## 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

### Keywords:

nonlinear evolution equations; monotone mappings

### Citations:

Zbl 0682.34010; Zbl 0806.34051
Full Text:

### References:

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