Infinite system of nonlinear tempered fractional order BVPs in tempered sequence spaces

Abstract

Abstract This paper deals with the existence results of the infinite system of tempered fractional BVPs D r ϱ , λ 0 R z j ( r ) + ψ j ( r , z ( r ) ) = 0 , 0 < r < 1 , z j ( 0 ) = 0 , 0 R D r m , λ z j ( 0 ) = 0 , b 1 z j ( 1 ) + b 2 0 R D r m , λ z j ( 1 ) = 0 , $$\begin{aligned}& {}^{\mathtt{R}}_{0}\mathrm{D}_{\mathrm{r}}^{\varrho , \uplambda} \mathtt{z}_{\mathtt{j}}(\mathrm{r})+\psi _{\mathtt{j}}\bigl(\mathrm{r}, \mathtt{z}(\mathrm{r})\bigr)=0,\quad 0< \mathrm{r}< 1, \\& \mathtt{z}_{\mathtt{j}}(0)=0,\qquad {}^{\mathtt{R}}_{0} \mathrm{D}_{ \mathrm{r}}^{\mathtt{m}, \uplambda} \mathtt{z}_{\mathtt{j}}(0)=0, \\& \mathtt{b}_{1} \mathtt{z}_{\mathtt{j}}(1)+\mathtt{b}_{2} {}^{ \mathtt{R}}_{0}\mathrm{D}_{\mathrm{r}}^{\mathtt{m}, \uplambda} \mathtt{z}_{\mathtt{j}}(1)=0, \end{aligned}$$ where j ∈ N $\mathtt{j}\in \mathbb{N}$ , 2 < ϱ ≤ 3 $2<\varrho \le 3$ , 1 < m ≤ 2 $1<\mathtt{m}\le 2$ , by utilizing the Hausdorff measure of noncompactness and Meir–Keeler fixed point theorem in a tempered sequence space.

Keywords

• Tempered fractional derivative
• Measure of noncompactness
• Meir–Keeler fixed point theorem
• Tempered sequence spaces