Existence results for some Kirchhoff–Carrier problems

Analyzing the viscoelastic problem for small vibrations of elastic strings, Kirchhoff and Carrier proposed two different models of nonlinear partial differential equations. By combining these two models, we deal here with some nonlocal hyperbolic problems that cover a large class of Kirchhoff and Carrier type problems. The existence of local solutions of degenerate problems as well as local and nonlocal solutions of nondegenerate problems is established. The proofs are based on the combination of the Schauder fixed point theorem with some asymptotic method.