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Nonlinear evolution equations with nonmonotonic perturbations. (English) Zbl 0682.34010
In this interesting paper the author considers the existence of solutions of the initial value problem \[ (1)\quad \frac{du}{dt}+Au+G(u)=f,\quad 0<t<T,\quad u(0)=u_ 0, \] where A is a monotone operator from V into U, V is a reflexive Banach space densely and compactly imbedded in a real Hilbert space H, G: \(V\to H\) is a continuous mapping and \(f(0,T)\to V'\) is a measurable function. The author’s argument is based on the previous results for pseudo monotonic mappings. The assumption of the problem implies neither that A generates a compact semigroups, nor that G is a completely continuous mapping. The author proves two main theorems applying three proved prepositions. So it is difficult to reproduce the five proofs here. The author draws the attention to the fact that the monotone mapping A can be replaced by a family of mappings A(t) say. Finally some examples and remarks are given at the end of the paper where he also presents a number of recent references for relevant further discussions of the topics covered in this paper.
Reviewer: F.M.Ragab

MSC:
34A34 Nonlinear ordinary differential equations and systems
47A99 General theory of linear operators
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