L 1 -Solutions for Implicit Fractional Order Differential Equations with Nonlocal Conditions
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1 Filoma 3:6 (26), DOI.2298/FIL66485B Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp:// L -Soluions for Implici Fracional Order Differenial Equaions wih Nonlocal Condiions Mouffak Benchohra a,b, Mohammed Said Souid c a Laboraory of Mahemaics, Universiy of Sidi Bel Abbès, P.O. Box 89, Sidi Bel Abbès 22, Algeria b Deparmen of Mahemaics, Faculy of Science, King Abdulaziz Universiy, P.O. Box 823, Jeddah 2589, Saudi Arabia c Déparemen de Science Economique, Universié de Tiare, Algérie Absrac. In his paper we sudy he exisence of inegrable soluions of he nonlocal problem for fracional order implici differenial equaions wih nonlocal condiion. Our resuls are based on Schauder s fixed poin heorem and he Banach conracion principle fixed poin heorem.. Inroducion The opic of fracional calculus (inegraion and differeniaion of fracional-order), which concerns singular inegral and inegro-differenial operaors, is enjoying ineres among mahemaicians, physiciss and engineers. Indeed, we can find numerous applicaions of differenial equaions of fracional order in viscoelasiciy, elecrochemisry, conrol, porous media, elecromagneic, ec. (see [5, 3, 6, 7, 9]). There has been a significan developmen in ordinary and parial fracional differenial equaions in recen years; see he monographs of Abbas e al. [3, 4], Kilbas e al. [4], Lakshmikanham e al. [5], and he papers by Agarwal e al [, 2], Belarbi e al. [6], Benchohra e al. [7], and he references herein. To our knowledge, he lieraure on inegral soluions for fracional differenial equaions is very limied. El-Sayed and Hashem [2] sudied he exisence of inegral and coninuous soluions for quadraic inegral equaions. El-Sayed and Abd El Salam considered L p -soluions for a weighed Cauchy problem for differenial equaions involving he Riemann-Liouville fracional derivaive. Moivaed by he above papers, in his paper we deal wih he exisence of soluions of he nonlocal problem, for fracional order implici differenial equaion c D α y() = f (, y(), c D α y()), a.e, J =: (, T], () a k y( k ) = y, (2) 2 Mahemaics Subjec Classificaion. 26A33, 34A8. Keywords. Implici fracional-order differenial equaion, Capuo fracional derivaive, Banach conracion fixed poin heorem, Nonlocal condiion. Received: 23 April 24; Acceped: 4 Augus 24 Communicaed by Dragan S. Djordjević addresses: benchohra@yahoo.com (Mouffak Benchohra), souimed28@yahoo.com (Mohammed Said Souid)
2 M. Benchohra, M. S. Souid / Filoma 3:6 (26), where f : J IR IR IR is a given funcion, y IR, a k IR, c D α is he Capuo fracional derivaive, and < < 2 <..., m < T, k =, 2,..., m. This paper is organized as follows. In Secion 2, we will recall briefly some basic definiions and preliminary facs which will be used hroughou he following secion. In Secion 3, we give wo resuls, he firs one is based on Schauder s fixed poin heorem (Theorem 3.3) and he second one on he Banach conracion principle (Theorem 3.4). An example is given in Secion 4 o demonsrae he applicaion of our main resuls. Le us menion ha mos of he exising resuls for fracional order differenial equaions are devoed o coninuous or Carahéodory soluions. Thus, he main resuls of he presen paper consiue a conribuion o his emerging field. 2. Preliminaries In his secion, we inroduce noaions, definiions, and preliminary facs which are used hroughou his paper. Le L (J) denoes he class of Lebesgue inegrable funcions on he inerval J = [, T], wih he norm u L = u() d. J Definiion 2...([4, 8]). The fracional (arbirary) order inegral of he funcion h L ([a, b], R + ) of order α R + is defined by Ia α h() = ( s) α h(s)ds, a where Γ(.) is he gamma funcion. When a =, we wrie I α h() = h() ϕ α (), where ϕ α () = α f or >, and ϕ α () = f or, and ϕ α δ() as α, where δ is he dela funcion. Definiion ([4, 8]). The Riemann-Liouville fracional derivaive of order α > of funcion h L ([a, b], R + ), is given by (D α d ) n a+h)() = ( s) Γ(n α)( n α h(s)ds, d Here n = [α] + and [α] denoes he ineger par of α. If α (, T], hen (D α a+h)() = d d I α a+ h() = a d Γ( α) ds a ( s) α h(s)ds. Definiion ([4]). The Capuo fracional derivaive of order α > of funcion h L ([a, b], R + ) is given by ( c D α a+h)() = ( s) n α h (n) (s)ds, Γ(n α) a where n = [α] +. If α (, T], hen ( c D α a+h)() = I α a+ d d h() = a ( s) α d Γ( α) ds h(s)ds. The following properies are some of he main ones of he fracional derivaives and inegrals. Proposiion 2.4. [4] Le α, β >. Then we have
3 M. Benchohra, M. S. Souid / Filoma 3:6 (26), (i) I α : L (J, R + ) L (J, R + ), and if f L (J, R + ), hen I α I β f () = I β I α f () = I α+β f (). (ii) If f L p (J, R + ), p +, hen I α f Lp Tα Γ(α+) f L p. (iii) lim α n I α f () = I n f (), n =, 2,... uniformly. The following heorems will be needed. Theorem 2.5. (Schauder fixed poin heorem []) Le E a Banach space and Q be a convex subse of E and T : Q Q is compac, and coninuous map. Then T has a leas one fixed poin in Q. Theorem 2.6. (Kolmogorov compacness crierion []) Le Ω L p ([, T], IR), p. If (i) Ω is bounded in L p ([, T], R), and (ii) u h u as h uniformly wih respec o u Ω, hen Ω is relaively compac in L p ([, T], R), where u h () = h +h u(s)ds. 3. Exisence of Soluions Le us sar by defining wha we mean by an inegrable soluion of he nonlocal problem () (2). Definiion 3... A funcion y L ([, T], R) is said o be a soluion of problem () (2) if y saisfies () and (2). In wha follows, we assume ha m a k. Se a = m a. k For he exisence of soluions for he nonlocal problem () (2), we need he following auxiliary lemma. Lemma 3.2. The nonlocal problem () (2) is equivalen o he inegral equaion ( k s) α ( s) α y() = ay a a k x(s)ds + x(s)ds, (3) where x is he soluion of he funcional inegral equaion x() = f, ay ( k s) α ( s) α a a k x(s)ds) + x(s)ds, x(). (4)
4 M. Benchohra, M. S. Souid / Filoma 3:6 (26), Proof. Le c D α y() = x()) in equaion (), hen and x() = f (, y(), x()) (5) y() = y() + I α x()) = y() + Le = k in (6), we obain and a k y( k ) = ( s) α x(s)ds. (6) k y( k ) = y() + ( k s) α x(s)ds, ( k s) α a k y() + a k x(s)ds. (7) Subsiue from (2) ino (7), we ge and y = ( k s) α a k y() + a k x(s)ds, k y() = a y ( k s) α a k x(s)ds. (8) Subsiue from (8) ino (6) and (5), we obain (3) and (4). For complee he proof, we prove ha equaion (3) saisfies he nonlocal problem () (2). Differeniaing (3), we ge c D α y() = x() = f (, y(), c D α y()). Le = k in (3), we obain ( k s) α k ( k s) α y( k ) = ay a a k x(s)ds) + x(s)ds = ay + a ( k s) a k α x(s)ds. Then a k y( k ) = ( k s) a k ay + a k a a k α x(s)ds = y. This complee he proof of he equivalen beween he nonlocal problem ()-(2) and he inegral equaion (3). Leu us inroduce he following assumpions: (H) f : [, T] R 2 R is measurable in [, T], for any (u, u 2 ) R 2 and coninuous in (u, u 2 ) R 2, for almos all [, T]. (H2) There exis a posiive funcion a L [, T] and consans, b i > ; i =, 2 such ha: f (, u, u 2 ) a() + b u + b 2 u 2, (, u, u 2 ) [, T] R 2.
5 M. Benchohra, M. S. Souid / Filoma 3:6 (26), Our firs resul is based on Schauder fixed poin heorem. Theorem 3.3. Assume ha he assumpions (H) (H2) are saisfied. If 2b T α Γ(α + ) + b 2 <, (9) hen he problem () (2) has a leas one soluion y L ([, T], IR). Proof. Transform he nonlocal problem () (2) ino a fixed poin problem. Consider he operaor defined by: Le H : L ([, T], IR) L ([, T], IR) (Hx)() = f, ay ( k s) α ( s) α a a k x(s)ds) + x(s)ds, x(). () r = Tab y + a L ( 2b T α Γ(α+) + b ), 2 and consider he se B r = {x L ([, T], IR) : x L r}. Clearly B r is nonempy, bounded, convex and closed. Now, we will show ha HB r B r, indeed, for each x B r, from (9) and () we ge Hx L = = Hx() d f, ay ( k s) α ( s) α a a k x(s)ds) + x(s)ds, x() d a() + b ay a a k I α x() =k + I α x() + b 2 x() d Tab y + a L + b a m a k α k Γ(α + ) ( 2b T α Tab y + a L + Γ(α + ) + b 2 r. x L + ) x L b T α Γ(α + ) x L + b 2 x L Then HB r B r. Assumpion (H) implies ha H is coninuous. Now, we will show ha H is compac, his is HB r is relaively compac. Clearly HB r is bounded in L ([, T], IR), i.e condiion (i) of Kolmogorov compacness crierion is saisfied. I remains o show (Hx) h (Hx) in L ([, T], IR) for each x B r.
6 Le x B r, hen we have (Hx) h (Hx) L M. Benchohra, M. S. Souid / Filoma 3:6 (26), = (Hx) h () (Hx)() d = +h (Hx)(s)ds (Hx)() h d ( +h ) (Hx)(s) (Hx)() ds d h +h sk f h, ay (s k τ) α s (s τ) α a a k x(τ)dτ) + x(τ)dτ, x(s) f, ay ( k s) α ( s) α a a k x(s)ds) + x(s)ds, x() dsd. a k k Since x B r L ([, T], IR) and assumpion (H2) ha implies f L ([, T], IR), i follows ha +h h f (, ay a m sk (s k τ) α x(τ)dτ + s (s τ) α x(τ)dτ, x(s) ) f (, ay a m a k ( k s) α x(s)ds + ( s) α x(s)ds, x() ) ds as h. Hence (Hx) h (Hx) uniformly as h. Then by Kolmogorov compacness crierion, HB r is relaively compac. As a consequence of Schauder s fixed poin heorem he nonlocal problem () (2) has a leas one soluion in B r. The following resul is based on he Banach conracion principle. Theorem 3.4. Assume ha (H) and he following condiion hold. (H3) There exis consans k, k 2 > such ha If 2k T α Γ(α + ) + k 2 <, f (, x, y ) f (, x 2, y 2 ) k x x 2 + k 2 y y 2, [, T], x, x 2, y, y 2 IR. hen he problem () (2) has a unique soluion y L ([, T], IR). Proof. We shall use he Banach conracion principle o prove ha H defined by () has a fixed poin. Le x, y L ([, T], IR), and [, T]. Then we have, = (Hx)() (Hy)() f (, ay a a k I α x() =k + I α x(), x()) f (, ay a k a a k I α y() =k + I α y(), y()) ( k s) α a k x(s) y(s) ds ( s) α +k x(s) y(s) ds + k 2 x y. ()
7 Thus M. Benchohra, M. S. Souid / Filoma 3:6 (26), (Hx) (Hy) L k α k a m a k x() y() d + k T α x() y() d Γ(α + ) Γ(α + ) +k 2 x() y() d 2k T α Γ(α + ) x y L + k 2 x y L ( ) 2k T α Γ(α + ) + k 2 x y L. Consequenly by () H is a conracion. As a consequence of he Banach conracion principle, we deduce ha H has a fixed poin which is a soluion of he nonlocal problem () (2). 4. Example Le us consider he following fracional nonlocal problem, c D α y() = (e + 5)( + y() + c D α, J := [, ], α (, ], (2) y() ) a k y( k ) =, where a k IR, < < 2 <... <. Se f (, y, z) = (e, (, y, z) J [, + ) [, + ). + 5)( + y + z) Le y, z [, + ) and J. Then we have f (, y, z ) f (, y 2, z 2 ) = ( ) e y + z + y 2 + z 2 y y 2 + z z 2 (e + 5)( + y + z )( + y 2 + z 2 ) (e + 5) ( y y 2 + z z 2 ) (3) 6 y y z z 2. Hence condiion (H3) holds wih k = k 2 = 6. We shall check ha condiion () is saisfied. Indeed 2k Γ(α + ) + k 2 = 3Γ(α + ) + <. (4) 6 Then by Theorem 3.2, he nonlocal problem (2) (3) has a unique inegrable soluion on [, ]. References [] R. P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differenial equaions and inclusions involving Riemann-Liouville fracional derivaive. Adv Differ. Equa. 29(29) Aricle ID 98728, -47.
8 M. Benchohra, M. S. Souid / Filoma 3:6 (26), [2] R.P Agarwal, M. Benchohra and S. Hamani, A survey on exisence resul for boundary value problems of nonlinear fracional differenial equaions and inclusions, Aca. Appl. Mah. 9 (3) (2), [3] S. Abbas, M. Benchohra and G.M. N Guérékaa, Topics in Fracional Differenial Equaions, Springer, New York, 22. [4] S. Abbas, M. Benchohra and G.M. N Guérékaa, Advanced Fracional Differenial and Inegral Equaions, Nova Science Publishers, New York, 25. [5] D. Baleanu, K. Diehelm, E. Scalas, J.J. Trujillo, Fracional Calculus Models and Numerical Mehods, World Scienific Publishing, New York, 22. [6] A. Belarbi, M. Benchohra and A. Ouahab, Uniqueness resuls for fracional funcional differenial equaions wih infinie delay in Fréche spaces, Appl. Anal. 85 (26), [7] M. Benchohra, J. Henderson, S.K. Nouyas and A. Ouahab, Exisence resuls for funcional differenial equaions of fracional order, J. Mah. Anal. Appl. 338 (28), [8] M. Benchohra, S. Hamani, and S.K. Nouyas, Boundary value problems for differenial equaions wih fracional order and nonlocal condiions, Nonlinear Anal. 7 (29), [9] M. Benchohra, S. Hamani and S.K. Nouyas, Boundary value problems for differenial equaions wih fracional order, Surveys Mah. Appl. 3 (28), -2. [] K. Deimling, Nonlinear Funcional Analysis, Springer-Verlag, Berlin, 985. [] A. M. A. El-Sayed, Sh. A. Abd El-Salam, L p -soluion of weighed Cauchy-ype problem of a differr-inegral funcional equaion, Inern. J. Nonlinear Sci. 5 (28) [2] A.M.A. El-Sayed, H.H.G. Hashem, Inegrable and coninuous soluions of a nonlinear quadraic inegral equaion, Elecron.J. Qual. Theory Differ. Equ. No [3] R. Hilfer, Applicaions of Fracional Calculus in Physics, World Scienific, Singapore, 2. [4] A.A. Kilbas, H.M. Srivasava, and J.J. Trujillo, Theory and Applicaions of Fracional Differenial Equaions. Norh-Holland Mahemaics Sudies, 24. Elsevier Science B.V., Amserdam, 26. [5] V. Lakshmikanham, S. Leela and J. Vasundhara, Theory of Fracional Dynamic Sysems, Cambridge Academic Publishers, Cambridge, 29. [6] F. Mainardi, Fracional Calculus and Waves in Linear Viscoelasiciy. An inroducion o mahemaical models. Imperial College Press, London, 2. [7] M. D. Origueira, Fracional Calculus for Scieniss and Engineers. Lecure Noes in Elecrical Engineering, 84. Springer, Dordrech, 2. [8] I. Podlubny, Fracional Differenial Equaions, Academic Press, San Diego, 999. [9] V. E. Tarasov, Fracional Dynamics: Applicaion of Fracional Calculus o Dynamics of Paricles, Fields and Media, Springer, Heidelberg; Higher Educaion Press, Beijing, 2.
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