Existence of local solutions of the Kirchhoff–Carrier equation in Banach spaces

This paper is concerned with the study of the existence of a local solution of the problem (∗)|Bu″(t)+M(‖u(t)‖Wβ)Au(t)=0,in V′,t>0,u(0)=u0,u′(0)=u1(u0≠0), where VV is a Hilbert space with dual V′V′; AA and BB symmetric linear operators from VV into V′V′ with 〈Av,v〉≥0〈Av,v〉≥0 and 〈Bv,v〉>0,v≠0〈Bv,v〉>0,v≠0; WW a Banach space with VV continuously embedding in WW; ββ a real number with β≥1β≥1; and M(ξ)M(ξ) a function with M(ξ)≥0,M(‖u0‖Wβ)>0 and smooth in a neighborhood of ‖u0‖Wβ.The characterization of the derivative of the nonlinear term of the equation of (∗) and the Arzela–Ascoli Theorem allow us to obtain a solution uu of (∗) defined in [0,T0][0,T0] where T0T0 depends on u0,u1u0,u1 and M(ξ)M(ξ).