EXISTENCE AND UNIQUENESS THEOREM FOR FRACTIONAL DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITION

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1 Journal of Fractional Calculu and Application, Vol. 3, July 212, No. 6, pp ISSN: EXISTENCE AND UNIQUENESS THEOREM FOR FRACTIONAL DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITION SHAYMA ADIL MURAD AND SAMIR BASHIR HADID Abtract. We have invetigated the exitence and uniquene olution of the nonlinear fractional differential equation of an arbitrary order with integral boundary condition. The reult i an application of the Schauder fixed point theorem and the Banach contraction principle. 1. Introduction In recent year, fractional differential equation have been of great interet to many mathematician. Thi i due to the development of the theory of fractional calculu itelf and the application of uch contruction in variou field of cience and engineering uch a: Control Theory, Phyic, Mechanic, Electrochemitry,... etc. There are many paper dicuing the olvability of nonlinear fractional differential equation and the exitence of poitive olution of nonlinear fractional differential equation, ee the monograph of Kilba et al. 1], Samko et al 11], and the paper 2, 3, 7, 8, 9, 1] and the reference therein. In 5], Benchohra and Ouaar conidered the boundary value problem of the fractional differential equation: D α y(t = f(t, y(t, α (, 1] y( µ T y(d = y(t where Dt α i the Caputo fractional derivative, and f :, 1] R R i continuou. In 6], Hu and Wang invetigated the exitence of olution of the nonlinear fractional differential equation with integral boundary condition: D α y(t = f(t, y(t, D β y(t, t (, 1, y( =, y(1 = g(y(d where 1 < α 2, < β < 1, and Dt α i the Riemann-Liouville fractional derivative, f :, 1] R R R, and g L 1, 1]. 2 Mathematic Subject Claification. 26A33. Key word and phrae. Fractional differential equation; Integral boundary condition; Schauder fixed point theorem; Banach contraction principle. Submitted: Jan. 27, 212. Accepted: May 15, 212. Publihed: July 1,

2 2 SHAYMA ADIL MURAD AND SAMIR BASHIR HADID JFCA-212/3 In thi paper, we conider the following boundary value problem for the nonlinear fractional differential equation with integral boundary condition: D α y(t = f(t, y(t, D β y(t, t (, 1, (1 y( =, y(1 = I 1 γ y( (2 where 1 < α 2, < β < 1, < γ 1, and Dt α i the Riemann-Liouville fractional derivative f :, 1] R R R. We hall prove that there exit a olution of the boundary value problem (1 and (2, by uing the Schauder fixed point theorem and Banach contraction principle. Firtly, we tarted by driving the correponding Green function of (1 and (2. Moreover, we add a certain condition to the above equation in order to obtain a unique olution. The reult i more general and contain uniquene olution. We verify the reult by contracting an intereting example. 2. PRELIMINARIES Firt of all, we recall ome baic definition Definition 2.111] Let p, q >, then the beta function β(p, q i defined a: β(p, q = x p 1 (1 x q 1 dx. Remark 2.211] For p, q >, the following identity hold: β(p, q = Γ(pΓ(q Γ(p q. Definition 2.311] Let f be a function which i defined almot everywhere (a.e on a, b], for α >, we define: b ad α f = 1 b provided that the integral (Lebegue exit. a (b α 1 f(d, Definition 2.41] The Riemann-Liouville fractional derivative of order α for a function f(t i defined by ( n t ad α 1 d t f(t = (t n α 1 f(d, Γ(n α dt a where α >, n = α] 1 and α] denote the integer part of α. Lemma 2.54] Let α >, then D α t D α t y(t = y(t c 1 t α 1 c 2 t α 2... c n t α n, for ome c i R, i =, 1,..., n 1, n = α] 1.

3 JFCA-212/3 EXISTENCE AND UNIQUENESS THEOREM 3 Lemma 2.6 Given y C(, 1, 1 < α 2, < β < 1 and < γ 1. Then, the unique olution of the boundary value problem (1 and (2, i given by: y(t = 1 (t α 1 f(, y(, D β y(d tα ] (1 r γ 1 (r α 1 dr (1 α 1 f(, y(, D β y(d. Γ(γ Proof. By applying Lemma 2.5, equation (1 can be reduced to the equivalent integral equation: y(t = 1 (t α 1 f(, y(, D β y(d c 1 t α 1 c 2 t α 2, (3 for c 1, c 2 R and y( =, we can obtain c 2 =. Then, we can write (3 a y(t = 1 and it follow from y(1 = I 1γ y(, that c 1 = where = c 1 = Γ(γ 1 Γ(γ 1 Γ(α γ (t α 1 f(, y(, D β y(d c 1 t α 1, (1 γ 1 ( r α 1 f(r, y(r, D β y(rdrd (1 α 1 f(, y(, D β y(d, (1 r γ 1 (r α 1 dr (1 α 1 ] f(, y(, D β y(d, ] 1. Therefore, the olution of (1 and (2 i given by y(t = 1 (t α 1 f(, y(, D β y(d tα 1 1 Γ(γ (1 r γ 1 (r α 1 dr (1 α 1 ] f(, y(, D β y(d, y(t = G(t, f(, y(, D β y(d. (4 Where G(t, i the Green function defined by: (t ] α 1 tα 1 1 (1 r γ 1 (r α 1 dr Γ(γ (1 α 1, if < t G(t, = t ] α 1 1 (1 r γ 1 (r α 1 dr Γ(γ (1 α 1, if t < 1 Thu, we complete the proof. Next, we define the pace X = { y(t C, 1] : D β y(t C, 1] },

4 4 SHAYMA ADIL MURAD AND SAMIR BASHIR HADID JFCA-212/3 endowed with the norm : i a Banach pace. y X = max t,1] y(t max t,1] D β y(t Lemma 2.712] (X,. X i a Banach pace. Let u et the following notation for convenience: w 1 = max t,1] G(t, a( d, w 2 = (t α β 1 a(d Γ(γ (1 α 1 a(d, (1 r γ 1 (r α 1 a(drd ( (1 Γ(α γ 1 Γ(α 1 Λ 1 =, Γ(α 1Γ(α γ 1 ( αγ(α γ 1 (α βγ(α 1 (α βγ(α γ 1 Λ 2 =. α(α βγ(α γ 1 3. Main Reult Our main reult of thi paper i a follow: Theorem 3.1 Let f :, 1] R R R be a continuou and aume that the following condition i atified: (H1 There exit a nonnegative function a(t L 1 (, 1] uch that: f(t, x, y a(t b 1 x ρ1 b 2 y ρ2, where b 1, b 2 and < ρ i < 1 for i = 1, 2. Then the problem(1 and (2 ha a olution. Proof: In order to ue Schauder fixed point theorem, to prove our main reult, we define U = {y(t X : y X R, t, 1]}. A firt tep, we hall prove that: T : U U. We have (T y(t = G(t, f(, y(, D β y( d, G(t, a( d G(t, d,

5 JFCA-212/3 EXISTENCE AND UNIQUENESS THEOREM 5 ( G(t, a( d tα 1 Γ(α 1 (t α 1 d tα 1 (1 r γ 1 (r α 1 drd Γ(γ tα 1 (1 α 1 d (b 1 R ρ1 b 2 R ρ2, ( t α G(t, a( d Γ(α 1 tα 1 (1 r γ 1 (r α 1 drd Γ(γ. Let u = r and dr = (1 du, we get 1 ( 1 G(t, a( d Γ(α 1 Γ(γ (1 αγ 1 d ( 1 G(t, a( d Γ(α 1 β(γ, α, (α γγ(γ ( 1 G(t, a( d Γ(α 1 Γ(α γ 1 ( (1 Γ(α γ 1 Γ(α 1 G(t, a( d Γ(α 1Γ(α γ 1, (1 u γ 1 u α 1 du,, and (D β T y(t (T y(t w 1 Λ 1 (b 1 R ρ1 b 2 R ρ2, (t α β 1 f(, y(, D β y( d tα β 1 Γ(γ (1 r γ 1 (r α 1 f(, y(, D β y( drd tα β 1 (t α β 1 a(d tα β 1 Γ(γ tα β 1 Γ(γ tα β 1 (1 α 1 f(, y(, D β y( d, (1 r γ 1 (r α 1 drd tα β 1 (1 r γ 1 (r α 1 a(drd ( (1 α 1 (t α β 1 a(d d (1 α 1 d,

6 6 SHAYMA ADIL MURAD AND SAMIR BASHIR HADID JFCA-212/3 (t α β 1 a(d Γ(γ 1 ( (1 α 1 a(d (t α β 1 a(d Γ(γ (D β T y(t ( (1 α 1 a(d (t α β 1 (1 r γ 1 (r α 1 a(drd t α β Γ(α β 1 t α β 1 β(α, γ (α γγ(γ tα β 1 α 1 Γ(α β 1 a(d Γ(γ (1 r γ 1 (r α 1 a(drd Γ(α γ 1 (1 r γ 1 (r α 1 a(drd (1 α 1 ( αγ(α γ 1 (α βγ(α 1 (α βγ(α γ 1, α(α βγ(α γ 1 Therefore (D β T y(t w 2 Λ 2, T y X = max T y max t,1] t,1] Dβ y(t, T y X w 3 Λ 3 R 3 R 3 R 3 R. Where w 3 = w 1 w 2, and Λ 3 = Λ 1 Λ 2. So, we conclude that T y X R. Since T y, D β T y are continuou on,1], therefor, T : U U. Now, we how that T i completely continuou operator. For thi purpoe we fix M = max t,1] f(, y(, D β y(, where y U, and t, τ, 1], with t < τ, then tα 1 T (y(t T (y(τ = M M (t α 1 d (1 α 1 d M ( 1 Γ(γ τ G(t, G(τ, f(, y(, D β y( d, G(t, G(τ, d, tα 1 Γ(γ α 1 τ (τ α 1 d Γ(γ α 1 τ 1 τ (1 α 1 d, (1 r γ 1 (r α 1 drd (1 r γ 1 (r α 1 drd (t α 1 d (τ α 1 d M tα 1 τ α 1 (1 r γ 1 (r α 1 drd (1 α 1 d, (b 1 R ρ1 b 2 R ρ2,, α

7 JFCA-212/3 EXISTENCE AND UNIQUENESS THEOREM 7 and tα β 1 M tα τ α M tα 1 τ α 1 ( β(α, γ Γ(α 1 (α γγ(γ 1, α M tα τ α ( M t α 1 τ α 1 1 Γ(α 1 Γ(α γ 1 1, α D β T (y(t D β T (y(τ = tα β 1 Γ(γ α β 1 τ (1 α 1 f(, y(, D β y(d Γ(γ τ α β 1 (t α β 1 f(, y(, D β y(d (1 r γ 1 (r α 1 f(, y(, D β y(drd τ (τ α β 1 f(, y(, D β y(d (1 r γ 1 (r α 1 f(, y(, D β y(drd (1 α 1 f(, y(, D β y(d, τ M (t α β 1 d (τ α β 1 d M tα β 1 τ α β 1 ( 1 (1 r γ 1 (r α 1 drd (1 α 1 d, Γ(γ M tα β τ α β M tα β 1 τ α β 1 ( β(α, γ Γ(α β 1 (α γγ(γ 1, α M tα β τ α β M tα β 1 τ α β 1 ( Γ(α β 1 Γ(α γ 1 1. α Since the function t α β 1, t α β, t α 1, and t α are uniformly continuou on, 1]. Therefore, it follow from the above etimate that T U i an equicontinuou et. Alo, it i uniformly bounded a T U U. Thu, we conclude that T i completely continuou operator. Theorem 3.2 Letf :, 1] R R R be a continuou. Aume that one of the following condition i atified: (H2 There exit a contant k > uch that f(t, x, y f(t, x, ȳ k ( x x y ȳ for each t, 1] and all x, y, x, ȳ R. (H3 ε = max {kλ 1, kλ 2 }. Then the problem (1 and (2 ha unique olution. Proof: Let x, y X, by (H2, we have T (x(t T (y(t = k G(t, f(, x(, D β x( f(, y(, D β y( d, G(t, x( y( D β x( D β y( ] d,

8 8 SHAYMA ADIL MURAD AND SAMIR BASHIR HADID JFCA-212/3 and ( 1 k Γ(α 1 (x y(t D β (x y(t ], Γ(α γ 1 T (x(t T (y(t kλ 1 (x y(t D β (x y(t ], D β T (x(t D β T (y(t tα β 1 Γ(γ tα β 1 (t α β 1 f(, x(, D β x( f(, y(, D β y( d (1 r γ 1 (r α 1 f(, x(, D β x( f(, y(, D β y( drd (1 α 1 f(, x(, D β x( f(, y(, D β y( d, t α β k Γ(α β 1 t α β 1 ] β(α, γ (α γγ(γ tα β 1 (x y(t D β (x y(t ], α ( αγ(α γ 1 (α βγ(α 1 (α βγ(α γ 1 (x k y(t D β (x y(t ], α(α βγ(α γ 1 then D β T (x(t D β T (y(t kλ 2 (x y(t D β (x y(t ], T (x(t T (y(t kλ 1 (x y(t D β (x y(t ], D β T (x(t D β T (y(t kλ 2 (x y(t D β (x y(t ], T (x(t T (y(t X = max T (x(t T (y(t max t,1] t,1] Dβ T (x(t D β T (y(t, T (x(t T (y(t X ε x y. Hence, we conclude that the problem (1 and (2 ha a unique olution by the contraction mapping principle. Thi end the proof. Next we give an example that atify the above theorem: 4. Example Conider the following boundary value problem: Then Let = { 3 D 2 y(t = f(t, y(t, D 1 2 y(t, t (, 1, y( =, y(1 = I y( (5 1 ] 1 = 1 Γ( 3 2 ] 1 Γ(α γ Γ( = f(t, y(t, D 1 2 y(t = t 6 e t 1(1 e t y(t D 1 2 y(t,

9 JFCA-212/3 EXISTENCE AND UNIQUENESS THEOREM 9 and k = 1 15, by a direct calculation, we get ( (1 Γ(α γ 1 Γ(α 1 Λ 1 = = , Γ(α 1Γ(α γ 1 and kλ 1 = < 1. Similarly ( αγ(α γ 1 (α βγ(α 1 (α βγ(α γ 1 Λ 2 = = , α(α βγ(α γ 1 and kλ 2 = < 1. ε = max {kλ 1, kλ 2 } = max { , } = < 1. Therefore, by Theorem 3.2, the boundary value problem (5 ha a unique olution. Reference 1] A. A. Kilba, H. M. Srivatava, and J. J. Trujillo, Theory and Application of Fractional Differential Equation, North-Holland Mathematic Studie, Elevier Science B.V., Amterdam, The Netherland,Vol. 24,(26. 2] A. M. EL-Sayed and E.O. Bin-Taher, Poitive Nondecreaing olution for a Multi-Term Fractional-Order Functional Differential Equation With Integral Condition, Electronic Journal of Dierential Equation, Vol. 211, No. 166, pp. 18, (211. 3] B. Ahmed and J. Nieto, Exitence Reult for nonlinear boundary value problem of fractional integrodifferential equation with integral boundary condition, Hindawi Publihing Corporation Boundary Value problem., Volume29, ArticleID 78576,pp.1-11, (29. 4] I. Podlubny, Fractional differential equation, Academic Pre, ] M. Benchohra and F. Ouaar,Exiyence Reult for nonlinear fractional differential equation with integral boundary condition, Bullten Of Mathematical analyi and Application, Vol.2 Iue4, pp.7-15, (21. 6] Meng. Hu and L. Wang, Exitence of olution for a nonlinear fractional differential equation with integral boundary condition, International Journal of Mathematical and Computer Science, (7(1 (211. 7] R. W. Ibrahim, Exitence and uniquene of holomorphic olution for fractional Cauchy problem, J. Math. Anal. Appl. 38, pp , (211. 8] S. B. Hadid, Local and global exitence theorem on differential equation of non-integer order, J. Fractional Calculu, Vol. 7, pp.11-15, May ( ] Shaher M. Momani, Local and global exitence theorem on fractional integro-differential equation, Journal of Fractional Calculu, Vol. 18, pp.81-86, (2. 1] Shayma A. Murad, Huein J.Zekri and Samir Hadid, Exitence and Uniquene Theorem of Fractional Mixed Volterra-Fredholm Integrodifferential Equation with Integral Boundary Condition, International Journal of Differential Equation, Volume 211, Article ID 3457, 15 page. 11] S. G. Samko, A. A. Kilba, and O. I. Marichev, Fractional Integral and Derivative: Theory and Application, Gordon and Breach Science, Yverdon, Switzerland, ( ] X. W. Su, Boundary value problem for a coupled ytem of nonlinear fractional differential equation, Appl. Math. Lett. 22, pp.64-69, (29. Shayma Adil Murad Department of Mathematic, Faculty of Science, Duhok Univerity, Kurditan, Iraq. Samir Bahir Hadid Department of Mathematic and Baic Science, College of Education and Baic Science, Ajman Univerity of Science and Technology, UAE.