On a class of second order differential inclusions driven by the scalar \(p\)-Laplacian
The authors study a class of nonlinear boundary value problems for second order differential inclusions. In particular they consider the problem: \[ (|x'(t)|^{p-2}x'(t))'\in Ax(t)+F(t,x(t),x'(t))+\theta(x(t)) \text{ a.e. on } T=[0,b], \] \[ x'(0)\in \xi_1(x(0)), -x'(b)\in \xi_2(x(b)), 2 \leq p <+\infty, \] where \(A:D(A)\subset \mathbb R \to 2^{\mathbb R}\backslash {\emptyset}\), \(\xi_{i}:D(\xi_{i})\subset \mathbb R \to 2^{\mathbb R}\backslash {\emptyset}\), \(i=1,2\), are maximal monotone operators, \(\theta:\mathbb R \to \mathbb R\) is a function and \(F:T\times \mathbb R^2 \to 2^{\mathbb R}\backslash {\emptyset}\) is a multivalued map upper semicontinuous in the second variable. By using fixed point techniques combined with the method of lower and upper solutions, the authors prove some existence results of extremal (greatest and smallest) solutions in the ordered interval delimited by the lower and the upper solution.
scientific article; zbMATH DE number 5656697