Bull. Math. Soc. Sci. Math. Roumanie Tome 60 (108) No. 1, 2017, 3 18

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1 Bull. Math. Soc. Sci. Math. Roumanie Tome 6 8 No., 27, 3 8 On a coupled system of sequential fractional differential equations with variable coefficients and coupled integral boundary conditions by Bashir Ahmad a, Ahmed Alsaedi a, Shorog Aljoudi a and Sotiris K. Ntouyas b,a Abstract In this paper, we investigate the existence and uniqueness of solutions for a coupled system of Caputo Liouville-Caputo type sequential fractional differential equations with variable coefficients supplemented with coupled nonlocal Riemann-Liouville integral boundary conditions. We make use of standard tools of the fixed-point theory to obtain the desired results. Our results are new and give more insight into the study of coupled systems of fractional differential equations with non-constant coefficients. Examples are included for the illustration of main results. Key Words: Caputo derivative; coupled system; variable coefficients; Riemann- Liouville integral boundary conditions; fixed point. 2 Mathematics Subject Classification: Primary 34A8 Secondary 34A2, 34B5 Introduction Fractional-order derivatives are found to be of great value in modelling many real world phenomena. In contrast to integer-order derivatives, these operators can trace the past history of the processes and materials involved in the phenomena. In particular, fractional Laplacian provides a paradigm of the vast family of nonlocal linear operators, and appears in the formulation of anomalous diffusion process a diffusion process involving nonlinear relationship to time. Further details on anomalous diffusion phenomena can be found in a recent paper [ and the text [2. Due to the widespread applications of fractional calculus in biomedical and chemical processes, control theory, biomathematics, signal and image processing, wave propagation, etc., the mathematical community has shown a great interest in this subject, though its basics were known in the period of Riemann and Liouville. For more examples and explanations, we refer the reader to the texts [3. The topic of boundary value problems BVPs of fractional differential equations supplemented with a variety of boundary conditions has attracted a significant attention in recent years. In particular, the literature on fractional order BVPs involving nonlocal and integral boundary conditions is now much enriched, for instance, see [4-[9 and the references cited therein. Coupled systems of fractional-order differential equations have also been extensively studied due to their occurrence in several diverse disciplines, for example, nonlocal thermoelasticity [, synchronization phenomena [, 2, anomalous diffusion [3, etc. Some recent works on coupled systems of fractional-order differential equations equipped

2 4 B. Ahmad, A. Alsaedi, S. Aljoudi and S. K. Ntouyas with different kinds of boundary conditions can be found in [4-[8 and the references cited therein. In a recent work [8, the authors studied the following problem of a coupled system of sequential fractional differential equations and nonlocal uncoupled integral boundary conditions: c D q k c D q xt = ft, xt, yt, t [,, 2 < q 3, k >, c D p k c D p yt = gt, xt, yt, 2 < p 3, x =, x =, xζ = a η y =, y =, yz = b η s β Γβ xsds, β >, < η < ζ <, θ s γ Γγ ysds, γ >, < θ < z <, where c D. denotes the Caputo fractional derivatives of order., f, g : [, R 2 R are given continuous functions and a, b are real constants. To explore further in this direction, we replace the parameter k in the above system by different variable coefficients k t and k 2 t in the equations of the system. We also allow the nonlinearities in the system to depend on the unknown functions together with their fractional derivatives. Furthermore, we consider nonlocal coupled integral boundary conditions in contrast to the uncoupled ones in the above system. Precisely, for 2 < p, q 3, < α, δ <, we consider the following coupled system: c D q k t c D q xt = ft, xt, yt, c D α yt, t [,, c D p k 2 t c D p yt = gt, xt, c D δ xt, yt,. subject to coupled Riemann-Liouville type integral boundary conditions: η x =, x =, xζ = a y =, y =, yz = b η s β ys ds, β >, Γβ θ s γ xs ds, γ >,.2 Γγ where c D. denotes the Caputo derivatives of fractional order., f, g : [, R 3 R are given continuous functions, k t, k 2 t are increasing functions with k t, k 2 t C[,, R, < η < ζ <, < θ < z <, and a, b are real constants. In the rest of the paper, we organize the content as follows. We recall some basic facts of fractional calculus and establish an auxiliary lemma in Section 2. The main existence and uniqueness results, relying on contraction mapping principle and Leray-Schauder alternative, are presented in Section 3. Though the methods of proofs are the standard ones, yet their application in the context of problem.-.2 contribute further to the development of the subject of coupled systems of fractional differential equations with Riemann-Liouville integral boundary conditions. We also illustrate the existence and uniqueness result with the aid of an example.

3 On a coupled system of sequential fractional differential equations 5 2 Background material This section is devoted to some fundamental concepts of fractional calculus [3 and a basic lemma related to the linear variant of the given problem. Definition. The fractional integral of order r with the lower limit zero for a function f is defined as I r ft = fs ds, t >, r >, Γr t s r provided the right hand-side is point-wise defined on [,, where Γ is the gamma function, which is defined by Γr = t r e t dt. Definition 2. The Riemann-Liouville fractional derivative of order r >, n < r < n, n N, is defined as D r ft = n d t t s n r fsds, Γn r dt where the function ft has absolutely continuous derivative up to order n. Definition 3. The Caputo derivative of order r for a function f : [, R can be written as c D r ft = D r ft n k= t k k! f k, t >, n < r < n. Remark. If ft C n [,, then c D r ft = t f n s Γn r t s r n ds = In r f n t, t >, n < r < n. To define the solution for problem.-.2, we consider the following lemma dealing with the linear variant of.-.2. Lemma. Let h, h 2 C[,, R and x, y C 3 [,, R. Then the integral solution for the linear system of fractional differential equations: c D q k t c D q xt = h t, c D p k 2 t c D p yt = h 2 t, 2.

4 6 B. Ahmad, A. Alsaedi, S. Aljoudi and S. K. Ntouyas supplemented with the boundary conditions.2 is given by [ xt = t η s e µ t µ s η s ds {B β 2 a Γβ s ζ e µ 2s µ 2 m I p h 2mdmds e µ ζ µ s I q h s ds B [b z θ s γ s e µ s µ m I q h mdm Γγ } e µ 2z µ 2 s I p h 2sds ds e µ t µ s I q h sds, 2.2 [ yt = t s e µ 2t µ 2 s θ s ds {A γ b Γγ s z e µ s µ m I q h mdm ds e µ 2z µ 2 s I p h 2sds A 2 [a ζ η η s β s e µ 2s µ 2 m I p h 2mdmds Γβ } e µ ζ µ s I q h s ds e µ 2t µ 2 s I p h 2sds, 2.3 where = A B 2 A 2 B, µ i t = ζ k i sds, i =, 2, 2.4 A = s e µζ µs ds, 2.5 η η s β s B = a u e µ2s µ2u du ds, 2.6 Γβ θ s γ s A 2 = b u e µs µu du ds, 2.7 Γγ B 2 = z s e µ2z µ2s ds. 2.8

5 On a coupled system of sequential fractional differential equations 7 Proof: As argued in [9, the general solution of the system 2. can be written as xt = b e µt µs ds b s e µt µs ds b 2 e µt e µt µs s s τ q 2 Γq h τ dτ ds, 2.9 yt = c e µ2t µ2s ds c s e µ2t µ2s ds c 2 e µ2t e µ2t µ2s s s τ p 2 Γp h 2τ dτ ds, 2. where µ i t = k isds, i =, 2 and b j, c j, j =,, 2 are unknown arbitrary constants. Using the boundary conditions.2 in 2.9 and 2., we find that b =, b 2 =, c =, c 2 = and A b B c = J, A 2 b B 2 c = J 2, 2. where A i and B i i =, 2 are respectively given by , and η J = a J 2 = b ζ z η s β s Γβ e µ2s µ2u I p h 2 u du ds e µζ µs I q h sds, 2.2 θ s γ s Γγ e µs µu I q h u du ds e µ2z µ2s I p h 2 sds. 2.3

6 8 B. Ahmad, A. Alsaedi, S. Aljoudi and S. K. Ntouyas Solving the system 2. for b and c, we get b = [ η η s {B β s 2 a Γβ ζ e µζ µs I q h s ds θ s B [b γ s Γγ z } e µ2z µ2s I p h 2 sds, e µ2s µ2u I p h 2 u du ds e µs µu I q h u du ds c = [ θ s {A γ s b e µs µu I q h u du ds Γγ z e µ2z µ2s I p h 2 sds η η s A 2 [a β s e µ2s µ2u I p h 2 u du ds Γβ ζ } e µζ µs I q h s ds, 2.4 where is given by 2.4. Substituting the values of b j, c j j =,, 2 in 2.9 and 2., we obtain the solution given by 2.2 and 2.3. Note that the converse follows by direct computation. This completes the proof. 3 Main results Let X = { x : x C[,, R and c D δ x C[,, R } { and Y = y : y C[,, R and } c D α y C[,, R denote the spaces equipped respectively with the norms x X = x c D δ x = sup t [, xt sup t [, c D δ xt and y Y = y c D α y = sup t [, yt sup t [, c D α yt. 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. Using Lemma, we introduce an operator H : X Y X Y associated with the problem.-.2 as follows: Hu, vt := H u, vt, H 2 u, vt, 3.

7 On a coupled system of sequential fractional differential equations 9 where H u, vt = t { [ η s e µ t µ s η s β ds B 2 a Γβ s ζ e µ 2s µ 2 m I p ĝmdm ds e µ ζ µ s I q fs ds B [b z θ s γ Γγ } e µ 2z µ 2 s I p ĝsds s e µ s µ m I q fmdm ds e µ t µ s I q fsds, 3.2 H 2u, vt = [ s e µ 2t µ 2 s θ s ds {A γ b Γγ s z e µ s µ m I q fmdm ds e µ 2z µ 2 s I p ĝsds A 2 [a ζ η η s β Γβ e µ ζ µ s I q fs ds } s e µ 2s µ 2 m I p ĝmdm ds e µ 2t µ 2 s I p ĝsds, 3.3 with ft = ft, ut, vt, c D α vt, ĝt = gt, ut, c D δ ut, vt. Observe that existence of a fixed point of the operator H implies the existence of a solution of the problem.-.2. In the forthcoming analysis, we need the following assumptions: A Let f, g : [, R 3 R be continuous functions and there exist real constants µ j, λ j j =, 2, 3 and µ >, λ > such that ft, x, x 2, x 3 µ µ x µ 2 x 2 µ 3 x 3, gt, x, x 2, x 3 λ λ x λ 2 x 2 λ 3 x 3, x j R, j =, 2, 3. A 2 There exist positive constants l, l such that ft, u, u 2, u 3 ft, v, v 2, v 3 l u v u 2 v 2 u 3 v 3, gt, u, u 2, u 3 gt, v, v 2, v 3 l u v u 2 v 2 u 3 v 3, t [,, u j, v j R.

8 B. Ahmad, A. Alsaedi, S. Aljoudi and S. K. Ntouyas For computational convenience, we set the following notations: ρ i = sup k i t, i =, 2, 3.4 t [, [ M = B 2 ζ q θ qγ b B 2 Γq Γγ 2, 3.5 [ η pβ M 2 = a B 2 2 Γp Γβ B z p, 3.6 M = ρ B 2 ζ q 2 Γq b B θ qγ ρ q Γγ Γq Γq, 3.7 M 2 = ρ η pβ a B 2 2 Γβ Γp B z p, 3.8 Γp [ θ qγ N = b A 2 Γq Γγ A 2 ζ q, 3.9 [ η pβ N 2 = a A 2 2 Γp Γβ A z p 2, 3. N = ρ 2 ζ q A 2 2 Γq b A θ qγ, 3. Γγ Γq N 2 = ρ 2 a A 2 η pβ 2 Γβ Γp A z p ρ 2 p Γp Γp, 3.2 M σ = µ M Γ2 δ N N Γ2 α M 2 λ M 2 Γ2 δ N N 2 2, 3.3 Γ2 α σ 2 = M µ M N Γ2 δ N Γ2 α M 2 max{λ, λ 2 } M 2 N 2 Γ2 δ N2 Γ2 α σ 3 = M 2 λ 3 M 2 N 2 Γ2 δ N2 Γ2 α M 2 max{µ, µ 2 } M N Γ2 δ N Γ2 α, Now we present our first result which deals with the existence of solution of the problem at hand and is based on Leray-Schauder alternative [2. Lemma 2. Leray-Schauder alternative Let G : E E be a completely continuous operator. Let εg = {x E : x = λgx for some < λ < }. Then either εg is unbounded or G has at least one fixed point.

9 On a coupled system of sequential fractional differential equations Theorem. Assume that A holds and that max{σ 2, σ 3 } <, where σ 2 and σ 3 are given by 3.4 and 3.5 respectively. Then the boundary value problem.-.2 has at least one solution [,. Proof: In the first step, we show that the operator H : X Y X Y is completely continuous. By continuity of the functions f and g, it follows that the operators H and H 2 are continuous. In consequence, the operator H is continuous. Next we show that the operator H is uniformly bounded. For that, let Ω X Y be a bounded set. Then there exist positive constants L and L 2 such that ft, ut, vt, c D α vt L, gt, ut, c D δ ut, vt L 2, u, v Ω. Then, for any u, v Ω, we have L H u, vt B 2 ζ q θ qγ b B 2 Γq Γγ 2 L 2 η pβ a B 2 2 Γp Γβ B z p L M L 2 M 2, 3.6 which, on taking the norm for t [,, yields H u, v L M L 2 M 2, where M, M 2 are respectively given by 3.5 and 3.6. Using 3.2 together with 3.9 and 3., we obtain H u, vt L [ Γq ρ B 2 ζ q b B θ qγ 2 Γγ L 2 ρ a B 2 η pβ B z p Γp 2 Γβ L M L 2 M2. By definition of Caputo fractional derivative with < δ <, we get ρ q Hence c D δ H u, vt t s δ Γ δ H u, vs ds L t s δ M L 2 M2 Γ δ ds L M L 2 M2. Γ2 δ H u, v X = H u, v c D δ H u, v L M L 2 M 2 L M L 2 M Γ2 δ Similarly, we can find that H 2 u, v Y = H 2 u, v c D α H 2 u, v L N L 2 N 2 L N L 2 N2, 3.8 Γ2 α

10 2 B. Ahmad, A. Alsaedi, S. Aljoudi and S. K. Ntouyas where N, N 2, N, N 2 are respectively given by 3.9, 3., 3. and 3.2. From the inequalities 3.7 and 3.8, we deduce that H and H 2 are uniformly bounded and hence the operator H is uniformly bounded. Next, we show that H is equicontinuous. Let t, t 2 [, with t < t 2. Then we have H u, vt 2 H u, vt [ s e µs e µt2 e µt ds [ ζ s {L B 2 e µζ µs B b L 2 [ B B 2 a L [ 2 t z η 2 t s τ q 2 Γq dτ s e µt2 µs ds ds θ s γ s m e µs µm Γγ s e µ2z µ2s s τ p 2 Γp dτ ds η s β s m e µ2s µ2m Γβ s s τ q 2 Γq dτ ds e µt2 µs e µs e µt2 e µt s s τ q 2 Γq dτ m τ q 2 Γq dτ dm ds m τ p 2 } Γp dτ dm ds ds. Evidently, H u, vt 2 H u, vt independent of u, v as t 2 t. Also c D δ H u, vt 2 c D δ H u, vt Γ δ 2 t s δ t 2 s δ t s δ t 2 s δ H u, vs ds t 2 s δ H Γ δ u, vs ds t L M L 2 M2 Γ δ { t s δ t 2 s δ t2 } t s δ t 2 s δ ds t 2 s δ ds, t independent of u, v as t 2 t. In a similar manner, one can obtain that H 2 u, vt 2 H 2 u, vt, c D α H 2 u, vt 2 c D α H 2 u, vt, independent of u, v as t 2 t. Thus the operator H is equicontinuous in view of equicontinuity of H and H 2. Therefore, by Arzelá-Ascoli s theorem, we deduce that the operator H is completely continuous. Finally, it will be established that the set εh = {u, v X Y : u, v = λhu, v ; λ } is bounded. Let u, v εh. Then u, v = λhu, v. For any t [,, we have ut = λh u, vt, vt = λh 2 u, vt. Using A in 3.2, we can find that u µ µ u X max{µ 2, µ 3 } v Y M λ max{λ, λ 2 } u X λ 3 v Y M 2.

11 On a coupled system of sequential fractional differential equations 3 Similarly, we can obtain u µ µ u X max{µ 2, µ 3 } v Y M λ max{λ, λ 2 } u X λ 3 v Y M 2, which leads to Thus we have c D δ u { µ µ u X max{µ 2, µ 3 } v Y Γ2 δ M λ max{λ, λ 2 } u X λ 3 v Y M 2 }. Likewise, we can have u X = u c D δ u µ µ u X max{µ 2, µ 3 } v Y M λ max{λ, λ 2 } u X λ 3 v Y M 2 { µ µ u X max{µ 2, µ 3 } v Y Γ2 δ M λ max{λ, λ 2 } u X λ 3 v Y M 2 }. 3.9 v Y µ µ u X max{µ 2, µ 3 } v Y N λ max{λ, λ 2 } u X λ 3 v Y N 2 Γ2 α {µ µ u X max{µ 2, µ 3 } v Y N λ max{λ, λ 2 } u X λ 3 v Y N 2 }. 3.2 From 3.9 and 3.2 together with the notations , we find that u, v X Y = u X v Y σ max{σ 2, σ 3 } u, v X Y, 3.2 σ which yields u, v X Y. This shows that εh is bounded. Thus, the max{σ 2, σ 3 } conclusion of Lemma 2 applies and the operator H has at least one fixed point. Consequently, the problem. and.2 has at least one solution on [,. This completes the proof. Example. Consider the following coupled system of fractional differential equations c D 4/3 D txt = 7 sin xt 3 yt 2 c D /3 yt, t [, c D 6/5 D t yt = 4 t 2 6 xt 2 c D /3 xt 4 yt 6 yt, 3.22

12 4 B. Ahmad, A. Alsaedi, S. Aljoudi and S. K. Ntouyas supplemented with nonlocal coupled integral boundary conditions: x =, x =, x3/4 = 2 /2 y =, y =, y/4 = /3 ys ds, /5 xs ds Here, k t = t, k 2 t = t/4, ρ =, ρ 2 = /4, q = 7/3, a = 2, b = /3, p = /5, ζ = 3/4, η = /2, β =, z = /4, θ = /5, γ =, α = /3, δ = /2, ft, xt, yt, c D α yt = sin xt 7 3 yt 2 c D /3 yt and gt, xt, c D δ xt, yt = 4 yt 6 yt t 2 6 xt 2 c D /3 xt. Clearly, the functions f and g satisfy the condition A with µ =, µ =, µ 2 = 7 3, µ 3 = 2, λ =, λ = 6, λ 2 = 2, λ 3 = 4 6. Using the given data, we find that A.2456, A 2.442, B.4279, B 2.328,.765, M.73462, M , M , M , N.42834, N , N.32376, N2.7925, σ , σ With max{σ 2, σ 3 } <, all the conditions of Theorem are satisfied. Therefore, the problem has a solution on [,. Notice that x =, y = is also a solution of the problem In the following result, we establish the uniqueness of solutions for the problem. and.2 by means of Banach s contraction mapping principle. In the sequel, we use the notations: Λ = lm l M 2, M = r M r 2 M 2, Λ = l M l M2, M = r M r 2 M2, Λ = ln l N 2, N = r N r 2 N 2, Λ = l N l N2, N = r N r 2 N2, r = sup t [, ft,,, <, r 2 = sup t [, gt,,, < Theorem 2. Assume that H 2 holds. Further, we suppose that Λ Λ Γ2 δ Λ Λ <, 3.25 Γ2 α where Λ, Λ, Λ and Λ are given by Then the problem.-.2 has a unique solution on [,. [ M Proof: Let us define r M Γ2 δ N N Λ Λ Γ2 α Γ2 δ Λ Λ, where Λ, Λ, Λ, Λ and Γ2 α M, M, N and N are given by Then we show that HB r B r, where B r = {u, v X Y : u, v X Y r}. For u, v B r, we have ft, ut, vt, c D α vt ft, ut, vt, c D α vt ft,,, ft,,, l[ ut vt c D α vt r l[ u X v Y r l u, v X Y r lr r,

13 On a coupled system of sequential fractional differential equations 5 where r is defined by Similarly, we have gt, ut, c D δ ut, vt l r r 2, where r 2 is defined by Then and H u, vt lr r M l r r 2 M 2 = lm l M 2 r r M r 2 M 2 Λr M, H u, vt lr r M l r r 2 M 2 = l M l M2 r r M r 2 M2 which implies that Λ r M, Therefore, c D δ H u, vt t s δ Γ δ H u, vs ds Γ2 δ Λ r M. H u, v X = H u, v c D δ H u, v Λ Λ r M Γ2 δ M Γ2 δ In similar manner, we obtain H 2 u, vt Λ r N, H 2u, vt Λ r N and c D α H 2 u, vt In consequence, we get t s α Γ α H 2u, vs ds Γ2 α Λ r N. H 2 u, v Y = H 2 u, v c D α H 2 u, v Λ Λ N r N Γ2 α Γ2 α Thus, it follows from 3.26 and 3.27 that Hu, v X Y = H u, v X H 2 u, v Y r, which implies HB r B r. Now we prove that the operator H is a contraction. For u i, v i B r ; i =, 2 and for each t [,, by virtue of the condition H 2, we obtain H u, v t H u 2, v 2 t Λ [ u u 2 X v v 2 Y, which implies that H u, v t H u 2, v 2 t Λ [ u u 2 X v v 2 Y, c D δ H u, v t c D δ H u 2, v 2 t t s δ Γ δ H u, v s H u 2, v 2 s ds Γ2 δ Λ [ u u 2 X v v 2 Y.

14 6 B. Ahmad, A. Alsaedi, S. Aljoudi and S. K. Ntouyas From the above inequalities, we get H u, v H u 2, v 2 X = H u, v H u 2, v 2 c D δ H u, v c D δ H u 2, v 2 [ Λ Λ [ u u 2 X v v 2 Y Γ2 δ Similarly, we can find that H 2 u, v H 2 u 2, v 2 Y Consequently, it follows from 3.28 and 3.29 that [ Λ Λ [ u u 2 X v v 2 Y Γ2 α Hu, v Hu 2, v 2 X Y [ Λ Λ Λ Γ2 δ Λ Γ2 α [ u u 2 X v v 2 Y. By the assumption 3.25, it follows that the operator H is a contraction. Hence, by Banach s fixed point theorem, the operator H has a unique fixed point, which corresponds to a unique solution of problem.-.2. This completes the proof. Example 2. Consider the coupled system of fractional differential equations: c D 5/2 D 2txt = ft, xt, yt, c D /2 yt, c D 9/4 D t yt = gt, xt, c D /3 xt, yt, equipped with nonlocal coupled integral boundary conditions: 32t 2 x =, x =, x3/4 = y =, y =, y/2 = yt c D /2 yt e t yt c D /2 yt /4 /3 ys ds, xs ds. 3.3 Here, k t = 2t, k 2 t = t/2, ρ = 2, ρ 2 = /2, q = 5/2, a = b =, p = 9/4, ζ = 3/4, η = /4, β =, z = /2, θ = /3, γ =, α = /2, δ = /3, ft, xt, yt, c D α yt = tan xt e t and gt, xt, c D δ xt, yt = 2πt 2 sinxt sin c D /3 xt yt logt. Clearly l = /32, and l = /2π. Using the given data, we find that A.2469, A , B.2592, B 2.286,.25938, M.64297, M , M , M , Λ.2376, Λ.85972, N , N , Λ.3466, N.8839, N2.9337, Λ and [Λ Λ /Γ3/2 Λ Λ /Γ7/ <. Thus all the conditions of Theorem 2 are satisfied. In consequence, by the conclusion of Theorem 2, there exists a unique solution for the problem on [,.

15 On a coupled system of sequential fractional differential equations 7 References [ B. Ahmad, S.K. Ntouyas, J. Tariboon, Fractional differential equations with nonlocal integral and integer fractional-order Neumann type boundary conditions, Mediterr. J. Math. 3 26, [2 J. Klafter, S. C Lim, R. Metzler Editors, Fractional Dynamics in Physics, World Scientific, Singapore, 22. [3 A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 24. Elsevier Science B.V., Amsterdam, 26. [4 J.R. Graef, L. Kong, Q. Kong, Application of the mixed monotone operator method to fractional boundary value problems, Fract. Calc. Differ. Calc. 2 2, [5 Z.B. Bai, W. Sun, Existence and multiplicity of positive solutions for singular fractional boundary value problems, Comput. Math. Appl , [6 G. Wang, B. Ahmad, L. Zhang, R. P. Agarwal, Nonlinear fractional integrodifferential equations on unbounded domains in a Banach space, J. Comput. Appl. Math , [7 J.S. Duan, Z. Wang, Y.L. Liu, X. Qiu, Eigenvalue problems for fractional ordinary differential equations, Chaos Solitons Fractals 46 23, [8 D. O Regan, S. Stanek, Fractional boundary value problems with singularities in space variables, Nonlinear Dynam. 7 23, [9 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. 9 24, [ Y.Z. Povstenko, Fractional Thermoelasticity, Springer, New York, 25. [ Z.M. Ge, W.R. Jhuang, Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor, Chaos Solitons Fractals 33 27, [2 F. Zhang, G. Chen C. Li, J. Kurths, Chaos synchronization in fractional differential systems, Phil Trans R Soc A 37 23, [3 R. Metzler, J. Klafter, The random walk s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep , -77. [4 J. Sun, Y. Liu, G. Liu, Existence of solutions for fractional differential systems with antiperiodic boundary conditions, Comput. Math. Appl , [5 B. Senol, C. Yeroglu, Frequency boundary of fractional order systems with nonlinear uncertainties, J. Franklin Inst ,

16 8 B. Ahmad, A. Alsaedi, S. Aljoudi and S. K. Ntouyas [6 B. Ahmad, S.K. Ntouyas, A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal. 7 24, no. 2, [7 J. Henderson, R. Luca, A. Tudorache, On a system of fractional differential equations with coupled integral boundary conditions, Fract. Calc. Appl. Anal. 8 25, [8 B. Ahmad, S.K. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput , [9 B. Ahmad, J.J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces Appl. 23, Art. ID 49659, 8 pp. [2 A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 23. Received: Revised: Accepted:.2.26 a Nonlinear Analysis and Applied Mathematics NAAM-Research Group Department of Mathematics Faculty of Science, King Abdulaziz University P.O. Box 823, Jeddah 2589, Saudi Arabia bashirahmad qau@yahoo.com B. Ahmad aalsaedi@hotmail.com A. Alsaedi sh-aljoudi@hotmail.com S. Aljoudi b,a Department of Mathematics University of Ioannina 45 Ioannina, Greece sntouyas@uoi.gr