Singularly perturbed second order semilinear boundary value problems with interface conditions
- Hongxu Lin1 and
- Feng Xie1Email author
https://doi.org/10.1186/s13661-015-0309-5
© Lin and Xie; licensee Springer. 2015
Received: 14 October 2014
Accepted: 19 February 2015
Published: 3 March 2015
Abstract
In this paper we study a class of singularly perturbed interface boundary value problems with discontinuous source terms. We first establish a lemma of lower-upper solutions by using the Schauder fixed point theorem. By the method of boundary functions and the lemma of lower-upper solutions we obtain the existence, asymptotic estimates, and uniqueness of the solution with boundary and interior layers for the proposed problem.
Keywords
MSC
1 Introduction
In this paper, we are devoted to the study of the existence, uniqueness, and asymptotics of the singularly perturbed interface boundary value problem (1.1)-(1.3), whose solution exhibits an interior layer due to the discontinuity of the source term. By using the Schauder fixed point theorem, we first establish a lower and upper solutions lemma which is an extension of classical theory of lower and upper solutions (see [9], for instance). By the method of boundary functions (see [10, 11], for example) and the lemma of lower-upper solutions we obtain the existence, asymptotic estimates, and uniqueness of the solution with boundary and interior layers for the proposed problem.
The remainder of this paper is organized as follows. In Section 2 we establish the lemma of lower-upper solutions for a class of two-point boundary value problems with interface conditions by the Schauder fixed point theorem, which will be used to prove our main result. With the asymptotic expansions and the lemma of lower and upper solutions established in Section 2, the asymptotic estimates, existence, and uniqueness of the solution for the problem (1.1)-(1.3) are obtained in Section 3.
2 Basic lemmas
It is easily verified that \(Q^{1}[a,b]\) is a Banach space endowed with the norm \(\|u\|=\|u\|_{\infty}+\|u'\|_{\infty}\). The following lemma is a generalized Arzelà-Ascoli theorem on families of functions in \(Q^{1}[a,b]\).
Lemma 2.1
Assume that a bounded set E in \(Q^{1}[a,b]\) is piecewise equicontinuous, that is, \(E|_{[a,d)}=\{u|_{[a,d)}: u\in E\}\) is equicontinuous on \([a,d)\) and \(E|_{(d,b]}=\{u|_{(d,b]}: u\in E\}\) is equicontinuous on \((d,b]\). Then E is a relatively compact subset of \(Q^{1}[a,b]\).
Proof
The proof follows almost the same lines as that of the classical Arzelà-Ascoli theorem (see, for instance, [12]), noting that for \(u\in Q^{1}[a,b]\), u, and \(u'\) both have left and right limits at \(x=d\). Thus, details are omitted here. □
Definition 2.1
Lemma 2.2
Proof
Observe that \(f(x,\gamma(x,u))-\gamma(x,u): ([a,d)\cup (d,b] )\times\mathbb{R}\rightarrow\mathbb{R}\) is uniformly bounded in \(u\in Q^{1}[a,b]\). It follows that from Lemma 2.1 that the set \(T (Q^{1}[a,b] )\) is a relatively compact subset of \(Q^{1}[a,b]\). Moreover, T is continuous. Hence, it follows from the Schauder fixed point theorem that T has at least one fixed point \(u(x)\in Q^{1}[a,b]\).
We are now ready to prove that each solution \(u(x)\) of the problem (2.6)-(2.8) satisfies \(\alpha(x)\leq u(x)\leq\beta (x)\) for \(x\in[a,d)\cup(d,b]\).
In a similar way, we can prove that \(\alpha(x)\leq u(x)\) for all \(x\in [a,d)\cup(d,b]\).
Therefore, the solution of (2.6)-(2.8) is also that of (2.1)-(2.3) and satisfies \(\alpha(x)\leq u(x)\leq\beta (x)\) for \(x\in[a,d)\cup(d,b]\). The proof of the lemma is completed. □
Lemma 2.3
Assume that function \(f(x,u)\) is given in (1.1)-(1.3), and \(f(x,u)\) is strictly increasing with respect to u. Then the problem (2.1)-(2.3) has at most one solution.
Proof
3 Main results
In this section, we are interested in the asymptotic behavior of solution with respect to the small parameter ε, as well as the existence and uniqueness for the problem (1.1)-(1.3). For the sake of simplicity, we only consider the approximation of zero order.
- (H1)The functions \(f_{1}\in C^{2} ([a,d]\times \mathbb{R} )\), \(f_{2}\in C^{2} ([d,b]\times\mathbb{R} )\), and$$\begin{aligned}& f_{1}(d,u)\neq f_{2}(d,u), \quad \mbox{for } u\in\mathbb{R}, \\& \frac{\partial f_{1}}{\partial u}(x,u)\geq\sigma_{0}>0,\quad \mbox {for } (x,u) \in[a,d]\times\mathbb{R}, \\& \frac{\partial f_{2}}{\partial u}(x,u)\geq\sigma_{0}>0,\quad \mbox {for } (x,u) \in[d,b]\times\mathbb{R}. \end{aligned}$$
- (H2)The reduced problem \(f(x,u)=0\) has a solutionsuch that \(\varphi(x)\in C^{2}[a,d]\) and \(\psi(x)\in C^{2}[d,b]\).$$ u(x)=\left \{ \begin{array}{l@{\quad}l} \varphi(x), &x\in[a,d), \\ \psi(x), &x\in(d,b], \end{array} \right . $$
The following two lemmas are concerned with the asymptotic behavior of the boundary layer terms for the left problem, whose proofs are essentially similar to that of Lemma 3.2 in [13] and thus are omitted here.
Lemma 3.1
Lemma 3.2
- (H3)
Likewise, \(\gamma_{i}^{(\pm)}\) (\(i\geq1\)) can be determined recursively.
Thus, we obtain the formal asymptotic solution of the problem (1.1)-(1.3). Now we are in a position to state our main result.
Theorem 3.1
Proof
In order to prove our main result we need to construct suitable upper and lower solutions. To this end, we need to make some necessary preparations.
- (i)
\(Q_{1}^{(-)}(\tau)\), \(V_{1}^{(-)}(\eta)\), \(Q_{1}^{(+)}(\eta)\) and \(V_{1}^{(+)}(\xi )\) are nonnegative functions;
- (ii)$$ {\biggl.\frac{dV_{1}^{(-)}}{d\eta}\biggr|_{\eta =0}}=-\frac{M}{4\sigma_{0}}< 0,\qquad {\biggl.\frac{dQ_{1}^{(+)}}{d\eta}\biggr|_{\eta=0}}=\frac {M}{4\sigma_{0}}>0; $$
- (iii)\(V_{1}^{(-)}(\eta)\) and \(V_{1}^{(+)}(\xi)\) are solutions of the equationand \(Q_{1}^{(-)}(\tau)\), \(Q_{1}^{(+)}(\eta)\) are solutions of the equation$$- \frac{d^{2}y(t)}{dt^{2}}+\sigma_{0}y(t)+Mt{\mathrm{e}}^{\sqrt {\sigma_{0}}t}=0, $$$$- \frac{d^{2}y(t)}{dt^{2}}+\sigma_{0}y(t)-Mt{\mathrm{e}}^{-\sqrt {\sigma_{0}}t}=0. $$
Declarations
Acknowledgements
The authors are grateful to the referees for their valuable comments. The second author was supported by the Natural Science Foundation of Shanghai (No. 15ZR1400800), in part by the National Natural Science Foundation of China (No. 11371087) and by the Fundamental Research Funds for the Central Universities.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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