Generation theorems for fractional resolvent families

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Yun-Yi Mu ◽  
Miao Li

2018 ◽  
Vol 99 (2) ◽  
pp. 293-302
Author(s):  
Jie Mei ◽  
Chuang Chen ◽  
Miao Li


2011 ◽  
Vol 384 (2) ◽  
pp. 453-467 ◽  
Author(s):  
Chuang Chen ◽  
Miao Li ◽  
Fu-Bo Li


2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Nan-Ding Li ◽  
Ru Liu ◽  
Miao Li

This paper is concerned with positive α -times resolvent families on an ordered Banach space E (with normal and generating cone), where 0 < α ≤ 2 . We show that a closed and densely defined operator A on E generates a positive exponentially bounded α -times resolvent family for some 0 < α < 1 if and only if, for some ω ∈ ℝ , when λ > ω , λ ∈ ρ A , R λ , A ≥ 0 and sup λ R λ , A : λ ≥ ω < ∞ . Moreover, we obtain that when 0 < α < 1 , a positive exponentially bounded α -times resolvent family is always analytic. While A generates a positive α -times resolvent family for some 1 < α ≤ 2 if and only if the operator λ α − 1 λ α − A − 1 is completely monotonic. By using such characterizations of positivity, we investigate the positivity-preserving of positive fractional resolvent family under positive perturbations. Some examples of positive solutions to fractional differential equations are presented to illustrate our results.



2017 ◽  
Vol 11 (1) ◽  
pp. 39-61 ◽  
Author(s):  
Marko Kostic

In the paper under review, we investigate a class of abstract degenerate fractional differential inclusions with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.



2010 ◽  
Vol 259 (10) ◽  
pp. 2702-2726 ◽  
Author(s):  
Miao Li ◽  
Chuang Chen ◽  
Fu-Bo Li


1985 ◽  
Vol 188 (4) ◽  
pp. 449-474 ◽  
Author(s):  
Walter R. Bloom ◽  
Herbert Heyer


2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Rodrigo Ponce

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases0<α<1and1<α<2.



2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Luciano Abadias ◽  
Pedro J. Miana

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.



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