ABSTRACT.

This paper is concerned with the existence of two nontrivial positive solutions to a class of boundary value problems involving a \( p \)-Laplacian of the form

\[
(\Phi_p(x’))’ + g(t)\,f(t,x,x’) = 0, \qquad t \in (0,1),
\]

\[
x(0) – a x'(0) = \alpha[x],
\]

\[
x(1) + b x'(1) = \beta[x],
\]

where \( \Phi_p(x) = |x|^{p-2}x \) is a one dimensional \( p \)-Laplacian operator with \( p > 1 \), \( a \) and \( b \) are real constants, and \( \alpha \) and \( \beta \) are given by the Riemann–Stieltjes integrals

\[
\alpha[x] = \int_{0}^{1} x(t)\, dA(t),
\qquad
\beta[x] = \int_{0}^{1} x(t)\, dB(t),
\]

with \( A \) and \( B \) functions of bounded variation. The approach used is based on fixed point theorems.