Open AccessExistence and uniqueness of some Cauchy type problems in fractional q-difference calculus
Author(s) -
Serikbol Shaimardan,
Lars Erik Persson,
N.S. Tokmagambetov
Publication year - 2020
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil2013429s
Subject(s) - mathematics , fractional calculus , uniqueness , type (biology) , equivalence (formal languages) , cauchy's integral formula , generalizations of the derivative , derivative (finance) , operator (biology) , cauchy–riemann equations , pure mathematics , mathematical analysis , cauchy distribution , cauchy problem , initial value problem , ecology , biochemistry , chemistry , repressor , gene , transcription factor , financial economics , economics , biology
In this paper we derive a sufficient condition for the existence of a unique solution of a Cauchy type q-fractional problem (involving the fractional q-derivative of Riemann-Liouville type) for some nonlinear differential equations. The key technique is to first prove that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. Moreover, we define the q-analogue of the Hilfer fractional derivative or composite fractional derivative operator and prove some similar new equivalence, existence and uniqueness results as above. Finally, some examples are presented to illustrate our main results in cases where we can even give concrete formulas for these unique solutions.