Existence theorem for multivalued hyperbolic equation in Banach spaces (Q1262463)

The multivalued differential equations \[ (1)\quad \partial^ 2u/\partial x\partial y\in F(x,y,u(x,y)) \] are studied with initial conditions \[ u(x,y_ 0)=\sigma (x),\quad u(x_ 0,y)=\tau (y). \] Assume that E is a Banach space, \(z_ 0\in E\), \(B=\{x\in E:\quad \| x-x_ 0\| \leq r\},\quad F: P\times P\to CB(E)\) where \(P=\{(x,y):\quad x_ 0\leq x\leq x_ 0+a,\quad y_ 0\leq y\leq y_ 0+b\}\) and CB(E) denote the metric space of non-empty closed bounded subset of E with the Hausdorff metric H. It is assumed that \(\sigma\), \(\tau\) are defined in \(<x_ 0,x_ 0+a>\), \(<y_ 0,y_ 0+b>\) respectively, with continuous derivatives of the first order and \(\sigma (x_ 0)=\tau (y_ 0)=z_ 0.\) Let C, D and M such that \(\| \sigma (x)-\sigma (\bar x)\| \leq C\| x-\bar x\|,\quad \| \tau (y)-{\bar \tau}(\bar y)\| \leq D\| y-\bar y\|,\) \(\| F(x,y,z)\| \leq M\) for (x,y)\(\in P\), \(z\in B\). Let \(\alpha\) denote the Kuratowski measure of noncompactness, then a constructive proof of the existence of solutions of (1) is given in this main theorem: Theorem. Let the multifunction F be ``absolutely continuous'' and bounded by M and \(\alpha (F(P\times V))\leq \sup \{h(x,y,\alpha (V)):\quad x,y\in P'\}\) \(\forall V\subseteq B.\) Then for any \(f_{\infty}\in F(x_ 0,y_ 0,z_ 0)\) there exists on the rectangle \(P'=\{(x,y):\quad x_ 0\leq x\leq x_ 0+h,\quad y_ 0\leq y\leq y_ 0+h'\}\) a classical solution z(x,y) of (1) such that \(z(x_ 0,y_ 0)=z_ 0\) and \(\partial^ 2z(x_ 0,y_ 0)/\partial x\partial y=f_{\infty}\) and h, \(h'\) are such that \(h<a\), \(h'<b\), \(hC+hD+hh'M\leq r.\)