Existence, uniqueness and determination of solution of certain piecewise linear resistive networks

Abstract

For an equation of the form F(x)=y, it is shown that there is at least one solution for every y if F is eventually P0 passive or the effective Jacobian matrix in all the unbounded regions is a P matrix. In addition to this, if the Jacobian determinant has the same sign in all the regions, then F is a homeomorphism. For equations of the form F(x)=g(x)+Hx=y, F (x) is onto if H is P0 and F(x) is norm coercive where g(x) is diagonal. This statement is true for equations of the form F( x)=Ag(x)+Bx=y also where ( A,B) is W0. In these results g 1(x1) is allowed to saturate without requiring additional conditions on H or (AB). It is also shown that, roughly under these conditions, the generalized Katzenelson's method converges to a solution. Homeomorphism of these two forms is guaranteed if the Jacobian determinant has the same sign in all the regions in addition to the above conditions