In this paper, we prove the existence of multiple solutions for the following sixth-order ▫$p(x)$▫-Kirchhoff-type problem ▫$$\begin{cases} -M\left( \int\limits_{\it \Omega} \frac{1}{p(x)}|\nabla ...{\it\Delta} u|^{p(x)}dx\right){\it\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) &\mbox{in}\quad {\it\Omega}, \\ u = {\it\Delta} u = {\it\Delta}^2 u = 0, \quad &\mbox{on}\quad \partial{\it\Omega}, \end{cases}$$▫ where ▫${\it\Omega} \subset \mathbb{R}^N$▫ is a smooth bounded domain, ▫$N>3$▫, ▫${\it\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div} \Big({\it\Delta}(|\nabla {\it\Delta} u|^{p(x)-2}\nabla {\it\Delta} u)\Big)$▫ is the ▫$p(x)$▫-triharmonic operator, ▫$p, q, r \in C(\overline{\it\Omega}), 1 < p ( x ) < \frac{N}{3}$▫ for all ▫$x \in (\overline{\it \Omega}$▫, ▫$M(s) = a-bs^\gamma, \;a, b, \gamma > 0, \lambda > 0$▫, ▫$g \colon {\it\Omega} \times \mathbb{R} \to \mathbb{R}$▫ is a nonnegative continuous function while ▫$f, h \colon {\it\Omega} \times \mathbb{R} \to \mathbb{R}$▫ are sign-changing continuous functions in ▫${\it \Omega}$▫. To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order ▫$p(x)$▫-Kirchhoff type problems with sign changing Kirchhoff functions.
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