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Monotone iterative technique for causal differential equations with upper and lower solutions in the reversed order
 Wenli Wang^{1}Email author
 Received: 18 December 2015
 Accepted: 25 July 2016
 Published: 2 August 2016
Abstract
In this paper, we use monotone iterative technique in the presence of (coupled) upper and lower solutions in the reversed order to discuss the existence of extremal solutions (quasisolutions) for causal differential equations with nonlinear boundary conditions. Two examples are provided to illustrate the efficiency of the obtained results.
Keywords
 monotone iterative technique
 upper and lower solutions
 reversed order
 causal operators
MSC
 39A10
 34B37
1 Introduction
Causal differential equations are recognized as an excellent model for real world problems; compared with the traditional model [1], one has wider realtime applications in a variety of disciplines. Its theory also has the powerful quality of unifying ordinary differential equations, integro differential equations, differential equations with finite or infinite delay, Volterra integral equations, and neutral equations. For more information, the reader can refer to the monograph by Lakshmikantham [2] and to [3–5].
Our boundary conditions is given by a nonlinear function, and more general than ones given before. This paper is organized as follows. In Section 2, we prove a new comparison principle. In Section 3, we show the existence and uniqueness of the solutions for the linear problem of (1). Then by using the monotone iterative technique coupled with the upper and lower solutions in the reversed order, we obtain the existence of extremal solutions for problem (1). In Section 4, using the notion of coupled upper and lower solutions in the reversed order, the existence of coupled minimal and maximal quasisolutions for (1) is established. Finally, two examples are given to illustrate our results.
2 Comparison results
In this section, we present a definition and a lemma which help to prove our main results.
Definition 2.1
Lemma 2.1
Proof
Suppose that \(m(t)\leq0,t\in J\) is not true, then we have the following two cases:
Case 1: there exists \(\bar{t}\in J\) such that \(m(\bar{t})>0\) and \(m(t)\leq0\) for all \(t\in J\backslash\{\bar{t}\}\).
Case 2: there exist \(t_{*}\) and \(t^{*}\) such that \(m(t_{*})<0\) and \(m(t^{*})>0\).
3 Extremal solutions
In this section, we shall establish the existence of extremal solutions of problem (1).
Definition 3.1
Theorem 3.1
Proof
In the following paper, we denote \(\xi=\Vert G(t,s)\Vert =\max \{\vert \frac {M_{1} e^{\int_{0}^{T}M(t)\,dt}}{M_{1}M_{2}e^{\int_{0}^{T}M(t)\,dt}}\vert ,\vert \frac{M_{2} e^{\int_{0}^{T}M(t)\,dt}}{M_{1}M_{2}e^{\int_{0}^{T}M(t)\,dt}}\vert \}\).
Theorem 3.2
Proof
Theorem 3.3
 (H_{1}):

the functions \(\alpha, \beta\in\Omega\) are lower and upper solutions of problem (1), respectively, such that \(\beta\leq \alpha\);
 (H_{2}):

\(\mathcal{L}\in C(E,E)\) is a positive linear operator and \(M\in C(\mathbb{R},\mathbb{R}^{+})\) such that$$(Qu) (t)(Qv) (t)\leq M(t) \bigl(u(t)v(t) \bigr)+ \bigl(\mathcal{L}(uv) \bigr) (t),\quad \textit{for } \beta\leq v\leq u\leq\alpha; $$
 (H_{3}):

there exist \(M_{2}\geq M_{1}>0\) satisfying \(M_{1}\neq M_{2}e^{\int _{0}^{T}M(s)\,ds}\), andwhenever \(\beta(0)\leq u(0)\leq \bar{u}(0)\leq\alpha(0),\beta(T)\leq v(T)\leq\bar{v}(T)\leq\alpha(T)\).$$g(\bar{u},\bar{v})g(u,v)\geq M_{1}(\bar{u}\bar{v})M_{2}(uv), $$
Then there exist monotone sequences \(\{\alpha_{n}(t)\}\) and \(\{\beta_{n}(t)\}\) with \(\alpha_{0}=\alpha,\beta_{0}=\beta\), which converge to the extremal solutions of problem (1) in the sector \([\beta,\alpha]=\{u\in C^{1}(J,\mathbb{R}):\beta(t)\leq u(t)\leq\alpha(t),t\in J\}\).
Proof
It follows from Theorem 3.2 that both (11) and (12) have a unique solution, respectively. Then we complete the proof by four steps.
Step 1 We prove that \(\beta_{n1}\leq\beta_{n}\) and \(\alpha_{n}\leq \alpha_{n1}\), \(n=1,2,\ldots\) .
By mathematical induction, we obtain the sequence \(\alpha_{n}\) is a nonincreasing sequence. Analogously, we can show \(\beta_{n}\) is a nondecreasing sequence.
Step 2 We show that \(\beta_{1}\leq\alpha_{1}\) if \(\beta\leq\alpha\).
Step 3 We prove that there exists a solution of problem (1) that satisfies \(\beta(t)\leq u(t)\leq\alpha(t)\) in J.
Step 4 We prove that there exist extremal solutions of problem (1) in \([\beta,\alpha]\).
4 Coupled lower and upper solutions
Definition 4.1
Definition 4.2
Definition 4.3
Coupled quasisolution \(\rho,r\in C^{1}(J,\mathbb{R})\) are called coupled minimal and maximal coupled quasisolution of problem (1), if for any coupled quasisolution \(u,v\), we have \(\rho(t)\leq u(t),v(t)\leq r(t)\) on J.
Theorem 4.1
 (H_{4}):

\(\alpha, \beta\in\Omega\) are coupled lower and upper solutions of problem (1) such that \(\beta\leq\alpha\);
 (H_{5}):

the function \(g(u,v)\in C(\mathbb{R}^{2},\mathbb{R})\) is nonincreasing in the second variable andwhere \(M_{2}\geq M_{1}>0\) and \(M_{1}\neq M_{2}e^{\int_{0}^{T}M(s)\,ds}\).$$g(\bar{u},v)g(u,v)\leq M_{1}(\bar{u}u), \quad\textit{for } \beta(0)\leq u(0) \leq \bar{u}(0)\leq\alpha(0), $$
Then there exist monotone sequences \(\{\alpha_{n}(t)\}\) and \(\{\beta _{n}(t)\}\) with \(\alpha_{0}=\alpha,\beta_{0}=\beta\), such that \(\lim_{n\to\infty}\beta_{n}(t)=\rho(t), \lim_{n\to\infty}\alpha _{n}(t)=r(t)\), uniformly and monotonically on J and such that \(\rho, r\) are coupled minimal and maximal quasisolutions of (1) in the sector \([\beta,\alpha]\).
Proof
5 Example
In this section, we give two simple but illustrative examples, thereby validating the proposed theorems.
Example 5.1
Then all conditions of Theorem 3.3 are satisfied. Therefore, via Theorem 3.3, there exist monotone iterative sequences \(\{\alpha_{n}(t)\} ,\{\beta_{n}(t)\}\) which converge uniformly on J to the extremal solutions of (14) in \([\beta_{0},\alpha_{0}]\).
Example 5.2
Then all conditions of Theorem 4.1 are satisfied. So problem (20) has coupled minimal and maximal quasisolutions of (20) in the segment \([\beta_{0},\alpha_{0}]\).
Declarations
Acknowledgements
The authors sincerely thank the reviewers and the editors for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106), the Natural Science Foundation of Hebei Province, China (A2013201232) and the Science and Technology Research Projects of Higher Education Institutions of Hebei Province, China (Z2013038).
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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