Abstract
This paper is concerned with the two-point boundary value problems of a nonlinear fractional q-difference equation with dependence on the first order q-derivative. We discuss some new properties of the Green function by using q-difference calculus. Furthermore, by means of Schauder’s fixed point theorem and an extension of Krasnoselskii’s fixed point theorem in a cone, the existence of one positive solution and of at least one positive solution for the boundary value problem is established.
Highlights
The term “q-difference” refers to quantum difference
The topic of the fractional quantum difference equation has attracted the attention of many researchers in recent years
By Lemma 2.4, we find that T has a fixed point y in ( 2 \ ̄ 1) ∩ P, that is, problem (1.1) has at least one positive solution y(t) satisfying c < α(y) < b, Dqy(t) < L
Summary
The term “q-difference” refers to quantum difference. Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. Definition 2.1 ([24]) Let α ≥ 0 and f be a function defined on [0, 1].
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