  In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff
problem, which involves the $p(x)$-fractional Laplacian equations of variable
order. The problem is stated as follows: \begin{eqnarray*} \left\{
  \begin{array}{ll}
  M\Big(\sigma_{p(x,y)}(u)\Big)(-\Delta)^{s(\cdot)}_{p(\cdot)}u(x) =\lambda
|u|^{q(x)-2}u\left(\int_\O\frac{1}{q(x)} |u|^{q(x)}dx
\right)^{k_1}+\beta|u|^{r(x)-2}u\left(\int_\O\frac{1}{r(x)} |u|^{r(x)}dx
\right)^{k_2}
  \quad \mbox{in }\Omega,
  \\ u=0 \quad \mbox{on }\partial\Omega, \end{array} \right. \end{eqnarray*}
where the nonlocal term is defined as $$ \sigma_{p(x,y)}(u)=\int_{\Omega\times
\Omega}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+s(x,y)p(x,y)}}
\,dx\,dy. $$ Here, $\Omega\subset\mathbb{R}^{N}$ represents a bounded smooth
domain with at least $N\geq2$. The function $M(s)$ is given by $M(s) = a -
bs^\gamma$, where $a\geq 0$, $b>0$, and $\gamma>0$. The parameters $k_1$,
$k_2$, $\lambda$ and $\beta$ are real parameters, while the variables $p(x)$,
$s(\cdot)$, $q(x)$, and $r(x)$ are continuous and can change with respect to
$x$. To tackle this problem, we employ some new methods and variational
approaches along with two specific methods, namely the Fountain theorem and the
symmetric Mountain Pass theorem. By utilizing these techniques, we establish
the existence and multiplicity of solutions for this problem separately in two
distinct cases: when $a>0$ and when $a=0$. To the best of our knowledge, these
results are the first contributions to research on the variable-order
$p(x)$-fractional Laplacian operator.
