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

Authors: MAHAMMAD KHUDDUSH, KAPULA RAJENDRA PRASAD

Abstract: In this paper, we establish infinitely many positive solutions for the iterative system of conformable fractional order dynamic equations on time scales $$ \begin{aligned} &\mathcal{T}_α^{\Delta}\big[\mathcal{T}_β^{\Delta}\big(\vartheta_\mathtt{n}(t)\big)\big]=\varphi(t)\mathtt{f}_\mathtt{n}\left(\vartheta_{\mathtt{n}+1}(t)\right),~t\in(0,1)_\mathbb{T},~1<α, β ≤ 2,\\ &\hskip1.3cm\vartheta_1(t)=\vartheta_{\ell+1}(t),~t\in(0,1)_\mathbb{T},~n=1,2,\cdots,\ell, \end{aligned} $$ satisfying two-point Riemann--Stieltjes integral boundary conditions $$ \begin{aligned} &\vartheta_\mathtt{n}(0)=0,~\vartheta_\mathtt{n}(1)=\int_0^1\vartheta_\mathtt{n}(τ)\Box\mathtt{g}(τ),~n=1,2,\cdots,\ell,\\ (\mathcal{T}_β^{\Delta}&\vartheta_\mathtt{n})(0)=0,~(\mathcal{T}_β^{\Delta}\vartheta_\mathtt{n})(1)=\int_0^1(\mathcal{T}_β^{\Delta}\vartheta_\mathtt{n})(τ)\Box\mathtt{g}(τ),~n=1,2,\cdots,\ell, \end{aligned} $$ where $\mathcal{T}_\star^{\Delta}$ denotes the conformable fractional derivative of order * ∈ α, β on time scale $\mathbb{T},$ by an application of Krasnoselskii's fixed point theorem on a Banach space.

Keywords: Conformable fractional derivative, time scale, positive solution, fixed point theorem, cone

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