Iterative methods for the Darboux problem for partial functional differential equations (Q1125209)

Let \(B=[-a_o,0]\times [-b_o,0],\;E=[0,a]\times [0,b],\;E^o=[-a_o,a]\times [-b_o,b]\backslash (0,a]\times (0,b],\;E^*=E^o\cup E\), where \(a_o,\;b_o\in \mathbb{R}_+, a,b>0\), and \(C^1(E^*;X)\) be the set of all functions \(z\in C(E^*;X)\) for which the derivations \(D_xz,\;D_yz\) exist and are continuous; \(C^{1,*}(E^*;X)\) denotes the subset of \(C^1(E^*;X)\) consisting of functions for which additionally the mixed derivative \(D_{xy}z\) exists and is continuous on \(E\). For any \(z:[-a_o,a]\times [-b_o,b]\rightarrow \mathbb{R}\), any point \((x,y)\in E\), define \(z_{(x,y)}:B\rightarrow B\) by \(z_{(x,y)}(t,s)=z(x+t,y+s),\;(t,s)\in B.\) For given \(\phi:E^o\rightarrow \mathbb{R},\;f:E\times C^1(B;\mathbb{R})\rightarrow \mathbb{R}\), the author considers the Darboux problem \[ D_{xy}z(x,y)=f(x,y,z_{(x,y)}),\quad (x,y)\in E,\tag{1} \] \[ z(x,y)=\phi(x,y),\quad (x,y)\in E^o.\tag{2} \] Sufficient conditions for the existence of two monotone sequences \(\{u^{(m)}\},\;\{v^{(m)}\}\) are given such that if \(z\) is a solution of (1), (2) then \(u^{(m)}\leq z\leq v^{(m)}\) on \(E\) and \(\{u^{(m)}\},\;\{v^{(m)}\}\) are uniformly convergent to \(z\) on \(E\). The convergence is of the Newton type, which means that \[ 0\leq z(x,y)-u^{(m)}(x,y)\leq \frac{2A}{2^{2^m}},\quad 0\leq v^{(m)}(x,y)-z(x,y)\leq \frac{2A}{2^{2^m}} \quad \text{on} E, \] where \(A\) is some constant independent on \(m\). The monotonicity of sequences \(\{u^{(m)}\},\;\{v^{(m)}\}\) is proved by a theorem on functional differential inequalities. If \(X\) is a general Banach space, the author gives an analogous theorem on the convergence of the Newton method.