Nonlocal evolution inclusions under weak conditions
The authors investigate evolution inclusions given by multivalued perturbations of \(m\)-dissipative operators with nonlocal initial conditions. The system has the form \[ \dot{y}(s)\in Ay(s)+F(s,y(s)),\, s\in I=[t_0,T],\, y_0(t_0)=g(y(\cdot)), \] where \(A:D(A)\subset X\rightrightarrows X\) is a \(m\)-dissipative operator, \(F:I\times X\rightrightarrows X\) a multifunction, mapping \(g: C(I,X)\to \overline{D(A)}\), and \(X\) is a Banach space with uniformly convex dual. The authors prove the existence of solutions assuming that \(F(s,\cdot)\) is one-sided Lipschitz, which is weaker than the commonly used Lipschitz condition. To this end the corresponding Cauchy problem is studied. Next, they relax the growth conditions on \(g(\cdot)\) and it seems that the assumptions are weaker and more flexible. Notice also that even if \(F(s,\cdot )\) is Lipschitz, then either the one-sided Lipschitz constant or some function is in general smaller than the Lipschitz one. The existence result concerns the multipoint problems that cover periodic and antiperiodic cases. Some examples illustrate the applicability of the results.
scientific article; zbMATH DE number 7012793
scientific article