Sentence examples for basic results of fractional from inspiring English sources

Here we recall definition and basic results of fractional calculus; for more details, we refer to [32].

We recall the following definitions of a fractional order integral and a fractional order derivative in Caputo's sense, and some basic results of fractional calculus [2, 13].

We recall the following definitions of fractional order integral, fractional order derivative in Caputo's sense and some basic results of fractional calculus [2, 3].

In this paper we will need the definitions of a fractional order integral and the fractional order derivative in the Riemann-Liouville sense and some basic results of fractional calculus.

For some basic results on fractional calculus, we refer to [26 28, 44].

Concerning the history and basic results on fractional calculus theory, we refer to [1 5].

We first review some basic results about fractional sums and differences.

In Section 2, we introduce some preliminary results, including basic definitions of fractional integrals and derivatives, some properties and a fixed point theorem.

In this paper, we initiate the basic theory of fractional hybrid differential equations of mixed perturbations of second type involving three nonlinearities and prove the basic result such as the strict and nonstrict fractional differential inequalities, an existence theorem and maximal and minimal solutions etc.

To conclude the paper, the feasibility of some of the results is illustrated with concrete examples in Section 4. We first introduce some basic definitions of fractional derivative for the readers' convenience.

In Section 2, we recall some basic definitions of fractional calculus and present an auxiliary lemma, which plays a pivotal role in obtaining the main results presented in Section 3.