On a class of boundary value problems for singular equations (Q1921682)

Under some conditions the boundary value problem \(Lx=Fx\), \(lx=\varphi x\) for the equation \(Lx=Fx\) has a solution. Here \(L:D\to B\) is a bounded linear operator, \(F:D\to B\) a continuous operator, \(l:D\to\mathbb{R}^n\) a bounded linear functional, \(\varphi:D\to\mathbb{R}^n\) a continuous functional, \(B\) and \(D\) are Banach spaces, whereby \(D\) is isomorphic to the direct product \(B\times\mathbb{R}^n\). The result is applied to the boundary value problem \(\psi(t)\ddot x(t)+\beta\dot x(t)=f(t,x(t))\), \(t\in [a,b]\), \(x(a)=\alpha_1\), \(x'(b)=\alpha_2\) representing a mathematical model of processes in chemical reactors.