( J, R ) is given by I g ( ) = ( s ) g ( s ), J, where Γ denoes he Gamma funcion. Definiion 2.2. [] Le n < < n. The Capuo's fracional operaor of he funcion n g C ( J, R ) is defined as c d d D g s g s I g J Γ( n ) d d n n n n ( ) = ( ) ( ) = ( ),. n n In paricular, if < <, hen c D g = I g J. ( ) d ( ), d Lemma 2.. [,5] For, β > and g, h are appropriae funcions hen, for J, we have, ) c D I g ( ) = g ( ). c 2) I D g ( ) = g ( ) g (), < <. 346
3 Some New Uniqueness Resuls of Soluions o Nonlinear Fracional Inegro-Differenial Equaions 3) 4) 5) I I g ( ) = I g ( ) = I I g ( ). β β β I ( g ( ) h( )) = I g ( ) I h( ). c D C =, where C is a consan. Lemma 2.2. [7] (Banach conracion principle). Le K be a non-empy closed subse of a Banach space X, hen each conracion mapping T : K K has a unique fixed poin. Lemma 2.3. [4] (Bihari's inequaliy). Le g :[ a, b ] R be a coninuous funcion ha saisfies he inequaliy g ( ) A K ( s ) ψ ( g ( s )), [ a, b ], a where A, K :[ a, b ] R is coninuous and ψ :[, ) (, ) is coninuous and monoone-increasing. Then he following inequaliy hold g ( ) Ψ Ψ ( A ) K ( s ), [ a, b ], a x where Ψ : R R is given by Ψ ( x ) =, x R. x ψ ( s ) 3. Main resuls In his secion, we shall exhibi and demonsraor he uniqueness resuls of problem (.)- (.2). Before beginning and proving our main resuls, we presen he following lemma: Lemma 3.. The iniial value problem (.)-(.2) is equivalen o he nonlinear inegral equaion y ( ) = y ( s ) a( s ) y ( s ) ( s ) f ( s, y ( s )) ( s ) K ( τ, s, y ( s )) dτ, J. (3.) s In oher hand, each soluion of he inegral equaion (3.) is likewise a soluion of iniial value problem (.)-(.2) and vice versa. Proof: I can be proved efforlessly by effec he inegral operaor I described in he Definiion 2.o boh sides of Eq.(.), so according of Lemma 2. and swich he order of inegraion (Fubini's heorem) o ge he inegral equaion (3.). Le us lis he hypoheses: (A) a : J R is coninuous. (A2) f : J R R is coninuous such ha f (, u ) f (, v ) ψ ( u v ), J, u, v R. (A3) K : J J R R, is coninuous on D such ha 347
4 M. S. Abdo and S. K. Panchal K s u s s K s v s dτ M u (,, ( )) (,, ( )) ( ), for J, u, v R, where ψ : R R is nondecreasing coninuous funcion wih R ψ () = and dx ψ ( x ) =, < x < R, D = {(, s ) : s b} and M is a posiive consan. Our firs resul depen on Bihari's inequaliy. Theorem 3.. Assume ha he hypoheses (A),(A2) and (A3) hold. If Γ ( ) <, hen he fracional inegro-differenial equaion (.)-(.2) has a unique soluion in C ( J, R ). Proof: By Lemma 3., we know ha he funcion y is a soluion o (.)-(.2) iff y saisfies y ( ) = y ( s ) a( s ) y ( s ) ( s ) f ( s, y ( s )) ( s ) K ( τ, s, y ( s )) dτ, J. s Le y, y 2 C ( J, R ) and for any J such ha y ( ) = y ( s ) a( s ) y ( s ) ( s ) f ( s, y ( s )) ( s ) K ( τ, s, y ( s )) dτ s and y 2( ) = y ( s ) a( s ) y 2( s ) ( s ) f ( s, y 2( s )) ( s ) K ( τ, s, y 2( s )) dτ. s Consequenly, by (A), (A2) and (A3), hen for J, we have y ( ) y2( ) ( s) a( s) y ( s) y2( s) ( s) f ( s, y ( s)) f ( s, y2( s)) ( s) K( τ, s, y ( s)) K( τ, s, y2( s)) dτ s ( s) sup a( s) y ( s) y2( s) s J a b 348
5 Some New Uniqueness Resuls of Soluions o Nonlinear Fracional Inegro-Differenial Equaions where ( s) ψ ( y ( s) y2( s) ) ( s) M ψ ( y ( s) y2( s) ) a b M y ( ) y2( ) ( s) ψ ( y ( s) y2( s) ) Γ ( ) M ( s) ψ ( y ( s) y2( s) ) a b Γ ( ) M < ε a b Γ ( ) a b Γ ( ) ( s) ψ ( y ( s) y2( s) ), ε = >. We can now apply Lemma 2.3 o obain M y ( ) y 2( ) Ψ Ψ ( ε ) ( s ) a b ( ) Γ Γ ( ) where Ψ (.) is a primiive of M b ( ε ), a b Γ ( ) Γ ( ) and Ψ (.) is called he inverse of Ψ (.). Ψ Ψ, ψ (.) M b I follows ha Ψ Ψ ( ε ) ( ). a Γ (see [2]). Consequenly, y ( ) = y 2( ), for [, b ]. So, y ( ) C ( J, R ) is he unique soluion o fracional inegro differenial equaion (.)-(.2) and he proof is compleed. Our second resul depend on he Banach conracion principle. For sake of convenien, we inroduce he following assumpions (B) a : J R is coninuous. (B2) f : J R R is coninuous and here exiss a consan L > such ha f (, u ) f (, v ) L u v, J, u, v R. (B3) K : J J R R is coninuous and here exiss a consan L > such ha K ( τ, s, u ( s )) K ( τ, s, v ( s )) L u v, (, s ) D, u, v R, where D is defined as in Theorem
6 M. S. Abdo and S. K. Panchal Theorem 3.2. Assume ha he hypoheses (B)-(B3) are saisfied. And le ζ and be wo posiive real numbers such ha < ζ < and a L L b b = ζ, Γ ( ) ( ) q q b y b = ( ζ ). Γ ( ) ( ) Then he fracional inegro-differenial equaion (.)-(.2) has a unique soluion coninuous on [, b ], where q = max f (,) : J } and q = max K ( τ, s,) : ( τ, s ) D}. Proof: Le he operaor T : C ( J, R) C ( J, R ) be defined by ( Ty )( ) = y ( s ) a( s ) y ( s ) ( s ) f ( s, y ( s )) ( s ) K ( τ, s, y ( s )) dτ. s and define B = { y C ( J, R ) : y } for some >. Now, we need o prove ha he operaor T has a fixed poin on B C ( J, R ). This fixed poin is he unique soluion of (.)-(.2). In order ha, we presen he proof in wo seps: Sep : We show ha TB B. By he hypoheses, hen for any y and for J, we have B Ty y s a s y s s f s y s ( )( ) ( ) ( ) ( ) ( ) (, ( )) ( s) K( τ, s, y( s)) dτ s y ( s) a y ( ) Γ ( s) ( f ( s, y( s)) f ( s,) f ( s,) ) ( s) ( K( τ, s, y( s)) K( τ, s,) K( τ, s,) ) dτ s b a b b y L q L q Γ ( ) Γ ( ) ( ) Γ( ) ( ) ( ) 35
7 Some New Uniqueness Resuls of Soluions o Nonlinear Fracional Inegro-Differenial Equaions q q b a L L b = y b b ( ) ( ) ( ) ( ) ( ) ( ) Γ Γ Γ Γ = ( ζ ) ζ =. I follows ha Ty, his implise ha Ty B which lea o TB B. Sep 2: We shall show ha T : B B is a conracion mapping. Indeed, hrough he assumpions, hen for any y, y ( Ty )( ) ( Ty )( ) B and for J, we can wrie ( s ) a( s ) y ( s ) y ( s ) ( s ) f ( s, y ( s )) f ( s, y ( s )) ( s ) K ( τ, s, y ( s )) K ( τ, s, y ( s )) dτ s b a b L b L y y y y y y Γ ( ) Γ ( ) ( ) a L L b = b y y Γ ( ) ( ) = ζ y y Since ζ <, we ge. Ty Ty y y This implies ha T is conracion mapping. As consequence of Lemma 2.2, here exiss a fixed poin y C ( J, R ) such ha Ty = y which is he unique soluion of (.)-(.2) on J. This proves he required. 5. Conclusion and fuure work The main purpose of his paper was o presen new uniqueness resuls for nonlinear fracional inegro-differenial wih respec o he Capuo fracional operaor, highlighing he mehod which has been used by Momani e al. [2] o some resuls. The echniques used o prove our resuls are a variey of ools such as fracional calculus and is properies, Bihari's inequaliy and Banach fixed poin heorem. The possible generalizaion is o consider he problem (.)-(.2) on Banach space and deermine he condiions ha be fi closer o obain he bes resuls. As anoher proposal, considering he Capuo ype fracional derivaive wih respec o anoher funcion. These suggesions will be reaed in he fuure.. 35
8 M. S. Abdo and S. K. Panchal Acknowledgemens. The auhors are very graeful o referees for he valuable remarks and commens ha improved his paper. REFERENCES. M.S.Abdo and S.K.Panchal, Exisence and coninuous dependence for fracional neural funcional differenial equaions, J. Mah. Model., 5 (27) B.Ahmad and J.J.Nieo, Exisence resuls for nonlinear boundary value problems of fracional inegro-differenial equaions wih inegral boundary condiions, Boundary Value Problem, 29 (29) B.Ahmad and S.Sivasundaram, Some exisence resuls for fracional inegrodifferenial equaions wih nonlinear condiions, Communicaions Appl. Anal., 2 (28) A.Anguraj, P.Karhikeyan and G.M.N'guer ekaa, Nonlocal Cauchy problem for some fracional absrac inegro-differenial equaions in Banach spaces, Communicaions Mah. Anal., 6 (29) K.Balachandran and F.P.Samuel, Exisence of soluions for quasilinear delay inegrodifferenial equaions wih nonlocal condiion, Elecronic J. Diff. Equ., 29 (29) K.Balachandran and J.J.Trujillo, The nonlocal Cauchy problem for nonlinear fracional inegro-differenial equaions in Banach spaces, Nonlinear Anal. Theory Meh. Applic., 72 (2) T.G.Bhaskar, V.Lakshmikanham and S.Leela, Fracional differenial equaions wih Krasnoselskii-Krein-ype condiion, Nonlinear Anal. Hybrid Sys., 3 (29) D.Delboso and L.Rodino, Exisence and uniqueness for a nonlinear fracional differenial equaion, J. Mah. Anal. Appl., 24 (996) K.Diehelm, The Analysis of Fracional Differenial Equaions, Lecure Noes in Mahemaics 24, Springer, Berlin, 2.. A.A.Kilbas, H.M.Srivasava and J.J.Trujillo, Theory and Applicaions of Fracional Differenial Equaions, Norh-Holland Mah. Sud., 24, Elsevier, Amserdam, 26.. K.S.Miller and B.Ross, An Inroducion o he Fracional Calculus and Differenial Equaions, John Wiley, New York, S.Momani, A.Jameel and S.Al-Azawi, Local and global uniqueness heorems on fracional inegro-differenial equaions via Bihari's and Gronwall's inequaliies, Soochow Journal of Mahemaics, 33 (27) K.Oldham and J.Spanier, The fracional calculus heory and applicaions of differeniaion and inegraion o arbirary order,, Elsevier, M.R.Rao, Ordinary Differenial Equaions, Eas-Wes Press PVT Limied, S.G.Samko, A.A.Kilbas and O.I.Marichev, Fracional Inegrals and Derivaives. Theory and Applicaions, Gordon and Breach, Yverdon, J.Wu and Y.Liu, Exisence and uniqueness of soluions for he fracional inegrodifferenial equaions in Banach spaces. Elecronic Journal of Differenial Equaions, 29 (29), Y.Zhou, Basic heory of fracional differenial equaions, 6, Singapore: World Scienific,
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