Abstract
In this paper, we consider nthorder twopoint right focal boundary value problems
where is a Carathéodory () function and satisfies superlinear growth conditions. The existence and uniqueness of solutions for the above right focal boundary value problems are obtained by LeraySchauder continuation theorem and analytical technique. Meanwhile, as an application of our results, examples are given.
MSC: 34B15.
Keywords:
right focal boundary value problem; LeraySchauder continuation theorem; existence; uniqueness1 Introduction
In this paper, we shall discuss the existence and uniqueness of solutions of right focal boundary value problems for nthorder nonlinear differential equation
subject to the boundary conditions ()
where satisfies the Carathéodory () conditions, that is,
(i) for each , the function is measurable on ;
(ii) for a.e. , the function is continuous on ;
(iii) for each , there exists an such that for a.e. and all with .
As it is well known, the right focal boundary value problems have attracted many scholars’ attention. Among a substantial number of works dealing with right focal boundary value problems, we mention [116,1825].
Recently, using the LeraySchauder continuation theorem, Hopkins and Kosmatov [16] have obtained sufficient conditions for the existence of at least one signchanging solution for thirdorder right focal boundary value problems such as
and
where satisfies the Carathéodory () conditions and the linear growth conditions.
Motivated by [16], in this paper we study the solvability for general nthorder right focal boundary value problems (1.1), (1.2). The existence and uniqueness of signchanging solutions for the problems are obtained by LeraySchauder continuation theorem and analytical technique. We note that the nonlinearity of f in our problem allows up to the superlinear growth conditions.
The rest of this paper is organized as follows. In Section 2, we give some lemmas which help to simplify the proofs of our main results. In Section 3, we discuss the existence and uniqueness of signchanging solutions for nthorder right focal boundary value problems (1.1), (1.2) by LeraySchauder continuation theorem and analytical technique, and give two examples to demonstrate our results. Our results improve and generalize the corresponding results in [16].
2 Preliminary
In this section, we give some lemmas which help to simplify the presentation of our main results.
Let denote the space of absolutely continuous functions on , and denote the Banach space of times continuously differentiable functions defined on with the norm , where . Let be the usual Lebesgue space on with norm , .
For , we introduce the Sobolev space
with the norm . Let us consider a special subspace
Then it is clear that is closed in and hence is itself a Banach space with the norm .
Lemma 2.1 ([21])
Letbe the Green’s function of the differential equationsubject to the boundary conditions (1.2). Then
and
Lemma 2.2Let. Then the solution of the differential equation
subject to the boundary conditions (1.2) satisfies
Proof Firstly, let us show the lemma for case . Since
It follows by Hölder’s inequality that, for each ,
It follows by (2.4) that for ,
For , by Lemma 2.1, is nondecreasing in t, and thus
In summary,
Next, we show the lemma for the case . It is easy to see that for ,
Also by Lemma 2.1, we have for ,
so that for each , is nondecreasing in t, it follows that
Let
Then
Since
in particular
so that is nondecreasing on . Hence by (2.5), we have
In summary,
□
Lemma 2.3 ([17] LeraySchauder continuation theorem)
LetXbe a real Banach space and let Ω be a bounded open neighbourhood of 0 inX. Letbe a completely continuous operator such that for all, and, . Then the operator equation
3 Main results
Now we are ready to establish our existence theorems of solutions for nthorder right focal boundary value problems (1.1), (1.2). The LeraySchauder continuation theorem plays key roles in the proofs.
Theorem 3.1LetsatisfyCarathéodory’s conditions. Suppose that
(i) there exist functions, , and a constantsuch that
(ii)
where the constants, are given in Lemma 2.2;
(iii)
Then BVP (1.1), (1.2) has at least one solution in.
Proof We define a linear mapping , by setting for ,
We also define a nonlinear mapping by setting for ,
Then, we note that N is a bounded continuous mapping by Lebesgue’s dominated convergence theorem. It is easy to see that the linear mapping is a onetoone mapping. Also, let the linear mapping for be defined by
where is the Green’s function of BVP in Lemma 2.1.
Then K satisfies that for , and , and also for , . Furthermore, it follows easily by using ArzelàAscoli theorem that is a completely continuous operator.
Here we also note that is a solution of BVP (1.1), (1.2) if and only if is a solution of the operator equation
which is equivalent to the operator equation
We now apply the LeraySchauder continuation theorem to the operator equation . To do this, it is sufficient to verify that the set of all possible solutions of the family of equations
with boundary conditions
is, a priori, bounded in by a constant independent of .
Suppose is a solution of BVP (3.4), (3.5) for some . Then from (3.4), (3.1) and (2.2) in Lemma 2.2, we obtain
Consequently we obtain
Now we have two cases to consider:
Case 1. . In this case (3.6) becomes , i.e. . Thus from (2.1) in Lemma 2.2, we have that there exists a constant which is independent of such that
Now, let
Then estimate (3.7) show that has no fixed point on ∂Ω. Hence KN has a fixed point in by the LeraySchauder continuation theorem.
Case 2. . When in (3.1), it is easy to see that BVP (1.1), (1.2) has the trivial solution . Thus assume and let , . Then from (3.6), . It is easy to see that has a unique positive solution , say . By (3.3), we have and thus has a minimum positive solution, say which is less than and independent of . Hence it follows that if , then
From (2.1) in Lemma 2.2, we get
Now, we let
where . Then estimates (3.8) and (3.9) show that has no fixed point on ∂Ω. Consequently, KN has a fixed point in by the LeraySchauder continuation theorem. This completes the proof of the theorem. □
Corollary 3.1Let conditions (i) and (ii) of Theorem 3.1 hold. Iforis small enough, then BVP (1.1), (1.2) has at least one solution in.
Corollary 3.2Let conditions (i) and (ii) of Theorem 3.1 hold. Ifis small enough, then BVP (1.1), (1.2) has at least one solution in.
Remark 3.1 Theorem 3.13.4 in [16] are special cases of above Theorem 3.1.
Next, we give some results on the uniqueness of solutions for BVP (1.1), (1.2).
Theorem 3.2LetsatisfyCarathéodory’s conditions. Suppose that
(i) there exist functions, , and a constantsuch that
(ii)
where the constants, are given in Lemma 2.2;
(iii)
Then BVP (1.1), (1.2) has at least one solutionand in particular has at most one solutionwith.
Proof We note that assumption (3.10) implies
for a.e. and all . Accordingly from Theorem 3.1, BVP (1.1), (1.2) has at least one solution in .
Now, suppose that , are two solutions of BVP (1.1), (1.2) with , . Let . Then satisfies the boundary condition (1.2) and
Similarly to the proof of Theorem 3.1, we can show easily that
which gives
Now consider two cases. If , then from (3.13). Since , we have on , i.e., on .
If , let . Then from (3.13). It follows that and on . Since , we get . Consequently, on . This completes the proof of the theorem. □
Corollary 3.3Let conditions (i) and (ii) of Theorem 3.2 hold. If, then BVP (1.1), (1.2) has exactly one solution in.
Finally, we give two examples to which our results can be applicable.
Example 3.1 Consider the boundary value problem
Let . Then it is easy to see that f satisfies Carathéodory’s conditions. By the inequality for any with and , we get
Let , , , , , , . Then we have
It is easy to compute that
Consequently, we have
and
Thus by Theorem 3.1, the above boundary value problem has at least one solution in .
Example 3.2 Consider the boundary value problem
where
Let . Then it is easy to see that f satisfies Carathéodory’s conditions and
Let , , , , . Then it is easy to compute that
Consequently, we have
Thus by Theorem 3.2, the above boundary value problem has at least one solution and in particular has at most one solution with .
Also, since from the equation of the boundary value problem we have
it follows that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MP carried out most of calculations and manuscript preparation. SKC carried out literature survey and conceived ideas. YSO participated in discussions and coordination. All authors read and approved the final manuscript.
Acknowledgement
SKC was supported by Yeungnam University Research Grants 2012. YSO was supported by Daegu University Research Grants 2010.
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