In this paper, we consider nth-order two-point 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 Leray-Schauder continuation theorem and analytical technique. Meanwhile, as an application of our results, examples are given.

MSC: 34B15.

In this paper, we shall discuss the existence and uniqueness of solutions of right focal boundary value problems for nth-order 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 [1-16,18-25].

Recently, using the Leray-Schauder continuation theorem, Hopkins and Kosmatov [16] have obtained sufficient conditions for the existence of at least one sign-changing solution for third-order 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 nth-order right focal boundary value problems (1.1), (1.2). The existence and uniqueness of sign-changing solutions for the problems are obtained by Leray-Schauder 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 sign-changing solutions for nth-order right focal boundary value problems (1.1), (1.2) by Leray-Schauder continuation theorem and analytical technique, and give two examples to demonstrate our results. Our results improve and generalize the corresponding results in [16].

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

where for (),

and for,

Proof Firstly, let us show the lemma for case . Since

we have that for ,

where, for ,

and for ,

It follows by Hölder’s inequality that, for each ,

and consequently, for each ,

But for ,

It follows by (2.4) that for ,

For , by Lemma 2.1, is nondecreasing in t, and thus

Hence, by (2.4) we have for ,

In summary,

Next, we show the lemma for the case . It is easy to see that for ,

and thus for ,

Also by Lemma 2.1, we have for ,

so that for each , is nondecreasing in t, it follows that

Let

Then

Since

we have for each ,

in particular

so that is nondecreasing on . Hence by (2.5), we have

Thus for ,

In summary,

 □

Lemma 2.3 ([17] Leray-Schauder 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

has a solution.

Now we are ready to establish our existence theorems of solutions for nth-order right focal boundary value problems (1.1), (1.2). The Leray-Schauder continuation theorem plays key roles in the proofs.

Theorem 3.1Letsatisfy-Carathéodory’s conditions. Suppose that

(i) there exist functions, , and a constantsuch that

for a.e. and all;

(ii)

where the constants, are given in Lemma 2.2;

(iii)

where.

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 one-to-one 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 Arzelà-Ascoli 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 Leray-Schauder 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 Leray-Schauder 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 Leray-Schauder 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 3.1-3.4 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 3.2Letsatisfy-Carathéodory’s conditions. Suppose that

(i) there exist functions, , and a constantsuch that

for a.e. and all;

(ii)

where the constants, are given in Lemma 2.2;

(iii)

where.

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

Since and , 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

Hence above boundary value problem has a unique solution .

The authors declare that they have no competing interests.

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.

SKC was supported by Yeungnam University Research Grants 2012. YSO was supported by Daegu University Research Grants 2010.

1. Agarwal, RP: Focal Boundary Value Problems for Differential and Difference Equations, Kluwer, Dordrecht (1998)

2. Agarwal, RP, Usmani, RA: Iterative methods for solving right focal point boundary value problems. J. Comput. Appl. Math.. 14, 371–390 (1986). Publisher Full Text

3. Agarwal, RP, Usmani, RA: Monotone convergence of iterative methods for right focal point boundary value problems. J. Math. Anal. Appl.. 130, 451–459 (1988). Publisher Full Text

4. Anderson, DR, Davis, JM: Multiple solutions and eigenvalues for third-order right focal boundary value problems. J. Math. Anal. Appl.. 267, 135–157 (2002). Publisher Full Text

5. Anderson, DR: Green’s function for a third-order generalized right focal problem. J. Math. Anal. Appl.. 288, 1–14 (2003). Publisher Full Text

6. Cheung, WS, Wong, PJY: Fixed-sign solutions for a system of singular focal boundary value problems. J. Math. Anal. Appl.. 329, 851–869 (2007). Publisher Full Text

7. Ehme, J, Hankerson, D: Existence of solutions for right focal boundary value problems. Nonlinear Anal.. 18, 191–197 (1992). Publisher Full Text

8. Ehme, J: Uniqueness and existence for nth-order right focal boundary value problems. Appl. Math. Lett.. 13, 7–11 (2000)

9. Ehme, J, Lanz, A: Uniqueness implies existence of solutions for nonlinear focal-like boundary value problem. Appl. Math. Lett.. 22, 1325–1329 (2009). Publisher Full Text

10. Goecke, DM, Henderson, J: Uniqueness of solutions of right focal problems for third order differential equations. Nonlinear Anal.. 8, 253–259 (1984). Publisher Full Text

11. Graef, JR, Henderson, J, Wong, PJY, Yang, B: Three solutions of an nth order three-point focal type boundary value problem. Nonlinear Anal.. 69, 3386–3404 (2008). Publisher Full Text

12. Henderson, J: Existence of solutions of right focal point boundary value problems for ordinary differential equations. Nonlinear Anal.. 5, 989–1002 (1981). Publisher Full Text

13. Henderson, J: Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations. J. Differ. Equ.. 41, 218–227 (1998)

14. Henderson, J: Right focal point boundary value problems for ordinary differential equations and variational equations. J. Math. Anal. Appl.. 98, 363–377 (1984). Publisher Full Text

15. Henderson, J, Yin, W: Singular boundary value problems between conjugate and right focal. J. Comput. Appl. Math.. 88, 57–69 (1998). Publisher Full Text

16. Hopkins, B, Kosmatov, N: Third-order boundary value problems with sign-changing solutions. Nonlinear Anal.. 67, 126–137 (2007). Publisher Full Text

17. Ladde, GS, Lakshmikantham, V, Vatsala, AS: Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman Advanced Publishing, Boston (1985)

18. Lin, Y, Pei, M: Positive solution for two-point semipositone right focal eigenvalue problem. Bound. Value Probl.. 2007, (2007)

19. Liu, Z, Debnath, L, Kang, SM: Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem. Comput. Math. Appl.. 55, 356–367 (2008). Publisher Full Text

20. Ma, R: Multiple positive solutions for a semipositone fourth-order boundary value problem. Hiroshima Math. J.. 33, 217–227 (2003)

21. Wong, PJY, Agarwal, RP: Multiple positive solutions of two-point right focal boundary value problems. Math. Comput. Model.. 28, 41–49 (1998)

22. Wong, PJY: Constant-sign solutions for a system of third-order generalized right focal problems. Nonlinear Anal.. 63, 2153–2163 (2005). Publisher Full Text

23. Wong, PJY: Multiple fixed-sign solutions for a system of generalized right focal problems with deviating arguments. J. Math. Anal. Appl.. 323, 100–118 (2006). Publisher Full Text

24. Wong, PJY: Eigenvalues of a system of generalized right focal problems with deviating arguments. J. Comput. Appl. Math.. 218, 459–472 (2008). Publisher Full Text

25. Zhou, C, Ma, D: Existence and iteration of positive solutions for a generalized right-focal boundary value problem with p-Laplacian operator. J. Math. Anal. Appl.. 324, 409–424 (2006). Publisher Full Text