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- W2049586583 abstract "Several theorems are proved concerning the existence and uniqueness of solutions of the equation $AF( x ) + Bx = c$, in which A and B are $n times n$ matrices of real numbers, c is a real n-vector, and $F( x ) equiv [ f_1 ( x_1 ), cdots ,f_n ( x_n ) ]^T $ is a “diagonal” nonlinear mapping of real n-dimensional Euclidean space $E^n $ into itself. For the case in which the functions $f_k $ are continuous, strictly monotone increasing mappings of $E^1 $ into $E^1 $ (but not necessarily onto$E^1 $) and for a large and important class of pairs of matrices $( A,B )$, we give necessary and sufficient conditions for the existence of the global inverse of the mapping $AF( cdot ) + B$. For not-necessarily-monotone $f_k $ (tunnel-diode type functions, for example) we give conditions which guarantee the existence of at least one solution of $AF( x ) + Bx = c$ for each $c in E^n $. Several significant network-theoretic implications of the results are discussed. For example, we present necessary and sufficient conditions for the existence of the global inverse of the operator $AF( cdot ) + B$ associated with an arbitrary network containing independent sources and nonlinear resistors characterized by arbitrary, continuous, strictly monotone increasing conductance or resistance functions. In particular, the resistors may be described by exponential-type semiconductor diode functions." @default.
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- W2049586583 date "1972-03-01" @default.
- W2049586583 modified "2023-09-26" @default.
- W2049586583 title "Existence and Uniqueness of Solutions for the Equations of Nonlinear DC Networks" @default.
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- W2049586583 doi "https://doi.org/10.1137/0122020" @default.
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