Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions

Mawhin, Jean ; Szymańska-Dębowska, Katarzyna

Mathematica Bohemica, Tome 141 (2016) no. 2, pp. 239-259.

Résumé

A couple ($\sigma ,\tau $) of lower and upper slopes for the resonant second order boundary value problem $$ x'' = f(t,x,x'), \quad x(0) = 0,\quad x'(1) = \int _0^1 x'(s) {\rm d}g(s), $$ 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) {\rm d}g(s),\nonumber \\ \tau '(t) \leq f(t,x,\tau (t)), \quad \tau (1) \geq \int _0^1 \tau (s) {\rm d}g(s),\nonumber \end {gather} in the stripe $\int _0^t\sigma (s) {\rm d}s \leq x \leq \int _0^t \tau (s) {\rm d}s$ 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.

Détail

DOI : 10.21136/MB.2016.17

Classification : 34B10, 34B15, 47H11
Keywords: nonlocal boundary value problem; lower solution; upper solution; lower slope; upper slope; Leray-Schauder degree