Upper and Lower Solution Method for Fractional Boundary Value Problems on the Half-Line

where D 0 denotes the Riemann Liouville fractional derivative of order q, 1 < q < 2, η > 0, f : [0,∞) × R → R and e : [0,∞) → R are given functions satisfying some conditions. The problem (1.1)–(1.2) happens to be at resonance in the sense that the dimension of the kernel of the linear operator Lx = D 0+x is not less than one under boundary conditions (1.2). It should be noted that in recent years, there have been many works related to boundary value problems at resonance for ordinary differential equations. We refer the reader to [7–9,11,13,14,16] and the references therein.