Positive solutions of periodic boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative

摘要

In this article,we study the existence of positive solutions of periodic boundary value problems for the second-order differential equation with the nonlinearity depends on the derivative {(Lu) (t) = h (t) f (u (t), u'(t)), 0 ≤ t ≤ ω, R_1 (u) ≡ u (0) ? u (ω) = 0, R_2 (u) ≡ p (0) u'(0) ? p (ω) u'(ω) = 0, where (Lu) (t) = ?(p (t) u')' + q (t) u. By applying coincidence degree theorem, some conditions guaranteeing the existence of at least one positive solution are given in terms of the relative behaviors of the quotient (f (u, v))/(|u| + |v|) for |u|+|v| near 0 and +∞. The result discussed in the paper is a generalization of recent one and the difference is that a nonlinear term depends on the derivative of unknown function.