# Existence of positive solutions for a $p$-Laplacian equation with applications to Hematopoiesis

Document Type : Research Article

Authors

1 Department of Mathematics, Birla Institute of Technology, Mesra, Ranchi, India

2 Department of Mathematics, Florida Gulf Coast University FortMyres, Florida, USA

3 Department of Mathematics, National Institute of Technology Rourkela, India

Abstract

This paper is concerned with the existence of at least one   positive solution for a boundary value problem (BVP), with  $p$-Laplacian, of the form
\begin{equation*}
\begin{split}
(\Phi_p(x^{'}))^{'} + g(t)f(t,x)  &= 0, \quad t     \in (0,1),\\
x(1)+bx^{'}(1) = \beta[x],
\end{split}
\end{equation*}
where $\Phi_{p}(x) = |x|^{p-2}x$ is a one dimensional $p$-Laplacian operator with $p>1, a,b$ are real constants and $\alpha,\beta$ are  the Riemann-Stieltjes integrals
\begin{equation*}
\begin{split}
\alpha[x] = \int \limits_{0}^{1} x(t)dA(t), \quad  \beta[x] = \int \limits_{0}^{1} x(t)dB(t),
\end{split}
\end{equation*}
with $A$ and $B$ are functions of bounded variation. A Homotopy version of  Krasnosel'skii fixed point theorem is used to prove our results.

Keywords