Abstract
This paper is mainly concerned with a class of fractionalp,q-difference equations underp,q-integral boundary conditions. Multiple positive solutions are established by using the topological degree theory and Krein–Rutman theorem. Finally, two examples are worked out to illustrate the main results.
Highlights
Is paper is mainly concerned with a class of fractional (p, q)-difference equations under (p, q)-integral boundary conditions
The theory of quantum calculus based on two-parameter (p, q)-integer has been studied since it can be used efficiently in many fields such as difference equations, Lie group, hypergeometric series, and physical sciences. e (p, q)-calculus was first studied by Chakrabarti and Jagannathan [2] in the field of quantum algebra in 1991
We shall establish the existence and multiplicity results of BVP equation (1), which is based on the topological degree theory
Summary
We list some basic definitions and lemmas that will be used in this paper. Assuming that f: I ⟶ R is a given function, we define (p, q)-integral of f from a to b by b b a. E properties of (p, q)-difference operator are as follows:. Β ≥ 0 and 0 < q < p ≤ 1, (p, q)-integral and (p, q)-difference operators have the following properties:. For 0 ≤ t ≤ s ≤ 1 and 2 < α < 3, we know that G(t, qs) tα− 1(1 − qs)(α− 2) is obviously an increasing function with respect to t. Let Ω be a bounded open set in a Banach space E with 0 ∈ Ω, and T: Ω ⟶ E is a continuous compact operator. From Lemma 4, we can define operator T: E ⟶ E as follows:. From Krein–Rutman theorem, we know that L has a positive value λ1 e ig(ern(fLu)n)c− t1io, ni.eφ.,1φc1o rrλe1sLpoφn1.ding to its first eigen-
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