We define a nonoriented coincidence index for a compact, fundamentally restrictible, and condensing multivalued perturbations of a map which is nonlinear Fredholm of nonnegative index on the set of coincidence points. As an application, we consider an optimal controllability problem for a system governed by a second-order integro-differential equation.

We study a multiplicity result for the perturbed $p$-Laplacian equation $-{\Delta }_{p}u-\lambda g\left(x\right){\left|u\right|}^{p-2}u=f\left(x,u\right)+h\left(x\right)\text{in}{ℝ}^N$, where and $\lambda$ is near ${\lambda }_1$, the principal eigenvalue of the weighted eigenvalue problem $-{\Delta }_{p}u=\lambda g\left(x\right){\left|u\right|}^{p-2}u$ in ${ℝ}^N$. Depending on which side $\lambda$ is from ${\lambda }_1$, we prove the existence of one or three solutions. This kind of result was firstly obtained by Mawhin and Schmitt (1990) for a semilinear two-point boundary value problem.

We study perturbations and continuity of the Drazin inverse of a closed linear operator $A$ and obtain explicit error estimates in terms of the gap between closed operators and the gap between ranges and nullspaces of operators. The results are used to derive a theorem on the continuity of the Drazin inverse for closed operators and to describe the asymptotic behavior of operator semigroups.