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Characterization of Lyapunov pairs in the nonlinear case and applications. (English) Zbl 1172.34039

The authors study Lyapunov pairs for the nonlinear problem
\[ y'(t)+ Ay(t)\ni f(t,y(t)),\qquad y(0)= y_0 \]
in a Banach space, where \(A\) is an \(m\)-accretive multivalued operator and \(f\) is locally Lipschitz. The concept of a Lyapunov pair is based on the work in [A. Pazy, J. Anal. Math. 40, 239–262 (1981; Zbl 0507.47042)], and includes the standard definition of a Lyapunov function as a special case. In Theorem 5, given lower semicontinuous and proper \(V\) and \(g\), the authors give a necessary and sufficient condition (involving the J\(A\)-contingent derivative of \(V\)) for \(V\) and \(g\) to be a Lyapunov pair, extending results from [M. Kocan and P. Soravia, J. Funct. Anal. 192, No. 2, 342–363 (2002; Zbl 1040.93062)]. In Theorem 6, it is shown that the condition is still necessary and sufficient if the requirements on \(V\) are weakened, by assuming that \(g\) is locally Lipschitz. These results are then applied to a variety of problems, including a priori estimates of solutions, existence of global solutions, controllability and a problem in partial differential equations.

MSC:

34G25 Evolution inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34H05 Control problems involving ordinary differential equations
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