Nonlinear Problems With Asymmetric Principal Part

Abstract

<jats:p>The boundary value problem is considered provided that f : [0, +∞) → [0, +∞) is Lipschitzian and is continuous and Lipschitzian in xand x′. We assume that f is bounded by two linear functions kx and lx, where k &gt; l &gt; 0, and h is bounded. We find the conditions on (λ, µ) which guarantee the existence of a solution to the problem. These conditions are of geometrical nature.</jats:p>

The Fucík spectrum for nonlocal BVP with Sturm–Liouville boundary condition

“…These solutions correspond to the spectrum branch F ± 1 . Using the geometrical type arguments we obtain that the solutions of problem (1), (6) with one zero in the interval (0, 1) may be of the three different types.…”

“…Taking into account that the solutions of the problem (1), (6) have no zeroes in the interval (0, 1), follows that λ < π 2 /4. Consider the left side of equation (7) as a function.…”

“…The first type of them can be analytically described with the sine function, the next one with the linear function and the third type of solutions of problem (1), (6) without zeroes in the interval (0, 1) can be analytically described with the hyperbolic sine function. Let us denote the branches of the spectrum of the problem (1), (6), which correspond to the solutions of the first type with F Proof. Consider the solutions of the problem (1), (6) without zeroes in the interval (0, 1) for positive λ values.…”

“…Since the spectrum of problem (1), (6) coincides with the classical one in the case α = 1 than the notation compatible with the classical spectrum has been chosen. Therefore, branches, which relate to solutions x(t) with x(0) < 0, x (0) > 0, have been denoted by F + i , but F − i is used for branches related to solutions with x(0) > 0, x (0) < 0.…”

“…Later nonlinear equations of the type x + f (x + ) − g(x − ) = ϕ(t, x, x ) were considered in [5], where f and g are positive valued continuous functions, ϕ is a bounded continuous nonlinearity (often ϕ = 0), λ and µ are spectral parameters. Notions of spectral surfaces were introduced in [6] and [7]. Solvability of Dirichlet and Neumann boundary value problems was considered in [8].…”