Positive solutions for a new class of Hadamard fractional differential equations on infinite intervals

In the present article, the following nonlinear problem of new Hadamard fractional differential equations on an infinite interval { D ν H x ( t ) + b ( t ) f ( t , x ( t ) ) + c ( t ) = 0 , 1 < ν < 2 , t ∈ ( 1 , ∞ ) , x ( 1 ) = 0 , H D ν − 1 x ( ∞ ) = ∑ i = 1 m γ i H I β i x ( η ) , $$ \textstyle\begin{cases} {}^{H}\!D^{\nu }x(t)+ b(t)f(t,x(t))+c(t)=0,\quad 1< \nu < 2, t\in (1,\infty), \\ x(1)=0,\qquad {}^{H}\!D^{\nu -1}x(\infty )=\sum_{i=1}^{m}\gamma _{i} {}^{H}\!I^{\beta _{i}}x(\eta ), \end{cases} $$ is studied, where D ν H ${}^{H}\!D^{\nu }$ denotes the Hadamard fractional derivative of order ν, I H ( ⋅ ) ${}^{H}\!I(\cdot )$ is the Hadamard fractional integral, β i , γ i ≥ 0 $\beta _{i}, \gamma _{i}\geq 0$ ( i = 1 , 2 , … , m $i=1,2,\ldots , m$ ), η ∈ ( 1 , ∞ ) $\eta \in (1,\infty )$ are constants and Γ ( ν ) > ∑ i = 1 m γ i Γ ( ν ) Γ ( ν + β i ) ( log η ) ν + β i − 1 . $$ \varGamma (\nu )>\sum_{i=1}^{m} \frac{\gamma _{i}\varGamma (\nu )}{ \varGamma (\nu +\beta _{i})}(\log \eta )^{\nu +\beta _{i}-1}. $$ By making use of a fixed point theorem for generalized concave operators, the existence and uniqueness of positive solutions is established. Moreover, an application of the established results is also presented via an interesting example.