On the Existence and Uniqueness of a Nonlinear Q-Difference Boundary Value Problem of Fractional Order

Abstract

In this research collection, we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional q-difference equation having the given form (c)D(q)(zeta)v(t) - h(t,v(t)) = 0, 0 <= t <= 1, alpha(1)v(0) + beta(1)D(q)v(0)=v(eta(1)), alpha(2)v(1) - beta(2)D(q)v(1) = v(eta(2)), where 1 < zeta <= 2, (eta(1), eta(2)) is an element of (0, 1)(2), alpha(i), beta(i) is an element of R(i = 1, 2), h is an element of C([0, 1] x R, R) and D-c(q)zeta represents the Caputo-type nonclassical q-derivative of order zeta. We use well-known principal of Banach contraction, and Leray-Schauder nonlinear alternative to vindicate the unique solution existence of the given problem. Regarding the applications, some examples are solved to justify our outcomes.