THE EXISTENCE NONTRIVIAL AND EXTREMAL SOLUTIONS FOR HADAMARD FRACTIONAL DIFFERNENTIAL EQUATION WITH INTEGRAL AND INFINITE-POINT BOUNDARY VALUE CONDITION ON AN INFINITE DOMAIN

It is a qualitative leap to extend the boundary value problem from finite interval to infinite interval, this is because the fractional-order equation on infinite interval is more widely referenced, but the compactness of the interval on infinite interval is more difficult to prove, so there are not many studies on the upper value problem of infinite interval. In this paper, derivation of the expression of Green's function of fractional differential system on infinite interval, fractional derivative and its properties, then the problem of proving the compactness of operators on infinite intervals is overcome by establishing a suitable cone. Subsequently, the existence, uniqueness of nontrivial solution and positive extremal solutions are obtained by monotone iterative technique for Hadamard fractional differential system with integral and infinite-point boundary value condition on an infinite domain, moreover, the maximum and minimum solutions are obtained, and two iterative sequences which converge to the extremal solution pairs on an infinite interval. Finally, we give several examples to prove the validity of the theory.