Existence result and conservativeness for a fractional order non-autonomous fragmentation dynamics

Volume 9, Issue 11, pp 5850--5861 http://dx.doi.org/10.22436/jnsa.009.11.13

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Authors

Emile Franc Doungmo Goufo - Department of Mathematical Sciences, University of South Africa, Florida Sciences Campus, 003, South Africa. Morgan Kamga Pene - Department of Mathematical Sciences, University of South Africa, Florida Sciences Campus, 003, South Africa. Jeanine N. Mwambakana - Department of Science, Mathematics and Technology Education, University of Pretoria, Pretoria, South Africa.

Abstract

We use the subordination principle together with an equivalent norm approach and semigroup perturbation theory to state and set conditions for a non-autonomous fragmentation system to be conservative. The model is generalized with the Caputo fractional order derivative and we assume that the renormalizable generators involved in the perturbation process are in the class of quasi-contractive semigroups, but not in the class \(\mathcal{G}(1; 0)\) as usually assumed. This, thenceforth, allows the use of admissibility with respect to the involved operators, Hermitian conjugate, Hille-Yosida's condition and the uniform boundedness to show that the operator sum is closable, its closure generates a propagator (evolution system) and, therefore, a \(C_0\)-semigroup, leading to the existence result and conservativeness of the fractional model. This work brings a contribution that may lead to the full characterization of the infinitesimal generator of a \(C_0\)-semigroup for fractional non-autonomous fragmentation and coagulation dynamics which remain unsolved.

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Keywords

• Evolution system
• propagator
• semigroup perturbation
• renormalization
• fractional non-autonomous fragmentation
• conservativeness.

MSC

•  26A33
•  37N25
•  49K40

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