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First we consider the existence of non-trivial solutions of problem $(P_\lambda)$;
\begin{equation*}\tag*{$(P_\lambda)$}
\begin{cases}
\psi_p(u')'+ \lambda h(t) \cdot f(u) =0, \quad t\in (0,1), \\
u(0)= 0 = u(1),
\end{cases}
\end{equation*}
where $\lambda>0$ a parameter, $\psi_{p}:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is
defined by $\psi_{p}(x)=|x|^{p-2}x.$
$h:(0,1)\rightarrow \mathbb{R}^{N}$ may be
singular at 0 or/and 1 and changes a sign. Moreover $h$ may not be in $L^1(0,1).$
$f \in C(\mathbb{R}^N,\mathbb{R}^N)$ and $x \cdot y := (x_1y_1, x_2y_2, \cdots, x_Ny_N).$
Mainly motivated by the pioneering work of Man\'asevich-Mawhin [1],
we derive a new integral operator of $(P_\lambda)$ given as
\begin{equation*}
T_{\lambda} (u)(t) =
\begin{cases}
\int_{0}^{t}\psi_{p}^{-1}(\alpha(\lambda h \cdot f(u))+\lambda \int_{s}^{\frac{1}{2}} h(r) \cdot f(u(r))dr)ds, & \textrm{$0\leq t\leq\frac{1}{2}$,}\\
\int_{t}^{1}\psi_{p}^{-1}(-\alpha(\lambda h \cdot f(u))+ \lambda \int_{\frac{1}{2}}^{s} h(r) \cdot f(u(r))dr)ds,& \textrm{$\frac{1}{2}\leq t\leq 1$},
\end{cases}
\end{equation*}
where $\alpha=\alpha(\lambda h \cdot f(u)) \in \mathbb{R}^{N}$
is the unique zero of the following integral equation
$$\int_{0}^{\frac{1}{2}}\psi_{p}^{-1}\left(\alpha+\lambda \int_{s}^{\frac{1}{2}}h(r) \cdot f(u(r))dr\right)ds= \int_{\frac{1}{2}}^{1}\psi_{p}^{-1}\left(-\alpha+\int_{\frac{1}{2}}^{s}h(r) \cdot f(u(r))dr\right)ds.$$
This operator $T$ extends cases introduced by Man\'asevich-Mawhin [1] of $g\in L^1(0,1)$ and
Agarwal-L\"u-O'Regan [2] of $g\ge 0$ to case of $g\notin L^1(0,1)$ and sign-changing.
Second, we apply similar argument to the following $\varphi$-Laplacian systems;
\begin{equation*}\tag*{$(P)$}
\begin{cases}
-{\Phi(u')}'= {h}(t)\cdot {f}({u}),\quad t\in (0,1),\\
{u}(0)= 0 = {u}(1),
\end{cases}
\end{equation*}
where
${\Phi(u')}=(\varphi(u_1'),\cdots,\varphi(u_N'))$ with
$\varphi:\mathbb{R} \to \mathbb{R}$ an odd increasing homeomorphism, \\
${h}(t)=$ $(h_1(t),\cdots,h_N(t))$ with
$h_i :(0,1) \to \mathbb{R_+}$, $h_i \not \equiv 0$ on any subinterval in $(0,1)$ and ${f(u)}=(f^{1}({u}),\cdots,f^{N}({u}))$ with $f^{i}: \mathbb{R}_+^N\to \mathbb{R_+}$.
Under suitable assumption on $\varphi$ and a singularity condition on ${h}$, we introduce several existence results of positive solutions of problem $(P)$.\\
{\it References}
\begin{itemize}
\item[{[1]}] {\it R. Man\'asevich and J. Mawhin}: Periodic solutions of nonlinear systems with $p$-Laplacian-like
operators. JDE, {\it 145} (1998), 367-393.
\item[{[2]}] {\it R. P. Agarwal and H. L\"u and D. O'Regan}: Eigenvalues and the one-dimensional $p$-{L}aplacian. J. Math. Anal. Appl. {\it 266} (2002), 383-400.
\end{itemize}
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