^{1}Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran.

^{2}Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.

Abstract

In this paper, first, we investigate the construction of compact sets of $ C^k$ and $ C_0^k$ by proving ``$C^k, C_0^k-version$" of Arzel\`{a}-Ascoli theorem, and then introduce new measures of noncompactness on these spaces. Finally, as an application, we study the existence of entire solutions for a class of the functional integral-differential equations by using Darbo's fixed point theorem associated with these new measures of noncompactness. Further, some examples are presented to show the efficiency of our results.

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