# Positive Solutions for Four-Point Boundary Value Problem Involving the $$p(t)$$
p
(
t
) -Laplacian

Research paper by **Dehong Ji**

Indexed on: **13 Aug '16**Published on: **01 Apr '16**Published in: **Qualitative Theory of Dynamical Systems**

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#### Abstract

Abstract
In this paper, by means of the computation of fixed point index, we obtain the existence of positive solutions for four-point boundary value problem involving the
\(p(t)\)
-Laplacian
$$\begin{aligned}&(\phi (t,u'(t)))'+ f(t,u(t))=0,\quad t\in (0,1), \\&u'(0)=\alpha u'(\xi ),\quad u(1)=\beta u(\eta ), \end{aligned}$$
where
\(\phi (t,x)= x ^{p(t)-2} \ x,\ p\in C([0,1],(1,+\infty )),\ \alpha ,\beta ,\xi ,\eta \in (0,1),\ \xi >\eta ,\ p(0)=p(\xi ),\ f\in C([0,1]\times [0,+\infty ),[0,+\infty ))\)
.AbstractIn this paper, by means of the computation of fixed point index, we obtain the existence of positive solutions for four-point boundary value problem involving the
\(p(t)\)
-Laplacian
$$\begin{aligned}&(\phi (t,u'(t)))'+ f(t,u(t))=0,\quad t\in (0,1), \\&u'(0)=\alpha u'(\xi ),\quad u(1)=\beta u(\eta ), \end{aligned}$$
where
\(\phi (t,x)= x ^{p(t)-2} \ x,\ p\in C([0,1],(1,+\infty )),\ \alpha ,\beta ,\xi ,\eta \in (0,1),\ \xi >\eta ,\ p(0)=p(\xi ),\ f\in C([0,1]\times [0,+\infty ),[0,+\infty ))\)
.
\(p(t)\)
\(p(t)\)
$$\begin{aligned}&(\phi (t,u'(t)))'+ f(t,u(t))=0,\quad t\in (0,1), \\&u'(0)=\alpha u'(\xi ),\quad u(1)=\beta u(\eta ), \end{aligned}$$
$$\begin{aligned}&(\phi (t,u'(t)))'+ f(t,u(t))=0,\quad t\in (0,1), \\&u'(0)=\alpha u'(\xi ),\quad u(1)=\beta u(\eta ), \end{aligned}$$
\(\phi (t,x)= x ^{p(t)-2} \ x,\ p\in C([0,1],(1,+\infty )),\ \alpha ,\beta ,\xi ,\eta \in (0,1),\ \xi >\eta ,\ p(0)=p(\xi ),\ f\in C([0,1]\times [0,+\infty ),[0,+\infty ))\)
\(\phi (t,x)= x ^{p(t)-2} \ x,\ p\in C([0,1],(1,+\infty )),\ \alpha ,\beta ,\xi ,\eta \in (0,1),\ \xi >\eta ,\ p(0)=p(\xi ),\ f\in C([0,1]\times [0,+\infty ),[0,+\infty ))\)