Positive solutions of third-order nonlocal boundary value problems at resonance
© Zhang and Sun; licensee Springer 2012
Received: 21 April 2012
Accepted: 3 September 2012
Published: 18 September 2012
In this paper, we investigate the existence of positive solutions for a class of third-order nonlocal boundary value problems at resonance. Our results are based on the Leggett-Williams norm-type theorem, which is due to O’Regan and Zima. An example is also included to illustrate the main results.
Keywordsthird-order nonlocal at resonance positive solution
has nontrivial solutions. Clearly, the resonant condition is . Third-order differential equations arise in a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity-driven flows and so on .
by means of the Leggett-Williams norm-type theorem due to O’Regan and Zima , where , () and .
However, third-order or higher-order derivatives do not have the convexity; to the best of our knowledge, no results are available for the existence of positive solutions for third-order or higher- order BVPs at resonance. The main purpose of this paper is to fill the gap in this area. Motivated greatly by the above-mentioned excellent works, in this paper we will investigate the third-order nonlocal BVP (1.1) at resonance, where , () and . Some new existence results of at least one positive solution are established by applying the Leggett-Williams norm-type theorem due to O’Regan and Zima . An example is also included to illustrate the main results.
2 Some definitions and a fixed point theorem
For the convenience of the reader, we present here the necessary definitions and a new fixed point theorem due to O’Regan and Zima.
KerL has a finite dimension, and
ImL is closed and has a finite codimension.
Throughout the paper, we will assume that
1∘L is a Fredholm operator of index zero, that is, ImL is closed and .
for all and , and
Note that every cone induces a partial order ≤ in X by defining if and only if . The following property is valid for every cone in a Banach space.
Lemma 2.3 ()
Let P be a cone in X. Then for every, there exists a positive numbersuch thatfor all.
Our main results are based on the following theorem due to O’Regan and Zima.
Theorem 2.4 ()
Let C be a cone in X and let, be open bounded subsets of X withand. Assume that 1∘is satisfied and if the following assumptions hold:
(H1) is continuous and bounded, andis compact on every bounded subset of X;
(H2) for alland;
(H3) γ maps subsets ofinto bounded subsets of C;
(H4) , wherestands for the Brouwer degree;
(H5) there existssuch thatfor, whereandis such thatfor all;
then the equationhas a solution in the set.
3 Main results
for every . We also let .
We can now state our result on the existence of a positive solution for the BVP (1.1).
there exists a constant such that for all ;
- (2)there exist , , , and continuous functions , such that for , is non-increasing on with
Then the resonant BVP (1.1) has at least one positive solution on.
Proof Consider the Banach spaces with .
which shows that , which together with implies that . Note that and thus . Therefore, L is a Fredholm operator of index zero.
Considering that f can be extended continuously to , it is easy to check that is continuous and bounded, and is compact on every bounded subset of X, which ensures that (H1) of Theorem 2.4 is fulfilled.
Moreover, . Let and for . Then γ is a retraction and maps subsets of into bounded subsets of C, which means that (H3) of Theorem 2.4 holds.
Let , where and will be defined in the following proof.
Let . Now, we verify that and .
First, we show . Suppose, on the contrary, that achieves maximum value M only at . Then in combination with yields that , which is a contradiction.
Suppose, on the contrary, that . The step is divided into two cases:
which implies is increasing close to 1. This together with induces (t close to 1), that is, is decreasing close to 1, which contradicts .
Case 2. Assume that has zero points on ; we may choose nearest to 1 with . Then there is a constant such that . Similar to the above arguments, we easily get a contradiction too.
Hence, we can choose so that . This gives and . By , we know and . Similarly, we also divide the part of the proof into two cases.
which is a contradiction.
which is a contradiction. Therefore, (H2) of Theorem 2.4 holds.
which shows that (H4) of Theorem 2.4 holds.
which shows that . These ensure that (H6), (H7) of Theorem 2.4 hold. It remains to show that (H5) is satisfied.
That is, for all , which shows that (H5) of Theorem 2.4 holds.
Summing up, all the hypotheses of Theorem 2.4 are satisfied. Therefore, the equation has a solution . And so, the resonant BVP (1.1) has at least one positive solution on . □
4 An example
for all ;
- (2)for , is non-increasing on with
Thus, all the conditions of Theorem 3.1 are satisfied. Then the resonant problem (4.1) has at least one positive solution on .
This work is supported by the National Natural Science Foundation of China (10801068) and the Education Scientific Research Foundation of Tangshan College (120186). The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.
- Gregus M Math. Appl. In Third Order Linear Differential Equations. Reidel, Dordrecht; 1987.View ArticleGoogle Scholar
- Anderson DR: Green’s function for a third-order generalized right focal problem. J. Math. Anal. Appl. 2003, 288: 1-14. 10.1016/S0022-247X(03)00132-XMATHMathSciNetView ArticleGoogle Scholar
- Anderson DR, Davis JM: Multiple solutions and eigenvalues for three-order right focal boundary value problems. J. Math. Anal. Appl. 2002, 267: 135-157. 10.1006/jmaa.2001.7756MATHMathSciNetView ArticleGoogle Scholar
- Du ZJ, Ge WG, Zhou MR: Singular perturbations for third-order nonlinear multi-point boundary value problem. J. Differ. Equ. 2005, 218: 69-90. 10.1016/j.jde.2005.01.005MATHMathSciNetView ArticleGoogle Scholar
- El-Shahed M: Positive solutions for nonlinear singular third order boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 424-429. 10.1016/j.cnsns.2007.10.008MATHMathSciNetView ArticleGoogle Scholar
- Li S: Positive solutions of nonlinear singular third-order two-point boundary value problem. J. Math. Anal. Appl. 2006, 323: 413-425. 10.1016/j.jmaa.2005.10.037MATHMathSciNetView ArticleGoogle Scholar
- Liu ZQ, Debnath L, Kang SM: Existence of monotone positive solutions to a third-order two-point generalized right focal boundary value problem. Comput. Math. Appl. 2008, 55: 356-367. 10.1016/j.camwa.2007.03.021MATHMathSciNetView ArticleGoogle Scholar
- Palamides AP, Smyrlis G: Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function. Nonlinear Anal. 2008, 68: 2104-2118. 10.1016/j.na.2007.01.045MATHMathSciNetView ArticleGoogle Scholar
- Sun JP, Zhang HE: Existence of solutions to third-order m -point boundary value problems. Electron. J. Differ. Equ. 2008., 2008: Article ID 125Google Scholar
- Sun Y: Positive solutions for third-order three-point nonhomogeneous boundary value problems. Appl. Math. Lett. 2009, 22: 45-51. 10.1016/j.aml.2008.02.002MATHMathSciNetView ArticleGoogle Scholar
- Yao Q: The existence and multiplicity of positive solutions for a third-order three-point boundary value problem. Acta Math. Appl. Sin. 2003, 19: 117-122. 10.1007/s10255-003-0087-1MATHView ArticleGoogle Scholar
- Bai C, Fang J: Existence of positive solutions for boundary value problems at resonance. J. Math. Anal. Appl. 2004, 291: 538-549. 10.1016/j.jmaa.2003.11.014MATHMathSciNetView ArticleGoogle Scholar
- Infante G, Zima M: Positive solutions of multi-point boundary value problems at resonance. Nonlinear Anal. 2008, 69: 2458-2465. 10.1016/j.na.2007.08.024MATHMathSciNetView ArticleGoogle Scholar
- Liang SQ, Mu L: Multiplicity of positive solutions for singular three-point boundary value problem at resonance. Nonlinear Anal. 2009, 71: 2497-2505. 10.1016/j.na.2009.01.085MATHMathSciNetView ArticleGoogle Scholar
- Yang L, Shen CF: On the existence of positive solution for a kind of multi-order boundary value problem at resonance. Nonlinear Anal. 2010, 72: 4211-4220. 10.1016/j.na.2010.01.051MATHMathSciNetView ArticleGoogle Scholar
- O’Regan D, Zima M: Leggett-Williams norm-type theorems for coincidence. Arch. Math. 2006, 87: 233-244. 10.1007/s00013-006-1661-6MATHMathSciNetView ArticleGoogle Scholar
- Mawhin J: Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces. J. Differ. Equ. 2010, 72: 4211-4220.Google Scholar
- Santanilla J: Some coincidence theorems in wedges, cones and convex sets. J. Math. Anal. Appl. 1985, 105: 357-371. 10.1016/0022-247X(85)90053-8MATHMathSciNetView ArticleGoogle Scholar
- Petryshyn WV: On the solvability of in quasinormal cones with T and F k -set contractive. Nonlinear Anal. 1981, 5: 585-591. 10.1016/0362-546X(81)90105-XMATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.