Khuddush MAHAMMAD, Rajendra Prasad KAPULA, Leela DODDI

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

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.

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