Cover: On Nonlinear Differential Equations, the Maximum Operation, and Monotone Convergence

On Nonlinear Differential Equations, the Maximum Operation, and Monotone Convergence

Published 1957

by Robert E. Kalaba


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A proof that the solutions to certain classes of nonlinear ordinary and partial differential equations may be represented in terms of the maximum operation applied to the solutions of associated linear equations. This, in effect, affords a new approach to the quasi-linearization of nonlinear differential equations. The representation readily yields uniform lower bounds for solutions, and, in the case of stochastic nonlinear differential equations, leads to representations for the distribution functions of the solutions. In addition, a technique is provided for constructing monotone sequences of functions which converge quadratically to the solution of the nonlinear equation, which is of value in machine computation.

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