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A relation between matrix-valued complete Bernstein functions andmatrix-valued Stieltjes functions is applied to prove that the solutions ofmatricial convolution equations with extended LICM kernels belong to specialclasses of functions. In particular the cases of the solutions of theviscoelastic duality relation and the solutions of the matricial Sonineequation are discussed, with applications in anisotropic linear viscoelasticityand a generalization of fractional calculus.  In the first case it is in particular shown that duality of completelymonotone relaxation functions and Bernstein creep functions in general requiresinclusion in the relaxation function of a Newtonian viscosity term in additionto the memory effects represented by the completely monotone kernel.  We define anisotropic generalized fractional derivatives (GFD) by replacingthe kernel $t^{-\alpha}/\Gamma(1-\alpha)$ of the Caputo derivatives withcompletely monotone matrix-valued kernels which are weakly singular at 0.

zeitschrift für angewandte mathematik und physik

On solutions of matrix-valued convolution equations, anisotropic  fractional derivatives and their applications in linear and non-linear  anisotropic viscoelasticity