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

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{aligned} (Φ_{p} (x^{^{'}}))^{^{'}} + g (t) f (t, x) & = 0, t \in (0, 1), \\ x (0) - a x^{^{'}} (0) = α [x], & x (1) + b x^{^{'}} (1) = β [x], \end{aligned}

where

Φ_{p} (x) = | x |^{p - 2} x

is a one dimensional

p

-Laplacian operator with

p > 1, a, b

are real constants and

α, β

are the Riemann-Stieltjes integrals

\begin{array}{r} α [x] = \int_{0}^{1} x (t) d A (t), β [x] = \int_{0}^{1} x (t) d B (t), \end{array}

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