On a class of nonlinear convolution equations
The main result is an existence and uniqueness statement for operator equations of the second kind \(u + G_0u + G_1 u*G_2u = g\), where \(G_i\) \((i = 0,1,2)\) are Lipschitz continuous operators on either \(L_p (0,T; X)\) or \(C(0,T; X)\), equipped with certain exponentially weighted norms. The proofs are based on the contraction mapping principle. Several corollaries are also given, dealing with generalized versions of the equation above. One motivation for these results comes from an inverse problem, where one tries to identify memory kernels in viscoelasticity and heat transfer. Some further examples involve equations of auto-convolution type, nonlinear stress-strain relations in viscoelasticity, and nonlinear Volterra integro-differential equations.
nonlinear Volterra integro-differential equations
scientific article; zbMATH DE number 800342