SOLVABILITY OF A NONLINEAR FOUR–POINT BOUNDARY VALUE PROBLEM FOR A FOURTH ORDER DIFFERENTIAL EQUATION∗

Abstract

The authors consider four-point boundary value problem for a fourth order ordinary differential equations of the form (φp(u00(t)))00 (E) = a(t)f(u(t)), t ∈ (0, 1), with one of the following boundary conditions (B1) u(0) − λu0(η) = u0(1) = 0, u000(0) = α1u000(ξ), u00(1) = β1u00(ξ), or (B2) u(1) + λu0(η) = u0(0) = 0, u000(0) = α1u000(ξ), u00(1) = β1u00(ξ). They impose growth conditions on f which guarantee existence of at least two positive solutions for the problems (E) − (B1) and (E) − (B2).