On the homotopy property of topological degree for maximal monotone mappings

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Abstract

Let E be a real reflexive Banach space, E the dual space of E, and ΩE an open bounded subset, and let Ti:D(Ti)2E, i=1,2, be two maximal monotone mappings such that Ω¯D(T1)D(T2) and 0t[0,1][tT1+(1-t)T2](Ω(D(T1)D(T2))). Under some additional assumptions we prove that deg(T1,D(T1)Ω,0)=deg(T2,D(T2)Ω,0).

Section snippets

Introduction and preliminaries

Topological degree theory is a very powerful tool used in proving various existence results for nonlinear partial differential equations and it has received a lot of attention in the literature, see the references in [17]. In 1983, Browder established a degree theory for single valued mappings of class (S+), (see also [18] for the separable space case). Based on this degree, Browder also constructed a degree theory for mappings of class (S+) with a perturbation of a maximal monotone mapping or

Main results

In the following, E is a real reflexive Banach spaces, E the dual space of E, and both E and E are locally uniform convex spaces.

Theorem 2.1

Let T:D(T)E2E, S:D(S)E2E be two maximal monotone mappings. Let ΩE be an open bounded subset, and x0D(T)D(S). Assume that 0Tx0Sx0, and 0T(ΩD(T)), 0S(ΩD(S)), and infxΩx>x0. Then deg(T,ΩD(T),0)=deg(S,ΩD(S),0).

Proof

For any ϵ>0, we claim that there exists λ0>0 such that0tTλx+(1-t)Sλx+ϵJx,for allt[0,1],xΩ,λ(0,λ0).Suppose this is not true. There

Acknowledgement

The first author was supported by a NSFC grant, Grant No. 10871052.

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