SUPERLINEAR SINGULAR FRACTIONAL BOUNDARY-VALUE PROBLEMS

In this article, we study the superlinear fractional boundary-value problem D(alpha)u(x) = u(x)g(x, u(x)), 0 < x < 1, u(0) = 0, lim D(alpha-3)u(x) = 0, lim D(alpha-2)u(x) = xi, u ''(1) = zeta xo+ xo+ where 3 < alpha <= 4, D-alpha is the Riemann-Liouville fractional derivative and xi, zeta >= 0 are such that xi + zeta > 0. The function g(x, u) epsilon C((0, 1) x [0, infinity), [0, infinity)) that may be singular at x = 0 and x = 1 is required to satisfy convenient hypotheses to be stated later. By means of a perturbation argument, we establish the existence, uniqueness and global asymptotic behavior of a positive continuous solution to the above problem.An example is given to illustrate our main results.