Existence of multiple positive solutions for fourth-order boundary value problems in Banach spaces

Transcription

1 Cui and Sun Boundary Value Problems 212, 212:17 R E S E A R C H Open Access Existence of multiple positive solutions for fourth-order boundary value problems in Banach spaces Yujun Cui 1* and Jingxian Sun 2 * Correspondence: cyj7221@163.com 1 Department of Mathematics, Shandong University of Science and Technology, Qingdao, 26659, P.R. China Full list of author information is available at the end of the article Abstract This paper deals with the positive solutions of a fourth-order boundary value problem in Banach spaces. By using the fixed-point theorem of strict-set-contractions, some sufficient conditions for the existence of at least one or two positive solutions to a fourth-order boundary value problem in Banach spaces are obtained. An example illustrating the main results is given. MSC: 34B15 Keywords: u -positive operator; boundary value problem; positive solution; fixed-point theorem; measure of noncompactness 1 Introduction In this paper, we consider the existence of multiple positive solutions for the fourth-order ordinary differential equation boundary value problem in a Banach space E x 4) t)=ft, xt), x t)), t, 1), 1.1) x) = x1) = x ) = x 1) = θ, where f : I E E E is continuous, I = [, 1], θ is the zero element of E.Thisproblem models deformations of an elastic beam in equilibrium state, whose two ends are simply supported. Owing to its importance in physics, the existence of this problem in a scalar space has been studied by many authors using Schauder s fixed-point theorem and the Leray-Schauder degree theory see [1 5] and references therein). On the other hand, the theory of ordinary differential equations ODE) in abstract spaces has become an important branch of mathematics in last thirty years because of its application in partial differential equations and ODEs in appropriately infinite dimensional spaces see, for example, [6 8]). For an abstract space, it is here worth mentioning that Guo and Lakshmikantham [9] discussed the multiple solutions of two-point boundary value problems of ordinary differential equations in a Banach space. Recently, Liu [1] obtained the sufficient condition for multiple positive solutions to fourth-order singular boundary value problems in an abstract space. In [11], by using the fixed-point index theory in a cone for a strict-setcontraction operator, the authors have studied the existence of multiple positive solutions for the singular boundary value problems with an integral boundary condition. 212 Cui and Sun; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Cui and Sun Boundary Value Problems 212, 212:17 Page 2 of 13 However,theaboveworksinaBanachspacewerecarriedoutundertheassumption that the second-order derivative x is not involved explicitly in the nonlinear term f.this is because the presence of second-order derivatives in the nonlinear function f will make the study extremely difficult. As a result, the goal of this paper is to fill up the gap, that is, to investigate the existence of solutions for fourth-order boundary value problems of 1.1) in which the nonlinear function f contains second-order derivatives, i.e., f depends on x. The main features of this paper are as follows. First, we discuss the existence results in an abstract space E, note = R. Secondly, we will consider the nonlinear term which is more extensive than the nonlinear term of [1, 11]. Finally, the technique for dealing with fourth-order BVP is completely different from [1, 11]. Hence, we improve and generalize the results of [1, 11] to some degree, and so, it is interesting and important to study the existence of positive solutions of BVP 1.1). The arguments are based upon the u -positive operator and the fixed-point theorem in a cone for a strict-set-contraction operator. The paper is organized asfollows.in Section2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, various conditions on the existence of positive solutions to BVP 1.1) are discussed. In Section 4,we give an example to demonstrate our result. 2 Preliminaries Let the real Banach space E with norm be partially ordered by a cone P of E, i.e., x y if and only if y x P. P is said to be normal if there exists a positive constant N such that θ x y implies x N y. We consider problem 1.1)inC[I, E]. Evidently, C[I, E] isa Banach space with norm x C = max t I xt) and Q = {x C[I, E]:xt) θ, fort I} is a cone of the Banach space C[I, E]. In the following, x C 2 [I, E] C 4 [, 1), E] is called a solution of problem 1.1)ifitsatisfies1.1). x is a positive solution of 1.1) if, in addition, x is nonnegative and nontrivial, i.e., x C[I, P]andxt) θ for t I. For a bounded set V in a Banach space, we denote by αv) thekuratowskimeasureof noncompactness see [6 8] for further understanding). In this paper, we denote by α ) the Kuratowski measure of noncompactness of a bounded set in E and in C[I, E]. Lemma 2.1 [6] Let D EandDbeaboundedset, f be uniformly continuous and bounded from I DintoE, then α f I V) ) = max α f t, V) ), V D. t I The key tool in our approach is the following fixed-point theorem of strict-setcontractions: Theorem 2.1 [8] Let P be a cone of a real Banach space E and P r,r = {u P r u R} with <r < R. Suppose that A : P r,r P is a strict set contraction such that one of the following two conditions is satisfied: i) Au u for u P, u = r and Au u for u P, u = R; ii) Au u for u P, u = r and Au u for u P, u = R; then the operator A has at least one fixed point u P r,r such that r < u < R. The following concept is due to Krasnosel skii [12, 13], with a slightly more general definition in [12].

3 Cui and Sun Boundary Value Problems 212, 212:17 Page 3 of 13 Definition 2.1 We say that a bounded linear operator T : C[I, R] C[I, R]isu -positive on a cone K = {v C[I, R] :vt), t [, 1]} if there exists u K\{θ} such that for every u K\{θ}, there are positive constants k 1 u), k 2 u)suchthat k 1 u)u Tu k 2 u)u. Lemma 2.2 Let T be u -positive on a cone K. If T is completely continuous, then rl), the spectral radius of T, istheuniquepositiveeigenvalueoftwithitseigenfunctionink. Moreover, if λ 1 Tu = u holds with λ 1 =rt)) 1, then for an arbitrary non-zero u K u ku ) the elements u and λ 1 Tu are incomparable λ 1 Tu u, λ 1 Tu u. In the following, the closed balls in spaces E and C[I, E] are denoted, respectively, by l = {x E : x l} l >)andb l = {x C[I, E]: x C l} l >). For convenience, let us list the following assumptions: H 1 ) f C[I P P, P], f is bounded and uniformly continuous in t on I P r ) 2 for any r >, and there exist two nonnegative constants l 1, l 2 with 2l 1 +2l 2 <1such that α f t D 1 D 2 ) ) l 1 αd 1 )+l 2 αd 2 ), t I, D 1, D 2 P r. H 2 ) There are three positive constants a 1, b 1, c 1 such that f t, x, y) a 1 x + b 1 y + c 1 N for all t, x, y) I P 2 and a 1 + b π 4 1 <1. π 2 H 3 ) There is a ϕ P * P * denotes the dual cone of P) withϕx) >for any x > θ, two nonnegative constants a 2, b 2 and a real number r 1 >such that ϕ f t, x, y) ) a 2 ϕx)+b 2 ϕy) for all t, x, y) I P r1 ) 2 and a 2 + b π 4 2 >1. π 2 H 4 ) Thereisaϕ P * with ϕx)>for any x > θ and two nonnegative constants a 3, b 3 and arealnumberr 1 >such that ϕ f t, x, y) ) a 3 ϕx)+b 3 ϕy), t I, x, y P, x, y R 1 and a 3 π 4 + b 3 π 2 >1. H 5 ) There are three positive constants a 4, b 4, r 2 such that f t, x, y) a 4 x + b 4 y N for all t, x, y) I P r2 ) 2 and a 4 π 4 + b 4 π 2 <1.

4 Cui and Sun Boundary Value Problems 212, 212:17 Page 4 of 13 H 6 ) Thereexistsη >such that sup f t, x, y) 4η < t I,x,y P η N. H 7 ) Thereisaϕ P * with ϕ =1and ϕx)>for any x > θ andarealnumberη >such that ϕ f t, x, y) ) αη, t [τ,1 τ], x, y P, τ1 τ)η 3N x, y η, where τ, 1 2 ), α =τ1 τ) τ τ s1 s) ds) 1. Now, let Gt, s) be the Green s function of the linear problem v =,t, 1) together with v) = v1) =, which can be explicitly given by t1 s), t s 1, Gt, s)= s1 t), s t 1. Obviously, Gt, s) have the following properties: t1 t)s1 s) Gt, s) t1 t), t, s [, 1]; Gt, s) t1 t)gτ, s), τ, t, s [, 1]. 2.1) Set Su)t)= Gt, s)us) ds, t I, u C[I, E]. Obviously, S : C[I, E] C[I, E] is continuous. Let u = x.sincex) = x1) = θ,wehave xt)=su)t)= Gt, s)us) ds. 2.2) Using the above transformation and 2.2), BVP 1.1)becomes u t)=f t,su)t), ut) ) 2.3) with u) = u1) = θ. 2.4) From 2.3)and2.4), we have ut)= Gt, s)f s,su)s), us) ) ds.

5 Cui and Sun Boundary Value Problems 212, 212:17 Page 5 of 13 Now, define an operator A on Q by Au)t)= Gt, s)f s, The followinglemma 2.3can be easily obtained. ) Gs, τ)uτ) dτ, us) ds. 2.5) Lemma 2.3 Assume that H 1 ) holds. Then A : Q C 2 [I, P] and i) A : Q Q is continuous and bounded; ii) BVP 1.1) has a solution in C 2 [I, P] C 4 [, 1), P] if and only if A has a fixed point in Q. Let Tψ)t)= Gt, s)ψs) ds, ψ C[I, R]. It is easy to see that T : C[I, R] C[I, R]isau -positive operator with u t)=sin πt and rt)= 1 π 2. Lemma 2.4 Suppose that H 1 ) holds. Then for any l >,A : Q B l Q is a strict set contraction. Proof For any u Q B l and t [, 1], by the expression of S,wehave 1 Su)t) Gt, s) us) ds l, and thus SQ B l ) Q B l is continuous and bounded. By the uniformly continuous f and H 1 ), and Lemma 2.1,wehave α f I SD) D )) = max α f t SD) D )) l 1 α SD) ) + l 2 αd). t I Since f is uniformly continuous and bounded on I Q l ) 2,weseefrom2.5)thatA is continuous and bounded on Q B l.letd Q B l, according to 2.5), it is easy to show that the functions {Au : u D} are uniformly bounded and equicontinuous, and so in [9] we have where α AD) ) = max α A Dt) )), t I A Dt) ) = { Aut):u D, t is fixed } P l. Using the obvious formula ut) dt co { ut):t I }, u C[I, E]

6 Cui and Sun Boundary Value Problems 212, 212:17 Page 6 of 13 and observing Gt, s) 1, we find α A Dt) )) α co { Gt, s)f s, Sus), us) ) : u D, s I }) α co { f s, Sus), us) ) θ : u D, s I }) = α { f s, Sus), us) ) θ : u D, s I }) α { f s, Sus), us) ) : u D, s I }) α f I SB) B )) l 1 α SB) ) + l 2 αb) l 1 αb)+l 2 αb), 2.6) where B = {us) s I, u D} P l, SB)={ Gt, s)us) ds t I, u D}. From the fact of [9], we know αb) 2αD). 2.7) It follows from 2.6)and2.7)that α A Dt) )) 2l 1 + l 2 )αd), and consequently, A is a strict set contraction on Q B l because 2l 1 + l 2 )<1. 3 Main results Theorem 3.1 Let a cone P be normal and condition H 1 ) be satisfied. If H 2 ) and H 3 ) or H 4 ) and H 5 ) are satisfied, then BVP 1.1) has at least one positive solution. Proof Set Q 1 = { u Q : ut) t1 t)us), t, s [, 1] }. It is clear that Q 1 is a cone of the Banach space C[I, E]andQ 1 Q. For any u Q 1,by2.1), we can obtain AQ) Q 1,then AQ 1 ) Q 1. We first assume that H 2 )andh 3 ) are satisfied. Let W = {u Q 1 Au u}. In the following, we prove that W is bounded. For any u W,wehaveθ u Au, thatis,θ ut) Aut), t I. And so ut) N Aut),setvt)= ut),byh 2 ) vt) N Aut) N Gt, s) s, f Gs, τ)uτ) dτ, us)) ds

7 Cui and Sun Boundary Value Problems 212, 212:17 Page 7 of 13 1 ) Gt, s) a 1 Gs, τ)uτ) dτ + b 1 us) + c1 ds Gt, s) a 1 Gs, τ) uτ) dτ + b 1 us) ) ds + c ) For ψ C[I, R], let L 1 ψ = a 1 T 2 ψ + b 1 Tψ, thenl 1 : C[I, R] C[I, R] is a bounded linear operator. From 3.1), one deduces that I L1 )v ) t) c 1. Since π 2 is the first eigenvalue of T,byH 2 ), the first eigenvalue of L 1, rl 1 )= a 1 π 4 + b 1 π 2 <1. Therefore, by [14], the inverse operator I L 1 ) 1 exists and I L 1 ) 1 = I + L 1 + L Ln 1 +. It follows from L 1 K) K that I L 1 ) 1 K) K. So, we know that vt) I L 1 ) 1 c 1, t [, 1] and W is bounded. Taking R 2 > max{r 1, sup W},wehave Au u, u B R2 Q ) Next, we are going to verify that for any r 3, r 1 ), Au u, u B r3 Q ) If this is false, then there exists u 1 B r3 Q 1 such that Au 1 u 1. This together with H 3 ) yields ϕ u 1 t) ) ϕ Au 1 )t) ) = ϕ Gt, s)f s,su 1 )s), u 1 s) ) ) ds = = Gt, s)ϕ f s,su 1 )s), u 1 s) )) ds Gt, s) a 2 ϕ Su 1 )s) ) + b 2 ϕ u 1 s) )) ds Gt, s) a 2 Gs, τ)ϕ u 1 τ) ) dτ + b 2 ϕ u 1 s) )) ds = a 2 T 2 + b 2 T ) ϕ u 1 t) ), t I. For ψ C[I, R], let L 2 ψ = a 2 T 2 ψ + b 2 Tψ, then the above inequality can be written in the form ϕ u 1 t) ) L 2 ϕ u1 t) )). 3.4) It is easy to see that ϕ u 1 t) ), t I.

8 Cui and Sun Boundary Value Problems 212, 212:17 Page 8 of 13 In fact, ϕu 1 t)) = implies u 1 t) =θ for t I, and consequently, u 1 C =incontradiction to u 1 C = r 3.Now,noticethatL 2 is a u -positive operator with u t) =sin πt, then by Lemma 2.2, wehaveϕu 1 t)) = μ sin πt for some μ >. This together with a 2 + b π 4 2 ) sin πt = L π 2 2 sin πt)and3.4)impliesthat μ sin πt = ϕ u 1 t) ) L 2 ϕ u1 t) )) a2 = L 2 μ sin πt)=μ π + b ) 2 sin πt, 4 π 2 which is a contradiction to a 2 + b π 4 2 >1.So,3.3)holds. π 2 By Lemma 2.4, A is a strict set contraction on Q r3,r 2 = {u Q 1 : r 3 u C R 2 }.Observing 3.2)and3.3)andusingTheorem2.1,weseethatA has a fixed point on Q r3,r 2. Next, in the case that H 4 )andh 5 ) are satisfied, by the method as in establishing 3.3), we can assert from H 5 ) that for any r 4, r 2 ), Au u, u B r4 Q ) Let L 3 v)t)= ) Gt, s) a 3 Gs, τ)vτ) dτ + b 3 vs) ds, v C[I, R]. It is clear that L 3 : C[I, R] C[I, R] is a completely continuous linear u -operator with u t)=sin πt and L 3 : K 1 K 1 in which K 1 = {v K C[I, R]:vt) t1 t)vs), t, s I}. In addition, the spectral radius rl 3 )= a 3 + b π 4 3 and sin πt is the positive eigenfunction of π 2 L 3 corresponding to its first eigenvalue λ 1 =rl 3 )) 1. Let L δ v)t)= δ δ ) Gt, s) a 3 Gs, τ)vτ) dτ + b 3 vs) ds, v C[I, R], where δ, 1 2 ). It is clear that L δ : C[I, R] C[I, R] is a completely continuous linear u -operator with u t)=t1 t) andl δ K 1 ) K 1. Thus, the spectral radius rl δ ) and L δ has a positive eigenfunction corresponding to its first eigenvalue λ δ =rl δ )) 1. Take δ n, 1/2) n =1,2,...) satisfying δ 1 δ 2 δ n and δ n n ). For m > n, v K 1,wehave L δn v)t) L δm v)t) L 3 v)t), t I. By [12], we have rl δn ) rl δm ) rl 3 ). Let λ δn =rt δn )) 1,byGelfand sformula,wehave λ δn λ δm λ 1.Letλ δn λ 1 as n. In the following, we prove that λ 1 = λ 1. Let v δn be the positive eigenfunction of T δn corresponding to λ δn, i.e., v δn t)=λ δn L δ v δn )t) δn ) = λ δn Gt, s) a 3 Gs, τ)v δn τ) dτ + b 3 v δn s) ds, 3.6) δ n

9 Cui and Sun Boundary Value Problems 212, 212:17 Page 9 of 13 satisfying v δn = 1. Without loss of generality, by standard argument, we may suppose by the Arzela-Ascoli theorem and λ δn λ 1 that v δn ṽ as n.thus, ṽ =1andby 3.6), we have ) ṽ t)= λ 1 Gt, s) a 3 Gs, τ)ṽ τ) dτ + b 3 ṽ s) ds, that is, ṽ t)= λ 1 L 3 ṽ )t). This together with Lemma 2.2 guarantees that λ 1 = λ 1. By the above argument, it is easy to see that there exists a δ, 1 2 )suchthat 1<rL δ )< a 3 π 4 + b 3 π 2. Choose { R 3 > max R 1, r 2, 3NR } ) δ1 δ) Now, we assert that Au u, u Q 1, u C = R ) If this is not true, then there exists u 2 Q 1 with u C = R 3 such that Au 2 u 2,then Au 2 t) u 2 t). Moreover, by the definition of Q 1,weknow u 2 t) t1 t)u 2 s), t, s I, Su 2 )t)= Gt, s)u 2 s) ds t1 t) s 2 1 s) 2 ds u 2 τ), t, τ I. Thus, N u 2 t) t1 t) u 2 C, N Su 2 )t) t1 t) 3 u 2 C, which implies by 3.7)wehave and min t [δ,1 δ] min t [δ,1 δ] ut) So, by H 4 ), we get δ1 δ) N u C R 1, Su)t) δ1 δ) 3N u C R 1. ϕ u 2 t) ) ϕ Au 2 )t) ) = ϕ Gt, s)f s,su 2 )s), u 2 s) ) ) ds δ δ δ δ Gt, s)ϕ f s,su 2 )s), u 2 s) )) ds Gt, s) a 3 ϕ Su 2 )s) ) + b 3 ϕ u 2 s) )) ds

10 Cui and Sun Boundary Value Problems 212, 212:17 Page 1 of 13 It is easy to see that δ Gt, s) a 3 Gs, τ)ϕ u 2 τ) ) dτ + b 3 ϕ u 2 s) )) ds δ = L δ ϕ u2 t) )). ϕ u 2 t) ), t I. In fact, ϕu 2 t)) = implies u 2 t) =θ for t I, and consequently, u 2 C =incontradiction to u 2 C = R 3. Now, notice that L δ is a u -positive operator with u t)=t1 t). Then by Lemma 2.2, wehaveϕu 2 t)) = μ v δ t) forsomeμ >,wherev δ is the positive eigenfunction of L δ corresponding to λ δ. This together with rl δ )v δ t)=l δ v δ t)) implies that μ v δ t)=ϕ u 2 t) ) L δ ϕ u2 t) )) = L 3 μ v δ t) ) = μ rl δ )v δ t), which is a contradiction to rl δ )>1.So,3.8)holds. By Lemma 2.4, A is a strict set contraction on Q r4,r 3 = {u Q 1 : r 4 u C R 3 }.Observing 3.5) and3.8) andusingtheorem2.1, weseethata has a fixed point on Q r4,r 3. This together with Lemma 2.3implies that BVP 1.1) hasat least one positive solution. Theorem 3.2 Let a cone P be normal. Suppose that conditions H 1 ), H 3 ), H 4 ) and H 6 ) are satisfied. Then BVP 1.1) has at least two positive solutions. Proof We can take the same Q 1 C[I, E] asintheorem3.1. AsintheproofofTheorem 3.1, we can also obtain that AQ 1 ) Q 1. And we choose r 5, R 4 with R 4 > η > r 5 > such that Au u, u B r5 Q 1, 3.9) Au u, u B R4 Q ) On the other hand, it is easy to see that Au u, u B η Q ) In fact, if there exists u 3 Q 1 with u 3 C = η such that Au 3 u 3,thenobserving max t,s I Gt, s)= 1 4 and Su 2 C η,weget and so θ u 3 t) 1 4 u 3 t) N 4 Gt, s)f s,su 3 )s), u 3 s) ) ds f s,su 3 )s), u 3 s) ) ds, t I, f s,su 3 )s), u 3 s) ) ds 1 NM, t I, 3.12) 4

11 Cui and Sun Boundary Value Problems 212, 212:17 Page 11 of 13 where, by virtue of H 5 ), M = sup f t, x, y) 4η < t I,x,y P η N. 3.13) It follows from 3.12)and3.13)that η = u 3 C 4η N M < η, a contradiction. Thus 3.11) istrue. By Lemma 2.4, A is a strict set contraction on Q r5,η = {u Q 1 r 5 u C η}, and also on Q η,r4 = {u Q 1 η u C R 4 }.Observing3.9), 3.1), 3.11) and applying, respectively, Theorem 2.1 to A, Q r5,η and Q η,r4,weassertthatthereexistu 1 Q r5,η and u 2 Q η,r4 such that Au 1 = u 1 and Au 2 = u 2 and, by Lemma 2.3 and 3.11), Su 1, Su 2 are positive solutions of BVP 1.1). Theorem 3.3 Let a cone P be normal. Suppose that conditions H 1 ), H 2 ) and H 5 ) and H 7 ) are satisfied. Then BVP 1.1) has at least two positive solutions. Proof We can take the same Q 1 C[I, E] asintheorem3.1. AsintheproofofTheorem 3.1, we can also obtain that AQ 1 ) Q 1. And we choose r 6, R 5 with R 5 > η > r 6 > such that Au u, u B r6 Q 1, 3.14) Au u, u B R5 Q ) On the other hand, it is easy to see that Au u, u B η Q ) In fact, if there exists u 4 Q 1 with u 4 C = η such that Au 4 u 4,then θ Au 4 )t) u 4 t), t I. Observing u 4 t) Q 1 and t1 t) 3 u 4s) Su 4 )t) u 4 s), we get η = ϕ u 4 C ϕ u 4 t) ) ϕ Au 4 )t) ) = τ τ Gt, s)ϕ f s,su 4 )s), u 4 s) )) ds > αητ1 τ) Gt, s)ϕ f s,su 4 )s), u 4 s) )) ds τ τ s1 s) ds = η, which is a contradiction. Hence, 3.16)holds.

12 Cui and Sun Boundary Value Problems 212, 212:17 Page 12 of 13 By Lemma 2.4, A is a strict set contraction on Q r6,η = {u Q 1 r 6 u C η} and also on Q η,r5 = {u Q 1 η u C R 5 }.Observing3.14), 3.15), 3.16) and applying, respectively, Theorem 2.1 to A, Q r6,η and Q η,r5,weassertthatthereexistu 1 Q r6,η and u 2 Q η,r5 such that Au 1 = u 1 and Au 2 = u 2 and, by Lemma 2.3 and 3.16), Su 1, Su 2 are positive solutions of BVP 1.1). 4 One example Now, we consider an example to illustrate our results. Example 4.1 Consider the following boundary value problem of the finite system of scalar differential equations: x 4) n t)=f n t, xt), x t)), t, 1), 4.1) x) = x1) = x ) = x 1) =, where x n f n t, x, y)=gy n )+1+sin 2πt), n =1,2,...,9, 4.2) 11 + x n + y n+1 ) x 1 f 1 t, x, y)=gy 1 )+1+sin 2πt) 11 + x 1 + y 1 ), x =x 1, x 2,...,x 1 ), y =y 1, y 2,...,y 1 ), 1u, u.2, 2,.2 u.51, gu)= 2u 1,.51 u 1, 1u, u ) Claim 4.1) has at least two positive solutions x * t) =x * 1 t), x* 2 t),...,x* 1 t)) and x** t) = x ** 1 t), x** 2 t),...,x** 1 t)) such that < max 1 i 1,t [,1] x * i t) <.51< max 1 i 1,t [,1] x ** i t). Proof Let E = R 1 = {x =x 1, x 2,...,x 1 ):x n R, n = 1,2,...,1} with the norm x = max 1 n 1 x n,andp = {x =x 1, x 2,...,x 1 ):x n, n = 1,2,...,1}. ThenP is a normal cone in E, and the normal constant is N =1.System4.1) can be regarded as a boundary value problem of 1.1)inE,wherex =x 1, x 2,...,x 1 ), y =y 1, y 2,...,y 1 ), f t, x, y)= f 1 t, x, y), f 2 t, x, y),...,f 1 t, x, y) ). Evidently, f : I P 2 P is continuous. In this case, condition H 1 ) is automatically satisfied. Since αf t, D 1, D 2 )) is identical zero for any t I and D 1, D 2 P l.obviously, P * = P,sowemaychooseϕ =1,1,...,1),thenforanyx, y P,wehave ϕ f t, x, y) ) = 1 n=1 f n t, x, y).

13 Cui and Sun Boundary Value Problems 212, 212:17 Page 13 of 13 Noticing x ϕx) 1 x, wehave ϕ f t, x, y) ) = 1 n=1 f n t, x, y) 1 y 1ϕy) for all t, x, y) I P 2 with x.2 and y.2, and ϕ f t, x, y) ) = 1 n=1 f n t, x, y) 1 y 1ϕy), for all t, x, y) I P 2 with x 1and y 1. So, the conditions H 3 )andh 4 )are satisfied with a 2 = a 3 =andb 2 = b 3 =1. Choosing η =.51fort I and x, y P with x, y.51, we have f t, x, y) 2.12 < 2.4 = 4η. So, condition H 6 ) is satisfied. Thus, our conclusion follows from Theorem 3.2. Competing interests The authors declare that they have no competing interests. Authors contributions All authors typed, read and approved the final manuscript. Author details 1 Department of Mathematics, Shandong University of Science and Technology, Qingdao, 26659, P.R. China. 2 Department of Mathematics, Xuzhou Normal University, Xuzhou, Jiangsu , P.R. China. Acknowledgements The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This work is supported by the Foundation items: NSFC ), NSF BS21SF23, BS212SF22) of Shandong Province. Received: 2 June 212 Accepted: 24 September 212 Published: 9 October 212 References 1. Aftabizadeh, AR: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 116, ) 2. Yang, Y: Fourth-order two-point boundary value problems. Proc. Am. Math. Soc. 14, ) 3. del Pino, MA, Manasevich, RF: Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition. Proc. Am. Math. Soc. 112, ) 4. Agarwal, RP: On fourth-order boundary value problems arising in beam analysis. Differ. Integral Equ. 2, ) 5. Cui, Y, Zou, Y: Existence and uniqueness theorems for fourth-order singular boundary value problems. Comput. Math. Appl. 58, ) 6. Deimling, K: Ordinary Differential Equations in Banach Spaces. Springer, Berlin 1977) 7. Lakshmikantham, V, Leela, S: Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford 1981) 8. Guo, D, Lakshmikantham, V, Liu, X: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht 1996) 9. Guo, D, Lakshmikantham, V: Multiple solutions of two-point boundary value problem of ordinary differential equations in Banach space. J. Math. Anal. Appl. 129, ) 1. Liu, Y: Multiple positive solutions to fourth-order singular boundary value problems in abstract space. Electron. J. Differ. Equ. 12, ) 11. Zhang, X, Feng, M, Ge, W: Existence results for nonlinear boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 69, ) 12. Krasnosel skii, MA: Positive Solutions of Operator Equations. Noordhoff, Groningen 1964) 13. Krasnosel skii, MA, Zabreiko, PP: Geometrical Methods of Nonlinear Analysis. Springer, Berlin 1984) 14. Xia, D, Wu, Z, Yan, S, Shu, W: Theory of Real Variable Functions and Functional Analysis, 2nd edn. Higher Education Press, Beijing 1985) doi:1.1186/ Cite this article as: Cui and Sun: Existence of multiple positive solutions for fourth-order boundary value problems in Banach spaces. Boundary Value Problems :17.