Sign-changing solution for a third-order boundary-value problem in ordered Banach space with lattice structure

where $f\in C([0,1]\times R,R)$.

where $f\in C(R,R)$, $a\in L(0,1)$ is nonnegative with ${\int }_0^1{a}^2(s)ds<1$. For the integral boundary-value problem (1.2), Li and Liu [5] also obtained the existence and multiplicity of the sign-changing solutions in ordered Banach space with the lattice structure.

For the third-order boundary-value problem, the existence and multiplicity of solutions have also been discussed in many papers (see [6–11] and the references therein). However, the research on the sign-changing solutions has been proceeded slowly. For the problem (1.1), Yao and Feng [10, 11] established several existence results for the solutions including the positive solutions using the lower and upper solutions and a maximum principle, respectively. To our knowledge, however, there are fewer papers considered the sign-changing solutions of the problem (1.1). Motivated by the work mentioned above, using the eigenvalues of linear operator, we give an existence result for the sign-changing solutions of the problem (1.1).

The main contribution of this paper are as follows: (a) for the sign-changing solutions of the problem (1.1), to our knowledge, there is no result using the eigenvalues of the linear operator until now; (b) we obtain the eigenvalues and the algebraic multiplicity of the linear problem corresponding the problem (1.1), which is one of the key points that we can use to prove our main result; (c) some conditions are given to guarantee the existence for a sign-changing solution of the problem (1.1).

The following results will be used throughout the paper.

Let $E=C[0,1]$, $\parallel u\parallel ={max}_{t\in [0,1]}|u(t)|$. Then E is a real Banach space with the norm $\parallel \cdot \parallel$. Let $P=\{u\in E:u(t)\ge 0,t\in [0,1]\}$, and P is a normal solid cone of E, $\{u\in E:u(t)>0,t\in (0,1]\}\subset \stackrel{\circ }{P}=\{x\in P|x\text{ is an interior point of }P\}$.

Definition 2.1 [12]

We call E a lattice under the partial ordering ≤, if $sup\{x,y\}$ and $inf\{x,y\}$ exist for arbitrary $x,y\in E$.

Remark 2 $E=C[0,1]$ is a lattice under the partial ordering ≤ that is deduced by the cone $P=\{u\in E:u(t)\ge 0,t\in [0,1]\}$ of E.

Definition 2.2 [12]

L is said to be the derived operator of A along $P_w$ and will be denoted by $A_{P_w}^{\prime }$. Similarly, we can also define a unilaterally asymptotically linear operator along $P^w=\{x\in E:x\le w,w\in E\}$. Specially, if $w=\theta$, We call that A is a unilaterally asymptotically linear operator along P and −P.

Definition 2.3 [12]

where $x_{+}=x^{+}=sup\{x,\theta \}$, $x_{-}=x^{-}=-sup\{-x,\theta \}$.

Remark 3 It is easy to see that the operators F and $A=KF$ defined by (2.1) are both quasi-additive on the lattice $E=C[0,1]$.

Let us list some conditions and preliminary lemmas to be used in this paper.

(H₁) $f\in C([0,1]\times R,R)$ is strictly increasing in u, and $f(t,u)u>0$ for all $t\in [0,1]$, $u\in R\mathrm{\setminus }\{0\}$.

(H₃) ${lim}_{u\to 0}\frac{f(t,u)}{u}={\beta }_0$ uniformly on $[0,1]$, and $0<{\beta }_0<{\lambda }_1^3$.

where $G(t,s)$ is defined by (2.2).

Moreover, we get $u(0)=u^{\prime }(0)=u^{\prime }(1)=0$. □

Remark 4 Considering Lemma 2.1, we find that u is a solution of the problem (1.1) if and only if u is a fixed point of the operator $A=KF$.

From the following lemma, we can obtain the eigenvalues and the algebraic multiplicity of the linear operator K.

and the algebraic multiplicity of each positive eigenvalue $\frac{1}{{\lambda }_n^3}$ is equal to 1, where $0<{\lambda }_1<{\lambda }_2<\cdots <{\lambda }_n<\cdots$ are the positive solutions of (2.3).

Applying the condition $y(0)=y^{\prime }(0)=0$, we obtain $C_2=-C_1$, $C_3=\sqrt{3}{C}_1$, where $C_1\ne 0$.

where C is a nonzero constant.

Therefore, the algebraic multiplicity of the eigenvalue $\frac{1}{{\lambda }_n^3}$ is 1. □

Lemma 2.3 Suppose that (H₁) holds and $y\in P\mathrm{\setminus }\{\theta \}$ is a solution of the (1.1), then $y\in \stackrel{\circ }{P}$. Similarly, if $y\in (-P)\mathrm{\setminus }\{\theta \}$ is a solution of the (1.1), then $y\in -\stackrel{\circ }{P}$.

Proof The proof is obvious. □

Lemma 2.4 Suppose that (H₁)-(H₃) hold. Then the operator A is Fréchet differentiable at θ and ∞, and $A_{\theta }^{\prime }={\beta }_{0}K$, $A_{\mathrm{\infty }}^{\prime }={\beta }_{1}K$.

which means $A_{\theta }^{\prime }={\beta }_{0}K$.

Therefore, $A_{\mathrm{\infty }}^{\prime }={\beta }_{1}K$. □

Lemma 2.5 [13]

Then A has at least three nontrivial fixed points containing one sign-changing fixed point.

We state the main result of this paper.

Theorem 3.1 Suppose that (H₁)-(H₃) hold. Then the problem (1.1) has at least three solutions including a sign-changing solution.

We need only to prove that $A=KF$ satisfies the four conditions of Lemma 2.5.

Therefore, from Lemma 2.5, we see that A has at least three nontrivial fixed points including one sign-changing fixed point. Then, the problem (1.1) has at least three solutions, including one sign-changing solution. □

By simple calculations, we have ${\lambda }_1\approx 3.0167$, ${\lambda }_2\approx 6.6506$, ${\lambda }_3\approx 10.2782$, ${\beta }_0=3$, ${\beta }_1=300$. Then it is easy to see that $f(t,u)$ satisfies the conditions (H₁)-(H₃). Therefore, the boundary-value problem (3.1) has at least three solutions, including one sign-changing solution.