Abstract

Nonlinear evolution equations and corresponding nonlinear semigroups has been investigated by Kato, Philips, Lions, Komora and .... This theory which extends Hille- Yosida theory for Cauchy problem with monotone (possibly nonlinear) operator allowed a unified treatment of some different classes of nonlinear partial differential equations. In this lecture, first we introduced nonlinear evolution equations of monotone type and their corresponding semigroups and investigate their asymptotic behavior. Then we introduced second order evolution equation of monotone type. We show existence of solution and equivalence with a minimization problem as well as asymptotic behavior of solutions of these evolution equations. Finally, we introduce discrete analogue of these equations and we show some convergence results similar to continuous case for solutions to second order difference equations. These results also give an algorithm to approximate a zero of a maximal monotone operator.



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