Stability of a three-layer symmetric differential-difference scheme in the class of functions summable on a network-like domain

In the paper, the stability conditions of a three-layer symmetric differential-difference scheme with a weight parameter in the class of functions summable on a network-like domain are obtained. To analyze the stability of the differential-difference system in the space of feasible solutions $H$, a composite norm is introduced that has the structure of a norm in the space $H^2=H\oplus H.$ Namely, for $Y=\{Y_1,Y_2\}\in H^2,$ $Y_\ell\in H$ ($\ell=1,2$), $\| Y\|^2_H=\| Y_1\|^2_{1,H}+\| Y_2\|^2_{2,H},$ where $\|\cdot\|^2_{1,H}$ $\|\cdot\|^2_{2,H}$ are some norms in $H.$ The use of such a norm in the description of the energy identity opens the way for constructing a priori estimates for weak solutions of the differential-difference system, convenient for practical testing in the case of specific differential-difference schemes. The results obtained can be used to analyze optimization problems that arise when modeling network-like transfer processes with the help of formalisms of differential-difference systems.