On a set of solutions for differential inclusions with one sided Lipschitz condition
Consider the differential inclusion NEWLINE\[NEWLINE \dot{x}\in F(t,x)+ G(t,x),\quad x(t_{0})=x_{0}, \tag{1} NEWLINE\]NEWLINE where \(x \in\mathbb R^{n}\), \(t\in [t_{0}, t_{1}]\). The multimappings \(F(\cdot,\cdot), G(\cdot,\cdot)\) are compact and convex valued and bounded on \([t_{0}, t_{1}]\times\mathbb R^{n}\). Along with (1) it is considered the differential inclusion NEWLINE\[NEWLINE \dot{x}\in F_{\lambda}(t,x) + G(t,x),\quad x(t_{0})=x_{0}, \tag{2} NEWLINE\]NEWLINE where \(F_{\lambda}(\cdot,\cdot)\) is the Yosida approximation of \(F(\cdot,\cdot)\): \(\exists\lambda_{0} >0\;\forall t\in[t_{0}, t_{1}]\;\forall x,y\in\mathbb R^{n}\;\forall v\in F(t,x)\) such that NEWLINE\[NEWLINE \langle x-y,F_{\lambda}(t,x)-v\rangle\leq l\|x-y\|^{2} + \lambda L \quad \forall\lambda\in(0,\lambda_{0}], NEWLINE\]NEWLINE where \(l,L\) are constants. \newline { Theorem.} Let \(H(x_{0}), H_{\lambda}(x_{0})\) be the solution sets of (1) and (2), respectively, let \(h\) be the Hausdorff metric on the nonempty compact sets of \(C([t_{0}, t_{1}],\mathbb R^{n})\). Then there exist \(k\) and \(\lambda_{0}\) such that NEWLINE\[NEWLINE h(H(x_{0}), H_{\lambda}(x_{0}))\leq k\sqrt{\lambda}\quad \forall \lambda\in(0,\lambda_{0}]. NEWLINE\]
scientific article; zbMATH DE number 6077954