Abstract
In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form
(P-a,P- b) {D(alpha)u(x) + f(x, u(x)) = 0, x is an element of (0, 1),
u(0) = u(1) = 0, D alpha-3 u(0) = a, u'(1) = -b,
where 3 < alpha <= 4, D-alpha is the standard Riemann-Liouville fractional derivative and a, b are non-negative constants. First the authors suppose that f(x, t) = -p(x) t(alpha), with sigma is an element of(-1, 1) and p being a nonnegative continuous function that may be singular at x = 0 or x = 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Schauder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem (P-0,P-0). Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b > 0 and f(x, t) = t phi(x, t), with phi(x, t) being a nonnegative continuous function in (0, 1) x [0, 8) that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem (P-a,P- b), which behaves like the unique solution of the homogeneous problem corresponding to (P-a,P- b). Some examples are given to illustrate the existence results.