Solvability of \(m\)-point singular boundary value problems (Q2726141)

The author deals with the equation NEWLINE\[NEWLINE x''=q(t)f(t,x,x')+e(t), \tag{1}NEWLINE\]NEWLINE with \(q\in C(0,1)\), \(f\in C([0,1]\times \mathbb{R}^2)\), \(e\in L(0,1)\). He proves the existence of solutions to (1) satisfying the boundary conditions \( x'(0)=0\), \(x(1)=\alpha x(\eta)\), or \( x(0)=0\), \(x(1)=\alpha x(\eta)\), where \(\alpha \in \mathbb{R}\) and \(\eta \in (0,1)\) are given. The results are proved under the assumption, that \(f=g+h\) with \(h\) and \(g\) satifying both-sided and one-sided linear growth conditions, respectively. The proofs are based on the degree theory. Problems of this type were solved earlier by C. P. Gupta at al. Let us note that the author assumes in his main theorems that \(q\) and \(e\) are integrable on \([0,1]\) which means that the problem under consideration is not singular.