Constructive analysis for infinite‐dimensional nonlinear systems—infinite‐dimensional version of homotopy method

Related to the problem such as determination of the operating point for the nonlinear resistor circuit equation, remarkable progress has recently been observed in the global solution method for the finite‐dimensional nonlinear equation, called the homotopy method. This paper attempts to extend the homo‐topy method to the infinite dimensional nonlinear equation (functional equation) such as the equation for the semiconductor devices. First, two streams are pointed out as the theoretical frameworks for the homotopy method for the finite‐dimensional equation, which are: (1) the characterization of the solution structure for the nonlinear equation; and (2) the tracing of the solution trajectory. It is then pointed out that the Fredholm operator equation can be considered as a class of infinite‐dimensional equations, which can characterize the set of solutions as in item (1). The existence theorem for the solution in the finite‐dimensional homotopy method is extended to the class of Fredholm operator equations. Then it is pointed out that the A‐proper operator equation can be considered as a class of infinite‐dimensional equations, for which the computational algorithm for the solution can be described as in item (2). Based on the constructive theorem for the existence of the solution in the finite‐dimensional homotopy method, it is shown that the solution for the particular class of A‐proper operator equations can always be determined by numerical calculation. Finally, an application to the controllability problem is discussed.