By generalizing the extension of the continuous theorem of Ge and Ren and constructing suitable Banach spaces and operators, we investigate the existence of solutions for p-Laplacian boundary value problems at resonance. An example is given to illustrate our results.
Keywords:continuous theorem; resonance; p-Laplacian; boundary value problem
In this paper, we will study the boundary value problem
A boundary value problem is said to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. Mawhin’s continuous theorem  is an effective tool to solve this kind of problems when the differential operator is linear, see [2-10] and references cited therein. But it does not work for nonlinear cases such as boundary value problems with a p-Laplacian, which attracted the attention of mathematicians in recent years [11-15]. Ge and Ren extended Mawhin’s continuous theorem  and many authors used their results to solve boundary value problems with a p-Laplacian, see [16,17]. In this new theorem, two projectors P and Q must be constructed. But it is difficult to give the projector Q in many boundary value problems with a p-Laplacian. In this paper, we generalize the extension of the continuous theorem and show that the p-Laplacian problem is solvable when Q is not a projector. And we will use this new theorem to discuss problems (1.1) and (1.2), respectively.
In this paper, we will always suppose that
where domM denote the domain of the operator M.
Definition 2.2 Suppose , is a continuous and bounded operator. Denote by N. Let . is said to be M-quasi-compact in if there exists a vector subspace of Y satisfying and two operators Q, R with , , being continuous, bounded, and satisfying , continuous and compact such that for ,
Proof The proof is similar to the one of Lemma 2.1 and Theorem 2.1 in . □
We can easily get the following inequalities.
3 The existence of a solution for problem (1.1)
Lemma 3.2Mis a quasi-linear operator.
Proof The proof is simple. Therefore, we omit it. □
Theorem 3.1Assume that the following conditions hold.
(H3) There exists a nonnegative constantKsuch that one of (1) and (2) holds:
Then boundary value problem (1.1) has at least one solution.
In order to prove Theorem 3.1, we show two lemmas.
Lemma 3.5Suppose (H3) and (H4) hold. Then the set
is bounded inX.
Lemma 3.6Assume (H3) holds. Then
By the homotopy of degree, we get
Example Let us consider the following boundary value problem at resonance:
4 The existence of a solution for problem (1.2)
The proof is similar to Lemma 3.1.
Lemma 4.2Mis a quasi-linear operator.
Theorem 4.1Assume that the following conditions hold:
Then boundary value problem (1.2) has at least one solution.
In order to prove Theorem 4.1, we show two lemmas.
Lemma 4.4Suppose (H5)-(H7) hold. Then the set
is bounded inX.
Lemma 4.5Assume (H6) holds. Then
Since , we have either or . If , then . So, we have . This is a contradiction with the definition of ρ. If , then . Thus and . By , we get . This is a contradiction with the definition of ρ, too. So, , , .
By the homotopy of degree, we get
The author declares that she has no competing interests.
All results belong to WJ.
This work is supported by the National Science Foundation of China (11171088) and the Natural Science Foundation of Hebei Province (A2013208108).
The author is grateful to anonymous referees for their constructive comments and suggestions, which led to improvement of the original manuscript.
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