Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions
Continuing their previous work [J. Differ. Equations 147, No. 1, 123-154 (1998; Zbl 0912.34020)], the authors prove the existence of a solution \(x(.)\in C^1(T,\mathbb{R}^N), \;\|x'(.)\|^{p-2}x'(.)\in W^{1,q}(T,\mathbb{R}^N)\) to a boundary value problem of the form: \[ (\|x'(t)\|^{p-2}x'(t))'\in F(t,x(t),x'(t)) \quad \text{a.e. }(T=[0,b]), \] \[ (\|x'(0)\|^{p-2}x'(0),- \|x'(b)\|^{p-2}x'(b))\in \xi (x(0),x(b)), \] defined the convex-valued multifunction \(F(.,.,.)\), by the maximal monotone operator \(\xi (.,.)\) and by \(p\in [2,\infty)\). The rather technical proofs of the auxiliary results and of the main one rely on the construction of some approximate selectors of Carathéodory type and on the use of the Leray-Schauder fixed point theorem for compact single-valued operators. The authors show that their general formulation contains as particular cases the Dirichlet, Neumann, periodic solutions and some Sturm-Liouville-type problems.
scientific article; zbMATH DE number 1398395