On the solvability of boundary value problems for quasi-linear equations
Let \(D^n_p = D^n_p [a,b]\) denote the space of absolutely continuous functions \(x : [a,b] \to R^n\) such that \(\dot x \in L^n_p [a,b]\). Consider the quasilinear equation (1) \({\mathcal L} x = Fx\), where \({\mathcal L} : D^n_p \to L^n_p\) is a linear bounded operator, \(F : D^n_p \to L^n_p\) is a continuous operator, with boundary conditions (2) \(l_1x = \alpha\) or (3) \(l_2x = \varphi x\), where \(l_1, l_2 : D^n_p \to R^n\) are linear bounded vector functionals, \(\varphi : D^n_p \to R^n\) is a continuous vector functional. The authoress formulates conditions which assure that the boundary value problems (1), (2) and (1), (3), resp., have a solution. In particular she considers problem (1), (3) with \((Fx)(t) = f(t,(Tx) (t)\), \((S \dot x) (t))\) where \(T\) and \(S\) are linear operators.
scientific article; zbMATH DE number 815645