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**Fixed points, Volterra equations, and Becker’s resolvent.**
*(English)*
Zbl 1091.34040

Summary: In a recent paper we derived a stability criterion for a Volterra equation which is based on the contraction mapping principle. It turns out that this criterion has significantly wider application. In particular, when we use Becker’s form of the resolvent it readily establishes critical resolvent properties which have been very illusive when investigated by other techniques.

First, it enables us to show that the resolvent is \(L^1\). Next, it allows us to show that the resolvent satisfies a uniform bound and that it tends to zero. These properties are then used to prove boundedness of solutions of a nonlinear problem, establish the existence of periodic solutions of a linear problem, and to investigate asymptotic stability properties. We also apply the results to a Liénard equation with distributed delay and possibly negative damping so that relaxation oscillations may occur.

First, it enables us to show that the resolvent is \(L^1\). Next, it allows us to show that the resolvent satisfies a uniform bound and that it tends to zero. These properties are then used to prove boundedness of solutions of a nonlinear problem, establish the existence of periodic solutions of a linear problem, and to investigate asymptotic stability properties. We also apply the results to a Liénard equation with distributed delay and possibly negative damping so that relaxation oscillations may occur.

### MSC:

34K20 | Stability theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

47H10 | Fixed-point theorems |

34K12 | Growth, boundedness, comparison of solutions to functional-differential equations |