On a class of quasilinear differential equations: The Neumann problem (Q1289311)

The authors consider the quasilinear ordinary differential equation with Neumann boundary conditions \[ -\Bigl(a \biggl(\bigl| x' (t)\bigr |^2 \biggr)x'(t) \Bigr)'+f \bigl(t,x(t) \bigr)= v(t), \quad x'(0)= x'(b)=0. \tag{1} \] The existence of solutions to problem (1) is proved by methods of monotone operators under some assumptions with respect to \(a,f\) and \(v\).