L2‐absolute and input‐to‐state stabilities of equations with nonlinear causal mappings

Nonlinear scalar equations with causal mappings are considered. These equations include differential, difference, differential‐delay, integro‐differential and other traditional equations. Estimates for the L²‐norms of solutions are established. These estimates give us explicit conditions for the absolute and input‐to‐state stabilities of the considered equations. The Aizerman‐type problem from the theory of absolute stability is also discussed. The suggested approach enables us to consider various classes of systems from the unified point of view. Copyright © 2008 John Wiley & Sons, Ltd.