Computation of fixed point index and applications to superlinear periodic problem
 Feng Wang^{1, 2}Email author and
 Shengjun Li^{3}
https://doi.org/10.1186/s1366301504115
© Wang and Li 2015
Received: 23 February 2015
Accepted: 24 August 2015
Published: 1 September 2015
Abstract
In this paper we compute the fixed point index for Aproper semilinear operators under certain boundary conditions. The proof is based on the partial order method combined with the properties of the fixed point index. As an application, we use the abstract results presented above to study the existence conditions of positive solutions for superlinear firstorder and secondorder periodic problems.
Keywords
MSC
1 Introduction
Based on the approach of Fitzpatrick and Petryshyn [17], the concept of a fixed point index for Aproper maps related to (1.1) has been introduced by Cremins [18, 19]. Since then, some existence results for the semilinear equation (1.1) in cones have been established in [20, 21] using the properties of the fixed point index and partial order method. Recently, the computation of the fixed point index for Aproper semilinear operators was proved in [22] under the sublinear case, but to the best of the knowledge of the authors nothing has been published concerning the computation results available in the literature for the superlinear case. In this paper, we will continue this study and focus on the computation of the fixed point index with certain boundary conditions in the superlinear case.
The remaining part of the paper is organized as follows. Some preliminaries and a number of lemmas useful to the derivation of the main results are given in Section 2. In Section 3, we obtain some sufficient conditions that the fixed point index equals \(\{1\}\) or \(\{0\}\), values more easily applicable. As an application, in Section 4, we use the new results presented in Section 3 to study the existence conditions of positive solutions for firstorder and secondorder superlinear periodic boundary value problems.
2 Preliminaries
In this section, we recall some standard facts on Aproper mappings and Fredholm operators.
Let X and Y be Banach spaces with the zero element θ, D a linear subspace of X, \(\{X_{n}\}\subset D\), and \(\{Y_{n}\}\subset Y\) sequences of oriented finite dimensional subspaces such that \(Q_{n}y\rightarrow y\) in Y for every y and \(\operatorname{dist}(x,X_{n})\rightarrow0\) for every \(x\in D\) where \(Q_{n}: Y \rightarrow Y_{n}\) and \(P_{n}: X\rightarrow X_{n}\) are sequences of continuous linear projections. The projection scheme \(\Gamma=\{X_{n},Y_{n},P_{n},Q_{n}\}\) is then said to be admissible for maps from \(D\subset X\) to Y. A map \(T:D\subset X\rightarrow Y\) is called approximationproper (abbreviated Aproper) at a point \(y\in Y\) with respect to Γ if \(T_{n}\equiv Q_{n}T_{D\cap X_{n}}\) is continuous for each \(n\in\mathbb{N}\) and whenever \(\{x_{n_{j}}x_{n_{j}} \in D\cap X_{n_{j}}\}\) is bounded with \(T_{n_{j}} x_{n_{j}}\rightarrow y\), then there exists a subsequence \(\{x_{n_{j_{k}}}\}\) such that \(x_{n_{j_{k}}}\rightarrow x \in D\), and \(Tx=y\). T is said to be Aproper on a set D if it is Aproper at all points of Y.
\(L: \operatorname{dom} L\subset X \rightarrow Y\) is a Fredholm operator of index zero if ImL is closed and \(\operatorname{dim\,Ker}L=\operatorname{codim\,Im} L<\infty\). Then X and Y may be expressed as direct sums \(X=X_{0}\oplus X_{1}\), \(Y=Y_{0}\oplus Y_{1}\) with continuous linear projections \(P: X\rightarrow \operatorname{Ker} L=X_{0}\) and \(Q: Y\rightarrow Y_{0}\). The restriction of L to \(\operatorname{dom}L\cap X_{1}\), denoted \(L_{1}\), is a bijection onto \(\operatorname{Im}L=Y_{1}\) with continuous inverse \(L_{1}^{1}: Y_{1}\rightarrow \operatorname{dom}L\cap X_{1}\). Since \(X_{0}\) and \(Y_{0}\) have the same finite dimension, there exists a continuous bijection \(J: Y_{0}\rightarrow X_{0}\).
Cremins [18, 19] defined a fixed point index \(\operatorname{ind}_{K}([L,N],\Omega)\) for Aproper maps acting on cones, which has the usual properties of the classical fixed point index, that is, existence, normalization, additivity, and homotopy invariance. Let K be a cone in the Banach space X, then X becomes a partial ordered Banach space under the partial ordering ≤ which is induced by K. K is said to be normal if there exists a positive constant Ñ such that \(\theta\leq x\leq y\) implies \(\x\\leq\widetilde{N}\y\\). For the concepts and the properties as regards the cone we refer to [23–25]. Here we remark that the results in [18, 19, 22] hold in partial ordered Banach spaces. Let \(\Omega\subset X\) be open and bounded such that \(\Omega_{K}=\Omega\cap K\neq\emptyset\). If we let \(K_{1}=(L+J^{1}P)(K\cap\operatorname{dom} L)\), then \(K_{1}\) is a cone in Y and the linear operator \(L+J^{1}P\) is inversely positive by using [18], Proposition 1.
 (A_{1}):

\(L: \operatorname{dom} L\rightarrow Y\) is Fredholm of index zero.
 (A_{2}):

\(L\lambda N\) is Aproper for \(\lambda\in[0,1]\).
 (A_{3}):

N is bounded and \(P+JQN+L_{1}^{1}(IQ)N\) maps K to K.
Lemma 2.1
([26], Lemma 2(a))
If \(L_{1}^{1}\) is compact, then \(L\lambda N\) is Aproper for each \(\lambda\in[0,1]\).
To obtain some new methods of computing the fixed point index for the Aproper semilinear operator (1.1), we need the following two lemmas.
Lemma 2.2
[19]
Lemma 2.3
[22]
3 Main results
Theorem 3.1
 (i)there exists a positive bounded linear operator \(B: X\rightarrow X\), such thatwhere this partial order is induced by the cone \(K_{1}\) in Y.$$\bigl(N+J^{1}P \bigr)x\leq \bigl(L+J^{1}P \bigr)Bx, \quad \textit{for all } x \in\partial \Omega_{K}, $$
 (ii)
\(r(B)\leq1\), where \(r(B)\) is the spectral radius of B.
Proof
Theorem 3.2
Proof
Now the following two corollaries are immediate consequences of Theorems 3.1 and 3.2 and the facts that \((L+J^{1}P)^{1}(N+J^{1}P)=P+JQN+L_{1}^{1}(IQ)N\) (see [18], Lemma 2) and we have linearity and positivity of the operator \((L+J^{1}P)^{1}\) (see [18], Proposition 1).
Corollary 3.3
Corollary 3.4
Using Corollaries 3.3 and 3.4, we complete this section with a proof of the following important result to be used later.
Theorem 3.5
 (i)
\((P+JQN+L_{1}^{1}(IQ)N)x\geq Cxu_{0}\), \(\forall x \in K\),
 (ii)
\((P+JQN+L_{1}^{1}(IQ)N)x\leq Bx\), \(\forall x \in\partial B_{r} \cap K\), where \(B_{r}=\{x\in X: \x\< r\}\), then there exists \(x^{*} \in\operatorname {dom}L\cap K\setminus\{\theta\}\) such that \(Lx^{*}=Nx^{*}\).
Proof
4 Applications to superlinear periodic boundary value problem
4.1 Firstorder periodic boundary value problem
Lemma 4.1
 (H_{1}):

\(f(t,x)\geq\frac{1}{3}x\), for all \(t\in[0,1]\), \(x \geq0\),
Proof
Therefore, \((P+JQN)x+L_{1}^{1}(IQ)Nx\in K\). □
We can now state and prove our result on the existence of a positive solution for the PBVP (4.1).
Theorem 4.2
 (H_{2}):

\(\liminf_{x\rightarrow+\infty}\min_{t\in [0,1]} \frac{f(t,x)}{x}>4\),
 (H_{3}):

\(\limsup_{x\rightarrow0^{+}}\max_{t\in [0,1]} \frac{f(t,x)}{x}<0\),
Proof
First, we note that L, as defined, is Fredholm of index zero, \(L_{1}^{1}\) is compact by ArzelaAscoli theorem and thus \(L\lambda N\) is Aproper for \(\lambda\in[0,1]\) by Lemma 2.1.
We see that all assumptions of Theorem 3.5 are satisfied. The proof is finished. □
4.2 Secondorder periodic boundary value problem
Since some parts of the proof are in the same line as that of Theorem 4.2, we will outline the proof with the emphasis on the difference.
Lemma 4.3
 (H_{4}):

\(f(t,x)\geq3x\), for all \(t\in[0,1]\), \(x \geq0\),
Proof
Theorem 4.4
 (H_{5}):

\(\liminf_{x\rightarrow+\infty}\min_{t\in [0,1]} \frac{f(t,x)}{x}>12\),
 (H_{6}):

\(\limsup_{x\rightarrow0^{+}}\max_{t\in [0,1]} \frac{f(t,x)}{x}<0\),
Proof
Example 4.5
Proof
We will apply Theorem 4.4 with \(f(t,x)=a(t)x^{\alpha}kx\). Since \(k\in(0,3)\), it is easy to see that (H_{4}) holds.
One may easily see that \(\liminf_{x\rightarrow+\infty}\min_{t\in [0,1]} \frac{f(t,x)}{x}=\liminf_{x\rightarrow +\infty}\min_{t\in[0,1]}(a(t)x^{\alpha1}k)=+\infty>12\), which implies that (H_{5}) holds.
On the other hand, we have \(\limsup_{x\rightarrow0^{+}}\max_{t\in [0,1]} \frac{f(t,x)}{x}=\limsup_{x\rightarrow 0^{+}}\max_{t\in[0,1]}(a(t)x^{\alpha1}k)=k<0\), which implies that (H_{6}) holds. Now we have the desired result. □
Remark 4.6
In [27, 28], some existence results for the secondorder periodic problem were established by Graef, Kong, Wang and Torres, respectively. Their proofs are based on Krasnosel’skii fixed point theorem in cones for completely continuous operators and the proofs are simpler and more clear than the proof presented in our paper. However, for us it seems difficult to obtain the same results in our paper using the fixed point theorem in cones. The main reason is that our condition (H_{5}) is weaker than the classical superlinear condition near \(x=+\infty\) (that is, \(\liminf_{x\rightarrow+\infty}f(t,x)/x=+\infty\)), yet being used in [27], Theorem 2.1 and [28], Corollary 4.1. Therefore, in a sense, our results improve and generalize those in [27, 28].
Remark 4.7
The following natural question concerning the optimality of conditions (H_{2}) and (H_{5}) remains open to the authors: ‘Find an optimal constant \(\lambda^{*}\) such that if \(\liminf_{x\rightarrow+\infty}f(t,x)/x>\lambda^{*}\) then Theorems 4.2 and 4.4 remain valid.’ In other words, ‘Are (H_{2}) and (H_{5}) also necessary conditions in order to get at least one positive solution in Theorems 4.2 and 4.4 respectively?’
Declarations
Acknowledgements
The authors express their thanks to the referees and the editor for their valuable comments and suggestions. The research of Feng Wang is supported by the National Natural Science Foundation of China (Grant No. 11501055 and No. 11401166), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 15KJB110001). The research of Shengjun Li is supported by the National Natural Science Foundation of China (Grant No. 11461016) and Hainan Natural Science Foundation (Grant No. 113001).
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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