ON THE EXISTENCE OF SOLUTIONS FOR NONLINEAR FRACTIONAL q-DIFFERENCE EQUATIONS WITH q-INTEGRAL BOUNDARY CONDITIONS

In this paper, we investigate the existence of solutions for nonlinear fractional q-difference equation cDα q u(t) = f(t, u(t),Dqu(t)) with the boundary value conditions λ1u(0) +μ1Dqu(0) = η1I^β q u(ξ1) and λ2u(1)+μ2Dqu(1) = η2IW^β q u(ξ2), where 1 ≤ α ≤ 2, 0 q 1 and cDα q denotes the fractional q-derivative of the Caputo type of order α. By applying the some fixed point theorems, we prove our main results. Also, an example is presented.