EXISTENCE OF SOLUTIONS FOR RIEMANN-LIOUVILLLE TYPE COUPLED SYSTEMS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS

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1 Electronic Journal of Differential Equations, Vol. 216 (216, No. 211, pp ISSN: URL: or EXISTENCE OF SOLUTIONS FOR RIEMANN-LIOUVILLLE TYPE COUPLED SYSTEMS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS AHMED ALSAEDI, SHOROG ALJOUDI, BASHIR AHMAD Abstract. In this article, we study a boundary value problem of coupled systems of nonlinear Riemann-Liouvillle fractional integro-differential equations supplemented with nonlocal Riemann-Liouvillle fractional integro-differential boundary conditions. Our results rely on some standard tools of the fixed point theory. An illustrative example is also discussed. 1. Introduction Fractional calculus is regarded as an important mathematical modelling tool for describing dynamical systems involving phenomena such as fractal and chaos. The subject started with the speculations of Leibniz (1697 and Euler (173 about fractional-order derivatives and developed into an important branch of mathematical analysis with the passage of time. It deals with differential and integral operators of arbitrary (non-integer order. An important and useful feature characterizing fractional-order differential and integral operators (in contrast to integer-order operators is their nonlocal nature that accounts for the past and hereditary behavior of materials and processes involved in the real world problems. In addition to the extensive applications of fractional-order differential equations in various disciplines of technical and applied sciences, there has been a great focus on developing the theoretical aspects, and analytic and numerical methods for solving fractional order differential equations. For applications of fractional calculus in engineering and physics, we refer the reader to the texts [22, 24, 3], while some recent results on fractional differential equations can be found in [1, 2, 3, 8, 11, 14, 16, 19, 23, 29, 35, 37, 39]. Coupled systems of fractional-order differential equations appear in the mathematical formulation of several real world phenomena and processes. Examples of the occurrence of fractional systems include disease models [7, 9, 1, 26, 27], anomalous diffusion [25, 32], ecological models [18], synchronization of chaotic systems [12, 13, 38], nonlocal thermoelasticity [28], etc. For details concerning the 21 Mathematics Subject Classification. 34A8, 34B1, 34B15. Key words and phrases. Riemann-Liouvillle; fractional derivative; coupled system; nonlocal integral conditions; existence of solutions. c 216 Texas State University. Submitted June 3, 216. Published August 1,

2 2 A. ALSAEDI, S. ALJOUDI, B. AHMAD EJDE-216/211 theoretical development of coupled systems of fractional-order differential equations supplemented with a variety of boundary conditions, for instance, see [4, 5, 6, 17, 21, 31, 33, 34, 36]. In this article, we study the existence of solutions for a Riemann-Liouville coupled system of nonlinear fractional integro-differential equations given by D α u(t = f(t, u(t, v(t, (φ 1 u(t, (ψ 1 v(t, t [, T ], D β v(t = g(t, u(t, v(t, (φ 2 u(t, (ψ 2 v(t, 1 < α, β 2. subject to coupled Riemann-Liouville integro-differential boundary conditions: D α 2 u( =, D u( = νi v(η, < η < T, D β 2 v( =, D β 1 v( = µi β 1 u(σ, < σ < T, (1.1 (1.2 where D (, I ( denote the Riemann-Liouville derivatives and integral of fractional order (, respectively, f, g : [, T ] R 4 R are given continuous functions, ν, µ are real constants, and (φ 1 u(t = (ψ 1 v(t = γ 1 (t, su(sds, (φ 2 u(t = δ 1 (t, sv(sds, (ψ 2 v(t = γ 2 (t, su(sds, δ 2 (t, sv(sds, with γ i and δ i (i = 1, 2 being continuous function on [, T ] [, T ]. The rest of the article is organized as follows. In Section 2, we recall some preliminary concepts of Riemann-Liouville calculus and prove an auxiliary lemma. Section 3 contains the existence and uniqueness results. Though we use the standard methodology (Leray-Schauder alternative to prove the existence of solutions and Contraction mapping principle to obtain the uniqueness result, yet its exposition to the given problem is new. Indeed our results are new and contribute to the existing literature on fully Riemann-Liouville type nonlinear nonlocal coupled systems of fractional integro-differential equations and boundary conditions. 2. Preliminaries This section is devoted to some basic concepts of fractional calculus concerning Riemann-Liouville derivatives and integrals [2]. We also present an auxiliary lemma related to linear variant of the given problem. Definition 2.1. The Riemann-Liouville fractional integral of order ρ > for a continuous function u : (, R is defined as provided the integral exists. I ρ u(t = 1 Γ(ρ (t s ρ 1 u(sds, Definition 2.2. For a continuous function u : (, R, the Riemann-Liouville derivative of fractional order ρ, n = [ρ] 1 ( [ρ]denotes the integer part of the real number ρ is defined as D ρ u(t = provided it exists. 1 ( d Γ(n ρ dt n (t s n ρ 1 u(sds = ( d ni n ρ u(t, dt

3 EJDE-216/211 EXISTENCE OF SOLUTIONS FOR RIEMANN-LIOUVILLLE SYSTEMS 3 For ρ <, we use the convention that D ρ u = I ρ u. Also, for β [, ρ], we have D β I ρ u = I ρ β. Note that for λ > 1, λ ρ 1, ρ 2,..., ρ n, we have D ρ t λ = Γ(λ 1 Γ(λ ρ 1 tλ ρ and D ρ t ρ i =, i = 1, 2,..., n. In particular, for the constant function u(t = 1, we obtain D ρ 1 = 1 Γ(1 ρ t ρ, ρ / N. For ρ N, we obtain, of course, D ρ 1 = because of the poles of the gamma function at the points, 1, 2,.... For ρ >, the general solution of the homogeneous equation D ρ u(t = in C(, T L(, T is u(t = c t ρ n c 1 t ρ n 1 c n 2 t ρ 2 c n 1 t ρ 1, where c i, i = 1, 2,..., n 1, are arbitrary real constants. Further, we always have D ρ I ρ u = u, and I ρ D ρ u(t = u(t c t ρ n c 1 t ρ n 1 c n 2 t ρ 2 c n 1 t ρ 1. (2.1 To define the solution for problem (1.1-(1.2, we consider the following lemma. Lemma 2.3. Let h 1, h 2 C[, T ] L[, T ]. Then the integral solution for the linear system of fractional differential equations: D α u(t = h 1 (t, D β v(t = h 2 (t, (2.2 supplemented with the boundary conditions (1.2 is given by u(t = νγ(βt η (η s α 2 (s τ β 1 Γ(α 1 Γ(β where µηαβ 2 Γ(α β 1 (σ s β 2 Γ(β 1 (s τ Γ(α (t s h 1 (sds, Γ(α v(t = µγ(αtβ 1 σ (σ s β 2 (s τ Γ(β 1 Γ(α η νσαβ 2 (η s α 2 (s τ β 1 Γ(α β 1 Γ(α 1 Γ(β (t s β 1 h 2 (sds, Γ(β h 2 (τdτ ds h 1 (τdτ ds h 1 (τdτ ds h 2 (τdτ ds (2.3 (2.4 = Γ(αΓ(β νµγ(αγ(β(ησαβ 2 (Γ(α β 1 2. (2.5 Proof. Using the formula (2.1, the general solution of the system (2.2 can be written as u(t = a t α 2 a 1 t I α h 1 (t, (2.6 v(t = b t β 2 b 1 t β 1 I β h 2 (t, (2.7

4 4 A. ALSAEDI, S. ALJOUDI, B. AHMAD EJDE-216/211 where a i, b i, (i =, 1 are unknown arbitrary constants. From (2.6 and (2.7, we have D u(t = a 1 Γ(α Ih 1 (t, (2.8 D β 1 v(t = b 1 Γ(β Ih 2 (t, (2.9 D α 2 u(t = a Γ(α 1 a 1 Γ(αt I 2 h 1 (t, (2.1 D β 2 v(t = b Γ(β 1 b 1 Γ(βt I 2 h 2 (t. (2.11 Using the given conditions: D α 2 u( = = D β 2 v( in (2.1-(2.11, we find that a =, b =. Thus (2.6 and (2.7 take the form u(t = a 1 t I α h 1 (t, (2.12 v(t = b 1 t β 1 I β h 2 (t. (2.13 Using the coupled integral boundary conditions: D u( = νi v(η and D β 1 v( = µi β 1 u(σ in (2.8 and (2.9, we obtain Γ(αa 1 νγ(βηαβ 2 Γ(α β 1 b 1 = νi αβ 1 h 2 (η, µγ(ασ αβ 2 Γ(α β 1 a 1 Γ(βb 2 = µi αβ 1 h 1 (σ. (2.14 Solving the system (2.14, we find that a 1 = ν Γ(βI αβ 1 h 2 (η µγ(βηαβ 2 Γ(α β 1 Iαβ 1 h 1 (σ, (2.15 b 1 = µ Γ(αI αβ 1 h 1 (σ νγ(ασαβ 2 Γ(α β 1 Iαβ 1 h 2 (η, (2.16 where is given by (2.5. Substituting the values of a 1 and b 1 (from (2.15 and (2.16 in (2.12 and (2.13, we obtain the solution (2.3-(2.4. Note that the converse follows by direct computation. This completes the proof. The following lemma contains certain estimates that we need in the sequel. We do not provide the proof as it is based on simple computation. Lemma 2.4. For h 1, h 2 C[, T ] L[, T ] with h 1 = sup t [,T ] h 1 (t and h 2 = sup t [,T ] h 2 (t, we have (i (ii (iii (iv η (σ s β 2 Γ(β 1 (η s α 2 Γ(α 1 (t s h 1 (sds Γ(α (t s β 1 h 2 (sds Γ(β (s τ h 1 (τdτ ds σ αβ 1 Γ(α Γ(α β h 1. (s τ β 1 h 2 (τdτ ds ηαβ 1 Γ(β Γ(α β h 2. T α Γ(α 1 h 1. T β Γ(β 1 h 2.

5 EJDE-216/211 EXISTENCE OF SOLUTIONS FOR RIEMANN-LIOUVILLLE SYSTEMS 5 3. Existence and uniqueness of solutions Denote by X = x : x C([, T ], R and Y = y : y C([, T ], R the spaces equipped respectively with the norms x X = sup t [,T ] x(t and y Y = sup t [,T ] y(t. Observe that (X, X and (Y, Y are Banach spaces. In consequence, the product space (X Y, X Y is a Banach space endowed with the norm (x, y X Y = x X y Y for (x, y X Y. By Lemma 2.3, we define an operator F : X Y X Y associated with the problem (1.1-(1.2 as follows: F (u, v(t := (F 1 (u, v(t, F 2 (u, v(t, (3.1 where F 1 (u, v(t = νγ(βt η (η s α 2 Γ(α 1 (s τ β 1 g(τ, u(τ, v(τ, (φ 2 u(τ, (ψ 2 v(τdτ ds Γ(β µηαβ 2 (σ s β 2 Γ(α β 1 Γ(β 1 (s τ f(τ, u(τ, v(τ, (φ 1 u(τ, (ψ 1 v(τdτ ds Γ(α (t s f(s, u(s, v(s, (φ 1 u(s, (ψ 1 v(sds, Γ(α F 2 (u, v(t = µγ(αtβ 1 σ (σ s β 2 Γ(β 1 (s τ f(τ, u(τ, v(τ, (φ 1 u(τ, (ψ 1 v(τdτ Γ(α η ds νσαβ 2 (η s α 2 Γ(α β 1 Γ(α 1 (s τ β 1 g(τ, u(τ, v(τ, (φ 2 u(τ, (ψ 2 v(τdτ ds Γ(β (t s β 1 g(s, u(s, v(s, (φ 2 u(s, (ψ 2 v(sds. Γ(β For computational convenience, we set (3.2 (3.3 λ = 1 γ, λ = 1 γ, (3.4 θ = 1 δ, θ = 1 δ, (3.5, (3.6 ε 1 = T νµ Γ(β(ησ αβ 1 ηγ(α β 1Γ(α β T Γ(α 1 ε 2 = ν Γ(βηαβ 1 T, (3.7 Γ(α β ε 1 = µ Γ(ασαβ 1 T β 1, Γ(α β (3.8 ε 2 = T β 1 νµ (ησ αβ 1 Γ(α σγ(α βγ(α β 1 T, Γ(β 1 (3.9

6 6 A. ALSAEDI, S. ALJOUDI, B. AHMAD EJDE-216/211 Ω 1 = ϖ (ε 1 ε 1 κ (ε 2 ε 2, (3.1 Ω 2 = λ maxϖ 1, ϖ 3 (ε 1 ε 1 λ maxκ 1, κ 3 (ε 2 ε 2, (3.11 Ω 3 = θ maxϖ 2, ϖ 4 (ε 1 ε 1 θ maxκ 2, κ 4 (ε 2 ε 2. (3.12 Observe that problem (1.1-(1.2 has solutions if and only if the operator equation F (u, v = (u, v has a fixed point. Now we are ready to present our first existence result, which is based on Leray- Schauder alternative. Lemma 3.1 (Leray-Schauder alternative [15]. Let F : E E be a completely continuous operator. Let V (F = x E : x = λf (x for some < λ < 1. Then either the set V (F is unbounded or F has at least one fixed point. Theorem 3.2. Let f, g : [, T ] R 4 R be continuous functions and there exist real constants ϖ i, κ i (i = 1,..., 4 and ϖ, κ > such that (H1 f(t, u(t, v(t, (φ 1 u(t, (ψ 1 v(t ϖ ϖ 1 u ϖ 2 v ϖ 3 φ 1 u ϖ 4 ψ 1 v, g(t, u(t, v(t, (φ 2 u(t, (ψ 2 v(t κ κ 1 u κ 2 v κ 3 φ 2 u κ 4 ψ 2 v, for all (u, v X Y. Further it is assumed that maxω 2, Ω 3 < 1, where Ω 2 and Ω 3 are given by (3.11 and (3.12 respectively. Then problem (1.1-(1.2 has at least one solution on [, T ]. Proof. In the first step, we show that the operator F : X Y X Y defined by (3.1 is completely continuous. By continuity of the functions f and g, we deduce that the operators F 1 and F 2 respectively given by (3.2 and (3.3 are continuous. In consequence, the operator F is continuous. Next we show that the operator F is uniformly bounded. For that, let A X Y be a bounded set. Then, for any (u, v A, there exist positive constants L 1 and L 2 such that f(t, u(t, v(t, (φ 1 u(t, (ψ 1 v(t L 1, g(t, u(t, v(t, (φ 2 u(t, (ψ 2 v(t L 2, for all (u, v A. Then, for any (u, v A, we have F 1 (u, v(t Γ(β ν t η (η s α 2 Γ(α 1 (s τ β 1 g(τ, u(τ, v(τ, (φ 2 u(τ, (ψ 2 v(τ dτ ds Γ(β µ ηαβ 2 (σ s β 2 Γ(α β 1 Γ(β 1 (s τ f(τ, u(τ, v(τ, (φ 1 u(τ, (ψ 1 v(τ dτ ds Γ(α (t s f(s, u(s, v(s, (φ 1 u(s, (ψ 1 v(s ds, Γ(α ν Γ(βT η (η s L α 2 (s τ β 1 2 dτ ds Γ(α 1 Γ(β (s τ µ ηαβ 2 L 1 Γ(α β 1 T L 1 (σ s β 2 Γ(β 1 νµ (ησ αβ 1 Γ(β ηγ(α β 1Γ(α β Γ(α T Γ(α 1 dτ ds L 1 (t s ds, Γ(α ν Γ(βηαβ 1 T L 2, Γ(α β

7 EJDE-216/211 EXISTENCE OF SOLUTIONS FOR RIEMANN-LIOUVILLLE SYSTEMS 7 which, on taking the norm for t [, T ] and using the notation (3.7, yields In a similar manner, we can find that β 1 µ Γ(αT F 2 (u, v(t F 1 (u, v ε 1 L 1 ε 2 L 2. (3.13 L1 σ αβ 1 Γ(α β which together with (3.8 and (3.9 implies ν L 2 (ησ αβ 1 L 2T β σγ(α βγ(α β 1 Γ(β 1, F 2 (u, v ε 1 L 1 ε 2 L 2. (3.14 From the inequalities (3.13 and (3.14, we infer that F 1 and F 2 are uniformly bounded, and hence the operator F is uniformly bounded. Next, we show that F is equicontinuous. Let t 1, t 2 [, T ] with t 1 < t 2. Then we have F 1 (u, v(t 2 F 1 (u, v(t 1 ν Γ(β t 2 t1 L2 η αβ 1 Γ(α β µ (ησ αβ 1 L 1 ηγ(α β 1Γ(α β L 1 ( 2(t2 t 1 α t α 2 t α 1. Γ(α 1 Obviously F 1 (u, v(t 2 F 1 (u, v(t 1 as t 2 t 1. In a similar manner, one can show that F 2 (u, v(t 2 F 2 (u, v(t 1 as t 2 t 1. Thus the operator F is equicontinuous in view of equicontinuity of F 1 and F 2. Therefore, by Arzela-Ascoli s theorem, we deduce that the operator F is compact (completely continuous. Finally, we consider a set V (F = (u, v X Y : (u, v = λf (u, v ; λ 1 and show that it is bounded. Let (u, v V. Then (u, v = λf (u, v. For any t [, T ], we have u(t = λf 1 (u, v(t, v(t = λf 2 (u, v(t. Using the assumption (H1 together with the notation (3.4 and (3.5, we obtain u(t λ F 1 (u, v(t F 1 (u, v(t Γ(β ν t η (η s α 2 (s τ β 1 [κ κ 1 u(τ κ 2 v(τ Γ(α 1 Γ(β κ 3 (φ 2 u(τ κ 4 (ψ 2 v(τ ]dτ ds µ ηαβ 2 (σ s β 2 (s τ [ϖ ϖ 1 u(τ ϖ 2 v(τ Γ(α β 1 Γ(β 1 Γ(α ϖ 3 (φ 1 u(τ ϖ 4 (ψ 1 v(τ ]dτ ds (t s [ϖ ϖ 1 u(s ϖ 2 v(s ϖ 3 (φ 1 u(s ϖ 4 (ψ 1 v(s ]ds Γ(α Γ(β ν t η (η s α 2 (s τ β 1 [κ (κ 1 γ κ 3 u(τ Γ(α 1 Γ(β (κ 2 δ κ 4 v(τ ]dτ ds µ ηαβ 2 Γ(α β 1 (σ s β 2 Γ(β 1 (s τ [ϖ (ϖ 1 γ ϖ 3 u(τ Γ(α

8 8 A. ALSAEDI, S. ALJOUDI, B. AHMAD EJDE-216/211 (ϖ 2 δ ϖ 4 v(τ ]dτ ds (t s [ϖ (ϖ 1 γ ϖ 3 u(s (ϖ 2 δ ϖ 4 v(s ]ds, Γ(α ν Γ(βT [κ λ maxκ 1, κ 3 u X η θ (η s α 2 (s τ β 1 maxκ 2, κ 4 v Y ] dτ ds Γ(α 1 Γ(β µ ηαβ 2 (σ s β 2 ( [ϖ λ maxϖ 1, ϖ 3 u X Γ(α β 1 Γ(β 1 s (s τ θ maxϖ 2, ϖ 4 v Y ] dτ ds Γ(α (t s [ϖ λ maxϖ 1, ϖ 3 u X θ maxϖ 2, ϖ 4 v Y ] ds, Γ(α T νµ (ησ αβ 1 Γ(β ηγ(α β 1Γ(α β T [ϖ λ maxϖ 1, ϖ 3 u X Γ(α 1 θ maxϖ 2, ϖ 4 v Y ] ν Γ(βηαβ 1 T [κ Γ(α β λ maxκ 1, κ 3 u X θ maxκ 2, κ 4 v Y ], which, on taking the norm for t [, T ] and using (3.7, yields u X ε 1 [ϖ λ maxϖ 1, ϖ 3 u X θ maxϖ 2, ϖ 4 v Y ] ε 2 [κ λ maxκ 1, κ 3 u X θ maxκ 2, κ 4 v Y ]. (3.15 Similarly, with the aid of notation (3.4, (3.5 and (3.9, one can obtain v Y ε 1 [ϖ λ maxϖ 1, ϖ 3 u X θ maxϖ 2, ϖ 4 v Y ] ε 2 [κ λ maxκ 1, κ 3 u X θ maxκ 2, κ 4 v Y ]. (3.16 From (3.15 and (3.16, we find that u X v Y ϖ (ε 1 ε 1 κ (ε 2 ε 2 [ λ maxϖ 1, ϖ 3 (ε 1 ε 1 λ ] maxκ 1, κ 3 (ε 2 ε 2 u X [ θ maxϖ 2, ϖ 4 (ε 1 ε 1 θ ] maxκ 2, κ 4 (ε 2 ε 2 v Y, Ω 1 maxω 2, Ω 3 (u, v X Y, (3.17 which, in view of (u, v X Y = u X v Y, yields (u, v X Y Ω 1 1 maxω 2, Ω 3, where Ω 1, Ω 2, Ω 3 are respectively given by (3.1, (3.11 and (3.12. This shows that V (F is bounded. Thus, by Lemma 3.1, the operator F has at least one fixed point. Consequently, the problem (1.1-(1.2 has at least one solution on [, T ]. This completes the proof.

9 EJDE-216/211 EXISTENCE OF SOLUTIONS FOR RIEMANN-LIOUVILLLE SYSTEMS 9 Our next result deals with the uniqueness of solutions for problem (1.1-(1.2 and relies on Banach s contraction mapping principle. For computational convenience, we introduce the notation: ν Γ(βT Λ = ξ 1 (1 γ ξ 2 (1 δ [ µ ηαβ 2 ζ 1 (1 γ Γ(α β 1 ] (3.18 ζ 2 (1 δ ζ 1 (1 γ ζ 2 (1 δ, β 1 µ Γ(αT [ Λ 1 = ζ 1 (1 γ ζ 2 (1 δ ν σαβ 2 ξ 1 (1 γ Γ(α β 1 ξ 2 (1 δ ] ξ 1 (1 γ ξ 2 (1 δ, where γ = δ = (3.19 ζ 1 = max I αβ 1 M 1 (σ, I αβ 1 M 3 (σ, ζ 2 = max I αβ 1 M 2 (σ, I αβ 1 M 4 (σ, (3.2 ζ 1 = sup I α M 1 (t, I α M 3 (t, ζ 2 = t [,T ] sup t [,T ] I α M 2 (t, I α M 4 (t, (3.21 ξ 1 = max I αβ 1 N 1 (η, I αβ 1 N 3 (η, ξ 2 = max I αβ 1 N 2 (η, I αβ 1 N 4 (η, (3.22 ξ 1 = sup I β N 1 (t, I β N 3 (t, sup t [,T ] sup t [,T ] ξ 2 = t [,T ] sup t [,T ] I β N 2 (t, I β N 4 (t, (3.23 γ 1 (t, sds, γ = sup t [,T ] δ 1 (t, sds, δ = sup t [,T ] γ 2 (t, sds, (3.24 δ 2 (t, sds, (3.25 ε = ε 1 ϱ 1 ε 2 ϱ 2, ε = ε 1 ϱ 1 ε 2 ϱ 2, ϱ 1 = sup f(t,,,,, t [,T ] ϱ 2 = where ε i, ε i (i = 1, 2 are respectively given by (3.6 (3.9. sup g(t,,,,, (3.26 t [,T ] Theorem 3.3. Let f, g : [, T ] R 4 R be continuous functions and there exist positive functions M i (t, N i (t (i = 1,..., 4 such that f(t, u 1, u 2, u 3, u 4 f(t, v 1, v 2, v 3, v 4 g(t, u 1, u 2, u 3, u 4 g(t, v 1, v 2, v 3, v 4 4 M i (t u i v i, i=1 4 N i (t u i v i, i=1

10 1 A. ALSAEDI, S. ALJOUDI, B. AHMAD EJDE-216/211 for all t [, T ], u i, v i R. In addition, assume that Λ < 1 2 and Λ 1 < 1 2, where Λ and Λ 1 are given by (3.18and (3.19 respectively. Then boundary value problem (1.1-(1.2 has a unique solution on [, T ]. Proof. Let us fix r max2ε/(1 2Λ, 2 ε/(1 2Λ 1, where Λ, Λ 1, and ε, ε are respectively given by (3.18, (3.19 and (3.26. Firstly, we show that F B r B r, where B r = (u, v X Y : (u, v X Y r and F is given by (3.1. For (u, v B r, note that f(t, u(t, v(t, (φ 1 u(t, (ψ 1 v(t f(t, u(t, v(t, (φ 1 u(t, (ψ 1 v(t f(t,,,, f(t,,,, M 1 (t u(t M 2 (t v(t M 3 (t (φ 1 u(t M 4 (t (ψ 1 v(t ϱ 1 M 1 (t u(t M 2 (t v(t γ M 3 (t u(t δ M 4 (t v(t ϱ 1 [ ] [ ] M 1 (t γ M 3 (t u(t M 2 (t δ M 4 (t v(t ϱ 1 ( M 1 (t M 2 (t γ M 3 (t δ M 4 (t (u, v X Y ϱ 1 ( M 1 (t M 2 (t γ M 3 (t δ M 4 (t r ϱ 1. Similarly, one can obtain g(t, u(t, v(t, (φ 2 u(t, (ψ 2 v(t ( N 1 (t N 2 (t γ N 3 (t δ N 4 (t r ϱ 2. Then, using the notation (3.2, (3.21, (3.22 and (3.26, we have F 1 (u, v(t Γ(βT η (η s α 2 (s τ β 1 ( N 1 (τ N 2 (τ Γ(α 1 Γ(β γ N 3 (τ δ N 4 (τ r ϱ 2 dτ ds ν ηαβ 2 (σ s β 2 (s τ ( M 1 (τ M 2 (τ γ M 3 (τ Γ(α β 1 Γ(β 1 Γ(α δ M 4 (τ r ϱ 1 dτ ds (t s ( M 1 (s M 2 (s γ M 3 (s δ M 4 (s r ϱ 1 ds Γ(α ν Γ(βT I αβ 1( N 1 (η γ N 3 (η r I αβ 1( N 2 (η δ N 4 (η r ηαβ 1 ϱ 2 Γ(α β µ ηαβ 2 Γ(α β 1 ρ 1 µ (ησ αβ 1 ηγ(α βγ(α β 1 ϱ 1T α Γ(α 1 [ I αβ 1( M 1 (σ γ M 3 (σ r I αβ 1( ] M 2 (σ δ M 4 (σ r I α( M 1 (t γ M 3 (t r I α( M 2 (t δ M 4 (t r

11 EJDE-216/211 EXISTENCE OF SOLUTIONS FOR RIEMANN-LIOUVILLLE SYSTEMS 11 ν Γ(βT r (ξ 1 (1 γ ξ 2 (1 δ µ ζ 1η αβ 2 Γ(α β 1 (1 γ µ ζ 2η αβ 2 Γ(α β 1 (1 δ ζ 1 (1 γ ζ 2 (1 δ [ µν Γ(β(ησ αβ 1 T ϱ 1 ηγ(α βγ(α β 1 T α Γ(α 1 which, in view of (3.18 and (3.26, implies Analogously, using (3.19 and (3.26, we can obtain ] ν Γ(βη αβ 1 T ϱ 2, Γ(α β F 1 (u, v X Λr ε r 2. (3.27 F 2 (u, v Y = Λ 1 r ε r 2. (3.28 From the estimates (3.27 and (3.28, it clearly follows that F (u, v X Y = F 1 (u, v X F 2 (u, v Y r, and hence F B r B r. Now we show that the operator F is a contraction. For that, let u i, v i R, i = 1, 2. Then, for each t [, T ], it follows by (3.2-(3.22 that F 1 (u 1, v 1 (t F 1 (u 2, v 2 (t ν Γ(βT η (η s α 2 (s τ β 1 g(τ, u 1 (τ, v 1 (τ, (φ 2 u 1 (τ, Γ(α 1 Γ(β (ψ 2 v 1 (τ g(τ, u 2 (τ, v 2 (τ, (φ 2 u 2 (τ, (ψ 2 v 2 (τ dτ ds µ ηαβ 2 (σ s β 2 (s τ f(τ, u 1 (τ, v 1 (τ, (φ 1 u 1 (τ, Γ(α β 1 Γ(β 1 Γ(α (ψ 1 v 1 (τ f(τ, u 2 (τ, v 2 (τ, (φ 1 u 2 (τ, (ψ 1 v 2 (τ dτ ds (t s f(s, u 1 (s, v 1 (s, (φ 1 u 1 (s, (ψ 1 v 1 (s Γ(α f(s, u 2 (s, v 2 (s, (φ 1 u 2 (s, (ψ 1 v 2 (s ds ν Γ(βT I αβ 1( N 1 (η γ N 3 (η u 1 u 2 X I αβ 1( N 2 (η δ N 4 (η v 1 v 2 Y [ µ ηαβ 2 I αβ 1( M 1 (σ γ M 3 (σ u 1 u 2 X Γ(α β 1 I αβ 1( ] M 2 (σ δ M 4 (σ v 1 v 2 Y I α( M 1 (t γ M 3 (t u 1 u 2 X I α( M 2 (t δ M 4 (t v 1 v 2 Y ν Γ(βT [ ξ 1 (1 γ µ ηαβ 2 ζ ] 1 Γ(α β 1 (1 γ ν Γ(βT [ ξ 2 (1 δ µ ηαβ 2 ζ ] 2 Γ(α β 1 (1 δ ζ 1 (1 γ u 1 u 2 X ζ 2 (1 δ v 1 v 2 Y,

12 12 A. ALSAEDI, S. ALJOUDI, B. AHMAD EJDE-216/211 which yields F 1 (u 1, v 1 F 1 (u 2, v 2 X Λ[ u 1 u 2 X v 1 v 2 Y ], (3.29 where we have used (3.18. Similarly, we can find that F 2 (u 1, v 1 F 2 (u 2, v 2 Y Λ 1 [ u 1 u 2 X v 1 v 2 Y ], (3.3 where we have used (3.19. Thus, from (3.29 and (3.3, we have F (u 1, v 1 F (u 2, v 2 X Y = F 1 (u 1, v 1 F 1 (u 2, v 2 X F 2 (u 1, v 1 F 2 (u 2, v 2 Y (Λ Λ 1 [ u 1 u 2 X v 1 v 2 Y ], which implies that F is a contraction in view of the given condition Λ Λ 1 < 1. Hence, by Banach s fixed point theorem, the operator F has a unique fixed point which corresponds to the unique solution of the problem (1.1-(1.2 on [, T ]. This completes the proof. Example. Consider the boundary-value problem D 3/2 t2 1 u(t = t2 t2 u(t tan 1 v(t t (t s 1/2 25 Γ( 3 2 u(sds 1 (t s 1/3 25 Γ( 4 3 v(sds, D 5/4 v(t = et 1 ( sin u(t v(t 1 e (s t u(sds t e (s t/2 14 D 1/2 u( =, D 1/2 u( = I 1/2 (1/2, D 3/4 v( =, D 1/4 v( = 2I 1/4 (1/4. v(sds, < t < 1 (3.31 Here, α = 3/2, β = 5/4, v = 1, µ = 2, η = 1/2, σ = 1/4, T = 1, γ 1 = (t s 1/2 /Γ(3/2, δ 1 = (t s 1/3 /Γ(4/3, γ 2 = e (s t /5, δ 2 = e (s t/2 /14, M 1 (t = t 2 /15, M 2 (t = t 2 /1, M 3 (t = t/25, M 4 (t = 1/25, N 1 (t = e t /8, N 2 (t = 1/2, N 3 (t = 1/8, N 4 (t = t/2. Using the given data, we find that γ.75225, δ.83988, γ.3437, δ.927, , ζ 1.489, ζ 2.733, ζ 1.515, ζ , ξ 1.16, ξ 2.148, ξ , ξ Further, Λ < 1/2, and Λ < 1/2. Thus, by Theorem 3.3, problem (3.31 has a unique solution on [, 1]. References [1] B. Ahmad, J. J. Nieto; Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. (211, 211:36, 9 pp. [2] B. Ahmad, J. J. Nieto; Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed Point Theory 13 (212, [3] B. Ahmad, S. K. Ntouyas; Initial value problems of fractional order Hadamard-type functional differential equations, Electron. J. Differential Equations 215, No. 77, 9 pp. [4] B. Ahmad, J. J. Nieto; Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58 (29,

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14 14 A. ALSAEDI, S. ALJOUDI, B. AHMAD EJDE-216/211 [33] J. Sun, Y. Liu, G. Liu; Existence of solutions for fractional differential systems with antiperiodic boundary conditions, Comput. Math. Appl. 64 (212, [34] J. Tariboon, S. K. Ntouyas, W. Sudsutad; Coupled systems of Riemann-Liouville fractional differential equations with Hadamard fractional integral boundary conditions, J. Nonlinear Sci. Appl. 9 (216, [35] P. Thiramanus, S. K. Ntouyas, J. Tariboon; Existence of solutions for Riemann-Liouville fractional differential equations with nonlocal Erdlyi-Kober integral boundary conditions on the half-line, Bound. Value Probl. (215, 215:196, 15 pp. [36] J. R. Wang, Y. Zhang; Analysis of fractional order differential coupled systems, Math. Methods Appl. Sci. 38 (215, [37] C. Zhai, L. Xu; Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter, Commun. Nonlinear Sci. Numer. Simul. 19 (214, [38] F. Zhang, G. Chen C. Li, J. Kurths; Chaos synchronization in fractional differential systems, Phil Trans R Soc A 371 (213, [39] L. Zhang, B. Ahmad, G. Wang; Successive iterations for positive extremal solutions of nonlinear fractional differential equations on a half line, Bull. Aust. Math. Soc. 91 (215, Ahmed Alsaedi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 823, Jeddah 21589, Saudi Arabia address: aalsaedi@hotmail.com Shorog Aljoudi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 823, Jeddah 21589, Saudi Arabia address: sh-aljoudi@hotmail.com Bashir Ahmad Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box. 823, Jeddah 21589, Saudi Arabia address: bashirahmad qau@yahoo.com