Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition. (English) Zbl 0725.34020

del Pino, Manuel A.; Manásevich, Raúl F.

Proc. Am. Math. Soc. 112, No. 1, 81-86 (1991).

This paper deals with the existence of solutions of the boundary value problem \(y^{\text{(IV)}}=f(x,y,y'')\), \(0<x<1\), \(y(0)=y_ 0\), \(y(1)=y_ 1\), \(y''(0)=\bar y_ 0\), \(y''(1)=\bar y_ 1\), where \(f: [0,1]\times {\mathbb R}^ 2\to {\mathbb R}\) is continuous. The main theorem under a nonresonance condition involving a two-parameter linear eigenvalue problem contains a previous result of A. R. Aftabizadeh [J. Math. Anal. Appl. 116, 415–426 (1986; Zbl 0634.34009)]. This theorem also provides the uniqueness of the solution under a suitable Lipschitz-type condition, which recovers the uniqueness result due to Y. Yang in Theorem 2 of [Proc. Am. Math. Soc. 104, No. 1, 175–180 (1988; Zbl 0671.34016)]. The authors also give extensions of their results to some higher-order semilinear elliptic problems.

Reviewer: P.Pucci (Modena)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
35J50	Variational methods for elliptic systems

Keywords:

Leray-Schauder theorem; existence of solutions; nonresonance; two-parameter linear eigenvalue problem; uniqueness; higher-order semilinear elliptic problems

Citations:

Zbl 0634.34009; Zbl 0671.34016