Boundary Value Problems

, 2011:198598 | Cite as

Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales

Open Access
Research Article

Abstract

We discuss the existence of solutions for the first-order multipoint BVPs on time scale Open image in new window : Open image in new window , Open image in new window , Open image in new window , where Open image in new window is a parameter, Open image in new window is a fixed number, Open image in new window , Open image in new window is continuous, Open image in new window is regressive and rd-continuous, Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window . For suitable Open image in new window , some existence, multiplicity, and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.

Keywords

Positive Constant Real Line Closed Subset Main Tool Index Theory 

1. Introduction

Let Open image in new window be a time scale (a nonempty closed subset of the real line Open image in new window ). We discuss the existence of positive solutions to the first-order multipoint BVPs on time scale Open image in new window :

where Open image in new window is a fixed number, Open image in new window , Open image in new window is continuous, Open image in new window is regressive and rd-continuous, Open image in new window , and Open image in new window , Open image in new window is defined in its standard form; see [1, page 59] for details.

The multipoint boundary value problems arise in a variety of different areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross-section and composed of Open image in new window parts of different densities can be set up as a multipoint boundary value problem [2]; also many problems in the theory of elastic stability can be handled by a multipoint problem [3]. So, the existence of solutions to multipoint boundary value problems have been studied by many authors; see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and the reference therein. Especially, in recent years the existence of positive solutions to multipoint boundary value problems on time scales has caught considerable attention; see [10, 11, 12, 13, 14]. For other background on dynamic equations on time scales, one can see [1, 15, 16, 17, 18].

Our ideas arise from [13, 16]. In [13], Tian and Ge discussed the existence of positive solutions to nonlinear first-order three-point boundary value problems on time scale Open image in new window :

where Open image in new window is continuous, Open image in new window is regressive and rd-continuous, Open image in new window and Open image in new window . The existence results are based on Krasnoselskii's fixed-point theorem in cones and Leggett-Williams's theorem.

As we can see, if we take Open image in new window , Open image in new window , Open image in new window , and Open image in new window for Open image in new window , then (1.1) is reduced to (1.2). Because of the influence of the parameter Open image in new window , it will be more difficult to solve (1.1) than to solve (1.2).

In 2009, by using the fixed-point index theory, Sun and Li [16] discussed the existence of positive solutions to the first-order PBVPs on time scale Open image in new window :

For suitable Open image in new window , they gave some existence, multiplicity, and nonexistence criteria of positive solutions.

Motivated by the above results, by using the well-known fixed-point index theory [16, 19], we attempt to obtain some existence, multiplicity and nonexistence criteria of positive solutions to (1.1) for suitable Open image in new window .

The rest of this paper is arranged as follows. Some preliminary results including Green's function are given in Section 2. In Section 3, we obtain some useful lemmas for the proof of the main result. In Section 4, some existence and multiplicity results are established. At last, some nonexistence results are given in Section 5.

2. Preliminaries

Throughout the rest of this paper, we make the following assumptions:

Open image in new window is continuous and Open image in new window for Open image in new window ,

Open image in new window is rd-continuous, which implies that Open image in new window (where Open image in new window is defined in [16, 18, 20]).

Moreover, let

Our main tool is the well-known results from the fixed-point index, which we state here for the convenience of the reader.

Theorem 2.1 (see [19]).

Let Open image in new window be a Banach space and Open image in new window be a cone in Open image in new window . For Open image in new window , we define Open image in new window . Assume that Open image in new window is completely continuous such Open image in new window for Open image in new window .

Let Open image in new window be equipped with the norm Open image in new window . It is easy to see that Open image in new window is a Banach space.

For Open image in new window , we consider the following linear BVP:

Lemma 2.2.

For Open image in new window , the linear BVP (2.4)-(2.5) has a solution Open image in new window if and only if Open image in new window satisfies

Proof.

By (2.4), we have
Combining this with (2.5), we get

Lemma 2.3.

If the function Open image in new window is defined in (2.7), then Open image in new window may be expressed by

Proof.

When Open image in new window , Open image in new window ,

When Open image in new window ,

Lemma 2.4.

Green's function Open image in new window has the following properties.

(i) Open image in new window ,

(ii) Open image in new window where Open image in new window Open image in new window

(iii) Open image in new window , Open image in new window

Proof.

This proof is similar to [13, Lemma Open image in new window ], so we omit it.

where Open image in new window . For Open image in new window , let Open image in new window and Open image in new window .

Similar to the proof of [13, Lemma Open image in new window ], we can see that Open image in new window is completely continuous. By the above discussions, its not difficult to see that Open image in new window being a solution of BVP (1.1) equals the solution that Open image in new window is a fixed point of the operator Open image in new window .

3. Some Lemmas

4. Some Existence and Multiplicity Results

Theorem 4.1.

Assume that (H1) and (H2) hold and that Open image in new window . Then the BVP (1.1) has at least two positive solutions for

Proof.

Let Open image in new window . Then it follows from (4.1) and Lemma 3.3 that
In view of Theorem 2.1, we have
In view of (4.1), (4.4), (4.5), and Lemma 3.2, we have
It follows from Theorem 2.1 that
By (4.3) and (4.7), we get

This shows that Open image in new window has a fixed point in Open image in new window , which is a positive solution of the BVP (1.1).

In view of (4.1), (4.9), and Lemma 3.2, we have
It follows from Theorem 2.1 that
By (4.3) and (4.12), we get

This shows that Open image in new window has a fixed point in Open image in new window , which is another positive solution of the BVP (1.1).

Similar to the proof of Theorem 4.1, we have the following results.

Theorem 4.2.

Suppose that (H1) and (H2) hold and

Then,

(i)equation (1.1) has at least one positive solution if Open image in new window ,

(ii)equation (1.1) has at least one positive solution if Open image in new window ,

(iii)equation (1.1) has at least two positive solutions if Open image in new window .

Theorem 4.3.

Assume that (H1) and (H2) hold. If Open image in new window Open image in new window , then the BVP (1.1) has at least two positive solutions for

Proof.

Let Open image in new window . Then it follows from (4.15) and Lemma 3.3 that
In view of Theorem 2.1, we have
In view of (4.15), (4.18), and Lemma 3.1, we have
It follows from Theorem 2.1 that
By (4.17) and (4.20), we get

This shows that Open image in new window has a fixed point in Open image in new window , which is a positive solution of the BVP (1.1).

Combined with (4.22) and Lemma 3.1, we have
It follows from Theorem 2.1 that
By (4.17) and (4.25), we get

This shows that Open image in new window has a fixed point in Open image in new window , which is another positive solution of the BVP (1.1).

Similar to the proof of Theorem 4.3, we have the following results.

Theorem 4.4.

Suppose that (H1) and (H2) hold and that

Then,

(i)equation (1.1) has at least one positive solution if Open image in new window ,

(ii)equation (1.1) has at least one positive solution if Open image in new window ,
  1. (iii)

    equation (1.1) has at least two positive solutions if Open image in new window .

     

Theorem 4.5.

Suppose that (H1) and (H2) hold. If Open image in new window , then the BVP (1.1) has at least one positive solution for

Proof.

We only deal with the case that Open image in new window , Open image in new window . The other three cases can be discussed similarly.

Let Open image in new window satisfy (4.28) and let Open image in new window be chosen such that
From the definition of Open image in new window , we know that there exists a constant Open image in new window such that Open image in new window for Open image in new window and Open image in new window . So,
This combines with (4.29) and Lemma 3.2, we have
It follows from Theorem 2.1 that
Combined with (4.29) and Lemma 3.1, we have
It follows from Theorem 2.1 that
By (4.32) and (4.35), we get

which implies that the BVP (1.1) has at least one positive solution in Open image in new window .

Remark 4.6.

By making some minor modifications to the proof of Theorem 4.5, we can obtain the existence of at least one positive solution, if one of the following conditions is satisfied:

(i) Open image in new window , Open image in new window and Open image in new window .

(ii) Open image in new window , Open image in new window and Open image in new window .

(iii) Open image in new window , Open image in new window and Open image in new window .

(iv) Open image in new window , Open image in new window and Open image in new window .

Remark 4.7.

From Conditions (ii) and (iv) of Remark 4.6, we know that the conclusion in Theorem 4.5 holds for Open image in new window in these two cases. By Open image in new window and Open image in new window , there exist two positive constants Open image in new window such that, for Open image in new window ,
This is the condition of Theorem Open image in new window of [13]. By Open image in new window and Open image in new window , there exist two positive constants Open image in new window such that for Open image in new window ,

This is the condition of Theorem Open image in new window of [13]. So, our conclusions extend and improve the results of [13].

5. Some Nonexistence Results

Theorem 5.1.

Assume that (H1) and (H2) hold. If Open image in new window and Open image in new window , then the BVP (1.1) has no positive solutions for sufficiently small Open image in new window .

Proof.

We assert that the BVP (1.1) has no positive solutions for Open image in new window .

Suppose on the contrary that the BVP (1.1) has a positive solution Open image in new window for Open image in new window . Then from (5.3) and Lemma 3.2, we get

which is a contradiction.

Theorem 5.2.

Assume that (H1) and (H2) hold. If Open image in new window and Open image in new window , then the BVP (1.1) has no positive solutions for sufficiently large Open image in new window .

Proof.

We assert that the BVP (1.1) has no positive solutions for Open image in new window .

Suppose on the contrary that the BVP (1.1) has a positive solution Open image in new window for Open image in new window . Then from (5.7) and Lemma 3.1 we get

which is a contradiction.

Corollary 5.3.

Assume that (H1) and (H2) hold. If Open image in new window and Open image in new window , then the BVP (1.1) has no positive solutions for sufficiently large Open image in new window .

Notes

Acknowledgments

This work was supported by the NSFC Young Item (no. 70901016), HSSF of Ministry of Education of China (no. 09YJA790028), Program for Innovative Research Team of Liaoning Educational Committee (no. 2008T054), the NSF of Liaoning Province (no. L09DJY065), and NWNU-LKQN-09-3

References

  1. 1.
    Sun J-P, Li W-T: Existence of solutions to nonlinear first-order PBVPs on time scales. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(3):883-888. 10.1016/j.na.2006.06.046CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Moshinsky M: Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas. Boletin Sociedad Matemática Mexicana 1950, 7: l-25.MathSciNetGoogle Scholar
  3. 3.
    Timoshenko SP: Theory of Elastic Stability. 2nd edition. McGraw-Hill, New York, NY, USA; 1961:xvi+541.Google Scholar
  4. 4.
    Rodriguez J, Taylor P: Scalar discrete nonlinear multipoint boundary value problems. Journal of Mathematical Analysis and Applications 2007, 330(2):876-890. 10.1016/j.jmaa.2006.08.008CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Ma R: Positive solutions for a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations 1999, 34: 1-8.Google Scholar
  6. 6.
    Ma R: Multiplicity of positive solutions for second-order three-point boundary value problems. Computers & Mathematics with Applications 2000, 40(2-3):193-204. 10.1016/S0898-1221(00)00153-XCrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Ma R: Existence and uniqueness of solutions to first-order three-point boundary value problems. Applied Mathematics Letters 2002, 15(2):211-216. 10.1016/S0893-9659(01)00120-3CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Ma R:Positive solutions for nonhomogeneous Open image in new window-point boundary value problems. Computers & Mathematics with Applications 2004, 47(4-5):689-698. 10.1016/S0898-1221(04)90056-9CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Liu B: Existence and uniqueness of solutions to first-order multipoint boundary value problems. Applied Mathematics Letters 2004, 17(11):1307-1316. 10.1016/j.aml.2003.08.014CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Anderson DR, Wong PJY: Positive solutions for second-order semipositone problems on time scales. Computers & Mathematics with Applications 2009, 58(2):281-291. 10.1016/j.camwa.2009.02.033CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Luo H: Positive solutions to singular multi-point dynamic eigenvalue problems with mixed derivatives. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(4):1679-1691. 10.1016/j.na.2008.02.051CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Luo H, Ma Q: Positive solutions to a generalized second-order three-point boundary-value problem on time scales. Electronic Journal of Differential Equations 2005, 2005(17):-14.MathSciNetGoogle Scholar
  13. 13.
    Tian Y, Ge W: Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(9):2833-2842. 10.1016/j.na.2007.08.054CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Sun J-P: Existence of positive solution to second-order three-point BVPs on time scales. Boundary Value Problems 2009, 2009:-6.Google Scholar
  15. 15.
    Cabada A, Vivero DR: Existence of solutions of first-order dynamic equations with nonlinear functional boundary value conditions. Nonlinear Analysis: Theory, Methods & Applications 2005, 63(5–7):e697-e706.CrossRefMATHGoogle Scholar
  16. 16.
    Sun J-P, Li W-T: Positive solutions to nonlinear first-order PBVPs with parameter on time scales. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(3):1133-1145. 10.1016/j.na.2008.02.007CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Sun J-P, Li W-T: Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales. Computers & Mathematics with Applications 2007, 54(6):861-871. 10.1016/j.camwa.2007.03.009CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001.CrossRefGoogle Scholar
  19. 19.
    Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985.CrossRefMATHGoogle Scholar
  20. 20.
    Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xvi+541.CrossRefMATHGoogle Scholar

Copyright information

© C. Gao and H. Luo. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouChina
  2. 2.School of Mathematics and Quantitative EconomicsDongbei University of Finance and EconomicsDalianChina

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