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Received Sep 30, 2017; Accepted Jan 4, 2018

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1. Introduction

In recent years, there has been a great development in the study of fractional differential equations. This advancement is ranging from the theoretical analysis of the subject to analytical and numerical techniques (see [1–3] and the references cited therein). Among the theoretic approach, the existence theory of solutions for fractional differential models has gained attentions of many authors. Most of them have focused on using Riemann-Liouville and Caputo derivatives in representing the underlying fractional differential equation (see [4–10]). Another kind of fractional derivative is Hadamard type which was introduced in 1892 [11]. This derivative differs from aforementioned derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains logarithmic function of arbitrary exponent. A detailed description of Hadamard fractional derivative and integral can be found in [12, 13]. Recently, the existence and uniqueness of solution for fractional differential equations in Hadamard sense were introduced in many faces by several authors ([6, 7] and references therein). We add in this article a new idea concerning the sequential definition of Hadamard fractional operator with constant coefficients of order less than three. For q∈(2,3), the Hadamard fractional differential equation can be transformed to classical Euler-Cauchy second-order nonhomogeneous differential equation that can be solved by variation of parameter technique.

More precisely, we consider the nonlinear Hadamard fractional differential equations given by(1)DHa+qxt=f1t,xt, q∈0,1,xa+=0,(2)DHa+q+γ DHa+q-1xt=f2t,xt, q∈1,2,xa+=x1a+=0,(3)DHa+q+λ1DHa+q-1+λ2DHa+q-2xt=f3t,xt, q∈2,3,x

     

   

    

           

          

             

 

  

 

    

  

   

 

  

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