Existence theorem for hyperbolic equation in Banach space (Q1262462)

The hyperbolic equation \[ (1)\quad \partial^ 2z/\partial x\partial y=f(x,y,z,\partial z/\partial x,\partial z/\partial y), \] with the initial condition \[ z(x,y_ 0)=\sigma (x),\quad z(x_ 0,y)=\tau (y) \] is studied in a Banach space E, \(z_ 0\in E\), \(P=\{(x,y):\quad x_ 0\leq x\leq x_ 0+a,\quad y_ 0\leq y\leq y_ 0+b\},\quad f: P\times E\times E\times E\to E,\) \(\sigma\), \(\tau\) defined in \(<x_ 0,x_ 0+a>\), \(<y_ 0,y_ 0+b>\) respectively with continuous derivatives of first order and \(\sigma (x_ 0)=\tau (y_ 0)=z_ 0.\) The existence of a solution of (1) is stated under the assumptions i) f is bounded, uniformly continuous and \[ (2)\quad \alpha (f(P\times A\times B\times C))\leq k \max [\alpha (A),\alpha (B),\alpha (C)] \] for any bounded subset A, B, C of E, where \(\alpha\) is the Kuratowski measure of noncompactness. ii) \(\| f(x,y,z,p,q)-f(x,y,z,\bar p,\bar q)\| \leq L\{\| p-\bar p\| +\| q-\bar q\| \}\) \(\forall x,y\in P\), \(z,p,q,\bar p,\bar q\in E\) with \(k \max \{\mu (P),a,b\}<1,\quad \max \{2La,2Lb\}=R<1.\) If f satisfies a Lipschitz condition on p and q and \(\alpha (f(x,y,A,p,q))\leq k\alpha (A)\) for any \((x,y)\equiv P\), \(p,q\in E\) and for any bounded subset of E, then f satisfies (2). Also existence of solution of equation (1) with initial conditions \[ z(x,0)=0,\quad z(0,y)=0,\quad x\geq 0,\quad y\geq 0 \] are given using Sadowski fixed point theorem.