## 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
Full Text: