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Computation of fixed point index and applications to superlinear periodic problem
Fixed Point Theory and Applications volume 2015, Article number: 157 (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.
Introduction
The topological degree and fixed point index are important theories in nonlinear functional analysis as they have had significant applications as regards obtaining results on the existence and the number of solutions for differential equations [1], differential inclusions [2] and dynamical systems [3, 4]. In recent years, many authors have focused on the computation of the topological degree and fixed point index (see [5–12] and the references therein). It is noticed that the results cited above apply to an operator equation of the form \(x=Ax\), which is closely connected to fixed point theory. However, in this paper we will be mainly interested in studying a more general operator equation of the form
where L is not necessarily invertible. During the last three decades, the existence problem for (1.1) has been an interesting topic and has attracted the attention of many researchers [1, 13–16] because it has especially broad applications in the existence of periodic solutions of nonlinear differential equations.
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.
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.
Throughout this paper we assume that the following conditions are satisfied:
 (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]
Let \(\theta\in\Omega \subset X\). If \(Lx\neq\mu Nx(1\mu)J^{1}Px\) for all \(x\in\partial\Omega_{K}\) and \(\mu\in[0,1]\), then
Lemma 2.3
[22]
If there exists \(e\in K_{1}\backslash\{\theta\}\) such that
for all \(x\in\partial\Omega_{K}\) and all \(\mu\geq0\), then
Main results
Theorem 3.1
Let \(\theta\in\Omega\subset X\) and \(Lx\neq Nx\) for all \(x \in\partial\Omega_{K}\). Assume that the following hypotheses hold:

(i)
there exists a positive bounded linear operator \(B: X\rightarrow X\), such that
$$\bigl(N+J^{1}P \bigr)x\leq \bigl(L+J^{1}P \bigr)Bx, \quad \textit{for all } x \in\partial \Omega_{K}, $$where this partial order is induced by the cone \(K_{1}\) in Y.

(ii)
\(r(B)\leq1\), where \(r(B)\) is the spectral radius of B.
Then the fixed point index
Proof
We show that
Suppose the assertion of (3.1) is false. Then there exist \(x_{1}\in\partial\Omega_{K}\) and \(\mu_{1}\in[0,1]\) such that \(Lx_{1}=\mu_{1} Nx_{1}(1\mu_{1})J^{1}Px_{1}\). Since \(Lx\neq Nx \) \(\forall x\in\partial\Omega_{K}\), we see that \(\mu_{1}\in[0,1)\). This and condition (i) imply that
Applying \((L+J^{1}P)^{1}\) to the above inequality, we obtain \(x_{1}\leq \mu_{1}Bx_{1}\). Continuing this process, we get
Let \(\Sigma=\{yy\geq x_{1}\}\). Equation (3.2) yields \(\{\mu _{1}^{n}B^{n}x_{1}n=1,2,\ldots\}\subset\Sigma\). \(x_{1}\in\partial\Omega_{K}\) and \(\theta\in\Omega_{K}\) imply \(d=d(\theta,\Sigma)>0\). Consequently, we get by (3.2)
This shows
This contradicts the condition (ii). Hence (3.1) is true and we see from Lemma 2.2 that the conclusion is true. □
Theorem 3.2
Let K be a normal cone in X. If there exist a constant \(C>0\) and \(u_{1}\in K_{1} \backslash \{\theta\}\), such that
where this partial order is induced by the cone \(K_{1}\) in Y, then there exists \(R_{0}>0\), and for \(R>R_{0}\), the fixed point index
where \(B_{R}=\{x\in X :\x\_{X}< R\}\).
Proof
Setting
we claim that W is bounded. For \(x\in W\), there exists \(\lambda\geq0\) such that \(LxNx=\lambda u_{1}\). This and assumption (3.3) imply that
Operating on both sides of the latter inequality by \((L+J^{1}P)^{1}\), we obtain
This shows that \(x\leq\frac{1}{C1}(L+J^{1}P)^{1}u_{1}\). In view of the normality of K, there exists an \(\widetilde{N}>0\) such that
This shows that W is bounded.
Let \(R_{0}=\sup_{x\in W}\x\\). For \(R>R_{0}\), we have
Using Lemma 2.3, we infer by (3.4) that the conclusion is true. □
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
Let \(\theta\in\Omega_{K}\) and \(Lx\neq Nx\) for all \(x\in\partial\Omega_{K}\). If there exists a positive bounded linear operator \(B: X\rightarrow X\) with \(r(B)\leq1\), such that
where this partial order is induced by the cone K in X, then the fixed point index
Corollary 3.4
Let K be a normal cone in X. If there exist a constant \(C>1\) and \(u_{0}\in K\backslash \{\theta\}\), such that
where this partial order is induced by the cone K in X, then there exists \(R_{0}>0\), and for \(R>R_{0}\), the fixed point index
where \(B_{R}=\{x\in X :\x\_{X}< R\}\).
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
Let K be a normal cone in X, \(u_{0}\in K\backslash\{\theta\}\) and let there be a constant \(C>1\). If there exists a positive bounded linear operator \(B: X\rightarrow X\) with \(r(B)\leq1\), such that the following hypotheses hold:

(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
It follows from Corollary 3.4 and condition (i) that there exists \(R>0\) with \(R>r\) such that
We assume \(Lx\neq Nx\) on \(\partial B_{r}\cap K\cap\operatorname{dom}L\), otherwise the conclusion follows. Using Corollary 3.3 we get from condition (ii)
In view of (3.5), (3.6), and the additivity property, we obtain
which completes the proof from the existence property. □
Applications to superlinear periodic boundary value problem
Firstorder periodic boundary value problem
We shall apply Theorem 3.5 to obtain positive solutions to the following firstorder periodic boundary value problem (PBVP):
where \(f: [0,1]\times\mathbb{R}^{+} \rightarrow\mathbb{R}\) is a continuous function.
Let \(X=Y=C[0,1]\) with the usual norm \(\x\=\max_{t\in [0,1]}x(t)\). Define the linear operator \(L:\operatorname{dom}L\subset X\rightarrow Y\), \((Lx)(t)=x'(t)\), \(t\in [0,1]\), where
and \(N: X \rightarrow Y\) with
It is easy to check that
so that L is a Fredholm operator of index zero.
Next, define the projections \(P:X\rightarrow X\) by
and \(Q:Y\rightarrow Y\) by
Furthermore, we define the isomorphism \(J:\operatorname{Im}Q\rightarrow \operatorname{Im}P\) as \(Jy=\beta y\) with \(\beta=1\). We may easy verify that the inverse operator \(L_{1}^{1}:\operatorname{Im} L\rightarrow\operatorname{dom} L\cap\operatorname{Ker} P\) of \(L_{\operatorname{dom} L\cap\operatorname{Ker} P}:\operatorname{dom} L\cap\operatorname{Ker} P\rightarrow \operatorname{Im} L\) is \((L_{1}^{1}y)(t)= \int_{0}^{1}K(t,s)y(s)\,ds\), where
For notational convenience, we set \(G(t,s)=1+K(t,s) \int_{0}^{1}K(t,s)\,ds\). We can verify that
and \(\frac{1}{2}\leq G(t,s)\leq\frac{3}{2}\), \(t,s \in[0,1]\).
Define the cone K in X by
then K is a normal cone of X (see [25]).
Lemma 4.1
If
 (H_{1}):

\(f(t,x)\geq\frac{1}{3}x\), for all \(t\in[0,1]\), \(x \geq0\),
Proof
It follows from condition (H_{1}) that
for all \(x\in K\). Thus \((P+JQN+L_{1}^{1}(IQ)N)(x)\geq0\).
Now we are ready to prove
Using condition (H_{1}), we obtain
From the last inequality, we have from condition (H_{1})
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
Suppose that condition (H_{1}) is satisfied. If
 (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.
Condition (H_{2}) guarantees that there exist \(\epsilon>0\) and \(\tau >0\) such that
Hence,
where \(M=\max_{0\leq t\leq1, 0\leq x\leq\tau }f(t,x)(4+\epsilon)x+1\).
Set \(C=1+\frac{\varepsilon}{6}\), \(u_{0}=\frac{3M}{2}\). Then \(C>1\), \(u_{0}\in K\backslash\{\theta\}\). Using (4.2), we obtain
This implies that condition (i) of Theorem 3.5 is satisfied.
It follows from condition (H_{3}) that there exist \(\sigma\in(0,2)\) and \(r\in(0,\tau)\) such that
Take \(Bx=(1\frac{\sigma}{2}) \int_{0}^{1}x(s)\,ds\). One can see that \(B: X \rightarrow X\) is a positive bounded linear operator. It is clear that \(r(B)=1\frac {\sigma}{2}<1\). Thus, by (4.3), we have
This means that condition (ii) of Theorem 3.5 is verified.
We see that all assumptions of Theorem 3.5 are satisfied. The proof is finished. □
Secondorder periodic boundary value problem
We will discuss the existence of positive solutions of the secondorder periodic boundary value problem (PBVP)
where \(f: [0,1]\times\mathbb{R}^{+}\rightarrow\mathbb{R}\) is a continuous function.
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.
Let Banach spaces X, Y be as in Section 4.1. In this case, we may define
and let the linear operator \(L:\operatorname{dom}L\subset X\rightarrow Y\) be defined by
Then L is Fredholm of index zero,
and
Define \(N: X \rightarrow Y\) by
Thus it is clear that the PBVP (4.5) is equivalent to the operator equation (1.1).
We use the same projections P, Q as in Section 4.1 and define the isomorphism \(J:\operatorname{Im}Q\rightarrow\operatorname{Im}P\) as \(Jy=\beta y\) with \(\beta=1\). It is easy to verify that the inverse operator \(L_{1}^{1}:\operatorname{Im} L\rightarrow\operatorname{dom} L\cap\operatorname{Ker} P\) of \(L_{\operatorname{dom} L\cap\operatorname{Ker} P}:\operatorname{dom} L\cap\operatorname{Ker} P\rightarrow \operatorname{Im} L\) is
where
Set
We can verify that
and
Define a normal cone K in X by
Lemma 4.3
If
 (H_{4}):

\(f(t,x)\geq3x\), for all \(t\in[0,1]\), \(x \geq0\),
Proof
For each \(x\in K\), by condition (H_{4})
Thus \((P+JQN+L_{1}^{1}(IQ)N)(x)\geq0\).
We now show that
In fact, we get from condition (H_{4})
By the above inequality, we have from condition (H_{4})
Thus \((P+JQN)x+L_{1}^{1}(IQ)Nx\in K\). □
Theorem 4.4
Suppose that condition (H_{4}) is satisfied. If
 (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
It is again easy to show that \(L\lambda N\) is Aproper for \(\lambda \in[0,1]\) by Lemma 2.1. From condition (H_{5}), we know that there exist \(\epsilon>0\) and \(\tau>0\)
Thus
where \(M=\max_{0\leq t\leq1, 0\leq x\leq\tau }f(t,x)(12+\epsilon)x+1\).
From (4.6), we have
and condition (i) of Theorem 3.5 is satisfied with \(C=1+\frac {\varepsilon}{20}\) and \(u_{0}=\frac{M}{4}\).
By condition (H_{6}), we can find \(\sigma\in(0, 8)\) and \(r\in(0, \tau)\) such that
If we take \(Bx=(1\frac{\sigma}{8}) \int_{0}^{1}x(s)\,ds\), then \(B: X \rightarrow X\) is a positive bounded linear operator and \(r(B)=1\frac{\sigma}{8}<1\). Finally, similar to the proof of (4.4), it follows from (4.7) that
Thus condition (ii) of Theorem 3.5 is satisfied and the proof is complete. □
Example 4.5
Let the nonlinearity in (4.5) be
where \(\alpha>1\), \(a \in C[0,1]\) is positive 1periodic function and \(k\in(0,3)\). Then (4.5) has at least one positive 1periodic solution.
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?’
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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).
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Wang, F., Li, S. Computation of fixed point index and applications to superlinear periodic problem. Fixed Point Theory Appl 2015, 157 (2015). https://doi.org/10.1186/s1366301504115
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MSC
 34B10
 34B15
Keywords
 computation
 fixed point index
 partial order method
 superlinear periodic problem