Existence of weak solutions for a boundary value problem of a second order ordinary differential equation
The authors investigate the existence of weak solutions for a boundary value problem of second-order differential equations of the form \[ \left(\frac{x(t)-g(t,x(t))}{h(t,x(t))}\right)''+f(t,x(t)=0,\;\;\;0<t<1, \tag{1} \] \[ x(0)=g(0,x(0)),\;\;\;x(1)=f(1,x(1)), \tag{2} \] where $f, g:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}$, and $g:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}\setminus \{0\}$ satisfy the Carathéodory conditions. They transform (1) and (2) to the following perturbed quadratic integral equation: \[ x(t)=g(t,x(t))+h(t,x(t))\int_0^1 G(t,s)f(s,x(s))ds,\;\;0\leq t \leq 1. \] Upper and lower bounds of the Green function $G$ are given in order to apply the well-known Krasnosel'skii-type fixed point theorem in conjunction with the technique of measures of weak noncompactness in Banach spaces. Two examples are presented to illustrate the validity of the main result.
scientific article; zbMATH DE number 6996317
scientific article