We introduce a new construction of topological degree for densely defined mappings of monotone type. We also study the structure of the classes of mappings involved. Using the basic properties of the degree, we prove some abstract existence results that can be applied to elliptic problems.

We establish conditions that guarantee Fredholm solvability in the Banach space $L_p$ of nonlocal boundary value problems for elliptic abstract differential equations of the second order in an interval. Moreover, in the space $L_2$ we prove in addition the coercive solvability, and the completeness of root functions (eigenfunctions and associated functions). The obtained results are then applied to the study of a nonlocal boundary value problem for Laplace equation in a cylindrical domain.

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation $\left(d/dt\right)u\left(t\right)=Au\left(t\right)+B\left(t\right)u\left(t\right)+f\left(t\right),t\in ℝ$, where $\left(A,D\left(A\right)\right)$ is a Hille-Yosida operator on a Banach space $X,B\left(t\right),t\in ℝ$, is a family of operators in $ℒ\left(\overline{D\left(A\right)},X\right)$ satisfying certain boundedness and measurability conditions and $f\in L_{\text{loc}}^1\left(ℝ,X\right)$. The solutions of the corresponding homogeneous equations are represented by an evolution family ${\left(U_B\left(t,s\right)\right)}_{t\ge s}$. For various function spaces $ℱ$ we show conditions on ${\left(U_B\left(t,s\right)\right)}_{t\ge s}$ and $f$ which ensure the existence of a unique solution contained in $ℱ$. In particular, if ${\left(U_B\left(t,s\right)\right)}_{t\ge s}$ is $p$-periodic there exists a unique bounded solution $u$ subject to certain spectral assumptions on $U_B\left(p,0\right),f$ and $u$. We apply the results to nonautonomous semilinear retarded differential equations. For certain $p$-periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of ${\left(U_B\left(t,s\right)\right)}_{t\ge s}$.

We study semilinear elliptic boundary value problems of one parameter dependence where the number of positive solutions is discussed. Our main purpose is to characterize the critical value given by the infimum of such parameters for which positive solutions exist. Our approach is based on super- and sub-solutions, and relies on the topological degree theory on the positive cones of ordered Banach spaces. A concrete example is also presented.