On fractional integral operator over non-Newtonian calculus

Abstract

The definition of a non-Newtonian calculus is based on the homeomorphism which customary denoted by y = ?(x). In the mean of this function, elementary algebraic operations can be modified and we reach to the world of new calculus that is called a Non-Newtonian calculus. Nowadays, fractional operators role an important topic in mathematics because of their applications in many area of interest. In this paper we use an old technique of Cauchy iterated integrals to define bi?-fractional integral operator. The allocated method makes the new class of fractional integral operators which are successfully compatible with the non-Newtonian calculi and supported with several examples. Since the non-Newtonian calculi were introduced, the bigeometric calculus has been considered as a brilliant example of these kind of calculi. The definition of fractional integral operator in this calculus leads to Hadamard type fractional integral operator which answers many questions about the behavior of this operator. Classic property of fractional integral operator, semigroup property is stablished and this operator is studied. Moreover, Jensen’s inequality provide boundness theorem for general bi?-fractional integral operator.