POSITIVE SOLUTIONS FOR HADAMARD FRACTIONAL BOUNDARY VALUE PROBLEMS ON AN INFINITE INTERVAL

In this paper, we consider a boundary value problem of Hadamard fractional differential equations on an infinite interval {(H)D(alpha)u(t) + lambda a(t)F(t, u(t)) = 0,1 < alpha < 2, t is an element of(1,infinity), u(1) = 0, (H)D(alpha-1)u(infinity) = Sigma(m)(i-1) gamma(H)(i)I(beta i)u(eta), where D-H(alpha) is the Hadamard fractional derivative of order alpha, eta is an element of (1, infinity), I-H(center dot) denotes the Hadamard fractional integral, lambda > 0 is a parameter, beta(i), gamma(i) >= 0(i = 1, 2, ..., m) are constants and Gamma(alpha)>Sigma(m)(i=1) gamma i Gamma(alpha)/Gamma(alpha+beta(i)) (log eta)(alpha+beta i-1) . The existence and uniqueness of positive solutions is given for each fixed lambda > 0. The relations between the positive solution and the parameter l are presented. The results obtained in this paper show that the unique positive solution u(lambda)* has good properties: continuity, monotonicity, iteration and approximation. The method of this paper is based upon different fixed point theorems and properties for two types of operators: monotone operators and mixed monotone operators. Finally, two examples are also provided.