Some new results for singular problems with sign changing nonlinearities
The authors deal with second-order differential equations of the form \[ y''= q(t)f(t,y),\quad 0< t< 1,\tag{1} \] where \(q\in C(0,1)\) can have singularities at \(0\) and \(1\) and \(f:[0,1]\times (0,\infty)\to \mathbb{R}\) is continuous with a possible singularity at \(y= 0\). \(f\) is allowed to change sign. The authors prove the existence of a solution to (1) which satisfies either the Dirichlet conditions \(y(0)= y(1)= 0\) or the nonlinear conditions \(y(0)= 0\), \(\theta(y'(1)+ y(1))= 0\) with \(\theta: \mathbb{R}\to\mathbb{R}\) continuous, nondecreasing and satisfying \(\theta(0)= 0\). The first existence result, concerning the Dirichlet problem, is formulated in terms of the lower and upper solutions and its proof is based on the Schauder fixed point theorem. The existence of a solution to the above nonlinear boundary value problem is proved via the nonlinear alternative of Leray-Schauder type. In this case some growth restrictions are imposed on \(f\). These results generalize and complete earlier results obtained by the authors.
nonlinear alternative of Leray-Schauder type
scientific article; zbMATH DE number 1398391