Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

This paper investigates the second-order multipoint boundary value problem on the half-line ,, , , , where , , , , and is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition that is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.

In this paper, we consider the following second-order multipoint boundary value problem on the half-line

where , and is continuous, in which .

The study of multipoint boundary value problems (BVPs) for second-order differential equations was initiated by Bicadze and Samarskĭ [1] and later continued by II'in and Moiseev [2, 3] and Gupta [4]. Since then, great efforts have been devoted to nonlinear multi-point BVPs due to their theoretical challenge and great application potential. Many results on the existence of (positive) solutions for multi-point BVPs have been obtained, and for more details the reader is referred to [5–10] and the references therein. The BVPs on the half-line arise naturally in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium [11–13] and have been also widely studied [14–27]. When , BVP (1.1) reduces to the following three-point BVP on the half-line:

where . Lian and Ge [16] only studied the solvability of BVP (1.2) by the Leray-Schauder continuation theorem. When , , and nonlinearity is variable separable, BVP (1.1) reduces to the second order two-point BVP on the half-line

Yan et al. [17] established the results of existence and multiplicity of positive solutions to the BVP (1.3) by using lower and upper solutions technique.

Motivated by the above works, we will study the existence results of unbounded (positive) solution for second order multi-point BVP (1.1). Our main features are as follows. Firstly, BVP (1.1) depends on derivative, and the boundary conditions are more general. Secondly, we will study multi-point BVP on infinite intervals. Thirdly, we will obtain the unbounded (positive) solution to BVP (1.1). Obviously, with the boundary condition in (1.1), if the solution exists, it is unbounded. Hence, we extend and generalize the results of [16, 17] to some degree. The main tools used in this paper are Leray-Schauder nonlinear alternative and the fixed point index theory.

The rest of the paper is organized as follows. In Section 2, we give some preliminaries and lemmas. In Section 3, the existence of unbounded solution is established. In Section 4, the existence and uniqueness of positive solution are obtained. Finally, we formulate two examples to illustrate the main results.

Denote , where . Let

For any , define

then is a Banach space with the norm (see [17]).

The Arzela-Ascoli theorem fails to work in the Banach space due to the fact that the infinite interval is noncompact. The following compactness criterion will help us to resolve this problem.

Lemma 2.1 (see [17]).

Let . Then, is relatively compact in if the following conditions hold:

(a) is bounded in ;

(b)the functions belonging to and are locally equicontinuous on ;

(c)the functions from and are equiconvergent, at .

Throughout the paper we assume the following.

Suppose that , and there exist nonnegative functions with such that

, where

Denote

Lemma 2.2.

Supposing that with , then BVP

has a unique solution

where

in which , and for .

Proof.

Integrating the differential equation from to , one has

Then, integrating the above integral equation from to , noticing that and , we have

Since , it holds that

By using arguments similar to those used to prove Lemma 2.2 in [9], we conclude that (2.7) holds. This completes the proof.

Now, BVP (1.1) is equivalent to

Letting , (2.12) becomes

For , define operator by

Then,

Set

Remark 2.3.

is the Green function for the following associated homogeneous BVP on the half-line:

It is not difficult to testify that

Let us first give the following result of completely continuous operator.

Lemma 2.4.

Supposing that and hold, then is completely continuous.

Proof.

(1) First, we show that is well defined.

For any , there exists such that . Then,

so

Similarly,

Further,

On the other hand, for any and , by Remark 2.3, we have

Hence, by , the Lebesgue dominated convergence theorem, and the continuity of , for any , we have

So, for any .

We can show that . In fact, by (2.23) and (2.24), we obtain

Hence, is well defined.

(2) We show that is continuous.

Suppose , and . Then, as , and there exists such that . The continuity of implies that

as . Moreover, since

we have from the Lebesgue dominated convergence theorem that

Thus, is continuous.

(3) We show that is relatively compact.

(a)Let be a bounded subset. Then, there exists such that for all . By the similar proof of (2.20) and (2.22), if , one has

which implies that is uniformly bounded.

(b)For any , if , we have

Thus, for any there exists such that if , then

Since is arbitrary, then and are locally equicontinuous on .

(c)For , from (2.27), we have

which means that and are equiconvergent at . By Lemma 2.1, is relatively compact.

Therefore, is completely continuous. The proof is complete.

Lemma 2.5 (see [28, 29]).

Let be Banach space, be a bounded open subset of , and be a completely continuous operator. Then either there exist such that , or there exists a fixed point .

Lemma 2.6 (see [28, 29]).

Let be a bounded open set in real Banach space , let be a cone of , and let be completely continuous. Suppose that

Then,

In this section, we present the existence of an unbounded solution for BVP (1.1) by using the Leray-Schauder nonlinear alternative.

Theorem 3.1.

Suppose that conditions hold. Then BVP (1.1) has at least one unbounded solution.

Proof.

Since , by , we have , a.e. , which implies that . Set

From Lemmas 2.2 and 2.4, BVP (1.1) has a solution if and only if is a fixed point of in . So, we only need to seek a fixed point of in .

Suppose such that . Then

Therefore,

which contradicts . By Lemma 2.5, has a fixed point . Letting , boundary conditions imply that is an unbounded solution of BVP (1.1).

In this section, we restrict the nonlinearity and discuss the existence and uniqueness of positive solution for BVP (1.1).

Define the cone as follows:

Lemma 4.1.

Suppose that and hold. Then, is completely continuous.

Proof.

Lemma 2.4 shows that is completely continuous, so we only need to prove . Since , and from Remark 2.3, we have

Then,

Therefore, .

Theorem 4.2.

Suppose that conditions and hold and the following condition holds:

suppose that and there exist nonnegative functions with such that

Then, BVP (1.1) has a unique unbounded positive solution.

Proof.

We first show that implies . By (4.4), we have

By Lemma 4.1, is completely continuous. Let . Then, . Set

For any , by (4.5), we have

Therefore, for all , that is, for any . Then, Lemma 2.6 yields , which implies that has a fixed point . Let . Then, is an unbounded positive solution of BVP (1.1).

Next, we show the uniqueness of positive solution for BVP (1.1). We will show that is a contraction. In fact, by (4.4), we have

So, is indeed a contraction. The Banach contraction mapping principle yields the uniqueness of positive solution to BVP (1.1).

Example 5.1.

Consider the following BVP:

We have

Let

Then, , and it is easy to prove that is satisfied. By direct calculations, we can obtain that . By Theorem 3.1, BVP (5.1) has an unbounded solution.

Example 5.2.

Consider the following BVP:

In this case, we have

Let

Then, . By Theorem 4.2, BVP (5.4) has a unique unbounded positive solution.

The authors are grateful to the referees for valuable suggestions and comments. The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 10771117) and the Natural Science Foundation of Shandong Province of China (Y2007A23, Y2008A24). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

1. Bicadze, AV, Samarskiĭ, AA: Some elementary generalizations of linear elliptic boundary value problems. Doklady Akademii Nauk SSSR. 185, 739–740 (1969)

2. II'in, VA, Moiseev, EI: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations. 23, 803–810 (1987)

3. II'in, VA, Moiseev, EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential Equations. 23, 979–987 (1987)

4. Gupta, CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. Journal of Mathematical Analysis and Applications. 168(2), 540–551 (1992). Publisher Full Text

5. Feng, W, Webb, JRL: Solvability of -point boundary value problems with nonlinear growth. Journal of Mathematical Analysis and Applications. 212(2), 467–480 (1997). Publisher Full Text

6. Sun, Y, Liu, L: Solvability for a nonlinear second-order three-point boundary value problem. Journal of Mathematical Analysis and Applications. 296(1), 265–275 (2004). Publisher Full Text

7. Sun, Y: Positive solutions of nonlinear second-order -point boundary value problem. Nonlinear Analysis. Theory, Methods & Applications. 61(7), 1283–1294 (2005). PubMed Abstract | Publisher Full Text

8. Zhang, X, Liu, L, Wu, C: Nontrivial solution of third-order nonlinear eigenvalue problems. Applied Mathematics and Computation. 176(2), 714–721 (2006). Publisher Full Text

9. Cheung, W-S, Ren, J: Positive solution for -point boundary value problems. Journal of Mathematical Analysis and Applications. 303(2), 565–575 (2005). Publisher Full Text

10. Eloe, PW, Ahmad, B: Positive solutions of a nonlinear th order boundary value problem with nonlocal conditions. Applied Mathematics Letters. 18(5), 521–527 (2005). Publisher Full Text

11. Atkinson, FV, Peletier, LA: Ground states of () and the Emden-Fowler equation. Archive for Rational Mechanics and Analysis. 93(2), 103–127 (1986). Publisher Full Text

12. Erbe, L, Schmitt, K: On radial solutions of some semilinear elliptic equations. Differential and Integral Equations. 1(1), 71–78 (1988)

13. Kawano, N, Yanagida, E, Yotsutani, S: Structure theorems for positive radial solutions to in . Funkcialaj Ekvacioj. 36(3), 557–579 (1993)

14. Meehan, M, O'Regan, D: Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals. Nonlinear Analysis. Theory, Methods & Applications. 35, 355–387 (1999). PubMed Abstract | Publisher Full Text

15. Tian, Y, Ge, W: Positive solutions for multi-point boundary value problem on the half-line. Journal of Mathematical Analysis and Applications. 325(2), 1339–1349 (2007). Publisher Full Text

16. Lian, H, Ge, W: Solvability for second-order three-point boundary value problems on a half-line. Applied Mathematics Letters. 19(10), 1000–1006 (2006). Publisher Full Text

17. Yan, B, O'Regan, D, Agarwal, RP: Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity. Journal of Computational and Applied Mathematics. 197(2), 365–386 (2006). Publisher Full Text

18. Lian, H, Ge, W: Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line. Journal of Mathematical Analysis and Applications. 321(2), 781–792 (2006). Publisher Full Text

19. Baxley, JV: Existence and uniqueness for nonlinear boundary value problems on infinite intervals. Journal of Mathematical Analysis and Applications. 147(1), 122–133 (1990). Publisher Full Text

20. Zima, M: On positive solutions of boundary value problems on the half-line. Journal of Mathematical Analysis and Applications. 259(1), 127–136 (2001). Publisher Full Text

21. Bai, C, Fang, J: On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals. Journal of Mathematical Analysis and Applications. 282(2), 711–731 (2003). Publisher Full Text

22. Hao, Z-C, Liang, J, Xiao, T-J: Positive solutions of operator equations on half-line. Journal of Mathematical Analysis and Applications. 314(2), 423–435 (2006). Publisher Full Text

23. Wang, Y, Liu, L, Wu, Y: Positive solutions of singular boundary value problems on the half-line. Applied Mathematics and Computation. 197(2), 789–796 (2008). Publisher Full Text

24. Kang, P, Wei, Z: Multiple positive solutions of multi-point boundary value problems on the half-line. Applied Mathematics and Computation. 196(1), 402–415 (2008). Publisher Full Text

25. Zhang, X: Successive iteration and positive solutions for a second-order multi-point boundary value problem on a half-line. Computers & Mathematics with Applications. 58(3), 528–535 (2009). PubMed Abstract | Publisher Full Text

26. Sun, Y, Sun, Y, Debnath, L: On the existence of positive solutions for singular boundary value problems on the half-line. Applied Mathematics Letters. 22(5), 806–812 (2009). Publisher Full Text

27. Agarwal, PR, O'Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equation, Kluwer Academic, Dodrecht, The Netherlands (2001)

28. Guo, DJ, Lakshmikantham, V: Nonlinear problems in abstract cones,p. viii+275. Academic Press, New York, NY, USA (1988)

29. Deimling, K: Nonlinear Functional Analysis,p. xiv+450. Springer, Berlin, Germany (1985)