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

where $f:I\times E\times E\to E$ is continuous, $I=[0,1]$, θ is the zero element of E. This problem 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 [10] 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-set-contraction operator, the authors have studied the existence of multiple positive solutions for the singular boundary value problems with an integral boundary condition.

However, the above works in a Banach space were carried out under the assumption 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, not $E=\mathbb{R}$. Secondly, we will consider the nonlinear term which is more extensive than the nonlinear term of [10, 11]. Finally, the technique for dealing with fourth-order BVP is completely different from [10, 11]. Hence, we improve and generalize the results of [10, 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 $u_0$-positive operator and the fixed-point theorem in a cone for a strict-set-contraction operator.

The paper is organized as follows. In Section 2, 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.

Let the real Banach space E with norm $\parallel \cdot \parallel$ be partially ordered by a cone P of E, i.e., $x\le y$ if and only if $y-x\in P$. P is said to be normal if there exists a positive constant N such that $\theta \le x\le y$ implies $\parallel x\parallel \le N\parallel y\parallel$. We consider problem (1.1) in $C[I,E]$. Evidently, $C[I,E]$ is a Banach space with norm ${\parallel x\parallel }_C={max}_{t\in I}\parallel x(t)\parallel$ and $Q=\{x\in C[I,E]:x(t)\ge \theta ,\text{ for }t\in I\}$ is a cone of the Banach space $C[I,E]$. In the following, $x\in C^2[I,E]\cap C^4[(0,1),E]$ is called a solution of problem (1.1) if it satisfies (1.1). x is a positive solution of (1.1) if, in addition, x is nonnegative and nontrivial, i.e., $x\in C[I,P]$ and $x(t)\not\equiv \theta$ for $t\in I$.

For a bounded set V in a Banach space, we denote by $\alpha (V)$ the Kuratowski measure of noncompactness (see [6–8] for further understanding). In this paper, we denote by $\alpha (\cdot )$ the Kuratowski measure of noncompactness of a bounded set in E and in $C[I,E]$.

Lemma 2.1 [6]

The key tool in our approach is the following fixed-point theorem of strict-set-contractions:

Theorem 2.1 [8]

then the operator A has at least one fixed point $u\in P_{r,R}$ such that $r<\parallel u\parallel <R$.

The following concept is due to Krasnosel’skii [12, 13], with a slightly more general definition in [12].

Lemma 2.2 Let T be $u_0$-positive on a cone K. If T is completely continuous, then $r(L)$, the spectral radius of T, is the unique positive eigenvalue of T with its eigenfunction in K. Moreover, if ${\lambda }_{1}T{u}_0=u_0$ holds with ${\lambda }_1={(r(T))}^{-1}$, then for an arbitrary non-zero $u\in K$ ($u\ne k{u}_0$) the elements u and ${\lambda }_{1}Tu$ are incomparable

In the following, the closed balls in spaces E and $C[I,E]$ are denoted, respectively, by ${\mathrm{\Omega }}_l=\{x\in E:\parallel x\parallel \le l\}$ ($l>0$) and $B_l=\{x\in C[I,E]:{\parallel x\parallel }_C\le l\}$ ($l>0$).

For convenience, let us list the following assumptions:

for all $(t,x,y)\in I\times P^2$ and $\frac{a_1}{{\pi }^4}+\frac{b_1}{{\pi }^2}<1$.

for all $(t,x,y)\in I\times {(P\cap {\mathrm{\Omega }}_{r_1})}^2$ and $\frac{a_2}{{\pi }^4}+\frac{b_2}{{\pi }^2}>1$.

and $\frac{a_3}{{\pi }^4}+\frac{b_3}{{\pi }^2}>1$.

for all $(t,x,y)\in I\times {(P\cap {\mathrm{\Omega }}_{r_2})}^2$ and $\frac{a_4}{{\pi }^4}+\frac{b_4}{{\pi }^2}<1$.

where $\tau \in (0,\frac{1}{2})$, $\alpha ={(\tau (1-\tau ){\int }_{\tau }^{1-\tau }s(1-s)ds)}^{-1}$.

Obviously, $S:C[I,E]\to C[I,E]$ is continuous.

The following Lemma 2.3 can be easily obtained.

It is easy to see that $T:C[I,\mathbb{R}]\to C[I,\mathbb{R}]$ is a $u_0$-positive operator with $u_0(t)=sin\pi t$ and $r(T)=\frac{1}{{\pi }^2}$.

Lemma 2.4 Suppose that ($H_1$) holds. Then for any $l>0$, $A:Q\cap B_l\to Q$ is a strict set contraction.

where $B=\{u(s)|s\in I,u\in D\}\subset P_l$, $S(B)=\{{\int }_0^{1}G(t,s)u(s)ds|t\in I,u\in D\}$.

and consequently, A is a strict set contraction on $Q\cap B_l$ because $2(l_1+l_2)<1$. □

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.

In the following, we prove that W is bounded.

It follows from $L_1(K)\subset K$ that ${(I-L_1)}^{-1}(K)\subset K$. So, we know that $v(t)\le {(I-L_1)}^{-1}{c}_1$, $t\in [0,1]$ and W is bounded.

Next, we are going to verify that for any $r_3\in (0,r_1)$,

(3.3)

which is a contradiction to $\frac{a_2}{{\pi }^4}+\frac{b_2}{{\pi }^2}>1$. So, (3.3) holds.

By Lemma 2.4, A is a strict set contraction on $Q_{r_3,R_2}=\{u\in Q_1:r_3\le {\parallel u\parallel }_C\le R_2\}$. Observing (3.2) and (3.3) and using Theorem 2.1, we see that A has a fixed point on $Q_{r_3,R_2}$.

It is clear that $L_3:C[I,\mathbb{R}]\to C[I,\mathbb{R}]$ is a completely continuous linear $u_0$-operator with $u_0(t)=sin\pi t$ and $L_3:K_1\to K_1$ in which $K_1=\{v\in K\subset C[I,\mathbb{R}]:v(t)\ge t(1-t)v(s),\mathrm{\forall }t,s\in I\}$. In addition, the spectral radius $r(L_3)=\frac{a_3}{{\pi }^4}+\frac{b_3}{{\pi }^2}$ and $sin\pi t$ is the positive eigenfunction of $L_3$ corresponding to its first eigenvalue ${\lambda }_1={(r(L_3))}^{-1}$.

where $\delta \in (0,\frac{1}{2})$. It is clear that $L_{\delta }:C[I,\mathbb{R}]\to C[I,\mathbb{R}]$ is a completely continuous linear $u_0$-operator with $u_0(t)=t(1-t)$ and $L_{\delta }(K_1)\subset K_1$. Thus, the spectral radius $r(L_{\delta })\ne 0$ and $L_{\delta }$ has a positive eigenfunction corresponding to its first eigenvalue ${\lambda }_{\delta }={(r(L_{\delta }))}^{-1}$.

By [12], we have $r(L_{{\delta }_n})\le r(L_{{\delta }_m})\le r(L_3)$. Let ${\lambda }_{{\delta }_n}={(r(T_{{\delta }_n}))}^{-1}$, by Gelfand’s formula, we have ${\lambda }_{{\delta }_n}\ge {\lambda }_{{\delta }_m}\ge {\lambda }_1$. Let ${\lambda }_{{\delta }_n}\to {\tilde{\lambda }}_1$ as $n\to \mathrm{\infty }$.

In the following, we prove that ${\tilde{\lambda }}_1={\lambda }_1$.

that is, ${\tilde{v}}_0(t)={\tilde{\lambda }}_1(L_3{\tilde{v}}_0)(t)$. This together with Lemma 2.2 guarantees that ${\tilde{\lambda }}_1={\lambda }_1$.

Now, we assert that

(3.8)

which is a contradiction to $r(L_{\delta })>1$. So, (3.8) holds.

By Lemma 2.4, A is a strict set contraction on $Q_{r_4,R_3}=\{u\in Q_1:r_4\le {\parallel u\parallel }_C\le R_3\}$. Observing (3.5) and (3.8) and using Theorem 2.1, we see that A has a fixed point on $Q_{r_4,R_3}$. This together with Lemma 2.3 implies that BVP (1.1) has at 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 $Q_1\subset C[I,E]$ as in Theorem 3.1. As in the proof of Theorem 3.1, we can also obtain that $A(Q_1)\subset Q_1$. And we choose $r_5$, $R_4$ with $R_4>\eta >r_5>0$ such that

(3.9)

(3.10)

a contradiction. Thus (3.11) is true.

By Lemma 2.4, A is a strict set contraction on $Q_{r_5,\eta }=\{u\in Q_1|r_5\le {\parallel u\parallel }_C\le \eta \}$, and also on $Q_{\eta ,R_4}=\{u\in Q_1|\eta \le {\parallel u\parallel }_C\le R_4\}$. Observing (3.9), (3.10), (3.11) and applying, respectively, Theorem 2.1 to A, $Q_{r_5,\eta }$ and $Q_{\eta ,R_4}$, we assert that there exist $u_1\in Q_{r_5,\eta }$ and $u_2\in Q_{\eta ,R_4}$ such that $A{u}_1=u_1$ and $A{u}_2=u_2$ and, by Lemma 2.3 and (3.11), $S{u}_1$, $S{u}_2$ 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 $Q_1\subset C[I,E]$ as in Theorem 3.1. As in the proof of Theorem 3.1, we can also obtain that $A(Q_1)\subset Q_1$. And we choose $r_6$, $R_5$ with $R_5>\eta >r_6>0$ such that

(3.14)

(3.15)

On the other hand, it is easy to see that

(3.16)

which is a contradiction. Hence, (3.16) holds.

By Lemma 2.4, A is a strict set contraction on $Q_{r_6,\eta }=\{u\in Q_1|r_6\le {\parallel u\parallel }_C\le \eta \}$ and also on $Q_{\eta ,R_5}=\{u\in Q_1|\eta \le {\parallel u\parallel }_C\le R_5\}$. Observing (3.14), (3.15), (3.16) and applying, respectively, Theorem 2.1 to A, $Q_{r_6,\eta }$ and $Q_{\eta ,R_5}$, we assert that there exist $u_1\in Q_{r_6,\eta }$ and $u_2\in Q_{\eta ,R_5}$ such that $A{u}_1=u_1$ and $A{u}_2=u_2$ and, by Lemma 2.3 and (3.16), $S{u}_1$, $S{u}_2$ are positive solutions of BVP (1.1). □