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On the degree theory for densely defined mappings of class \((S_+)_L\). (English) Zbl 0991.47049

This article presents a new class of mappings of monotone type (class \((S_+)_L\)) for which the degree is defined with standard properties of additivity, invariance under homotopies, and normalization; the Leray-Schauder existence result for maps of this class with nonzero degre also holds.
A map \(A: X\to X^*\) is of class \((S_+)_L\) if
(a) there exists a subspace \(L\) such that \(L\subseteq D(A)\) and \(\overline L= X\),
(b) for any sequence \((F_j)\), \(F_j\in{\mathcal F}(L)\), such that \[ F_j\subseteq F_{j+1},\quad \dim F_j= j,\quad \overline{\bigcup^\infty_{j= 1} F_j}= X \] the conditions \[ (u_k)\subseteq L,\;u_k\rightharpoonup u_0,\;\limsup_{k\to\infty} \langle A(u_k), u_k\rangle\leq 0,\;\lim_{k\to\infty} \langle A(u_k)- h,v\rangle= 0\;\Biggl(v\in \bigcup^\infty_{j=1} F_j\Biggr) \] imply that \(u_k\to u_0\), \(u_0\in D(A)\), and \(A(u_0)= h\) for any \(h\in X^*\), and
(c) for every \(F\in{\mathcal F}(L)\), \(v\in L\) the mapping \(a(u,v)= \langle A(u), v\rangle: F\to \mathbb{R}\) is continuous (here \({\mathcal F}\) is the family of finite-dimensional subspaces of \(L\)).
In the end of the article, theorems about degree of mappings of class \((S_+)_L\) satisfying the Rothe condition and odd mappings of class \((S_+)_L\) are presented and some abstract existence results for equations with operators of class \((S_+)_L\) are proved.

MSC:

47H11 Degree theory for nonlinear operators
35J60 Nonlinear elliptic equations
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