The existence of solutions for p-Laplacian boundary value problem at resonance on the half-line is investigated. Our analysis relies on constructing the suitable Banach space, defining appropriate operators and using the extension of Mawhin’s continuation theorem. An example is given to illustrate our main result.
MSC: 70K30, 34B10, 34B15.
Keywords:p-Laplacian; resonance; half-line; multi-point boundary value problem; continuation theorem
A boundary value problem is said to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. Resonance problems can be expressed as an abstract equation , where L is a noninvertible operator. When L is linear, Mawhin’s continuation theorem  is an effective tool in finding solutions for these problems, see [2-10] and references cited therein. But it does not work when L is nonlinear, for instance, p-Laplacian operator. In order to solve this problem, Ge and Ren  proved a continuation theorem for the abstract equation when L is a noninvertible nonlinear operator and used it to study the existence of solutions for the boundary value problems with a p-Laplacian:
To the best of our knowledge, there are few papers that study the p-Laplacian boundary value problem at resonance on the half-line. In this paper, we investigate the existence of solutions for the boundary value problem
In order to obtain our main results, we always suppose that the following conditions hold.
For convenience, we introduce some notations and a theorem. For more details, see .
Let and be the complement space of in X, then . On the other hand, suppose that is a subspace of Y, and that is the complement of in Y, i.e., . Let and be two projectors and an open and bounded set with the origin .
3 Main result
In order to obtain our main results, we need the following additional conditions.
Lemma 3.4Assume that (H3) and (H4) hold. The set
is bounded inX.
Considering (H3), we have
By (3.5), (3.6) and (H3), we get
Lemma 3.5Assume that (H4) holds. The set
is bounded inX.
Theorem 3.1Suppose that (H1)-(H4) hold. Then problem (1.1) has at least one solution.
By the homotopy of degree, we get that
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 the improvement of the original manuscript.
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