Small solutions of certain boundary value problems (Q1386249)

The paper deals with boundary value problems (1) \(Ly + y^3=f(x)\), (2) \(M_i(y):=\alpha _i y(a)+\beta _i y(b) +\gamma _i y'(a)+ \delta _i y'(b)= 0\), \(i=1,2\), where \(Ly = y'' + p(x)y' + q(x)y\), \(p,q\) and \(f\in L^1(a,b)\), \(\alpha _i,\beta _i,\gamma _i\) and \(\delta _i\in \mathbb{R}\) and \(f\) is supposed to be small in the sense of the space \(L^1\). The authors prove several theorems on the existence and nonexistence of small solutions to problems (1), (2) in critical cases when for the corresponding linearized problem \(\mathcal Ly=0\) (i.e. \(Ly=0\), \(M_i(y)=0\), \(i=1,2)\) the relations \(\text{dim ker}(\mathcal L)= \text{codim im}(\mathcal L) = 1\) are satisfied. The results extend some recent investigations done by \textit{M. Fečkan} [Math. Slovaca 42, No. 2, 195-200 (1992; Zbl 0756.34028)] and \textit{L. Lefton} [J. Differ. Equations 85, No. 1, 171-185 (1990; Zbl 0699.34020)].