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Monotonicity methods for nonlinear evolution equations. (English) Zbl 0894.34055

The paper deals with evolution problems of the form \[ u'(t)+ A(t)u(t)= f(t),\quad 0<t<T,\quad u(0)= 0, \] where \(A(t): X\to X^*\) (\(X\) is a Banach space and \(X^*\) is its dual space), \(f\in V= L^p(0,T;X)^*\), \(u_0\in X\). The authors show that under suitable conditions the map \(\widetilde A:V\to V^*\) defined by \(\widetilde A(u)(t)= A(t)u(t)\) inherits the monotonicity properties of \(A(t)\). The results are applied to generalize previous papers by N. Hirano [Nonlinear Anal., Theory Methods Appl. 13, No. 6, 599-609 (1989; Zbl 0682.34010)] and N. U. Ahmed and X. Xiang [Nonlinear Anal., Theory Methods Appl. 22, No. 1, 81-89 (1994; Zbl 0806.34051)].

MSC:

34G20 Nonlinear differential equations in abstract spaces
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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