A fourth-order nonlinear boundary-value problem and an integral-type sign condition

摘要

The fourth-order nonlinear boundary-value problem –d4u/dx4 +g(x,u) = e(x), 0 < x < π, u′(0) = u′(π) = u′′′ (0) = u′′′ (π) = 0, associated to the unstable static equilibrium of an elastic beam which is supported by sliding clamps when the nonlinearity g(x,u) satisfies an integral sign condition instead of the usual asymptotic pointwise sign condition, is studied. A necessary and sufficient condition is given for the existence of a solution when g(x,u) is nondecreasing in u.