regulated functions space $R(\mathbb{R_+},\mathbb{R^\infty})$ and its application to some infinite systems of fractional differential equations via family of measures of noncompactness
Abstract
We study the solvability of following infinite systems of fractional boundary value problem
\begin{footnotesize}\begin{equation*}\begin{cases}^cD^{\rho}u_i(t)=f_i(t,u_i(t))),\ \rho\in(n-1,n),\ 0<t<+\infty,\ &\quad \\u_i(0)=0,\ u_i^q(0)=0,\ ^cD^{\rho-1}u_i(\infty)=\displaystyle\sum_{j=1}^{m-2}\beta_j u_i(\xi_j). &\quad \end{cases}\end{equation*}\end{footnotesize}
The purpose of this work is to present a new family of measures of noncompactness in the regulated function spaces $R(\mathbb{R_+},\mathbb{R^\infty})$ on unbounded interval and a fixed point theorem of Darbo type. Finally, we give an example to show the effectiveness of the obtained result.
\begin{footnotesize}\begin{equation*}\begin{cases}^cD^{\rho}u_i(t)=f_i(t,u_i(t))),\ \rho\in(n-1,n),\ 0<t<+\infty,\ &\quad \\u_i(0)=0,\ u_i^q(0)=0,\ ^cD^{\rho-1}u_i(\infty)=\displaystyle\sum_{j=1}^{m-2}\beta_j u_i(\xi_j). &\quad \end{cases}\end{equation*}\end{footnotesize}
The purpose of this work is to present a new family of measures of noncompactness in the regulated function spaces $R(\mathbb{R_+},\mathbb{R^\infty})$ on unbounded interval and a fixed point theorem of Darbo type. Finally, we give an example to show the effectiveness of the obtained result.
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