 Research
 Open Access
Signchanging solutions for asymptotically linear operator equations and applications
 Yanbin Sang^{1}Email author,
 Qian Meng^{1} and
 Zongyuan Zhu^{2}
 Received: 2 June 2015
 Accepted: 23 September 2015
 Published: 9 October 2015
Abstract
In this paper, by using the topological degree and fixed point index theory, the existence of three kinds of solutions (i.e., signchanging solutions, positive solutions, and negative solutions) for asymptotically linear operator equations is discussed. The abstract results obtained here are applied to nonlinear integral and differential equations.
Keywords
 signchanging solutions
 asymptotically linear operators
 topological degree
 fixed point index
MSC
 47H07
 47H10
 34B10
 34B15
 34B18
1 Introduction
 (\(\widetilde{\mathrm{A}}_{1}\)):

\(f: \mathbb{R}\rightarrow \mathbb{R}\) is a continuous and strictly increasing function, and \(f(0)=0\);
 (\(\widetilde{\mathrm{A}}_{2}\)):

\(\lim_{x\rightarrow \infty}\frac{f(x)}{x}=\beta_{\infty}\), where \(\lambda_{1}<\beta_{\infty}<\frac{8(1\alpha\eta)^{2}}{(1\alpha \eta^{2})^{2}}\), \(\beta_{\infty}\neq\lambda_{n}\), \(n=2,3,\ldots \) , andis the sequence of positive solutions for the equation \(\sin \sqrt{x}=\alpha\sin\eta\sqrt{x}\);$$\lambda_{1}< \lambda_{2}< \cdots< \lambda_{n}< \lambda_{n+1}< \cdots $$
 (\(\widetilde{\mathrm{A}}_{3}\)):

\(\lim_{x\rightarrow 0}\frac{f(x)}{x}=\beta_{0} >\lambda_{1}\).
Theorem 1.1
(See [11])
Let α and η be given numbers with \(0<\alpha<1\), \(0<\eta <1\). Suppose that (\(\widetilde{\mathrm{A}}_{1}\))(\(\widetilde{\mathrm{A}}_{3}\)) hold. Then problem (1.1) has at least one signchanging solution. Moreover, problem (1.1) has at least two positive solutions and two negative solutions.
The main purpose of this paper is to abstract more general conditions from (\(\widetilde{\mathrm{A}}_{1}\))(\(\widetilde{\mathrm{A}}_{3}\)) of Theorem 1.1 and to obtain some existence theorems of signchanging solutions for asymptotically linear operator equations. Then, we apply the abstract results obtained in this paper to nonlinear integral equations and elliptic partial differential equations. A feature of this paper is that we weaken the condition (\(\widetilde{\mathrm{A}}_{2}\)) of Theorem 1.1 (see Example 4.1). Compared with main results in [12–15], we use a different method consisting of the computation of LeraySchauder degree for asymptotically linear operators and lower and upper solutions conditions. In addition, the compressive conditions of abstract operator can be removed.
For the discussion of the following sections, we state here preliminary definitions and known results on cones, partial orderings, and topological degree theory, which can be found in [16–18].
Let E be a real Banach space. Given a cone \(P\subset E\), we define a partial ordering ≤ with respect to P by \(x\leq y\) iff \(yx\in P\). A cone P is said to be normal if there exists a constant \(N>0\) such that \(\theta\leq x\leq y\) implies \(\x\\leq N\y\\), the smallest N is called the normal constant of P. P is reproducing if \(PP=E\), i.e., for every \(x\in E\), we have that \(x=yz\), where \(y\in P\), \(z\in P\) and total if \(\overline{PP}=E\). Let \(B: E\rightarrow E\) be a bounded linear operator. B is said to be positive if \(B(P)\subset P\). A fixed point u of operator A is said to be a signchanging fixed point if \(u\notin P\cup(P)\). If \(x_{0}\in E\backslash\{\theta\}\) satisfies \(\lambda Ax_{0}=x_{0}\), where λ is some real number, then λ is called a characteristic value of A, and \(x_{0}\) is called a characteristic function belonging to the characteristic value λ.
Definition 1.1
(See [19])
2 Main results
Theorem 2.1
 (i)
there exist \(u_{1}\in(P)\backslash\{\theta\}\) and \(v_{1}\in P\backslash\{\theta\}\) such that \(u_{1}\leq Au_{1}\) and \(Av_{1}\leq v_{1}\);
 (ii)
there exist \(u_{2}\in(P)\backslash\{\theta\}\), \(v_{2}\in P\backslash\{\theta\}\), and \(\delta>0\) such that \(u_{1}< u_{2}<\theta<v_{2}<v_{1}\), \(Au_{2}\leq u_{2}\delta e, v_{2}+\delta e\leq Av_{2}\);
 (iii)
the Fréchet derivative \(A'_{\infty}\) at ∞ exists; \(A'_{\infty}\) is increasing; \(r(A'_{\infty})>1\); 1 is not a characteristic value of \(A'_{\infty}\).
Theorem 2.2
 (i)
there exist \(u_{1}\in(P)\backslash\{\theta\}\) and \(v_{1}\in P\backslash\{\theta\}\) such that \(u_{1}\leq Au_{1}\) and \(Av_{1}\leq v_{1}\), and there exists \(\alpha>0\) such that \(u_{1}\leq\alpha e\) and \(\alpha e\leq v_{1}\);
 (ii)
\(F(\theta)=\theta\), F is Fréchet differentiable at θ, and \(KF'_{\theta}\) has a characteristic value \(\lambda_{0}<1\) with a characteristic function ψ satisfying \(\mu_{1} e\leq\psi \leq \mu_{2} e\), where \(\mu_{1}>0\), \(\mu_{2}>0\).
In order to prove Theorems 2.1 and 2.2, we need to establish the following lemmas. In this section, we suppose that \(B_{R}=\{x\in E \x\< R\}\).
Lemma 2.1
 (i)
there exist \(u, v\in E\) such that \(u\leq Au\), \(Av\leq v\);
 (ii)
A is Fréchet differentiable at ∞; \(A'_{\infty}\) is increasing; \(r(A'_{\infty})>1\) and 1 is not a characteristic value of \(A'_{\infty}\).
Proof
We only prove that \(i(A,\widetilde{\Omega}_{1},S_{1})=0\), the proof of \(i(A,\widetilde{\Omega}_{2},S_{2})=0\) is similar.
Let \(\tilde{A}x=uA'_{\infty}(ux)\), \(x\in E\). By the compactness of \(A'_{\infty}\) ([16], Proposition 7.33, p.296), we can know that \(\tilde{A}x: E\rightarrow E\) is completely continuous. Since \(A'_{\infty}\) is increasing, we can find that Ã is also increasing, and \(\tilde{A}: S_{1}\rightarrow S_{1}\).
Lemma 2.2
Proof of Theorem 2.1
Proof of Theorem 2.2
It suffices to verify condition (ii) of Theorem 2.1 is satisfied. According to the chain rule for derivatives of composite operator [17], Proposition 4.10, we have \(A'_{\theta}=KF'_{\theta}\).
By condition (i) and Lemma 2.4 in [19], we know that there exists \(\beta>0\) such that \(t \psi+\delta e\leq A(t \psi), A(t \psi)\leq t \psi\delta e\) for all \(t\in(0,\beta)\), where \(\delta=\frac{t(\lambda_{0}^{1}1)\mu_{1}}{2}>0\).
3 Applications
Let \(E=C(G)\) denote the space consisting of all continuous functions on G. Then E is a real Banach space with the norm \(\\varphi\=\max_{x\in G} \varphi(x)\) for all \(\varphi\in E\). And let \(P=\{\varphi\in E: \varphi(x)\geq0, x\in G\}\). Then P is a normal and total cone in E. Let \(e(x)=\int_{G} k(x,y)\, dy\), \(x\in G\). Then \(e>0\).
 (C_{1}):

\(f(\cdot,0)=0\) on G, and for every \(x\in G\), \(f(x,\varphi)\) is nondecreasing in φ;
 (C_{2}):

there exists h with \(\mu e\leq h\), where μ is a positive number such that$$k(x,y)\geq h(x)k(z,y), \quad x, y, z\in G; $$
 (C_{3}):

\(\lim_{\varphi\rightarrow 0}\frac{f(x,\varphi)}{\varphi}=f_{0}\) uniformly for \(x\in G\), and \(f_{0}>\lambda_{1}\);
 (C_{4}):

\(\lim_{\varphi\rightarrow\infty}\frac{f(x,\varphi )}{\varphi}=f_{\infty}\) uniformly for \(x\in G\), \(f_{\infty}>\lambda_{1}\), \(f_{\infty}\neq \lambda_{k}\), \(k= 2, 3, \ldots \) ;
 (C_{5}):

for every \(x\in G\), there exist \(M, N>0\) such that$$\frac{f(x,M)}{M}< \frac{1}{\e\} \quad \mbox{and}\quad \frac{f(x,N)}{N}< \frac{1}{\e\}. $$
Lemma 3.2
(See [19])
Suppose that \(f(\cdot,0)=0\) on G, and \(\lim_{\varphi\rightarrow 0}\frac{f(x,\varphi)}{\varphi}=f_{0}\) uniformly for \(x\in G\). Then A is Fréchet differentiable at θ, and \(A'_{\theta}=f_{0} K\).
Lemma 3.3
Suppose that \(\lim_{\varphi\rightarrow\infty}\frac{f(x,\varphi )}{\varphi}=f_{\infty}\) uniformly for \(x\in G\). Then A is asymptotically linear, and \(A'_{\infty}=f_{\infty}K\).
Proof
Theorem 3.1
Suppose that (C_{1})(C_{5}) are satisfied. Then the integral equation (3.1) has at least five nontrivial solutions, two of which are positive, the two others are negative, the fifth one is signchanging.
Proof
By (C_{5}), we know that condition (i) of Theorem 2.2 holds. We only need to check condition (ii) of Theorem 2.2 and condition (iii) of Theorem 2.1.
By (C_{3}) and Lemma 3.2, we have that \(A'_{\theta}=f_{0} K\), and the characteristic values of the operator \(f_{0} K\) are \(\frac{\lambda_{1}}{f_{0}}, \frac{\lambda_{2}}{f_{0}}, \ldots \) . Since \(f_{0} >\lambda_{1}\), we know that \(A'_{\theta}\) has a characteristic value \(\frac{\lambda_{1}}{f_{0}}<1\). Moreover, from the proof of Theorem 4.1 in [19], we deduce that the characteristic function corresponding to the characteristic value \(\frac{\lambda_{1}}{f_{0}}\) satisfies \(\mu_{1} e\leq\psi\leq\mu_{2} e\), where \(\mu_{1}>0\) and \(\mu_{2}>0\). This implies that condition (ii) of Theorem 2.2 holds.
According to (C_{4}) and Lemma 3.3, we know that \(A'_{\infty}=f_{\infty}K\), \(r(A'_{\infty})=f_{\infty}r(K)=f_{\infty}\frac{1}{\lambda_{1}} >1\), and the characteristic values of the operator \(f_{\infty}K\) are \(\frac{\lambda_{1}}{f_{\infty}}, \frac{\lambda_{2}}{f_{\infty}}, \ldots \) . Noting that \(f_{\infty}\neq\lambda_{k}\), \(k\neq2, 3, \ldots \) , we get that 1 is not a characteristic value of \(A'_{\infty}\). Hence, condition (iii) of Theorem 2.1 holds. The proof is completed. □
Remark 3.1
From the proof of Theorem 11 of [14] and Theorem 4.2 of [19], we know that condition (C_{1}) can be replaced by the following one: \(f(\cdot,0)=0\) on G, and \(f(\cdot,\varphi)\varphi\geq0\) for all \(\varphi\in \mathbb{R}\). For nonlinear term f, the sublinear and superlinear cases were considered in [20].
 (i)
\(\delta=0\) and \(b(x)\equiv1\);
 (ii)
\(\delta=1\) and \(b(x)\equiv0\);
 (iii)
\(\delta=1\) and \(b(x)>0\).
 (D_{1}):

\(\lim_{\varphi\rightarrow \infty}\frac{f(x,\varphi)}{\varphi}=f_{\infty}\) uniformly for \(x\in \overline{\Omega}\);
 (D_{2}):

\(f(x,0)\equiv0\), \(\lim_{\varphi\rightarrow 0}\frac{f(x,\varphi)}{\varphi}=f_{0}\) uniformly for \(x\in \overline{\Omega}\).
Our main result is the following.
Theorem 3.2
 (i)
\(f(x,\varphi)\) is increasing in φ;
 (ii)
\(f_{0} >\lambda_{1}\);
 (iii)
\(f_{\infty}>\lambda_{1}\), \(f_{\infty}\neq\lambda_{k}\), \(k= 2, 3, \ldots \) ;
 (iv)for every \(x\in\overline{\Omega}\), there exist \(M, N>0\) such that$$\frac{f(x,M)}{M}< \frac{1}{\e\} \quad \textit{and} \quad \frac{f(x,N)}{N}< \frac{1}{\e\}. $$
Proof
Since the characteristic values of the operator \(A'_{\infty}\) are \(\frac{\lambda_{1}}{f_{\infty}}, \frac{\lambda_{2}}{f_{\infty}}, \ldots \) , noting that \(f_{\infty}\neq\lambda_{k}\), \(k=2,3,\ldots \) , we know that 1 is not a characteristic value of \(A'_{\infty}\). Therefore, all conditions of Theorem 2.2 are satisfied. The proof is completed. □
4 An example
In the section, we present an example to explain our results.
Example 4.1
Remark 4.1
Compared with Theorem 1.1, Theorem 13 of [23] and Theorem 2.1 of [24], the main contributions in this paper are that we change the range of \(\beta_{0}\) in (\(\widetilde{\mathrm{A}}_{3}\)) and \(\beta_{\infty}\) in (\(\widetilde{\mathrm{A}}_{2}\)) and add the numbers of positive solutions and negative solutions. Furthermore, for the lower and upper solutions conditions, we adopt condition (C_{5}) which is more easy to check than (H_{3}) in Theorem 2.1 of [24].
5 Conclusions
Declarations
Acknowledgements
The authors thank the editors and the referees for their valuable comments and suggestions which improved greatly the quality of this paper. This project is supported by the Scientific Research Foundation of North University of China.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
Authors’ Affiliations
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