New fixed point theorems on order intervals and their applications
- Wenying Feng^{1}Email author and
- Guang Zhang^{2}
https://doi.org/10.1186/s13663-015-0467-2
© Feng and Zhang 2015
Received: 14 August 2015
Accepted: 10 November 2015
Published: 25 November 2015
Abstract
In this paper, we prove the existence of fixed points for nonlinear and semilinear operators on order intervals. The abstract results unified some methods in studying the existence of positive solutions for boundary and initial value problems of nonlinear difference and differential equations. Applications are shown by examples.
Keywords
MSC
1 Introduction
Fixed point theory has been an important tool in the study of differential and integral equations [1, 2], economics [3], optimization and game theory [4] among others. The simplest theorem from elementary calculus considers the existence of positive roots for the equation: \(f(x)=x\) on \(\mathbb{R}_{+}=[0, +\infty)\). Clearly, if there exist \(b>a>0\) such that \(f\in C[a,b]\) and either \(f(a)\leq a\) and \(f(b)\geq b\) or \(f(a)\geq a\) and \(f(b)\leq b\), then there exists a \(x^{\star}\in [ a,b]\) such that \(x^{\star}=f(x^{\star})\), that is: the function \(f(x)\) has a fixed point \(x^{\star}\in[ a,b]\). Such result had been expanded to an abstract operator equation to obtain the Guo-Krasnoselskiĭ fixed point theorem concerning cone expansion and compression of norm type as follows (see [5] and [6]).
Lemma 1.1
Let X be a Banach space and P be a cone in X. Assume that \(\Omega_{1}\) and \(\Omega_{2}\) are open subsets of X with \(0\in\Omega_{1}\) and \(\overline{\Omega}_{1}\subset\Omega_{2}\). Let \(T:P\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\rightarrow P\) be completely continuous operator. If either \(\Vert Tu\Vert\leq\Vert u\Vert\) for \(u\in P\cap\partial\Omega_{1}\) and \(\Vert Tu\Vert\geq\Vert u\Vert\) for \(u\in P\cap\partial\Omega_{2}\) or \(\Vert Tu\Vert\leq\Vert u\Vert\) for \(u\in P\cap\partial\Omega_{2}\) and \(\Vert Tu\Vert\geq\Vert u\Vert\) for \(u\in P\cap\partial\Omega_{1}\) holds, then T has a fixed point in \(P\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\).
It is well-known that this abstract result can be applied to obtain an abundance of concrete results for some special problems [1, 7–10], for example, (a) Hammerstein integral equations, (b) boundary value problems for semilinear ordinary differential equations, (c) boundary value problems for semilinear elliptic differential equations, (d) initial-value problems for semilinear parabolic differential equations, (e) discrete boundary value problems or the nonlinear algebraic equations systems, (f) boundary value problems for semilinear fractional differential equations, (g) boundary value problems for semilinear time scale differential equations, (h) existence of periodic solutions for some functional differential equations, etc. Because these problems can be regarded as abstract operator equations.
\(P_{u_{0}}\) is a new and general cone. When we choose different \(u_{0}\), some known cones such as (1.1)-(1.4) can be obtained. When the obtained abstract results are applied to concrete cases (a)-(h), new results can be naturally obtained. Compared to the Guo-Krasnoselskiĭ’s result, our abstract results are established on order intervals rather than an annular region of the cone. Therefore, no conditions for the operator T outside the interval are necessary. This expands the recent idea in [9].
2 Main results
Theorem 2.1
Proof
The proof is similar if condition (2.2) holds. □
Theorem 2.2
In Theorem 2.2, let \(\delta=1\), we obtain the following corollary, which is a generalization of the fixed point theorem in finite dimensional spaces recently obtained in [9].
Corollary 2.3
Example 1
We now consider the semilinear operator equation (1.7).
Theorem 2.4
Proof
Remark 2.5
In Theorem 2.4, let \(u_{0}= \delta\varphi\), we can obtain the parallel theorem of Theorem 2.2 for the semilinear case.
Remark 2.6
Condition (2.7) is always true when K is bounded and invertible. For a bounded linear operator, \(M=\|K\|\). If K is invertible, \(m=\frac{1}{\|K^{-1}\|}\).
For a Banach space with a normal cone, the norm condition (2.7) can be reduced to an order condition. The definition of a normal cone is given below [6].
Definition 2.7
The following result for a Banach space with a normal cone can be easily applied in many cases.
Theorem 2.8
Example 2
It can be seen that Example 2 is true for any finite dimensional space with the dimension \(n> 2\).
Remark 2.9
Remark 2.10
Theorems proved in this section can also be extended to negative intervals to prove the existence of negative solutions.
3 Applications
The results obtained in Section 2 can be applied to existence of solutions for differential and difference equations. We will show some examples.
Example 3
The following definition of a fractional derivative is related to our next example on fractional boundary value problem.
Definition 3.1
Example 4
System (3.3)-(3.4) was recently studied in [17]. Applying Theorem 2.1, the following new result on the existence of a positive solution is obtained.
Theorem 3.2
Proof
Let \(u_{0}=q(t) \), \(\varphi= 1\). It was shown that, for any \(\lambda >0\), \(N: P_{u_{0}} \to P_{u_{0}} \) is completely continuous [15]. For \(u\in P_{u_{0}}\), \(\|u\| = u(1)\). So \(u_{0}\) and φ satisfy the conditions of Theorem 2.1.
Declarations
Acknowledgements
The authors thank the anonymous referees for helpful comments. The research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the National Natural Science Foundation of China (No. 11371277).
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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