First-order three-point boundary value problems at resonance (Q555174)

The authors investigate the three-point boundary value problem at resonance \[ \begin{aligned} & x' - A(t) x = \varepsilon F(t, x, \varepsilon) + E(t), \quad 0 \leq t \leq 1,\\ & M x(0) + N x(\eta) + R x(1) = l + \varepsilon g(x(0), x(\eta), x(1)), \end{aligned} \] where \(M\), \(N\) and \(R\) are constant square matrices of order \(n\), \(A(t)\) is an \(n\times n\) matrix with continuous entries, \(E\), \(F\) and \(g\) are continuous functions. They obtain the existence of solutions of this boundary value problem for \(\varepsilon\) sufficiently small by applying a version of Brouwer's fixed point theorem due to Miranda [see \textit{X. Yang}, Appl. Math. Comput. 137, No. 2--3, 413--422 (2003; Zbl 1090.34552)].