A subordination principle on Wright functions and regularized resolvent families

TL;DR

This paper introduces a vector-valued subordination principle for regularized resolvent families, linking solutions of fractional Cauchy problems using Wright functions related to Mittag-Leffler functions and stable Lévy processes.

Contribution

It establishes a new subordination principle for regularized resolvent families, unifying and extending previous results in fractional calculus and operator theory.

Findings

New subordination relations for fractional Cauchy problems

Properties of Wright functions including subordination and convolution

Applications to examples demonstrating the theoretical results

Abstract

We obtain a vector-valued subordination principle for $(g_{\alpha}, g_{\alpha})$-regularized resolvent families which unified and improves various previous results in the literature. As a consequence we establish new relations between solutions of different fractional Cauchy problems. To do that, we consider scaled Wright functions which are related to Mittag-Leffler functions, the fractional calculus and stable L\'evy processes. We study some interesting properties of these functions such as subordination (in the sense of Bochner), convolution properties, and their Laplace transforms. Finally we present some examples where we apply these results.