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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 -Laplacian equation , where and is near , the principal eigenvalue of the weighted eigenvalue problem in . Depending on which side is from , 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 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.