A sharper condition for the solvability of a three-point second order boundary value problem
Let \(f:[0,1] \times\mathbb{R}^2 \to\mathbb{R}\) be a function satisfying Carathéodory's conditions and \(e(t)\in L^1[0,1]\). Let \(\eta\in(0,1)\), \(\alpha\in\mathbb{R}\), \(\alpha>1\), \(\alpha\eta \neq 1\) be given. This paper is concerned with the problem of existence of a solution for the three-point boundary value problem \[ x''(t)=f \bigl(t,x(t), x'(t)\bigr)+ e(t),\;0<t<1,\;x(0)=0,\;x(1)= \alpha x(\eta). \tag{*} \] In the general case this problem was studied by the first author as a multipoint boundary value problem. In the present paper, sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case.
scientific article; zbMATH DE number 1005679
scientific article