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Solvability of n th-order Lipschitz equations with nonlinear three-point boundary conditions
Boundary Value Problems volume 2014, Article number: 239 (2014)
Abstract
In this paper, we investigate the solvability of n th-order Lipschitz equations , , with nonlinear three-point boundary conditions of the form , , , , where , . By using the matching technique together with set-valued function theory, the existence and uniqueness of solutions for the problems are obtained. Meanwhile, as an application of our results, an example is given.
MSC: 34B10, 34B15.
1 Introduction
As is well known, the differential equations with right hand sides satisfying the Lipschitz conditions (Lipschitz equations for short) are important, and thus their solvability has attracted much attention from many researchers. Among a substantial number of works dealing with higher order Lipschitz equations with three-point boundary conditions, we mention [1]–[14] and references therein. Most of these results are obtained via applying control theory methods (Pontryagin maximum principle), matching methods, and topological degree methods etc. To the best of our knowledge, most of the three-point boundary conditions in the above mentioned references are limited to simple boundary conditions.
In 1973, Barr and Sherman [2] showed by the matching technique that the third-order three-point boundary value problem with has a unique solution, under the following four conditions:
-
(A)
is continuous on ;
-
(B)
satisfies the monotonicity conditions, i.e., , implies
and , implies
-
(C)
for any ,
where , , and are nonnegative constants;
(D1) for each ,
where , .
In 1978, Moorti and Garner [12] by using the matching technique showed that BVP (∗) with and has a unique solution, under the conditions (A), (B), (C), and
(D2) for each ,
Since then, many authors improved the condition (D i ), . For example, in [4], Das and Lalli proved that BVP (∗) with has a unique solution, under the conditions of (A), (B), (C), and
(D3) for each ,
In [1], Agarwal showed that BVP (∗) with has a unique solution, under the conditions of (A), (B), (C), and
(D4) for each ,
In [14], Piao and Shi generalized the above results. They not only generalized the simple three boundary conditions to the nonlinear boundary conditions, but also they weakened the monotonicity condition (B) and removed the restriction (D i ) on the length of the interval.
Recently, Pei and Chang [13] generalized the results of Piao and Shi [14].
The purpose of this paper is to study the solvability of n th-order Lipschitz equations with more general nonlinear three-point boundary conditions of the form ()
where .
The paper is organized as follows. In Section 2, as a preliminary, we state some useful results as regards the solvability for the n th-order Lipschitz equation with the nonlinear two-point boundary conditions and a lemma of the differential inequality for n th-order differential equations. In Section 3, by using the matching technique together with set-valued function theory and nested interval theorem, we establish the existence and uniqueness theorems of solutions for BVP (1.1), (1.2). Our results improve and generalize widely the results of [1], [2], [4], [12]–[14].
We remark that the matching technique used in this paper is different from the classical one. In fact, by using the classical matching technique to obtain a matching solution of a three-point boundary value problem, it needs usually four two-point boundary value problems and among them two two-point boundary value problems need to have unique solutions, the other two two-point boundary value problems need to have at most one solution. However, our matching technique needs only two two-point boundary value problems and each of them needs to have at least one solution. For more about the three-point boundary value problems, we refer the readers to the references [15]–[19], with matching techniques, and to [20]–[35], with other techniques.
Throughout this paper, we make the following assumptions:
() is continuous on ;
() If and , , then
Also if and , , then
() For any ,
where , , are nonnegative Lipschitz constants;
() , , are continuously differentiable on , , , , , on , and for any bounded set , , the functions , , are bounded on ;
() The functions , are continuously differentiable on , and for each , , , , on ;
() , on ;
() , on ;
() , , on ;
() , , on .
In the above conditions, δ denotes a constant.
2 Preliminary results
In this section, we introduce some lemmas which will be useful in the proof of our main results.
Consider the following nonlinear two-point boundary value problems for the n th-order differential equation ():
with nonlinear two-point boundary conditions
where .
Let us list the following conditions for convenience.
(H1) is continuous on ;
(H2) for any , if , , then
(H3) for any ,
where , , are nonnegative constants;
() for any ,
where is a nonnegative constant;
(H4) , , are continuously differentiable on and is continuously differentiable on ;
(H5) , on , , , on ;
() , on , on , , , on , on ;
(H6) , on ;
(H7) , , on ;
() , , on ;
(H8) , , , on ;
() , , , on .
In the above conditions, δ denotes a constant.
Now we recall the results [36] of the existence and uniqueness of solutions for BVP (2.1), (2.2) and a lemma for a differential inequality for differential equation (2.1) of the n th order.
Lemma 2.1
(See [36], Theorem 3.1])
Assume that (H1), (H2), (H3), (H4), (H5), and (H8) hold. Then BVP (2.1), (2.2) has at least one solution.
Lemma 2.2
(See [36], Theorem 3.2])
Assume that (H1), (H2), (H3), (H4), (), and () hold. Then BVP (2.1), (2.2) has at least one solution.
Lemma 2.3
(See [36], Theorem 3.3])
Assume that (H1), (H2), (H3), (H4), (H5), (H6), and (H7) hold. Then BVP (2.1), (2.2) has exactly one solution.
Lemma 2.4
(See [36], Theorem 3.4])
Assume that (H1), (H2), (H3), (H4), (), (H6), and () hold. Then BVP (2.1), (2.2) has exactly one solution.
Lemma 2.5
(See [36], Lemma 2.4])
Assume that (H1), (H2), and () hold. Let, be solutions of the differential equation (2.1) on some intervalsatisfying
and
Thenfor.
3 Main results
In order to obtain the existence and uniqueness of solutions for BVP (1.1), (1.2) by using the matching technique, we need first to discuss the existence and uniqueness of solutions for the n th-order Lipschitz equation (1.1) with one of the following sets of two-point boundary conditions:
where .
Let and . Then BVP (1.1), (3.2) becomes an equivalent boundary value problem:
where
This shows that BVP (1.1), (3.2) on the interval can be transformed to the same type as BVP (1.1), (3.1) on the interval .
Lemma 3.1
Suppose that (), (), (), (), (), and () hold. Then each of BVP (1.1), (3.1), BVP (1.1), (3.2), and BVP (1.1), (3.3) has at least one solution.
Proof
It is easy to check that conditions (), (), (), (), (), and () imply conditions (H1), (H2), (H3), (H4), (H5), and (H8) for BVP (1.1), (3.1) as well as conditions (H1), (H2), (H3), (H4), (), and () for BVP (1.1), (3.3), respectively. Hence by Lemma 2.1 and 2.2, each of BVP (1.1), (3.1) and BVP (1.1), (3.3) has at least one solution.
Similarly, by Lemma 2.1 BVP (1.1′), (3.2′) has at least one solution. Hence BVP (1.1), (3.2) has at least one solution. □
Lemma 3.2
Suppose that (), (), (), (), (), and () hold. Then each of BVP (1.1), (3.1), BVP (1.1), (3.2), and BVP (1.1), (3.3) has exactly one solution.
Proof
Similarly to the proof of Lemma 3.1 by Lemma 2.3 and 2.4, the lemma follows. □
In order to prove our main results, we introduce some concepts as follows.
Definition 3.1
A set-valued function is said to be upper semi-continuous at if for any open set U with , there exists a neighborhood V of such that .
Definition 3.2
Let and be subsets of ℝ.
-
(1)
If for any and , holds, then we denote and say that is not greater than .
-
(2)
If for any and , holds, then we denote and say that is less than .
Definition 3.3
-
(1)
Define a set-valued function by
where ;
-
(2)
Define a set-valued function by
where .
Lemma 3.3
-
(1)
Suppose that (), (), (), (), () and () hold. If , then
-
(2)
Suppose that (), (), (), (), () and () hold. If , then
Proof
-
(1)
Let us show first the inequality with respect to . To do this, we take any , . Suppose that is false, i.e., . Then, for each , from (3.1) we have by the mean value theorem
and . By () we can inductively show that, for each , . Consequently by Lemma 2.5 we have for . Furthermore one can inductively get for each the result for . Now by () and () we get
This is a contradiction to (3.1). Thus we conclude that
i.e., for .
By similar arguments, we can show the inequality for .
-
(2)
Since () and () imply (), for any and , we have by (1), . Suppose . Then both and are solutions of BVP (1.1), (3.3) with . By Lemma 3.2 of the uniqueness, we conclude for , which implies
This is a contradiction. Thus , i.e., for . □
Lemma 3.4
Suppose that (), (), (), (), () and () (or ()) hold. Then, for any, bothandare compact and connected subsets of ℝ.
Proof
If (), hold, then by Lemma 3.2, each of BVP (1.1), (3.1) and BVP (1.1), (3.2) has exactly one solution. Consequently both and are single-point sets. Hence the theorem holds.
Now let (), hold. First, we prove that is an interval. To do this, let us take any with . We need to show that if , then . By (), it is easy to see inductively that , , and for any fixed there exist unique , , such that
Now let be the unique solution of (1.1) which satisfies the initial conditions , , where . Then by Lemma 2.5, for . Furthermore we have for , . Similarly we can show that for , . Hence by (), we have
and
Thus
Hence satisfies the boundary condition (3.1), which implies that is the solution of BVP (1.1), (3.1), and then .
Next, we show that is closed. To do this, for any sequence in with as , we need to show . By the definition of , corresponding to there exists a sequence of solutions of BVP (1.1), (3.1) such that . By (), it is easy to see that, for each , there exist , , such that
Furthermore we have, by (),
Now let us show that the sequences , , are convergent. In fact, when , for any positive integers we have
Consequently by (), we get
Since is a Cauchy sequence, so is the sequence . Hence converges to a number . Similarly we can show inductively that, for each , the sequence converges to a number .
We note that , . Then by Kamke’s standard convergence theorem [37], there exists a solution of (1.1) defined on satisfying initial conditions , , and there exists a subsequence of such that, for each , the sequence uniformly converges to on . It is easy to see that is the solution of BVP (1.1), (3.1). Hence .
Finally, we show that is bounded. To do this, we take with . Then from Lemma 3.3, we have
This implies the boundedness of .
By a similar argument for BVP (1.1′), (3.2′), we can show that is also a compact and connected subset of ℝ. □
Lemma 3.5
Suppose that (), (), (), (), (), and () hold. Then there exist sequencesandof solutions of BVP (1.1), (3.1) withand of BVP (1.1), (3.1) with, respectively, for which
Proof
Let us take a sequence with . Then, by Lemma 3.1, BVP (1.1), (3.3) with has a solution, denoted by . It is easy to see that is the solution of BVP (1.1), (3.1) with . Let and let . Then and
Similarly one can show that there exists , for which
□
Lemma 3.6
Suppose that (), (), (), (), () and () hold. Then
-
(1)
for any and , there exists such that if , then, for any , there exists satisfying ;
-
(2)
for any and , there exists such that if , then, for any , there exists satisfying .
Proof
Let us prove only (1), since (2) can be shown similarly.
Suppose the conclusion (1) is false. Then there exist and such that, for each , , there exist and such that, for any ,
Since , , we have by Lemma 3.3
Thus is bounded. Without loss of generality, we may assume that as . For any positive integers , we have, for each ,
Hence, for each , by () we have
Since and are convergent, is a Cauchy sequence, and thus is convergent. Similarly one can show inductively that, for each , is also convergent. Set , , where . Then by Kamke’s convergence theorem, there exists a solution of (1.1) defined on satisfying the initial conditions , and there exists a subsequence of such that, for each , the sequence uniformly converges to on . It is easy to see that is the solution of BVP (1.1), (3.1) with . Consequently , and hence
which is a contradiction to . Thus (1) holds. □
Lemma 3.7
Suppose that (), (), (), (), (), and () hold. Then bothandare upper semi-continuous on ℝ.
Proof
For any , let us show is upper semi-continuous at .
From Lemma 3.4, is a compact and connected subset of ℝ. Hence without loss of generality, we may assume that
Take any open set U with . Then there exists such that
Thus from Lemma 3.6, there exists such that if , then, for any , there exists for which
and so . Hence is upper semi-continuous at .
The upper semi-continuity of on ℝ can be shown similarly. □
Theorem 3.1
Suppose that (), (), (), (), (), and () hold. Then BVP (1.1), (1.2) has at least one solution.
Proof
We consider two cases as follows.
Case 1. Suppose there exists such that . Then BVP (1.1), (3.1) with and BVP (1.1), (3.2) with have solutions and , respectively, such that . Since , by () it is easy to see that , . Hence, if we let
then is a solution of BVP (1.1), (1.2).
Case 2. Suppose for any , . Then by Lemma 3.3 and 3.5, there exist and with , such that
In fact, let us take any and . Then by Lemma 3.5, there exists some such that . Take with . Then by Lemma 3.3, we have
Also by Lemma 3.5, there exists some such that
Again take . Then by Lemma 3.3, we have
and
Now we apply a bisection argument as follows. Set , . Then we have two cases, i.e.,
If , set and . If , set and . In summary, there exist such that
By continuing this bisection process, we can get sequences and with , , such that
Hence by the nested interval theorem, there uniquely exists such that , actually , squeeze to the common limit ξ.
Suppose . Then since both and are compact and connected subsets of ℝ and , there exist two open interval and such that , and . Consequently . Since both and are upper semi-continuous on ℝ by Lemma 3.7, there exists such that if then and , and thus . On the other hand since as , there exists such that , consequently , which is a contradiction.
If , then we can similarly obtain a contradiction. Hence the case 2 cannot occur. This completes the proof of the theorem. □
Theorem 3.2
Suppose that (), (), (), (), (), and () hold. Then BVP (1.1), (1.2) has exactly one solution.
Proof
Since () and () imply (), by Theorem 3.1, BVP (1.1), (1.2) has at least one solution.
Now we need to show the uniqueness. By Theorem 3.1, BVP (1.1), (1.2) has a solution , for which we denote
Let be any solution of BVP (1.1), (1.2), and let for , for and . Then and are the solutions of BVP (1.1), (3.1) with and BVP (1.1), (3.2) with , respectively.
If , then by Lemma 3.3 we have
which is a contradiction.
If , then by Lemma 3.3 we have
which is also a contradiction. Hence . Consequently by Lemma 3.2, we get for and for . Thus on . This completes the proof of the theorem. □
Remark 3.1
Theorem 3.2 includes the results of [1], [2], [4], [12]–[14] as particular cases.
It is easy to see that the linear boundary conditions in the next corollary satisfy (), (), and ().
Corollary 3.1
Suppose that (), (), and () hold. Suppose further that, , ; , , , , ; , , . Then, for any, , the three-point boundary value problem of (1.1) with linear boundary conditions
has exactly one solution.
By using the transformations and , from Theorem 3.1 we can easily obtain the following.
Theorem 3.3
Suppose that (), (), (), (), (), and () hold. Then BVP (1.1), (1.2) has at least one solution.
Similarly to the proof of Theorem 3.2, from Theorem 3.3 we can get the following.
Theorem 3.4
Suppose that (), (), (), (), (), and () hold. Then BVP (1.1), (1.2) has exactly one solution.
It is easy to see that the linear boundary conditions in the next corollary satisfy (), (), and ().
Corollary 3.2
Suppose that (), (), and () hold. Suppose further that, , ; , , , , ; , , . Then, for any, , the three-point boundary value problems of (1.1) with linear boundary conditions
has exactly one solution.
Finally, as an application, we give an example to demonstrate our results.
Example 3.1
Consider a third-order three-point boundary value problem
where , , are constants.
Let
Then it is easy to check that the assumptions (), (), and () are satisfied. Hence from either Corollary 3.1 or Corollary 3.2, BVP (3.4), (3.5) has exactly one solution under either of the following conditions:
(i):, , ;
(ii):, , ;
(iii):, , ,
or the following conditions:
(i′):, , ;
(ii′):, , ;
(iii′):, , .
We note that the results of [1], [2], [4], [12]–[14] cannot guarantee that the above third-order three-point boundary value problem has a unique solution, unless .
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Pei, M., Chang, S.K. Solvability of n th-order Lipschitz equations with nonlinear three-point boundary conditions. Bound Value Probl 2014, 239 (2014). https://doi.org/10.1186/s13661-014-0239-7
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DOI: https://doi.org/10.1186/s13661-014-0239-7