A Subordination Principle on Wright Functions and Regularized Resolvent Families

Luciano Abadias,Pedro J Miana

doi:10.1155/2015/158145

Luciano Abadias, Pedro J Miana

Open Access

https://doi.org/10.1155/2015/158145

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Abstract

We obtain a vector-valued subordination principle forgα,gβ-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 Lévy 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.

Highlights

A function f : (0, ∞) → R is a Bernstein function if f is of class C∞, f(λ) ≥ 0 for all λ > 0 and (−1)n−1 f(n) (λ) ≥ 0, λ > 0, n ∈ N. (1)The celebrated Bochner subordination theorem characterizes Bernstein functions: given f, a Bernstein function, there exists a unique convolution semigroup of subprobability measurest>0 on [0, ∞) such that ∞e−tf(λ) = ∫ e−λsdμt (s), Rλ > 0. (2)given a convolution semigroup of subprobability measurest>0 on [0, ∞), there exists a unique Bernstein function f such that (2) holds true; see, for example, [1, Theorem 5.2]
We introduce a new family of biparameter special functions ψα,β in two variables defined by scaling Wright functions (Definition 2). This family of functions ψα,β plays a fundamental role in the subordination principle for (tα−1/Γ(α), tβ−1/Γ(β))-regularized resolvent families; see formula (51)
We prove the main subordination principle, Theorem 12, and some consequences in Remark and Corollary

Summary

Introduction

A function f : (0, ∞) → R is a Bernstein function if f is of class C∞, f(λ) ≥ 0 for all λ > 0 and (−1)n−1 f(n) (λ) ≥ 0, λ > 0, n ∈ N. In [11, Theorem 3.1], using holomorphic functional calculus, the authors prove a subordination result for (tα−1/Γ(α), 1)-regularized resolvent families generated by fractional powers of closed operators, which extends both [4, Chapter IX, Section 11, Theorem 2] and [9, Theorem 3.1]. This family of functions ψα,β plays a fundamental role in the subordination principle for (tα−1/Γ(α), tβ−1/Γ(β))-regularized resolvent families; see formula (51). These functions satisfy a nice subordination formula, Theorem 8, which extends some known results for Wright M-function and stable Levy processes; see Remark 9. We denote by X an abstract Banach space, B(X) the set of linear and bounded operators on the Banach space X, and Cc(∞)(R+; X) the set of functions of compact support and infinitely differentiable on R+ into X

Mittag-Leffler and Wright Functions

Scaled Wright Functions

Subordination Principle for Regularized Resolvent Families

12. Let 0 generates

Multiplication Families multiplication semigroup in on Rn