Some variant of iteration method for infinite systems of parabolic differential-functional equations (Q2744593)

Consider the Fourier first boundary value problem: Find a regular solution to the system NEWLINE\[NEWLINE{\mathcal F}^{i}[z^{i}](t,x)={\mathcal F}^{i}(t,x,z^{i}(t,x),z(t,\cdot)),\quad i\in S,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE{\mathcal F}^{i}:=\frac{\partial }{\partial t}-{\mathcal A}^{i},\quad {\mathcal A}^{i}=\sum^m_{j,k=1} a_{jk}^{i}(t,x)\frac{\partial ^{2}}{\partial x_{j}\partial x_{k}},NEWLINE\]NEWLINE satisfying the boundary condition, NEWLINE\[NEWLINEz(t,x)=g(t,x)\quad\text{for }(t,x)\in \sum,NEWLINE\]NEWLINE where \(D_{0}:=\{(t,x):t=0,x\in \overline{G}\},\sigma :=(0,T]\times \partial G,\sum :=D_{0}\cup \sigma \) and \(\overline{D}:=D\cup \sum .\) Applying a monotone iterative method this problem is solved.