Institute of Mathematics

Abstract:  A couple ($\sigma,\tau$) of lower and upper slopes for the resonant second order boundary value problém $$x" = f(t,x,x'), \quad x(0) = 0,\quad x'(1) = \int_0^1 x'(s)  dg(s),$$ with $g$ increasing on $[0,1]$ such that $\int_0^1 dg = 1$, is a couple of functions $\sigma, \tau\in C^1([0,1])$ such that $\sigma(t) \leq\tau(t)$ for all $t \in[0,1]$, \begin{gather} \sigma'(t) \geq f(t,x,\sigma(t)), \quad\sigma(1) \leq\int_0^1 \sigma(s)  dg(s),\nonumber \tau'(t) \leq f(t,x,\tau(t)), \quad\tau(1) \geq\int_0^1 \tau(s)  dg(s),\nonumber\end{gather} in the stripe $\int_0^t\sigma(s)  ds \leq x \leq\int_0^t \tau(s)  ds$ and $t \in[0,1]$. It is proved that the existence of such a couple $(\sigma,\tau)$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.