\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage{natbib}




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%

\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}



\usepackage{float}








\begin{document}

\title{Multiple Solutions for a Kirchhoff-type Fractional Coupled Problem with
p-Laplacian}



\author[1]{yi wany}%
\author[1]{Lixin Tian}%
\author[2]{Dong Min-Jie}%
\affil[1]{School of Mathematical Sciences}%
\affil[2]{School of physical and mathematical sciences}%


\vspace{-1em}



  
  \date{May 21, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
In this paper, we look at a class of two-parameter coupled
Kirchhoff-type fractional differential equations. Two differentiated
methods are used to prove the existence of two solutions to the
equation. The fundamental difference between the two methods is that the
first provides asymptotic conditions for the non-linear terms on the
right-hand side of the equation, while the second provides algebraic
conditions; both methods combine substantial A-R conditions.%
\end{abstract}%



\sloppy


\textbf{Hosted file}

\verb`MMA.rar`  available at \url{https://authorea.com/users/491543/articles/574512-multiple-solutions-for-a-kirchhoff-type-fractional-coupled-problem-with-p-laplacian}

\selectlanguage{english}
\FloatBarrier
\end{document}

