Uniqueness of solutions to the coagulation–fragmentation equation with singular kernel

The existence of a solution to an important singular coagulation equation with a multiple fragmentation kernel has been recently proved in Jpn J Ind Appl Math 35(3):1283–1302, 2018. This paper proves the uniqueness of the solution to the same problem in the function space Ω.,r2(T)=⋃λ>0Ωλ,r2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega _{.,r_2} (T) = \bigcup _{\lambda >0 }\varOmega _{\lambda , r_2} (T)$$\end{document}, where Ωλ,r2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega _{\lambda , r_2} (T)$$\end{document} is the space of all continuous functions f such that ‖f‖λ,r2:=sup0≤t≤T∫0∞exp(λx)+1xr2|f(x,t)|dx<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert f\Vert _{\lambda , r_2} : = \sup \limits _{0\le t \le T} \int _0^{\infty } \left( \exp (\lambda x ) + \frac{1}{x^{r_2}}\right) |f(x,t)| dx ~<~ \infty \end{aligned}$$\end{document}and 0<r2<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< r_2 < 1$$\end{document}.