Existence and Hyers Ulam stability for three-point boundary value problems with Riemann Liouville fractional derivatives and integrals

Transcription

1 Xu et al. Advances in Difference Equations (28) 28:458 R E S E A R C H Open Access Existence and Hyers Ulam stability for three-point boundary value problems with Riemann Liouville fractional derivatives and integrals Lei Xu, Qixiang Dong * and Gang Li * Correspondence: qxdongyz@outlook.com; qxdong@yzu.edu.cn School of Mathematical Sciences, Yangzhou University, Yangzhou, China Abstract This paper is concerned with a class of two-term fractional differential equations. Three-point boundary value problems with mixed Riemann Liouville fractional differential and integral boundary conditions are discussed. The Green s function is investigated and the existence results are obtained based on some fixed point theorems. The Hyers Ulam stability is also studied for null boundary conditions. As an auxiliary result, a Gronwall type inequality of fractional order integral is obtained. Keywords: Fractional differential equation; Boundary value problem; Stability Introduction In this paper, we consider the following boundary value problem of nonlinear fractional differential equations: λd α x(t)dβ x(t)=f (t, x(t)), < t < T, x() =, D γ x(t)iγ 2 x(η)=γ 3, () where D α and Dβ are Riemann Liouville fractional derivatives with < α 2and β < α, <λ,, γ α β, γ 2, < η < T and f :[,T R R is a given function satisfying some assumptions that will be verified later. Fractional differential problems have attracted much attention in recent years due to its wide application in many fields of science and engineering, including fractal theory, potential theory, biology, chemistry, diffusion, etc. See, e.g., [, 2. Thereareseveral kindsoffractional derivatives used in different applied areas, such as Caputo derivative and Riemann Liouville derivative. Recently, Atangana et al. [3 6 presented the Caputo Fabrizio and the Atangana Baleanu fractional derivatives. In the literature, c D α u(t)f (t, u(t)) = is known as a single term equation. This kind of fractional differential equation has many applications and has been studied widely. See, e.g., [, 2, 7. Equations containing more than one fractional differential terms are called multi-term fractional differential equations; they have some concrete applications The Author(s) 28. This article is distributed under the terms of the Creative Commons Attribution 4. International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Xu et al. Advances in Difference Equations (28) 28:458 Page 2 of 7 in many fields. Due to the complexity of such a kind of equations, it seems that there has been no result for a general multi-term fractional differential equation. Only some special cases have been investigated. A classical example is the so-called Bagley Torvik equation (B T equation for short) [, Ay (t)b c D 3 2 y(t)cy(t)=f (t), where A, B and C are certain constants and f is a given function. This equation arises from the mathematical model of the motion of a thin plate in a Newtonian fluid. The B T equation, as well as various generalizations, has wide applications in fluid dynamics and hence hasattractedmuchattention. Forexample,Cermaket al. [2 investigated the twoterm fractional differential equation D α y(t)adβ y(t)yb(t)= with real coefficients a, b and positive real orders α > β, which contains some important casessuchastheb Tequationforα =2,β = 3/2 and the Basset equation for α =,β = /2. The analytic solution and the numerical solution for the B T equation were studied in [3 and [4, respectively. Various methods were introduced to investigate the approximate solutions such as the finite difference method [5, the variational iteration method [3, 6, the homotopy perturbation method [7 and the generalized differential transform method [8. BoundaryvalueproblemsfortheB Tequationswerestudiedin[9, 2and [8, 2 23for various boundary value conditions. In [8,the authors considered the approximate solution of B T equations with variable coefficients and three-point boundary value, y (x)p(x)d α ay q(x)y = g(x), x [a, b, y(a)=α, y(b)λy(ξ)=β, ξ [a, b, where < α <2,p(x), q(x) andg(x) areknownfunctions,α, β, λ, and ξ are given constants. In [9, 2, Ntouyas et al. studied the generalized B T equations with multiple integral and differential boundary conditions λd α x(t)( λ)dβ x(t)=f (t, x(t)), < t < T, x() =, D γ (T)( )Dγ 2 (T)=γ 3, and λd α x(t)( λ)dβ x(t)=f (t, x(t)), < t < T, x() =, I δ x(t)( )Iδ 2 x(t)=δ 3. Green functions for the corresponding problems were investigated and the existence results were obtained by using fixed point theorems. For more detailed information, we refer the reader to [2, and the references therein. Stability analysis is an important respect of differential equations. Hyers Ulam stability for differential equations was initialed in the 94s by Ulam [27and Hyers[28. Roughly

3 Xu et al. Advances in Difference Equations (28) 28:458 Page 3 of 7 speaking, the Hyers Ulam stability for a differential equation is the answer to the question whether there is an exact solution near an approximate solution (the solution to the approximate equation) to the differential equation. So it is obviously important for the study of numerical and approximate solutions and real world applications of differential equations. For this reason, many researchers investigate the Hyers Ulam stability for differential equations of both integer and fractional order [ There are only a few works on the Ulam stability of fractional differential equations. Recently, Wang, Lv and Zhou [33 presented some Ulam stability results of fractional differential equations by using Hery Gronwallinequality.Bythemethod oflaplacetransform,wang and Li [34 studied Ulam stability of a fractional order linear differential equation. Later, Wang and Li [35investigated the Hyers Ulam stability for a nonlinear fractional Langevin equation and its corresponding impulsive problem. The authors of [36 researched a class of new differential equations with no instantaneous impulses. However, to the best of our knowledge, few results can be found on the Hyers Ulam type stability for boundary value problems except that of Kumam [37. Inspired by the above comments, in this paper, we consider the three points boundary value problem of a two-term fractional differential equation, with mixed integral and differential boundary conditions (). Equation () with Riemann Liouville fractional derivatives of order α and β is a generalization of the B T equation. To compare with the discussion in [, 9, there is no coefficient ( λ)or( ) here, and the boundary conditions with mixed derivatives and integrals are different from the boundary value conditions in [, 9.So,our results can be regarded as an extension of B T equation and partially extend the results in [, 9, 38 and[2. We also investigate the Hyers Ulam stability for Eq. () with the boundary value conditions x() = and x(t)=γ.todealwiththehyers Ulam stability, we prove a fractional Gronwall-type integral inequality by the method of iteration, which is a generalization of the main result in [39and Lemma 3.4 in [. 2 Preliminaries Let C([, T, R) be the Banach space of all continuous functions from [, T intor with the norm x = max t [,T x(t),andl ([, T, R) denotes the Banach space of functions x :[,T R that are Lebesgue integrable with norm x L = x(t) dt. Definition 2. ([) The Riemann Liouville fractional integral of order α >ofafunction f :[a, b R at the point t is defined by Ia α f (t)= (t s) α f (s) ds, Γ (α) a provided the right side is point-wisely defined, where Γ is the Gammafunction. Definition 2.2 ([) The Riemann Liouville fractional derivative of order α >ofafunction f :[a, b R at the point t is defined by d n D α a f (t)= Γ (n α) dt n a (t s) n α f (s) ds, provided the right side is point-wisely defined, where n =[α,[α denotes the integer part of α.

4 Xu et al. Advances in Difference Equations (28) 28:458 Page 4 of 7 Lemma 2.3 ([) Let α >,and x C[, T L[, T. Then the fractional differential equation D α x(t)=has the unique solution x(t)=c t α c 2 t α 2 c n t α n, c i R, i =,,2,...,n,n =[α. Lemma 2.4 ([) Let α >.Then for x C(, T) L(, T), we have I α Dα x(t)=x(t)c t α c 2 t α 2 c n t α n, for some c i R, i =,,2,...,n,n =[α. Lemma 2.5 The solution of the boundary value problem () satisfies the integral equation where x(t)= γ 3 tα = Γ (α)t α γ Γ (α γ ) G (t, s)= λγ (α β) tα λ [ t α G (t, s)x(s) ds G 2 (t, s)f ( s, x(s) ) ds, Γ (α)ηαγ 2, <t < T, (2) Γ (α γ 2 ) (t s)α β Γ (α β γ ) (T s)α β γ Γ (α βγ 2 ) (η s)α βγ2, < s t η < T, λ [ Γ (α β γ ) (T s)α β γ Γ (α βγ 2 ) (η s)α βγ2, < t s η < T, t α λγ(α β γ ) (T s)α β γ, <t η s < T, (t s)α β λγ (α β) tα λ [ Γ (α β γ ) (T s)α β γ G 2 (t, s)= Γ (α βγ 2 ) (η s)α βγ2, < s η t < T, (4) λγ (α β) (t s)α β t α λγ(α β γ ) (T s)α β γ, <η s t < T, t α λγ(α β γ ) (T s)α β γ, <η t s < T, λγ (α) (t s)α tα λ [ Γ (α γ ) (T s)α γ G 2 (t, s)= Γ (αγ 2 ) (η s)αγ2, < s t η < T, tα λ [ tα Γ (α γ ) (T s)α γ Γ (αγ 2 ) (η s)αγ2, < t s η < T, λγ(α γ ) (T s)α γ, <t η s < T, λγ (α) (t s)α tα λ [ Γ (α γ ) (T s)α γ Γ (αγ G 22 (t, s)= 2 ) (η s)αγ2, < s η t < T, λγ (α) (t s)α tα λγ(α γ ) (T s)α γ, <η s t < T, tα λγ(α γ ) (T s)α γ, <η t s < T, (3) (5) (6)

5 Xu et al. Advances in Difference Equations (28) 28:458 Page 5 of 7 G (t, s), < t η < T, G (t, s)= G 2 (t, s), < η t < T, G 2 (t, s), < t η < T, G 2 (t, s)= G 22 (t, s), < η t < T, (7) (8) Proof From Eq. (), we have D α x(t)= λ Dβ x(t) λ f ( t, x(t) ), t [, T. (9) Taking the Riemann Liouville fractional integral of order α on both sides of (9), we get x(t)= (t s) α β x(s) ds λγ (α β) λγ (α) C t α C 2 t α 2. (t s) α f ( s, x(s) ) ds Then, x() = implies C 2 =.Hence x(t)= (t s) α β x(s) ds (t s) α f ( s, x(s) ) ds λγ (α β) λγ (α) C t α. () Applying the Riemann Liouville fractional derivative of order γ to both sides of (), we deduce that D γ x(t)= λγ (α β γ ) λγ (α γ ) C Γ (α) Γ (α γ ) tα γ. (t s) α β γ x(s) ds (t s) α γ f ( s, x(s) ) ds Taking the Riemann Liouville integral of order γ 2 to both sides of (), we obtain Let I γ 2 x(t)= λγ (α β γ 2 ) λγ (α γ 2 ) C Γ (α) Γ (α γ 2 ) tαγ 2. := Γ (α)t α γ Γ (α γ ) (t s) α βγ 2 x(s) ds (t s) αγ 2 f ( s, x(s) ) ds Γ (α)ηαγ 2 Γ (α γ 2 ).

6 Xu et al. Advances in Difference Equations (28) 28:458 Page 6 of 7 By using the second condition of (), we get γ 3 = λγ (α β γ ) λγ (α γ ) (T s) α β γ x(s) ds (T s) α γ f ( s, x(s) ) ds Γ (α) C Γ (α γ ) T α γ λγ (α β γ 2 ) λγ (α γ 2 ) η η (η s) α βγ 2 x(s) ds (η s) αγ 2 f ( s, x(s) ) ds C Γ (α) Γ (α γ 2 ) ηαγ 2, which leads to C = [ T γ 3 (T s) α β γ x(s) ds λγ (α β γ ) T (T s) α γ f ( s, x(s) ) ds λγ (α γ ) η (η s) α βγ2 x(s) ds λγ (α β γ 2 ) η (η s) αγ2 f ( s, x(s) ) ds. λγ (α γ 2 ) Substituting the value of C into (), we have x(t)= (t s) α β x(s) ds (t s) α f ( s, x(s) ) ds λγ (α β) λγ (α) [ tα T γ 3 (T s) α β γ x(s) ds λγ (α β γ ) T (T s) α γ f ( s, x(s) ) ds λγ (α γ ) η (η s) α βγ2 x(s) ds λγ (α β γ 2 ) η (η s) αγ2 f ( s, x(s) ) ds. λγ (α γ 2 ) In the two cases of < t η < T and < η t < T,wecanverify x(t)= γ 3 tα G (t, s)x(s) ds G 2 (t, s)f ( s, x(s) ) ds, which completes the proof. It is easily seen that G 2 is continuous, and hence is bounded on [, T [, T. When α β <orα β γ <,G is unbounded. Fortunately, we have G (t, s) ds λγ (α β) (t s) α β ds

7 Xu et al. Advances in Difference Equations (28) 28:458 Page 7 of 7 [ tα λ Γ (α β γ ) Γ (α β γ 2 ) η (T s) α β γ ds (η s) α βγ2 ds = λγ (α β ) tα β [ tα λ Γ (α β γ ) T α β γ Γ (α β γ 2 ) ηα βγ 2 λγ (α β ) T α β T α λ [ Γ (α β γ ) T α β γ Γ (α β γ 2 ) ηα βγ 2 which means G (t, s) ds is uniformly bounded for t [, T. We denote and M := max t (,T) M 2 := max t (,T) G (t, s) ds G 2 (t, s) ds. Next, we introduce an integral inequality which can be regarded as a generalization of the Gronwall inequality. It is also the generalization of the main result in [39 and Lemma 3.4 of [. Lemma 2.6 Suppose α >, a >, g(t, s) is a nonnegative continuous function defined on [, T [, T with g(t, s) M, and g(t, s) is nondecreasing w.r.t. the first variable and nonincreasing w.r.t. the second variable. Assume that u(t) is nonnegative and integrable on [, T with, u(t) a g(t, s)(t s) α u(s) ds, t [, T. Then u(t) a a (g(t, s)γ (α)) n (t s) nα ds. () Γ (nα) n= Proof Let Au(t)= g(t, s)(t s)α u(s) ds,whereu(t) is nonnegative and locally integrable on t [, T. It follows that u(t) a Au(t), which implies n u(t) A k a A n u(t) k= ( A a = a ). (2)

8 Xu et al. Advances in Difference Equations (28) 28:458 Page 8 of 7 We now prove that A n u(t) (g(t, s)γ (α)) n (t s) nα u(s) ds (3) Γ (nα) by induction. For n =,theproofistrivial.assumethateq.(3)holdsforn = k.then,for n = k,weobtain A k u(t) =A ( A k u(t) ) = = = = τ g(t, s)(t s) α A k u(s) ds s g(t, s)(t s) α (g(s, τ)γ (α)) k (s τ) kα u(τ) dτ ds Γ (kα) s (g(t, τ)γ (α)) k g(t, τ) (t s) α (s τ) kα u(τ) dτ ds Γ (kα) [ g(t, τ) k Γ (α) k Γ (kα) (t s)α (s τ) kα ds u(τ) dτ g(t, τ) k (Γ (α)) k B(α, kα)(t τ) (k)α u(τ) dτ Γ (kα) (g(t, s)γ (α)) k (t s) (k)α u(s) ds. Γ ((k )α) By mathematical induction, the inequality (3) holdsforalln N. Replacedu(t) bya in (3) wededucethata k a a (g(t,s)γ (α)) k (t s) kα ds, k =,2,...Similartotheproofof Γ (kα) Lemma 3.4 of [wecanverifythat A n u(t) (MΓ (α)) n (t s) nα u(s) ds Γ (nα) as n uniformly in t [, T. Finally, letting n in (2), we get u(t) n= A n a a a (g(t, s)γ (α)) n (t s) nα ds, Γ (nα) n= i.e., the inequality ()holds. The lemma is proved. 3 Existence results In this section, we investigate the existence for BVP (). For convenience we list the hypothesis. (H ) f :[,T R R is continuous. (H 2 ) There exists a constant L >such that f (t, x ) f(t, x 2 ) L x x 2 for all x, x 2 R and t [, T. (H 3 ) Thereexistsu C([, T, R ) such that f (t, x) u(t) for each t [, T and x R.

9 Xu et al. Advances in Difference Equations (28) 28:458 Page 9 of 7 (H 4 ) There exist a continuous function φ (t) C([, T, R ) and a nondecreasing function φ 2 C([, T, R ) such that f (t, x) φ (t)φ 2 ( x ) for each t [, T and x R. Theorem 3. Assume that (H ) and (H 2 ) hold. If M LM 2 <, then the boundary value problem () has a unique solution in C([, T,R). Proof We define an operator S : C([, T, R) C([, T, R)by Sx(t)= γ 3 tα G (t, s)x(s) ds G 2 (t, s)f ( s, x(s) ) ds, for each t (, T). By Lemma 2.3, x C([, T, R) is a solution of problem ()ifandonlyifx is a fixed point of S. We now prove that S has a fixed point. Taking x, x 2 C([, T, R) arbitrary, according to (H 2 ), we find Sx (t) Sx 2 (t) and hence G (t, s) x (s) x 2 (s) ds G2 (t, s) ( f s, x (s) ) f ( s, x 2 (s) ) ds x x 2 (M LM 2 ) x x 2, G (t, s) ds L x x 2 G 2 (t, s) ds Sx Sx 2 (M LM 2 ) x x 2. Since M LM 2 <,S is a contraction. By Banach contraction principle, S has a unique fixed point in C([, T,R), i.e., the boundary value problem () hasauniquesolution. This completes the proof. Theorem 3.2 Assume that (H ) and (H 3 ) hold. If M <, then the boundary value problem () has at least one solution in C([, T, R). Proof We define mappings A and B from C((, T), R)intoitselfby Ax(t)= γ 3 tα G (t, s)x(s) ds

10 Xu et al. Advances in Difference Equations (28) 28:458 Page of 7 and Bx(t)= G 2 (t, s)f ( s, x(s) ) ds. It is easy to verify that B is continuous on C([, T, R), since G 2 and f are continuous. Further, since M <, we can choose a constant r > large enough such that M M 2 u γ 3T α <, i.e., rm r r M 2 u γ 3T α < r, whereu belongs to C([, T, R ) such that f (t, x) u(t), according to (H 3 ). Set B r = {x C((, T), R): x r}.thenb r is nonempty, bounded, closed and convex. Moreover, for any x, y B r and t [, T, we have Ax(t) γ 3 t α G (t, s) x(s) ds and rm γ 3T α T By(t) G2 (t, s) ( ) f s, x(s) ds M 2 u. Thus, Ax(t)By(t) rm M 2 u γ 3T α < r, i.e., Ax By B r. Next we prove that A is a contraction. In fact, for any x, y C((, T), R)andt [,, Ax(t) Ay(t) G (t, s) x(t) y(t) ds M x y. It follows that Ax Ay M x y.sincem <,weknowthata is a contraction. Finally, we have to show that B is compact. Take any bounded subset U C([, T, R). Then there exists a constant r >suchthatu = {u U : u r }.WeprovethatBU is bounded and equicontinuous. In fact, for any x U, we have Bx(t) G 2 (t, s) f ( s, x(s) ) ds M 2 u M 2 r for each t (, T). Hence BU is bounded. Further, for any t < t 2 T and x U, we have Bx(t2 ) Bx(t ) T = G 2 (t 2, s)f ( s, x(s) ) G 2 (t, s)f ( s, x(s) ) ds ( t α t α ) 2 (T s) α γ f ( s, x(s) ) ds λγ(α γ ) η ( t α t α ) 2 (η s) αγ 2 f ( s, x(s) ) ds λγ(α γ 2 ) [ (t2 s) α (t s) α f ( s, x(s) ) ds λγ (α) 2 λγ (α) (t 2 s) α f ( s, x(s) ) ds t

11 Xu et al. Advances in Difference Equations (28) 28:458 Page of 7 (tα 2 t α ) λγ(α γ ) tα 2 t α λγ(α γ 2 ) λγ (α) λγ (α) 2 t η (T s) α γ ds u (η s) αγ 2 ds u (t2 s) α (t s) α ds u (t 2 s) α ds u = T α γ (t α 2 t α ) λγ(α γ ) r ηαγ 2(t α 2 t α ) λγ(α γ 2 ) r [ t α λγ (α ) 2 t α 2(t 2 t ) α r. We can see from the above inequality that Bx(t 2 ) Bx(t ) ast 2 t, and the convergence is independent on x, t and t 2. This shows that BU is equicontinuous. An employment of Arzelà Ascoli theorem shows that B is compact. Now, we apply Krasnoselskii s fixed point theorem on operator A and B to deduce that there exists at least one x such that Ax Bx = x, which is the solution of the boundary value problem (). Theorem 3.3 Assume that (H ) and (H 4 ) hold. If M M 2 φ lim sup r φ 2 (r) r <, then the boundary value problem () has at least one solution in C([, T, R). Proof Define a mapping S : C([, T, R) C([, T, R)by Sx(t)= γ 3 tα G (t, s)x(s) ds G 2 (t, s)f ( s, x(s) ) ds for t [, T. It is easy to prove that S is continuous. In order to apply Schauder s fixed point theorem, we only need to show that S is compact. Firstly, we take any bounded subset Q C([, T, R). Then there exists q (q >)satisfying that Q B q = {x C([, T, R) : x q}. NoticethatB q is a closed convex and bounded subset. For any x B q,wehave Sx(t) γ 3 t α G (t, s) T x(s) ds G2 (t, s) ( ) f s, x(s) ds γ 3T α x G (t, s) ( ) ds φ φ 2 x G 2 (t, s) ds, γ 3T α TM q TM 2 φ φ 2 (q),

12 Xu et al. Advances in Difference Equations (28) 28:458 Page 2 of 7 which means SB q, and therefore SQ,isuniformlyboundedinC((, T), R). For t < t 2 T and any x B q, Sx(t ) Sx(t 2 ) γ 3 ( t α 2 t α ) T G 2 (t 2, s)x(s) G (t 2, s)x(s) G (t, s)x(s) ds G 2 (t, s)x(s) ds = P P 2 P 3 ; P = γ 3 ( t α 2 t α ), P 2 = G (t 2, s)x(s) G (t, s)x(s) ds ( t α 2 t α ) (T s) α β γ x(s) ds λγ(α β γ ) η ( t α 2 t α ) (η s) α βγ 2 x(s) ds λγ(α β γ 2 ) [ (t2 s) α β (t s) α β x(s) ds λγ (α β) 2 λγ (α β) (t 2 s) α β x(s) ds t (tα 2 t α ) λγ(α β γ ) t2 α t α λγ(α β γ 2 ) λγ (α β) λγ (α β) 2 t η (T s) α β γ ds x (η s) α βγ 2 ds x (t 2 s) α β (t s) α β ds x (t 2 s) α β ds x (t2 α t α )T α β γ q = λγ(α β γ ) (tα 2 t α )η α β γ q λγ(α β γ 2 ) q [ α β t 2 t α β 2(t 2 t ) α β, λγ (α β) P 3 = G 2 (t 2, s)f ( s, x(s) ) G 2 (t, s)f ( s, x(s) ) ds ( t α t α ) 2 (T s) α γ f ( s, x(s) ) ds λγ(α γ ) η ( t α t α ) 2 (η s) αγ 2 f ( s, x(s) ) ds λγ(α γ 2 ) [ (t2 s) α (t s) α f ( s, x(s) ) ds λγ (α) 2 λγ (α) (t 2 s) α f ( s, x(s) ) ds t

13 Xu et al. Advances in Difference Equations (28) 28:458 Page 3 of 7 (tα 2 t α ) λγ(α γ ) (tα 2 t α ) λγ(α γ 2 ) λγ (α) λγ (α) [ 2 [ [ η (T s) α γ ( ) ds φ φ 2 x (η s) αγ2 ( ) ds φ φ 2 x (t2 s) α (t s) α ( ) ds φ φ 2 x t (t 2 s) α ds φ φ 2 ( x ) = T α γ (t α 2 t α ) λγ(α γ ) φ φ 2 (q) ηαγ 2(t2 α t α ) λγ(α γ 2 ) φ φ 2 (q) [ t α λγ (α ) 2 t α 2(t 2 t ) α φ φ 2 (q). It is trivial that P, P 2 and P 3 all tend to as t 2 t. Hence, Sx(t 2 ) Sx(t ) (t 2 t ). Notice that the convergence is independent on x, t and t 2. It follows that SB q is equicontinuous. An application of the Arzelà Ascoli Theorem yields that SB q is compact. Therefore, we have shown that S maps bounded subsets into compact subset, i.e., S is a compact mapping. Now, from the condition M M 2 φ lim r sup φ 2(r) <, we can choose a positive r r large enough, such that M M 2 φ φ 2(r) r γ 3T α r <, i.e., M r M 2 φ φ 2 (r) γ 3T α < r. Hence, we can take C >suchthatm C M 2 φ φ 2 (C) γ 3T α < C.LetU = {x C((, T), R): x C}.ThenS : U C((, T), R) is compact and continuous. If there exist λ (, ) and x U such that x = λsx, then,for every t (, T), x(t) = λsx(t) Sx(t) γ 3 t α T T G (t, s)x(s) ds γ 3T α M ( ) C M 2 φ φ 2 x M C M 2 φ φ 2 (C) γ 3T α. G 2 (t, s)f ( s, x(s) ) ds It then follows that C = x M C M 2 φ φ 2 (C) γ 3T α < C, which contradict to the fact that x U. Thus,x λsx for each λ (, ) and x U. BytheLeraySchauderalternative, S has at least one fixed point, which is the solution to the boundary value problem (). This completes the proof.

14 Xu et al. Advances in Difference Equations (28) 28:458 Page 4 of 7 4 Stability analysis In this section, we study Hyers Ulam stability of the boundary value problem λd α x(t)dβ x(t)=f (t, x(t)), t T, x() =, x(t)=γ. (4) Definition 4. The boundary value problem (4) ishyers Ulamstableifthereexistsa real constant c >, such that forany ε >, and forevery solution y(t) C([, T,R) of the inequality λd α y(t)d β y(t) f ( t, y(t) ) ε, t [, T, there exists a solution x(t) C([, T, R)ofproblem(4)with y(t) x(t) cε, t [, T. Theorem 4.2 Assume that (H ) and (H 2 ) hold. Then the solution of the boundary value problem (4) is Hyers Ulam stable. Proof For ε >, and each solution y(t) C([, T, R)oftheinequality λd α y(t)dβ y(t) f ( t, y(t) ) ε, t [, T, we can find a function g(t) satisfyingλd α y(t)dβ y(t)=f (t, y(t)) g(t) and g(t) ε. It follows that y(t)= λ Iα β y(t) λ Iα f ( t, y(t) ) λ Iα g(t) [ tα y(t) T α λ Iα β y(t) λ Iα f ( T, y(t) ) λ Iα. g(t) Let x(t) C([, T,R) betheuniquesolutionof (4). Then x(t) isgivenby x(t)= λ Iα β x(t) λ Iα f ( t, x(t) ) [ tα x(t) T α λ Iα β x(t) λ Iα f ( T, x(t) ). Then we have y(t) x(t) λ Iα β tα T α y(t) x(t) [ λ Iα β λ Iα ( ) ( ) f t, y(t) f t, x(t) y(t) x(t) λ Iα ( ) ( ) f T, y(t) f T, x(t) λ Iα g(t) t α λt α Iα g(t) [ (t s) α β (t s)α y(s) x(s) L ds λ Γ (α β) Γ (α) [ tα (T s) α β (T s)α L y(s) x(s) ds λt α Γ (α β) Γ (α)

15 Xu et al. Advances in Difference Equations (28) 28:458 Page 5 of 7 [ (T s)α β where λ on ε. Letg(t, s) = (t s) α g(s) t α (T s) α ds g(s) ds λ Γ (α) λt α Γ (α) [ (t s) α β (t s)α y(s) x(s) L ds λ Γ (α β) Γ (α) [ (T s) α β (T s)α L y(s) x(s) ds λ Γ (α β) Γ (α) 2ε (T s) α ds λ Γ (α) [ (t s) α β (t s)α L y(s) x(s) ds λ Γ (α β) Γ (α) Γ (α β) M(ε)ε y(t) x(t) Mε We note that g(t, s) 2T α λγ (α ) ε, L (T s)α y(s) x(s) ds M(ε)ε and M(ε) is a constant dependent Γ (α) and M = M(ε) 2Tα λγ (α β) L (t s)β λγ (α) y(t) x(t) Mε Mε λγ (α).then g(t, s)(t s) α β y(s) x(s) ds. λγ (α β) L Tβ (= M). Hence, in view of Lemma 3.4, λγ (α) Mε Mε = Mε Mε n= (g(t, s)γ (α β)) n (t s) n(α β) ds Γ (n(α β)) n= n= (MΓ (α β)) n Γ (n(α β)) (t s)n(α β) ds (MΓ (α β)) n Γ (n(α β)) T n(α β) MεE α β ( MT (α β) Γ (α β) ). Let c = ME α β (MT (α β) Γ (α β)). The inequality y(t) x(t) cε holds. Thus, the boundary value problem (4)isHyers Ulamstable. 5 Examples Example 5. Let us consider the following multiply three-term fractional differential equation 4 5 D 3 2 x(t)d 5 4 x(t)=t 2 sin(x(t)), < t < 4, x() =, 8 D 8 x( 4 )I 3 4 x( 8 )= 6. (5) Here α = 3 2, β = 5 4, λ = 4 5, = 8, γ = 8 < α β, γ 2 = 3 4, γ 3 = 6, T = 4, η = 8 and f (t, x) =t 2 sin x. It is evident that f (t, x ) f (t, x 2 ) <( 4 )2 x x 2 < x x 2,whichsatisfies condition (H 2 ). So, we can select L =.Wecancalculatethat = ,

16 Xu et al. Advances in Difference Equations (28) 28:458 Page 6 of 7 M , M So, M LM <. Therefore, by applying Theorem 3., we deduce that the boundary value problem (5)has auniquesolutionon(, 4 ). Let g(t, s) = λγ (α β) L (t s)β λγ (α).notingthatg(t, s) λγ (α β) L Tβ λγ (α) (= M). Let x( 4 )=.ByTheorem4.2,theproblem(5)withx( 4 )=ishyers Ulamstable. Example 5.2 Considering the following boundary value problem which contains Riemann Liouville fractional derivatives of two orders in a differential equation and the condition 4 5 D 3 2 x(t)d 5 4 x(t)=t 2 x 2, <t < 8, x() =, 6 D 8 x( 4 )I 3 4 x( 6 )= 24. (6) Here α = 3 2, β = 5 4, λ = 4 5, = 6, γ = 8 < α β, γ 2 = 3 4, γ 3 = 24, T = 8 and η = 6. By direct computation, we have , M , M Choosing φ (t) =t 2 and φ 2 = x 2, we can show that M M 2 φ lim r sup φ 2(r) <. Hence, by Theorem 3.3, theproblem(6) has at least one r solution on (, 8 ). 6 Conclusion In this paper, we study a class of two-term fractional differential equations. We first investigate the Green function of a three-point boundary value problems with mixed fractional differential and integral boundary conditions. Existence results are obtained based on some fixed point theorems. We also study the Hyers Ulam stability for null boundary conditions. Our results improve and generalize the results in [, 9, 38and[2. Moreover, we present a Gronwall type inequality of fractional order integral by the method of iteration. Acknowledgements The authors thank the anonymous referees for carefully reading the paper and for their comments and suggestions. Funding This research was partially supported by the National Natural Science Foundation of China (573) and the Natural Science Foundation for Universities in Jiangsu Province of China (6KJB23). Competing interests The authors declare that they have no competing interests. Authors contributions The authors have made the same contribution. All authors read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 4 August 28 Accepted: 2 November 28 References. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 24. Elsevier, Amsterdam (26) 2. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (999) 3. Atangana, A.: Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-markovian properties. Physica A 55, (28) 4. Atangana, A.: Blind in a commutative world: simple illustrations with functions and chaotic attractors. Chaos Solitons Fractals 4, (28)

17 Xu et al. Advances in Difference Equations (28) 28:458 Page 7 of 7 5. Atangana, A., Gómez-Aguilar, J.F.: Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 4, (28) 6. Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 33(4), 66 (28) 7. Graef, J.R., Kong, L., Wang, M.: Existence and uniqueness of solutions for a fractional boundary value problem on a graph. Fract. Calc. Appl. Anal. 7,499 5 (24) 8. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann Liouville fractional derivatives. Rheologica Acta 45(5), (26) 9. Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 9, (2). Dong, Q., Liu, C., Fan, Z.: Weighted fractional differential equations with infinite delay in Banach spaces. Open Math. 4, (26). Bagley, R.L., Torvik, P.J.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 5, (984) 2. Čermák, J., Kisela, T., Nechvatál, L.: Stability and asymptotic properties of a linear fractional difference equation. Adv. Differ. Equ. 22, 22 (22) Das, S.: Analytical solution of a fractional diffusion equation by variational iteration method. Comput. Math. Appl. 57(3), (29) 4. El-gamel, M., El-Hady, M.A.: Numerical solution of the Bagley Torvik equation by Legendre-collocation method. SeMA J. 74(4), (27) Meerschaert, M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56,8 9 (26) 6. Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7(), (26) 7. Ganji, Z., Ganji, D., Jafari, H., Rostamian, M.: Application of the homotopy perturbation method to coupled system of partial differential equations with time fractional derivatives. Topol. Methods Nonlinear Anal. 3(2), (28) 8. Huang, Q.A., Zhong, X.C., Guo, B.L.: Approximate solution of Bagley Torvik equations with variable coefficients and three-point boundary-value conditions. Int. J. Appl. Comput. Math. 2, (26) 9. Niyom, S., Ntouyas, S.K., Laoprasittichok, S., Taiboon, J.: Boundary value problem with four orders of Riemann Liouville fractional derivations. Adv. Differ. Equ. 26,65 (26). Ntouyas, S.K., Tariboon, J.: Fractional boundary value problems with mutiply orders of fractional derivatives and integrals. Electron. J. Differ. Equ. 27, (27) 2. Ahmad, B., Ntouyas, S.K., Alsaedi, A.: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2, ArticleID7384 (2) 22. Alsaedi, A., Ntouyas, S.K., Ahmad, B.: New existence resultss for fractional integro-differential equations with nonlocal integral boundary conditions. Abstr. Appl. Anal. 25, ArticleID25452 (25) 23. Stanek, S.: Two-point boundary value problems for the generalized Bagley Torvik fractional differential equation. Cent.Eur.J.Math.(3), (23) 24. Bai, Z.B., Sun, W.: Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 63, (22) 25. Ahmad, B., Ntoyas, S.K., Tariboon, J.: Fractional differential equations with nonlocal integral and integer-fractional-order Neumann type boundary conditions. Mediterr. J. Math. (25) Ahmad, B., Ntoyas, S.K.: Existence results for Caputo type sequential fractional differential inclusions with nonlocal integer boundary conditions. J. Appl. Math. Comput. 5, (26) 27. Ulam, S.M.: A Collection of the Mathematical Problems. Interscience, New York (96) 28. Hyers, D.H.: On the stability of the linear functional differential equations. J. Math. Anal. Appl. 426(2), 92 2 (25) 29. Jung, S.M.: Hyers Ulam stability of linear differentia equations of first order. Appl. Math. Lett. 9, (26) 3. Jung, S.M.: Hyers Ulam Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2) 3. Huang, J., Li, Y.: Hyers Ulam stability of linear functinal differential equations. J. Math. Anal. Appl. 426, 92 2 (25) 32. Wang, C., Xu, T.-Z.: Hyers Ulam stability of fractional linear differential equations involving Caputo fractional derivatives. Appl. Math. 69(4), (25) 33. Wang, J., Lv, L., Zhou, Y.: Ulam stability and data dependence for fractional differential equations with Caputo derivative.electron.j.qual.theorydiffer.equ.2, 63 (2) Wang, J., Li, X.: A uniform method to Ulam Hyers stability for some linear fractional equations. Mediterr. J. Math. 3(2), (26) 35. Wang, J., Li, X.: Ulam Hyers stability of fractional Langevin equations. Appl. Math. Comput. 258,72 83 (25) Wang, J., Zhou, Y., Lin, Z.: On a new class of impulsive fractional differential equations. Appl. Math. Comput. 242, (24) Kumam, P., Ali, A., Shah, K., Khan, R.A.: Existence results and Hyers Ulam stability to a class of nonlinear arbitrary order differential equations. J. Nonlinear Sci. Appl. (6), (27) 38. Kaufmann, E.R., Yao, K.D.: Existence of solutions for a nonlinear fractional order differential equation. Electron. J. Quali. Theory Diff. Equ. 29, 7 (29) 39. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 75 8 (27)