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Existence and Uniqueness of Solutions for the Initial Value Problem of Fractional $q_{k}$-Difference Equations for Impulsive with Varying Orders
  
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KeyWord:Impulsive fractional $q_{k}$-difference equation; Boundary value problem; Existence; Uniqueness
Author NameAffiliation
Lulu Zhang School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China 
Fanjun Li School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China 
Zhenlai Han School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China 
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Abstract:
      The paper studies the existence and uniqueness for impulsive fractional $q_{k}$-difference equations of initial value problems involving Riemann-Liouville fractional $q_{k}$-integral and $q_{k}$-derivative by defining a new $q$-shifting operator. In this paper, we obtain existence and uniqueness results for impulsive fractional $q_{k}$-difference equations of initial value problems by using the Schaefer's fixed point theorem and Banach contraction mapping principle. In addition, the main result is illustrated with the aid of several examples.