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The unique solution for periodic differential equations with upper and lower solutions in reverse order
Boundary Value Problems volume 2013, Article number: 88 (2013)
Abstract
In this paper, we obtain the unique solution for the following periodic problem
where g:[a,b]\times {\mathbb{R}}^{2}\to \mathbb{R} is a continuous function, by constructing an auxiliary system with bounded solutions. The upper and lower solution method and an antimaximum principle are employed to establish the monotone iterative sequences and obtain the extremal solutions for the auxiliary system.
MSC:34B15, 34C25.
1 Introduction
This paper is concerned with the unique solution to the following periodic problem:
where g:[a,b]\times {\mathbb{R}}^{2}\to \mathbb{R} is a continuous function.
It is well known that the upper and lower solution method together with the iterative technique is a powerful tool for proving the existence results for boundary value problems (see [1–12] and the references therein). Recently, the case when the upper solution and the lower solution are in the reversed order has received some attention (see [2–9] and the references therein). The monotone approximation method can be used in the case the lower and upper solutions are in the reversed order \beta \le \alpha. This method works for any boundary value problem such that a uniform antimaximum principle holds. This is the case for the Neumann and periodic problems.
For example, Cabada et al. in [3] used iteration schemes based on problems like
discussed a Neumann boundary value problem
In [4], Torres and Zhang considered a kind of 2πperiodic boundary value problem. The strategy of this paper was to exploit an antimaximum principle for the linear equation in order to construct a monotone approximation scheme converging to the solution.
In [8], Zuo et al. further developed the monotone method and invested the Tperiodic solution of
They used the monotone iterative technique with upper and lower solution in reversed order to define two sequences that converge uniformly to extremal solution of (1.2). However, they did not construct the explicit expression of the monotone iterative sequences.
We have investigated the periodic problems (1.1) in [13] by introducing the following auxiliary periodic system:
where f:[a,b]\times {\mathbb{R}}^{3}\to \mathbb{R} is a {L}^{1}Carathéodory function, and f(t,u,w,u)\equiv g(t,u,w).
Motivated by the above works, we are mainly concerned with the periodic problem (1.1). One might have noticed that computing the iterative sequences \{{\alpha}_{n}\} and \{{\beta}_{n}\} can be a difficult work in those existed papers. In this paper, by adopting an auxiliary periodic system and an antimaximum principle which are different from that in the reference [13], we also obtain the unique solution for the problem (1.1).
Here, we introduce the following auxiliary periodic system:
where f:[a,b]\times {\mathbb{R}}^{3}\to \mathbb{R} is a continuous function and f(t,u,w,u)=g(t,u,w), the constant K is defined in Section 3.
The organization of this paper is as follows. We shall introduce some useful lemmas in Section 2. The main results and their proof about auxiliary periodic (1.3) are given in Section 3. Section 4 gives the unique solution of (1.1) and an example is introduced.
2 Related lemmas
For the convenience of the reader, we give some lemmas which will be used in the next section. Firstly, recall the continuation theorem of Mawhin.
Lemma 2.1 [12]
Let X and Y be two Banach spaces with norms {\parallel \cdot \parallel}_{X} and {\parallel \cdot \parallel}_{Y}, respectively, and \mathrm{\Omega}\subset X an open and bounded set. Suppose L:X\cap domL\to Y is a Fredholm operator of index zero and {N}_{\lambda}:\overline{\mathrm{\Omega}}\to Y, \lambda \in [0,1] is Lcompact. In addition, if
(A1) Lx\ne \lambda Nx for \lambda \in (0,1), x\in (domL\setminus kerL)\cap \partial \mathrm{\Omega};
(A2) Nx\notin ImL for x\in kerL\cap \partial \mathrm{\Omega};
(A3) deg\{JQN{}_{\overline{\mathrm{\Omega}}\cap kerL},\mathrm{\Omega}\cap kerL,0\}\ne 0, where Q:Y\to Y is a projection such that ImL=kerQ and J:ImQ\to kerL is a homeomorphism.
Then the abstract equation Lx=Nx has at least one solution in \overline{\mathrm{\Omega}}.
Secondly, to prove the validity of the monotone iterative technique, we present the antimaximum comparison principle as follows.
Lemma 2.2 [1]
Let p\in \mathbb{R}, q>0, \sigma \in {L}^{1}(a,b) and A\in \mathbb{R}. Suppose u is a solution of
for some A\ge 0 and \sigma \ge 0. Then u\ge 0 provided that p\ge 0 and \theta (q,p)\ge \frac{ba}{2}, where
Finally, recall some classical integral inequalities.
Lemma 2.3 [14]
If u\in {C}^{1}[a,b] is such that \overline{u}=0, where \overline{u}=\frac{1}{ba}{\int}_{a}^{b}u(s)\phantom{\rule{0.2em}{0ex}}ds. Then
3 Extremal solution for (1.3)
Functions \alpha ,\beta \in {C}^{2}[a,b] are said to be a pair of lower and upper solutions to (1.3) if they satisfy
For given \alpha ,\beta \in C[a,b], we shall write \beta \le \alpha if \beta (t)\le \alpha (t) for all t\in [a,b]. In such a case, we shall denote
Assume that there exist constants N>0, M,L\ge 0 and K>0 such that LK\ge 0, we need the following hypotheses:
(H1) For given \alpha ,\beta \in C[a,b] with \beta \le \alpha,
(H2) For \beta \le {x}_{2}\le {x}_{1}\le \alpha, \beta \le {z}_{1}\le {z}_{2}\le \alpha,
(H3) 1\frac{ba}{\pi}N{(\frac{ba}{\pi})}^{2}K>0;
(H4) ba\le 2\theta (K,\frac{N}{2}), where θ is defined in (2.2).
In order to develop the monotone iterative technique for (1.3), we shall first consider the existence of solutions for the following periodic problems:
and
for each fixed \xi ,\eta \in [\beta ,\alpha ], here
Lemma 3.1 Assume that (H1), (H3) and (H4) hold. Then the problems (3.2) and (3.3) have unique solutions.
Proof In order to apply Lemma 2.1 to (3.2), we choose the Banach spaces X={C}^{1}[a,b] with the norm \parallel u\parallel =max\{{\parallel u\parallel}_{\mathrm{\infty}},{\parallel {u}^{\prime}\parallel}_{\mathrm{\infty}}\}, where {\parallel u\parallel}_{\mathrm{\infty}}={max}_{t\in [a,b]}u(t), and Y=C[a,b]. Define the linear operator
and the nonlinear operator
here,
Let
where \overline{u}=\frac{1}{ba}{\int}_{a}^{b}u(s)\phantom{\rule{0.2em}{0ex}}ds. Then, kerL=\{c\in domL:c\in \mathbb{R}\} and
Now, ImL is closed in Y. It is easy to see that ImL=kerQ and dimkerL=1=dimImQ=codimImL. Thus, L is a linear Fredholm operator with index zero.
Furthermore, let {L}_{p}=L{}_{domL\cap kerP} and {K}_{p}:ImL\to domL\cap kerP denoting the inverse of {L}_{p} be given by
where
Obviously, QN and {K}_{p}(IQ)N are continuous. By ArzelàAscoli theorem, we can show that {K}_{p}(IQ)N(\overline{\mathrm{\Omega}}) is relative compact for any open bounded set \mathrm{\Omega}\subset X. Moreover, QN(\overline{\mathrm{\Omega}}) is bounded. Thus, N is Lcompact on \overline{\mathrm{\Omega}}.
In the following, we complete the remainder proof by four steps.
Step 1. Define {\mathrm{\Omega}}_{1}=\{u\in domL\setminus kerL:Lu=\lambda Nu,\lambda \in (0,1)\}. For any u\in {\mathrm{\Omega}}_{1},
Integrating (3.4) on [a,b] to obtain
Set x=u\overline{u}, then x\in domL, \overline{x}=0, {x}^{\prime}={u}^{\prime}. It follows from (3.4) and (3.5) that
Notice that \overline{x}=0, by Lemma 2.3, we have
In view of (H1), we have
for some constant C. Multiplying (3.6) by x(t) and integrating it on [a,b], we can obtain
Notice the condition (H3), we can find a constant {M}_{1}>0 such that {\parallel {x}^{\prime}\parallel}_{2}\le {M}_{1} and also {\parallel x\parallel}_{\mathrm{\infty}}\le \sqrt{ba}{M}_{1}. Now,
implies
Since u(a)=u(b), there exists {t}_{0}\in (a,b) such that {u}^{\prime}({t}_{0})=0. Then
i.e. {\parallel {u}^{\prime}\parallel}_{\mathrm{\infty}}\le {M}_{3}. Hence, {\mathrm{\Omega}}_{1} is bounded.
Step 2. Let {\mathrm{\Omega}}_{2}=\{u\in kerL:QNx=0\}. For u\in {\mathrm{\Omega}}_{2}, we have u=c\in \mathbb{R}, and
Then
that is, {\mathrm{\Omega}}_{2} is bounded.
Step 3. Let \mathrm{\Omega}\supset {\mathrm{\Omega}}_{1}\cup {\mathrm{\Omega}}_{2} with Ω open and bounded. Clearly, conditions (A1) and (A2) in Lemma 2.1 are satisfied. The remainder is to verify (A3). To this end, we define an isomorphism J:ImQ\to kerL by J=I. Let
i.e.
It is easy to see that H(u,\mu )\ne 0 for (u,\mu )\in (\partial \mathrm{\Omega}\cap kerL)\times [0,1]. Hence,
Lemma 2.1 yields that Lu=Nu has at least one solution.
Step 4. We claim that the problem (3.2) has a unique solution. Suppose to the contrary, that there exist two solutions {u}_{1} and {u}_{2} of (3.2).
Let u={u}_{2}{u}_{1}, by (H1) and (H4),
Applying Lemma 2.2, we have u(t)\ge 0 on [a,b]. That is, {u}_{2}\ge {u}_{1}. Also, we can obtain {u}_{1}\ge {u}_{2} by the same method. Thus, {u}_{1}\equiv {u}_{2}.
By a similar argument, we can obtain the existence of unique solution for the problem (3.3). □
Now, we investigate the extremal solution of the periodic system (1.3). For given \alpha ,\beta \in X, we consider the approximation schemes
Now, we give the main result for (1.3).
Theorem 3.1 Assume that α, β are a pair of lower and upper solutions of (1.3) defined by (3.1) such that \beta \le \alpha. Suppose that (H1)(H4) hold.
Then there exist two monotone iterative sequences \{{\alpha}_{n}\},\{{\beta}_{n}\}\subset [\beta ,\alpha ], nonincreasing and nondecreasing, respectively, defined by (3.7) such that {\alpha}_{n}\to {u}^{\ast}, {\beta}_{n}\to {v}^{\ast} (n\to \mathrm{\infty}) uniformly on t\in [a,b], and the pair ({u}^{\ast},{v}^{\ast}) is a solution of (1.3) satisfying
Moreover, any solution (u,v) of (1.3) with \beta \le u\le \alpha, \beta \le v\le \alpha is such that
Proof Let R=\{(u,v)\in C[a,b]\times C[a,b]:u\ge 0,v\le 0\} be an ordered normal cone, the order is defined by
for any ({u}_{1},{v}_{1}),({u}_{2},{v}_{2})\in R. We define the operator T:[\beta ,\alpha ]\times [\beta ,\alpha ]\to X\times X by T(\xi ,\eta )=(u,v), where u and v are the unique solutions of problems (3.2) and (3.3), respectively, with given \xi ,\eta \in [\beta ,\alpha ]. Firstly, we claim that the mapping T has the following properties:

(i)
(\beta ,\alpha )\le T(\alpha ,\beta )\le (\alpha ,\beta );

(ii)
T({\xi}_{1},{\eta}_{1})\le T({\xi}_{2},{\eta}_{2}), when ({\xi}_{1},{\eta}_{1})\le ({\xi}_{2},{\eta}_{2}), {\xi}_{i},{\eta}_{i}\in [\beta ,\alpha ], i=1,2.
We start with (i). Set ({\alpha}_{1},{\beta}_{1})=T(\alpha ,\beta ), w=\alpha {\alpha}_{1}. From (H1) and (H2), we have
Obviously, w(a)=w(b) and {w}^{\prime}(a)\ge {w}^{\prime}(b). By Lemma 2.2, we have w\ge 0, and so \alpha \ge {\alpha}_{1}. A similar argument shows that {\beta}_{1}\ge \beta. Thus, T(\alpha ,\beta )\le (\alpha ,\beta ). Set w={\alpha}_{1}\beta, then
Lemma 2.2 implies that {\alpha}_{1}\ge \beta. Similarly, we can check that \alpha \ge {\beta}_{1}. Hence, (\beta ,\alpha )\le T(\alpha ,\beta ).
Now, we prove (ii). Let T({\xi}_{i},{\eta}_{i})=({u}_{i},{v}_{i}), i=1,2. Set w={u}_{2}{u}_{1},
Applying Lemma 2.2, we get {u}_{2}\ge {u}_{1}. A similar argument shows that {v}_{1}\ge {v}_{2}. Thus, T({\xi}_{2},{\eta}_{2})\ge T({\xi}_{1},{\eta}_{1}).
To prove the sequence \{{\alpha}_{n}\} and \{{\beta}_{n}\} are bounded, we define \{({\alpha}_{n},{\beta}_{n})\} by ({\alpha}_{n+1},{\beta}_{n+1})=T({\alpha}_{n},{\beta}_{n}). Notice that \{({\alpha}_{n},{\beta}_{n})\} is nonincreasing and \{({\beta}_{n},{\alpha}_{n})\} is nondecreasing. It is clear that \{{\alpha}_{n}\} and \{{\beta}_{n}\} are bounded. Hence, there exists a constant {C}_{1}>0 such that max\{{\parallel {\alpha}_{n}\parallel}_{\mathrm{\infty}},{\parallel {\beta}_{n}\parallel}_{\mathrm{\infty}}\}\le {C}_{1} for all n\in \mathbb{N}, and
Essentially, the same as in the proof of Lemma 3.1 guarantees that there exists a constant {C}_{2}>0 such that max\{{\parallel {\alpha}_{n}^{\prime}\parallel}_{\mathrm{\infty}},{\parallel {\beta}_{n}^{\prime}\parallel}_{\mathrm{\infty}}\}\le {C}_{2}. This shows that \{({\alpha}_{n},{\beta}_{n})\} converges in X, i.e. there exists ({u}^{\ast},{v}^{\ast})\in X so that ({\alpha}_{n},{\beta}_{n})\to ({u}^{\ast},{v}^{\ast}), {\beta}_{0}\le {u}^{\ast}\le {\alpha}_{0}, {\beta}_{0}\le {v}^{\ast}\le {\alpha}_{0}, and also {\beta}_{0}\le {v}^{\ast}\le {u}^{\ast}\le {\alpha}_{0} from (i).
Finally, we will prove that any other solution (u,v)\in X of (1.3) such that ({\beta}_{0},{\alpha}_{0})\le (u,v)\le ({\alpha}_{0},{\beta}_{0}) satisfies
We can proceed as in the proof of (i) to show that ({\beta}_{n},{\alpha}_{n})\le (u,v)\le ({\alpha}_{n},{\beta}_{n}), i.e. {\beta}_{n}\le u\le {\alpha}_{n} and {\beta}_{n}\le v\le {\alpha}_{n}, n=0,1,2,\dots . Hence, {v}^{\ast}\le u\le {u}^{\ast} and {v}^{\ast}\le v\le {u}^{\ast}, i.e. ({v}^{\ast},{u}^{\ast})\le (u,v)\le ({u}^{\ast},{v}^{\ast}). □
Remark 3.1 Comparing with the main result (Theorem 3.2) in [8], we construct the explicit expression of the monotone iterative sequences \{{\alpha}_{n}\} and \{{\beta}_{n}\} by (3.7) in Theorem 3.1, while reference [8] didn’t do so.
Remark 3.2 Notice that if (u,u) is a solution of the auxiliary system (1.3), then u is a solution of the given problem (1.1) under the assumption f(t,u,w,u)\equiv g(t,u,w). Thus, {u}^{\ast} and {v}^{\ast} are bounds on solutions of (1.1).
If the bounds {u}^{\ast} and {v}^{\ast} obtained in Theorem 3.1 are equal, then {u}^{\ast}={v}^{\ast} is a unique solution of the problem (1.1). In particular, under appropriate assumptions, this theorem provides an approximation scheme to the unique solution of (1.1).
4 Unique solution for (1.1)
Theorem 4.1 Suppose that the conditions (H1)(H4) hold. Then the periodic problem (1.1) has a unique solution provided that M+L>2K+\frac{{N}^{2}}{4}.
Proof If {u}^{\ast}>{v}^{\ast}, we compute
On the other hand,
If we denote {\gamma}_{1}={({\int}_{a}^{b}{{u}^{\ast}(t){v}^{\ast}(t)}^{2}\phantom{\rule{0.2em}{0ex}}dt)}^{\frac{1}{2}} and {\gamma}_{2}={({\int}_{a}^{b}{{({u}^{\ast}(t))}^{\prime}{({v}^{\ast}(t))}^{\prime}}^{2}\phantom{\rule{0.2em}{0ex}}dt)}^{\frac{1}{2}}, then
which is a contradiction. □
Example 4.1
Consider the problem
Corresponding to the problem (1.1), we take a=0, b=2\pi and
Let
Consider the following system:
with K=\frac{{\pi}^{2}}{400} corresponding to (1.3). Clearly, the system (4.3) has \alpha =\frac{400}{{\pi}^{2}} and \beta =\frac{400}{{\pi}^{2}} as the lower and upper solutions respectively according to the definition by (3.1). Take M=\frac{2{\pi}^{2}}{500}, N=\frac{\pi}{10\sqrt{2}} and L=K=\frac{{\pi}^{2}}{400}. It is easy to check that the conditions (H1)(H3) hold. Since K>\frac{{N}^{2}}{4},
Thus, (H4) is satisfied. Thanks to Theorems 3.1, the system (4.3) has a solution ({u}^{\ast},{v}^{\ast}). In view of the values of N, K, M, L, we have M+L>2K+\frac{{N}^{2}}{4}. Hence, Theorem 4.1 implies that {u}^{\ast}={v}^{\ast}, that is, the problem (4.1) has a unique solution.
Remark 4.1 When \alpha =\frac{400}{{\pi}^{2}}, \beta =\frac{400}{{\pi}^{2}}, and g is given by (4.3), Example 4.1 does not satisfy the conditions imposed in [[1], Theorem 3.3] because α, β are not a pair of lower and upper solutions of Eq. (3.7) in [1]. In fact, our definitions of the lower and upper α, β, and the monotone iterative sequences \{{\alpha}_{n}\}, \{{\beta}_{n}\} are all different from [[1], Theorem 3.3]. In addition, the ultimate goal of this paper is the unique solution of (1.1) by constructing an auxiliary system (1.3), which is one of the highlights of this article.
References
Coster CD, Habets P Mathematics in Science and Engineering 205. In TwoPoint Boundary Value Problems: Lower and Upper Solutions. Elsevier, Amsterdam; 2006.
Cabada A, Habets P, Pouso RL: Optimal existence conditions for p Laplacian equations with upper and lower solutions in the reverse order. J. Differ. Equ. 2000, 166: 385401. 10.1006/jdeq.2000.3803
Cabada A, Habets P, Lois S: Monotone method for the Neumann problem with lower and upper solutions in the reverse order. Appl. Math. Comput. 2001, 117: 114. 10.1016/S00963003(99)001496
Torres PJ, Zhang M: A monotone iterative scheme for a nonlinear second order equation based on a generalized antimaximum principle. Math. Nachr. 2003, 251: 101107. 10.1002/mana.200310033
Jiang D, Yang Y, Chu J, O’Regan D: The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order. Nonlinear Anal. 2007, 67: 28152828. 10.1016/j.na.2006.09.042
Cabada A, Grossinho MR, Minhos F: Extremal solutions for thirdorder nonlinear problems with upper and lower solutions in reversed order. Nonlinear Anal. 2005, 62: 11091121. 10.1016/j.na.2005.04.023
Li F, Jia M, Liu X, Li C, Li G: Existence and uniqueness of solutions of secondorder threepoint boundary value problems with upper and lower solutions in the reversed order. Nonlinear Anal. 2008, 68: 23812388. 10.1016/j.na.2007.01.065
Zuo WJ, Wang PX, Jiang DQ: Periodic boundary value problems and periodic solutions of second order FDE with upper and lower solutions. Southeast Asian Bull. Math. 2006, 30: 199207.
Wang W, Yang X, Shen J: Boundary value problems involving upper and lower solutions in reverse order. J. Comput. Appl. Math. 2009, 230: 17. 10.1016/j.cam.2008.10.040
Khan RA, Webb JRL: Existence of at least three solutions of a secondorder threepoint boundary value problem. Nonlinear Anal. 2006, 64: 13561366. 10.1016/j.na.2005.06.040
Khan RA, Webb JRL: Existence of at least three solutions of nonlinear three point boundary value problems with superquadratic growth. J. Math. Anal. Appl. 2007, 328: 690698. 10.1016/j.jmaa.2006.05.074
Khan RA, Webb JRL: Existence of at least two solutions of second order nonlinear three point boundary value problems. Dyn. Syst. Appl. 2006, 15: 119132.
Yang A, Ge W: The monotone method for periodic differential equations with the non wellordered upper and lower solutions. J. Comput. Appl. Math. 2009, 232: 632637. 10.1016/j.cam.2009.07.002
Mawhin J NSFCBMS Regional Conference Series in Mathematics. In Topological Degree Methods in Nonlinear Boundary Value Problems. Am. Math. Soc., Providence; 1979.
Acknowledgements
The authors would like to express their sincere gratitude to the anonymous reviewers for their insightful comments and constructive suggestions, which helped to enrich the content and considerably improved the presentation of this paper. This research is supported by the National Natural Sciences Foundation of People’s Republic of China (Grant No. 11226148 and No. 61273016), the Scientific Research Foundation of Zhejiang Province (Grant No. LY12F05006), and the Education Department Foundation of Zhejiang Province (Grant No. Y201121906).
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Yang, A., Wang, H. & Wang, D. The unique solution for periodic differential equations with upper and lower solutions in reverse order. Bound Value Probl 2013, 88 (2013). https://doi.org/10.1186/16872770201388
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DOI: https://doi.org/10.1186/16872770201388
Keywords
 reversed order upper and lower solutions
 monotone iterative method
 antimaximum principle
 unique solution