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Contraction conditions with perturbed linear operators and applications. (English) Zbl 1214.47051
Let $F$ be a nonempty closed subset of $BC\left(I,E\right)$ and $A:F\to F$ be an operator controlled by the contraction conditions with a perturbed linear operator. The authors establish a theorem which ensures that $A$ has a unique fixed point in $F$. The authors also show that some known fixed point theorems concerned with integral operators can be derived from their theorem. In addition, the authors obtain a multivalued version of their theorem. As applications, the existence and uniqueness of solutions of impulsive periodic boundary value problems and functional differential inclusions are exhibited in the last section.
##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 34B37 Boundary value problems for ODE with impulses 47N20 Applications of operator theory to differential and integral equations 47J22 Variational and other types of inclusions 47H04 Set-valued operators