Infinitely many positive solutions for an iterative system of conformable fractional order dynamic boundary value problems on time scales

Abstract

In this paper, we establish infinitely many positive solutions for the iterative system of conformable fractional order dynamic equations on time scales T ∆ α [ T ∆ β ( ϑn(t) )] = φ(t)fn (ϑn+1(t)), t ∈ (0, 1)T, 1 < α, β ≤ 2, ϑ1(t) = ϑℓ+1(t), t ∈ (0, 1)T, n = 1, 2, · · · , ℓ, satisfying two-point Riemann–Stieltjes integral boundary conditions ϑn(0) = 0, ϑn(1) = ∫ 1 0 ϑn(τ)✷g(τ), n = 1, 2, · · · , ℓ, (T ∆ β ϑn)(0) = 0, (T ∆ β ϑn)(1) = ∫ 1 0 (T ∆ β ϑn)(τ)✷g(τ), n = 1, 2, · · · , ℓ, where T ∆ ⋆ denotes the conformable fractional derivative of order ⋆ ∈ {α, β} on time scale T, by an application of Krasnoselskii’s fixed point theorem on a Banach space.