Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance

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1 Zhang and Liu Boundary Value Problems 218) 218:36 R E S E A R C H Open Access Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance Wei Zhang and Wenbin Liu * * Correspondence: liuwenbin-xz@163.com School of Mathematics, China University of Mining and Technology, Xuzhou, P.R. China Abstract In this paper, by using Mawhin s continuation theorem, we establish some sufficient conditions for the existence of at least one solution for a class of fractional infinite point boundary value problem at resonance. Moreover, an example is given to illustrate our results. MSC: 34A8; 34B15 Keywords: Fractional differential equation; Infinite point boundary value problem; Resonance; Mawhin s continuation theorem 1 Introduction In the past 2 years, fractional differential equations FDEs for short) as mathematical models have been successfully used in many fields of science and engineering see [1 7]), which have attracted considerable attention in studying FDEs. For example, Izhikevich neuron model can be described by two fractional-order differential equations as the form τ C D α vt)=fv 2 + gv + h u + RI, τ C D α ut)=abv u)i, where α, 1], C D α is the Caputo fractional derivative of order α, vt) representsthe membrane voltage and ut) express the recovery variable see [6]). Ates and Zegeling [7] studied the following fractional-order advection diffusion reaction boundary value problems BVPs for short): ε C D α u + γ u + f u)=sx), x [, 1], u) = u L, u1) = u R, where 1 < α 2, < ε 1, γ R, C D α is the Caputo fractional derivative of order α.the function Sx) represents a spatially dependent source term. Recently, fractional differential equations with various kinds of boundary conditions BCs for short) have been discussed widely and obtained numerous valuable results see The Authors) 218. 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 authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Zhangand LiuBoundary Value Problems 218) 218:36 Page 2 of 16 [8 31]). It is worth mentioning that the discuss of fractional differential equations with infinite point BCs have been attracted many scholars attention over the past two years see [18 31]). These work studied on many subjects, such as: existence of solutions, positive solutions, multiple solutions, unique solution. In [18, 19] Zhang and Zhai et al. considered the following fractional differential equation with infinite point BCs: D α + ut)+qt)f t, xt)) =, t, 1), u) = u ) = = u n 2) ) =, u i) 1) = j=1 α juξ j ), respectively, where α >2,n 1<α n, i [1, n 2] is a fixed integer, α j, < ξ 1 < ξ 2 < < ξ j 1 < ξ j < <1j =1,2...), D α + is the standard Riemann Liouville fractional derivative of order α. By employing the fixed-point theorem in cones, Zhang established the existence and multiplicity of positive solutions theorems and Zhai et al. obtained the existence and uniqueness result on positive solutions. In [2], Guo et al. investigated the following infinite point fractional BVPs: C D α + ut)+f t, ut), u t)) =, < t <1, u) = u ) =, u 1) = j=1 η juξ j ), where 2 < α 3, η j, < ξ 1 < ξ 2 < < ξ j 1 < ξ j < <1j =1,2,...), C D α + is the Caputo fractional derivative of order α. The authors obtained the existence of multiple positive solutions by means of Avery Peterson s fixed-point theorem. Although many papers dealing with fractional infinite points BVPs, only a few papers consider FDEs with infinite point BCs at resonance see [21 23]). In [21], Ge et al. discussed the following coupled FDEs with infinitely points BCs at resonance: D α + x 1t)=f 1 t, x 1 t), D β 1 + x 2t)), D β + x 2t)=f 2 t, x 2 t), D + x 1t)), x 1 ) =, lim t D + x 1t)= + γ ix 1 η i ), x 2 ) =, lim t D β 1 + x 2t)= + σ ix 2 ξ i ), where 1 < α, β 2, < η 1 < η 2 < < η i <,<ξ 1 < ξ 2 < < ξ i <, lim i η i =, lim i ξ i = and + γ i ηi α <, + σ i ξ β i <, D α + and Dβ + are standard Riemann Liouville fractional derivative. By using Mawhin s continuous theorem the authors obtained the existence result. Thus, motivated by the results mentioned, the purpose of this paper is to present the existence of solutions for the following infinite point BVPs by applying Mawhin s continuous theorem: D α + xt)=f t, xt), Dα 2 + xt), D + xt)), t, 1), x) =, D + x) = + α id + xξ i), D + x1) = + β id + xγ i), 1.1)

3 Zhangand LiuBoundary Value Problems 218) 218:36 Page 3 of 16 where 2 < α 3, D α + is the standard Riemann Liouville fractional derivative of order α, f [, 1] R 3 R is a Carathéodory function, ξ i, γ i, 1) and {ξ i } +, {γ i} + are two monotonic sequence with lim i + ξ i = a, lim i + γ i = b, a, b, 1), α i, β i R. Throughout this paper, we assume that the following condition holds: H 1 ) + α i =1, + β i =1, + α i <+, + β i <+,,where = a 11 a 22 a 12 a 21, a 11 = a 21 =1 β i γ i, a 22 =1 α i ξ i, a 12 = β i γ 2 i. α i ξ 2 i, Remark 1.1 Here, if we let α i, β i R +, a 2 ξ 1, γ 2 1 b and assume that {ξ i} +, {γ i} + are monotonically increasing sequence and monotonically decreasing sequence, respectively, then, by H 1 ), we have >. In fact, since {ξ i } + monotone increasing and {γ i } + monotone decreasing with lim i + ξ i = a, lim i + γ i = b, a, b, 1), then by H 1 )wehave = a 11 a 22 a 12 a 21 = α i ξ i 1 β i γ 2 i ) > ξ 1 1 γ 2 1 ) a 2 1 b). α i ξ 2 i 1 β i γ i ) A boundary value problem is called resonance if the corresponding homogeneous boundary value problem has a nontrivial solution. We point out that if condition H 1 ) holds, the BVP 1.1) happens to be at resonance in the sense that the following infinite point boundary value problem: D α + xt)=, t, 1), x) =, D + x) = + α id + uξ i), D + x1) = + β id + xγ i), has xt)=a 1 t +a 2 t α 2, a 1, a 2 R as a nontrivial solution. That means the linear operator Lx = D α + x is non-invertible. Even though in the papers [21, 22] and[23] authors have investigated the resonance case with dimker L = 2, all of them considered with the coupled fractional differential equations, in which the linear operator L defined as Lx, y) =L 1 x, L 2 y)anddimker L 1 = dimker L 2 = 1. In our paper, we study the one fractional differential equation resonance case with dimker L = 2, which is obviously different from papers [21, 22] and[23]. Compared with previous work the main difficulties in this paper are as follows. First, resonance BVPs cannot be dealt with fixed-point theorem directly. Second, the theory of Mawhin s continuation theorem is characterized by the higher dimension of kernel space on resonance BVPs, the more difficult to construct the projections P and Q. Third, wegivean example to support our main result, and we should point out that to give an example for BVPs 1.1) is very difficult.

4 Zhangand LiuBoundary Value Problems 218) 218:36 Page 4 of 16 The rest of this paper is organized as follows. In Sect. 2, we recall some preliminary definitions and lemmas. In Sect. 3, based on Mawhin s continuation theorem, we establish an existence theorem for problem 1.1). In Sect. 4, we present an example to demonstrate the results. In the last section, brief conclusions are given. 2 Preliminaries In this section, we recall some definitions and lemmas which are used throughout this paper. First, we present here the results of coincidence degree theory due to Mawhin which can be found in [32, 33]. Let X, X )andy, Y ) are two real Banach spaces. Define L : dom L X Y to be a Fredholm operator with index zero, then there exist two continuous projectors P : X X and Q : Y Y such that Im P = Ker L, Im L = Ker Q, X = Ker L Ker P, Y = Im L Im Q, and L dom L Ker P : dom L Im L is invertible. We denote its inverse by K p.let bean open bounded subset of X and dom L, then the map N : X Y is called L-compact on,ifqn ) is bounded and K P,Q N = K p I Q)N : X is compact. Theorem 2.1 Let L : dom L X Y be a Fredholm operator of index zero and N : X Y be L-compact on. Assume that the following conditions are satisfied: i) Lu λnu for any u dom L \ Ker L), λ, 1); ii) Nu / Im L for any u Ker L ; iii) degqn Ker L, Ker L,). Then the equation Lx = Nx has at least one solution in dom L. Next, we recall some basic knowledge about the fractional calculus. For more details we refer the reader to [2]. Definition 2.1 The Riemann Liouville fractional integral of order α > for a function x :,+) R is given by I α + xt)= 1 Ɣα) t t s) xs) ds provided that the right-hand side integral is pointwise defined on, +). Definition 2.2 The Riemann Liouville fractional derivative of order α > for a function x :,+) R is given by D α dn + xt)= dt n In α + xt)= 1 Ɣn α) d n dt n t t s) n xs) ds, where n =[α] + 1, provided that the right-hand side integral is pointwise defined on, +).

5 Zhangand LiuBoundary Value Problems 218) 218:36 Page 5 of 16 Lemma 2.1 Let α >.Assume that x, D α + x L1, 1), then I α + Dα + xt)=xt)+c 1t + c 2 t α c n t α n, where n =[α]+1,c i R i =1,2,...,n) are arbitrary constants. Lemma 2.2 Let α > β >.Assume that x L 1, 1), then I α + Iβ + xt)=iα+β + xt), Dβ + Iα + xt)=iα β + xt), in particular D α + Iα + xt)=xt). Lemma 2.3 Let α >,n NandD= d/dx. If the fractional derivatives D α + x)t) and D α+n + x)t) exist, then D n D α + x) t)= D α+n + x) t). Lemma 2.4 see [2, 34]) Assume that α >,λ > 1,t >,then I α + tλ = Ɣλ +1) Ɣλ +1+α) tα+λ, D α Ɣλ +1) + tλ = Ɣλ +1 α) tλ α, in particular D α + tα m =,m =1,2,...,n, where n =[α]+1. 3 Main result Let X = { x : x, D α 2 + x, D + x C[, 1]}, Y = L 1 [, 1]. It is easy to check that X is a Banach space with norm x X = max { x, D α 2 + x, D + x }, where x = sup t [,1] xt),andy is a Banach space with norm y Y = y 1 = 1 yt) dt. Define the linear operator L : dom L X Y and the nonlinear operator N : X Y as follows: Lxt)=D α + xt), xt) dom L, Nxt)=f t, xt), D α 2 + xt), D + xt)), xt) X, where dom L = { x X : D α + xt) Y, x satisfies boundary value conditions of 1.1)}. Then BVP 1.1) is equivalent to the operator equation Lx = Nx, x dom L.

6 Zhangand LiuBoundary Value Problems 218) 218:36 Page 6 of 16 Lemma 3.1 Assume that H 1 ) holds, then the operator L : dom L X Y satisfies Ker L = { x dom L : xt)=a 1 t + a 2 t α 2, a 1, a 2 R }, 3.1) Im L = {y Y : T 1 y = T 2 y =}, 3.2) where T 1 y = ξi α i ys) ds, T 2 y = Proof If Lx = D α + x =, by Lemma 2.1,wehave 1 β i ys) ds. γ i xt)=a 1 t + a 2 t α 2 + a 3 t α 3, a 1, a 2, a 3 R. Considering that boundary condition x) =, one has a 3 =,then xt)=a 1 t + a 2 t α 2. So, Ker L {x dom L : xt)=a 1 t + a 2 t α 2, a 1, a 2 R}. Conversely,foranya 1, a 2 R, take xt)=a 1 t + a 2 t α 2,itiseasytocheckthatD α + xt)=andxt) satisfies boundary value conditions of 1.1). Thus, 3.1) holds. For y Im L, thereexistsx dom L such that D α + xt)=yt). Again by Lemma 2.1 and combining with the boundary condition x) =, one gets xt)=i α + yt)+a 1t + a 2 t α 2. Noting that D + x) = + α i D + xξ i), D by Lemmas 2.2 and 2.4,weobtain and D + x) = a 1Ɣα)= α i D + xξ i) D = = + x1) = + [ ξi ] α i ys) ds + a 1 Ɣα) ξi α i ys) ds + a 1 Ɣα) 1 + x1) = ys) ds + a 1 Ɣα)= = [ γi ] β i ys) ds + a 1 Ɣα) = β i D + xγ i) β i D + xγ i), γi β i ys) ds + a 1 Ɣα).

7 Zhangand LiuBoundary Value Problems 218) 218:36 Page 7 of 16 Thus, T 1 y = T 2 y =, 3.3) that is, Im L {y Y : T 1 y = T 2 y =}. Conversely, let y Y satisfy 3.3) andtakext)=i+ α yt). Obviously, we have xt) dom L and Lxt)=yt). Then, {y Y : T 1 y = T 2 y =} Im L. Therefore, 3.2)holds. Let Q 1, Q 2 : Y Y betwolinearoperatorsdefinedasfollows: Q 1 y = 1 a 22T 1 y a 12 T 2 y), Q 2 y = 2 a 11T 2 y a 21 T 1 y). Lemma 3.2 Assume that H 1 ) holds, then L : dom L X Y is a Fredholm operator of index zero. The linear projector operator P : X XandQ: Y Y defined as follows: Px)t)= 1 1 Ɣα) D + x)t + Ɣα 1) Dα 2 + x)tα 2, Qyt)=Q 1 yt)+ Q 2 yt) ) t. Proof By the definition of P we can check that P is a continuous linear projector operator and satisfies Im P = Ker L, X = Ker P Ker L.ItisclearthatQ is a continuous linear operator and dim Im Q = 2. By the definitions of Q 1, Q 2, we can calculate the following equations hold: Q 1 Q1 yt) ) = Q 1 yt), Q 1 Q2 yt) ) t ) =, Q 2 Q1 yt) ) =, Q 2 Q2 yt) ) t ) = Q 2 yt). Thus, Q 2 yt)=q Qyt) ) = Q 1 Qyt) ) + Q2 Qyt) )) t = Q 1 [ Q1 yt)+ Q 2 yt) ) t ] + { Q 2 [ Q1 yt)+ Q 2 yt) ) t ]} t = Q 1 yt)+ Q 2 yt) ) t = Qyt). So, Q is a projector operator. From Lemma 3.1,wehaveIm L Ker Q.Now,weshowthe fact that Ker Q Im L. Infact,fory Ker Q, i.e., Qy =,thenwegetasystemoflinear equations with respect to T 1 y, T 2 y as follows: a 11 T 2 y a 21 T 1 y =, 3.4) a 22 T 1 y a 12 T 2 y =, Since the determinant of coefficient for 3.4)is,wegetT 1 y = T 2 y =,thusker Q Im L. Therefore, Ker Q = Im L.Fory Y,sety =y Qy)+Qy,theny Qy) Ker Q = Im L,

8 Zhangand LiuBoundary Value Problems 218) 218:36 Page 8 of 16 Qy Im Q.So,y = Im L + Im Q. Furthermore, for any y Im L Im Q, there exist constants c 1, c 2 R such that yt)=c 1 + c 2 t and T 1 y = T 2 y =.Thenwealsogetasystemoflinear equations with respect to c 1, c 2 as follows: 2a 11 c 1 + a 12 c 2 =, 3.5) 2a 21 c 1 + a 22 c 2 =. Because the determinant of coefficient for 3.5) is2.thus,c 1 = c 2 =.Itmeans that Im Q Im L = {}. Therefore, Y = Im Q Im L. Furthermore,dimKer L = dim Im Q = co dim Im L =2.So,L isafredholmoperatorofindexzero. Lemma 3.3 Assume that H 1 ) holds, define the linear operator K p : Im L dom L Ker P by K p y)t)= 1 Ɣα) t t s) ys) ds, y Im L, then K p is the inverse of L dom L Ker P and K p y X y 1, for all y Im L. Proof For y Im L, thent 1 y = T 2 y =, which is combined with the definition of K p and Lemma 2.2, wecancheckthatk p y dom L Ker P. So,K p is well defined on Im L. Obviously, LK p )yt)=yt), y Im L.Forxt) dom L, by Lemma 2.1,wehave K p L)xt)=I α + Dα + xt) = xt)+a 1 t + a 2 t α 2, a 1, a 2 R. It follows from P[K p L)xt)] = and a 1 t + a 2 t α 2 Ker L = Im P that c 1 t + c 2 t α 2 = Pxt). That is, K p L)xt) =xt) Pxt). Therefore, if xt) dom L Ker P, wehave K p L)xt) =xt). So, K p is the inverse of L dom L Ker P. By Lemma 2.2, we have the following inequalities: t K p y 1 Ɣα) D α 2 + K py t D + K py t 1 t s) ys) ds 1 Ɣα) t s) 1 ys) ds ys) ds = y 1, 1 ys) ds ys) ds = y 1. ys) ds y 1, So, K p y X y 1, for all y Im L. Lemma 3.4 Assume that H 1 ) holds and X is an open bounded subset with dom L, then N is L-compact on. Proof According to f [, 1] R 2 R satisfies the Carathéodory conditions, we can get QN ) andi Q)N ) are bounded almost everywhere on [, 1], that is, there exist constants m, m >suchthat QNxt) m, I Q)Nxt) m, x,a.e.t [, 1]. Next, we show that K p I Q)N : X is compact. In fact, by Lemma 3.3, K p I Q)N ) is

9 Zhangand LiuBoundary Value Problems 218) 218:36 Page 9 of 16 uniformly bounded. It follows from the Lebesgue dominated convergence theorem that K p I Q)N : X is continuous. For t 1 < t 2 1, x,wehave K p I Q)Nxt 1 ) K p I Q)Nxt 2 ) = 1 t1 t2 Ɣα) t 1 s) I Q)Nxs) ds t 2 s) I Q)Nxs) ds 1 t1 [ Ɣα) t1 s) t 2 s) ] I Q)Nxs) ds + 1 t2 Ɣα) t 2 s) I Q)Nxs) ds = m Ɣα) t1 t 1 m t α Ɣα +1) 2 t1 α ). [ t2 s) t 1 s) ] ds + m Ɣα) t2 t 1 t 2 s) ds Since t α is uniformly continuous on [, 1], K p I Q)N ) is equicontinuous. In addition, by Lemma 2.3 we see that the following equation holds: t2 D α 2 + K pi Q)Nxt 2 ) D α 2 + K pi Q)Nxt 1 )= D + K pi Q)Nxs) ds. t 1 So, it suffices to show that D + K pi Q)N ) is equicontinuous. In fact, D + K pi Q)Nxt 1 ) D + K pi Q)Nxt 2 ) t2 = I Q)Nxs) ds mt 2 t 1 ). t 1 Since t is uniformly continuous on [, 1], thus D + K pi Q)Nx ) is equicontinuous. By the Ascoli Arzelà theorem, we see that K p I Q)N : X is compact. In order to obtain our main results, we suppose that the following conditions are satisfied: H 2 ) There exist nonnegative functions pt), qt), rt), et) Y such that, for all u, v, w) R 3, t, 1), f t, u, v, w) pt) u + qt) v + rt) w + et) and p 1 + q 1 + r 1 < Ɣα 1) Ɣα 1)+2. H 3 ) There exist constants k 1, k 2 > such that, for all t, 1), x dom L,if D α 2 + xt) > k 1 or D + xt) > k 2, then either T 1 Nxt)) ort 2 Nxt)). H 4 ) There exists a constant g > such that, for all a 1, a 2 R, xt) =a 1 t + a 2 t α 2 Ker L,if a 1 > g or a 2 > g then either a 1 T 1 Nxt)+a 2 T 2 Nxt)>, 3.6)

10 Zhangand LiuBoundary Value Problems 218) 218:36 Page 1 of 16 or a 1 T 1 Nxt)+a 2 T 2 Nxt)<. 3.7) Lemma 3.5 Suppose that H 1 ) H 3 ) hold, set 1 = { x dom L \ Ker L : Lx = λnx, λ, 1) }. Then 1 is bounded. Proof For x 1,wehaveNx Im L = Ker Q.ThenT 1 Nxt)) = T 2 Nxt)) =. Thus, from H 3 ), there exist t, t 1 [, 1] such that D α 2 + xt ) k 1 and D + xt 1) k 2.Then,by Lemma2.3we have D + xt) = t + xt 1)+ D t 1 D α + xs) ds D + xt 1) t + D α + xs) ds t 1 k 2 + Nx 1 and D α 2 + x) = Dα 2 + xt ) t D + xs) ds D α 2 + xt ) t + D + xs) ds k 1 + D + x k 1 + k 2 + Nx 1. By the definition of P andlemma 2.4,we get D α 2 + Pxt)=D + x)t + Dα 2 + x), and D + Pxt)=D + x). Thus, D α 2 + Px D + x) + D α 2 + x) k1 +2k 2 +2 Nx 1, D + Px = D + x) k2 + Nx 1. We have Px 1 D + Ɣα) x) 1 + D α 2 + Ɣα 1) x) 1 ) k1 +2k 2 +2 Nx 1. Ɣα 1)

11 Zhangand LiuBoundary Value Problems 218) 218:36 Page 11 of 16 Therefore, Px X = max { Px, D α 2 + Px, D + Px } 1 ) k1 +2k 2 +2 Nx ) Ɣα 1) Also, for x 1,thenI P)x dom L Ker P, LPx =, from Lemma 3.3,wehave I P)x X = Kp LI P)x X = K p Lx X Lx 1 Nx ) Then, from 3.8)and3.9), we see that x X = Px +I P)x X Px X + I P)x X By H 2 ), we have 1 ) k1 +2k 2 +2 Nx 1 + Nx ) Ɣα 1) Nx 1 p 1 x + q 1 D α 2 + x + r 1 D + x + e 1 p 1 + q 1 + r 1 ) x X + e ) Substituting 3.11)into3.1), onegets x X k 1 +2k 2 +Ɣα 1)+2) e 1 Ɣα 1) [Ɣα 1)+2] p 1 + q 1 + r 1 ). So, 1 is bounded. Lemma 3.6 Suppose that H 3 ) holds, set 2 = {x Ker L : Nx Im L}. Then 2 is bounded. Proof For x 2,wehavext) =a 1 t + a 2 t α 2, a 1, a 2 R and T 1 Nxt) =T 2 Nxt) =. From H 3 ), there exist t 2, t 3 [, 1] such that D α 2 + xt 2) k 1 and D + xt 3) k 2,thatis, Thus, D + xt 3) = a 1 Ɣα) k 2, D α 2 + xt 2) = a1 Ɣα)t 2 + a 2 Ɣα 1) k1. a 1 k 2 /Ɣα), a 2 k 1 + k 2 )/Ɣα 1).

12 Zhangand LiuBoundary Value Problems 218) 218:36 Page 12 of 16 Therefore, x a 1 + a 2 k 2 Ɣα) + k 1 + k 2 Ɣα 1) = 1 [ k2 +α 1)k 1 + k 2 ) ], Ɣα) D α 2 + x Ɣα) a 1 + Ɣα 1) a 2 2k 2 + k 1, D + x Ɣα) a k 2. So, 2 is bounded. Lemma 3.7 Suppose that H 4 ) holds, we set 3 = { x Ker L : ϑλjx +1 λ)qnx =,λ [, 1] }. Then 3 is bounded, where ϑ =1,if 3.6) holds and ϑ = 1,if 3.7) holds, J : Ker L Im Q is the linear isomorphism defined by J a 1 t + a 2 t α 2) = 1 [ a22 a 1 a 12 a 2 )+2a 11 a 2 a 21 a 1 )t ], a 1, a 2 R. Proof Without loss of generality, we suppose that 3.7) holds,then,foranyx 3, there exist constants a 1, a 2 R, λ [, 1] such that xt) =a 1 t + a 2 t α 2 and λjx + 1 λ)qnx =. By Lemma 3.6, in order to prove Lemma 3.7, it suffices to show that a 1 g, a 2 g. Infact,ifλ =thenqnx =,whichmeanst 1 Nx = T 2 Nx =.From H 4 ), we get a 1 g, a 2 g.ifλ =1thenJx =,thatis,a 1 = a 2 =.Obviously, a 1 g, a 2 g.forλ, 1), by λjx =1 λ)qnx one has λa 22 a 1 a 12 a 2 )=1 λ)a 22 T 1 Nx a 12 T 2 Nx), λa 11 a 2 a 21 a 1 )=1 λ)a 11 T 2 Nx a 21 T 1 Nx). Because,wehave λa 1 =1 λ)t 1 Nx, λa 2 =1 λ)t 2 Nx. Then, if a 1 > g or a 2 > g,by3.7), we get a contradiction, <λ a a2 2) =1 λ)a1 T 1 Nx + a 2 T 2 Nx)<. Thus, a 1 g, a 2 g.so, 3 is bounded. If 3.6)holds,byasimilarmethod,wecansee that 3 is bounded. Theorem 3.1 Suppose that H 1 ) H 4 ) hold. Then problem 1.1) has at least on solution in X. Proof Let be a bounded open set of X such that 3 i. By Lemma 3.4, N is L- compact on. From Lemmas 3.5 and 3.6,weget

13 Zhangand LiuBoundary Value Problems 218) 218:36 Page 13 of 16 i) Lx λnx for any x, λ) [dom L \ Ker L) ], 1), ii) Nx Im L for any x Ker L. Thus, we only need to show that iii) of Theorem 2.1 is satisfied. Take Hx, λ)=ϑλjx +1 λ)qnx, where ϑ is defined as before. According to Lemma 3.7, wederivehx, λ) forallx Ker L. Thus, it follows from the homotopy of degree that deg{qn Ker L, Ker L,} = deg { H,), Ker L, } = deg { H,1), Ker L, } = degϑj, Ker L,). Then, by Theorem 2.1, we can see that the operator function Lx = Nx has at least one solution in dom L,which,equivalentlytoproblem1.1), has at least one solution in X. 4 Example Example 4.1 Consider the following fractional boundary value problem: D xt)=f t, xt), D.5 + xt), D1.5 + xt)), t, 1), x) =, D x) = + α id xξ i), D x1) = + β id xγ i), 4.1) where we take α i = 1 2 i, β i = 1 2 i, ξ i = f t, xt), D.5 + xt), D1.5 + xt)) [ = mt) 1, [, 1/2), mt)=, [1/2, 1]. Obviously, we have i 2i +1, γ i = i +1 2i +1, t cos2 xt)+ 1 ] 24 D.5 + xt) mt) ) D xt), α i, β i R +, a 2 = = ξ 1, γ 2 1 = = b, and {ξ i } + is a monotone increasing sequence and {γ i} + is a monotone decreasing sequence. Applying Remark 1.1 we conclude that >.Let pt)=1/8, qt) = 1/24, rt) = 1/12, et)=2. Then f t, u, v, w) pt) u + qt) v + rt) w + et)

14 Zhangand LiuBoundary Value Problems 218) 218:36 Page 14 of 16 and p 1 + q 1 + r 1 = 1 4 Ɣα 1) π < Ɣα 1)+2 =. π +4 Take k 1 =48,k 2 =24.Then,if D.5 + xt) > k 1,onehas ) + ξi T 1 Nxt) = α i and if D xt) > k 2,onegets f s, xs), D.5 + xs), D1.5 + xs)) ds, ) + 1 T 2 Nxt) = β i f s, xs), D.5 + xs), D1.5 + xs)) ds. γ i Let g = 261. Then if a 1 > g,wehave a 1 T 1 Nxt)+a 2 T 2 Nxt) + ξi = a 1 α i f s, xs), D.5 + xs), D1.5 + xs)) ds a 2 β i f s, xs), D.5 + xs), D1.5 + xs)) ds γ i + ξi = a 1 α i a ξi = a 1 α i [ s cos2 xs) a1 Ɣ1.5)s + a 2 Ɣ.5) )] ds 1 β i a 1 Ɣ1.5) ds γ i a 1a 2 Ɣ.5) + ξi = a 1 α i s ) 8 cos2 xs) ds Ɣ1.5)a2 1 α i ξi a a 1 >. α i ξ i 1 12 a 1a 2 Ɣ1.5) s cos2 xs) β i 1 γ i ) ) ds Ɣ1.5)a2 1 α i ξ 2 i In view of Theorem 3.1, boundary value problem 4.1) has at least one solution. Remark 4.1 By the example, we find that it is more difficult to give an infinite series + α i + α i =1, + α i <+) than to give a function gt) L 1 [, 1] 1 gt) dt =1).For example, for 1 gt)=1,wecantakegt) 1. 5 Conclusion In this paper, we are focused on investigating the existence of solutions for a class of FDEs with infinite point BCs. By using Mawhin s continuation theorem, an existence theorem

15 Zhangand LiuBoundary Value Problems 218) 218:36 Page 15 of 16 is established and we also give an example to illustrate the application of the theorem. To the best of our knowledge, the issue on the existence of solutions for infinite point BVPs was first studied by Ma see [35]) and discussed widely in recent years. It is an interesting question and there is some work to be done in the future, such as: discussing the existence of solutions for p-laplacian FDEs with infinite point BCs at resonance in the case dimker L = 2 for one fractional differential equation. Funding This research is supported by the National Natural Science Foundation of China ). Competing interests The authors declare that they have no competing interests. Authors contributions The authors contributed equally in this article. They have all 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: 27 December 217 Accepted: 11 March 218 References 1. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. Academic Press, New York 1999) 2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam 26) 3. Petráš, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin 211) 4. Sierociuk, D.,Dzieliński, A., Sarwas, G., Podlubny, I., Skovranek, T.: Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc., Math. Phys. Eng. Sci. 371, ) 5. Arenas, A.J., González-Parra, G., Chen-Charpentier, B.M.: Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order. Math. Comput. Simul. 121, ) 6. Teka, W.W., Upadhyay, P.K., Mondal, A.: Spiking and bursting patterns of fractional-order Izhikevich model. Commun. Nonlinear Sci. Numer. Simul. 56, ) 7. Ates, I., Zegeling, P.A.: A homotopy perturbation method for fractional-order advection-diffusion-reaction boundary-value problems. Appl. Math. Model. 47, ) 8. Ahmad, B., Ntouyas, S.K., Alsaedi, A.: On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. Chaos Solitons Fractals 83, ) 9. Agarwal, R.P., Ahmad, B., Garout, D., Alsaedi, A.: Existence results for coupled nonlinear fractional differential equations equipped with nonlocal coupled flux and multi-point boundary conditions. Chaos Solitons Fractals 12, ) 1. Günendi, M., Yaslan, İ.: Positive solutions of higher-order nonlinear multi-point fractional equations with integral boundary conditions. Fract. Calc. Appl. Anal. 19, ) 11. Zhang, X.Q., Zhong, Q.Y.: Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 8, ) 12. Zhang, X.Q., Zhong, Q.Y.: Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions. Fract. Calc. Appl. Anal. 2, ) 13. Zhang, W., Liu, W.B.: Existence of solutions for fractional multi-point boundary value problems at resonance with three-dimensional kernels. Adv. Differ. Equ. 218, ) 14. Zhang, W., Liu, W.B., Chen, T.Y.: Solvability for a fractional p-laplacian multipoint boundary value problem at resonance on infinite interval. Adv. Differ. Equ. 216, ) 15. Shen, T.F., Liu, W.B., Shen, X.H.: Existence and uniqueness of solutions for several BVPs of fractional differential equations with p-laplacian operator. Mediterr. J. Math. 13, ) 16. Kosmatov, N., Jiang, W.H.: Resonant functional problems of fractional order. Chaos Solitons Fractals 91, ) 17. Iqbal, M., Li, Y.J., Shah, K., Khan, R.A.: Application of topological degree method for solutions of coupled systems of multipoints boundary value problems of fractional order hybrid differential equations. Complexity 217,Article ID ) 18. Zhang, X.Q.: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett. 39, ) 19. Zhai, C.B., Wang, L.: Some existence, uniqueness results on positive solutions for a fractional differential equation with infinite-point boundary conditions. Nonlinear Anal., Model. Control 22, ) 2. Guo, L.M., Liu, L.S., Wu, Y.H.: Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions. Nonlinear Anal., Model. Control 21, ) 21. Ge, F.D., Zhou, H.C., Kou, C.H.: Existence of solutions for a coupled fractional differential equations with infinitely many points boundary conditions at resonance on an unbounded domain. Differ. Equ. Dyn. Syst. 216).

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