EXISTENCE RESULTS FOR NONLINEAR IMPULSIVE qk-INTEGRAL BOUNDARY VALUE PROBLEMS

u(T ) = ∑m i=0 ∫ ti+1 ti g(s, u(s)) dqis, where Dqk are qk-derivatives (k = 0, 1, 2, . . . ,m), f, g ∈ C(J ×R, R), Ik ∈ C(R,R), J = [0, T ](T > 0), 0 = t0 < t1 < · · · < tk < · · · < tm < tm+1 = T , J ′ = J\{t1, t2, . . . , tm}, and ∆u(tk) = u(t + k ) − u(t − k ), u(t + k ) and u(t − k ) denote the right and the left limits of u(t) at t = tk (k = 1, 2, . . . ,m) respectively. The study of q-difference equations, initiated with the pioneer work of Jackson [1], has been developed over the years. The concept of q-calculus corresponds to the classical calculus without the idea of limit. This subject is also known as quantum calculus and finds its applications in a variety of disciplines such as special functions, super-symmetry, control theory, operator theory, combinatorics, initial and boundary value problems of q-difference equations, etc. For the systematic development of q-calculus, we refer the reader to the books [2–4] and papers [5–10].