Asymptotic behavior of positive solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivative

In this paper we are concerned with the problem of asymptotic integration of positive solutions of higher order fractional differential equations with Caputo-type Hadamard derivative of the form C , H D r a x ( t ) = e ( t ) + f ( t , x ( t ) ) , a > 1 , where r = n + α − 1 , α ∈ ( 0 , 1 ) , n ∈ Z + . We shall apply our technique to investigate the oscillatory and asymptotic behavior of all solutions of the integral equation x ( t ) = e ( t ) + ∫ t a ( ln t s ) r − 1 k ( t , s ) f ( s , x ( s ) ) d s s , a > 1 , r is as above.