Existence of solutions for an infinite system of tempered fractional order boundary value problems in the spaces of tempered sequences

Abstract

This paper deals with infinite system of nonlinear two-point tempered fractional order boundary value problems $_{0}^{RL}textrm{}mathbb{D}_{mathbf{z}}^{delta _{2},ell}left [mathbf{p}_{j}(mathbf{z})_{0}^{RL}textrm{}mathbb{D}_{mathbf{z}}^{delta _{1},ell}vartheta _{j}(mathbf{z}) right ]=lambda _{j}varphi(mathbf{z},vartheta(mathbf{z})),mathbf{z}in left [ 0, mathbf{T}right ],delta _{1},delta _{2}in left ( 1,2 right ),$ $vartheta _{j}(0)=lim_{mathrm{z} to 0}left [ _{0}^{RL}textrm{}mathbb{D}_{mathrm{z}}^{delta _{1},ell}left ( e^{ellmathrm{z}}vartheta _{j}(mathrm{z}) right ) right ]=0,$ $e^{ellmathrm{T}}vartheta _{j}(mathbf{T})=lim_{mathrm{z} to mathrm{T} }left [ _{0}^{RL}textrm{}mathbb{D}_{mathrm{z}}^{delta _{1},ell}left ( e^{ellmathrm{z}}vartheta _{j}(mathrm{z}) right ) right ]=0,$ where $j inleft { 1,2,3,... right }, ellgeqslant 0,_{0}^{RL}textrm{}mathbb{D}_{mathrm{z}}^{star ,ell}$ denotes the Riemann–Liouville tempered fractional derivative of order $starinleft { delta _{1},delta _{2} right } , vartheta (mathbf{z})=(vartheta_{j}(mathbf{z}))_{j=1}^{infty },varphi _{j}:left [ 0,mathbf{T} right ]rightarrow left [ 0,mathbf{T} right ]$ are continuous and we derive sufficient conditions for the existence of solutions to the system via the Hausdorff measure of noncompactness and Meir–Keeler fixed point theorem in tempered sequence spaces.