Fučík spectrum for a third order equation
The Fučík spectrum of the third order boundary value problem \[ y''' + \mu y^+ - \nu y^- = 0,\;y(0) = y'(0) = y(1) = 0 \tag{1} \] is investigated, i.e. the set of all \((\mu, \nu) \in \mathbb{R}^2\) where (1) has a nontrivial solution \(y\). The eigenvalues of (1) are the numbers \(\lambda\) for which \((\lambda, \lambda)\) belongs to the Fučík spectrum. In Section 4 it is shown that the Fučík spectrum is the union of countably many curves whose graphs are of the form \(F_i^\pm (i = 1, 2, \dots)\) with \((\mu, \nu) \in F_i^+\) if and only if \((\nu, \mu) \in F^-_i\), \(F^+_1 = \{(\lambda_1, \nu) : \nu \in \mathbb{R}\}\), \(F^+_i = \{(\mu, f_i^+ (\mu))\}\), \(i \geq 2\), and the function \(f^+_i\) are continuous decreasing functions defined on \(([{i + 1 \over 2}]^3 \lambda_1, \infty)\), and map onto \(([{i \over 2}]^3 \lambda_1, \infty)\), where \(\lambda_1 > 0\) is the smallest eigenvalue of (1). Furthermore properties of the functions \(f_i^\pm\) are derived. For the solvability of the nonlinear boundary value problem \[ x''' + f(x,t) = 0, \quad x(0) = x'(0) = x(1) = 0 \tag{2} \] it is important to introduce property \(P\): a set function \(E(t) \subset \mathbb{R}^2\), \(t \in [0,1]\), is said to have property \(P\) if for all \(p,q \in L^1 (0,1)\) such that \((p(t), q(t)) \in E(t)\) for almost all \(t \in [0,1]\) the boundary value problem \(y''' + p(t) y^+ - q(t)y^- = 0\), \(y(0) = y'(0) = y(1) = 0\) has only the trivial solution. Conditions for a set function to have property \(P\) are derived in Section 5, and a class of counterexamples is obtained in Section 6. In Theorem 7.1 some sufficient conditions for (2) having at least one solution are given.
scientific article; zbMATH DE number 910003