### ${\mathcal{C}}^{m}$-smoothness of invariant fiber bundles for dynamic equations on measure chains.

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Laplace transform and some of the author’s previous results about first order differential-recurrence equations with discrete auto-convolution are used to solve a new type of non-linear quadratic integral equation. This paper continues the author’s work from other articles in which are considered and solved new types of algebraic-differential or integral equations.