Existence and exact asymptotic behaviour of positive solutions for fractional boundary value problem with P-Laplacian operator

This paper deals with existence, uniqueness and global behaviour of a positive solution for the fractional boundary value problem D-beta(psi(x)phi(p)(D(alpha)u)) = a(x)u(sigma) in (0, 1) with the condition lim(x -> 0) D beta-1(psi(x)Phi(p)(D(alpha)u(x))) = lim(x -> 1)psi(x)Phi(p)(D(alpha)u(x)) = 0 and lim(x -> 0) D(alpha-1)u(x) = u(1) = 0, where beta, alpha. (1, 2], Phi(p)(t) = t vertical bar t vertical bar p(-2), p> 1, sigma is an element of(1 - p, p - 1), the differential operator is taken in the Riemann-Liouville sense and psi, a : (0, 1) -> R are non-negative and continuous functions that may are singular at x = 0 or x = 1 and satisfies some appropriate conditions.