- Research Article
- Open Access
Existence of Positive Solutions for Nonlocal Fourth-Order Boundary Value Problem with Variable Parameter
© Xiaoling Han et al. 2011
- Received: 26 November 2010
- Accepted: 14 January 2011
- Published: 23 January 2011
By using the Krasnoselskii's fixed point theorem and operator spectral theorem, the existence of positive solutions for the nonlocal fourth-order boundary value problem with variable parameter , , , is considered, where is a parameter, and , .
- Variable Parameter
- Point Theorem
- Fixed Point Theorem
- Lower Solution
- Fixed Point Theory
The existence of positive solutions for nonlinear fourth-order multipoint boundary value problems has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point theory, and the method of upper and lower solutions (see, e.g., [1–15] and references therein). The multipoint boundary value problem is in fact a special case of the boundary value problem with integral boundary conditions.
under the assumption:
(A1) and ,
(A2) , , , , , .
where , is a parameter.
Obviously, BVP(1.1) can be regarded as the special case of BVP(1.2) with . Since the parameters is variable, we cannot expect to transform directly BVP(1.2) into an integral equation as in . We will apply the cone fixed point theory, combining with the operator spectra theorem to establish the existence of positive solutions of BVP(1.2). Our results generalize the main result in .
Let , and we assume that the following conditions hold throughout the paper:
(H1) and ,
(H2) , , , and , .
Lemma 2.1 (see ).
if and only if
Let , and , for . , , , , .
It is easy to show that are norms on .
Lemma 2.2 (see ).
and ( ) is a Banach space.
Lemma 2.3 (see ).
Assume that (A1), (A2) hold. Then,
(i) , for , ; , for , ,
(ii) , for , ,
where , ; , .
Computations yield the following results.
Lemma 2.4 (see ).
(i)when , ,
(ii)when , ,
(iii)when , .
Lemma 2.5 (see ).
Suppose that (A1), (A2) hold and , , are given as above. Then,
(ii) , .
By Lemmas 2.4 and 2.5, .
Take , by Lemma 2.5, .
is completely continuous, and .
It is similar to Lemma 6 of , so we omit it.
Lemma 2.7 (see ).
Let E be a Banach space, a cone, and , be two bounded open sets of with . Suppose that is a completely continuous operator such that either
(i) and , , or
(ii) and ,
holds. Then, has a fixed point in .
Assume that (H1), (H2) hold and . Then BVP(1.2) has at least one positive solution if one of the following cases holds:
(i) , ,
(ii) , .
and so , .
and so .
It is easy to know that is a cone in , . Now, we show .
The proof is similar to (i), so we omit it.
Assume that (H1), (H2) hold, and . Then that (1.2) has at least two positive solution, if satisfy
(i) , ,
(ii)There exists such that , for , .
By the proof of Theorem 3.1, we know that (1) from the condition , there exists , such that , , (2) from the condition , there exists , , such that , , (3) from the condition (ii), there exists , , such that , . By the use of Krasnoselskii's fixed point theorem, it is easy to know that (1.2) has at least two positive solutions.
Assume (H1), (H2) hold, and . Then problem (1.2) has at least two positive solution, if satisfy
(i) , ,
(ii)There exists such that , for , .
The proof is similar to Corollary 3.2, so we omit it.
Thus, , , , and satisfy the conditions of Theorem 3.1, and there exists at least a positive solution of the above problem.
This work is sponsored by the NSFC (no. 11061030), NSFC (no. 11026060), and nwnu-kjcxgc-03-69, 03-61.
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