Van Hien Pham, The second-order Cauchy problem in a scale of Banach spaces with vector-valued measures of noncompactness and an application to Kirchhoff equations, Vol. 2023 (2023), Article ID 10, pp. 1-11

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DOI: 10.23952/jnfa.2023.10

Received January 8, 2022; Accepted August 26, 2022; Published March 22, 2023

Abstract. In the paper, by using Darbo-Sadovskii fixed point theorem for condensing operators on Fréchet spaces with respect to vector-valued measure of noncompactness, we prove the existence results for the second-order Cauchy problem $u''(t)=f(t,u(t)),$ $t\in(0,T),$ $u(0)=u_0,$ $u'(0)=u_1$ , in a scale of Banach spaces. The result is applied to the Kirchhoff equations in Gevrey class.

How to Cite this Article:
V.H. Pham, The second-order Cauchy problem in a scale of Banach spaces with vector-valued measures of noncompactness and an application to Kirchhoff equations, J. Nonlinear Funct. Anal. 2023 (2023) 10.