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In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by p · -fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results.

In this article, we consider the following nonlocal fractional Kirchhoff-type elliptic systems \begin{document}$ \begin{equation*} \left\{\begin{array}{l} -M_{1}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\eta(x)-\eta(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}} \ \ \ \ \ dxdy +\int_{\Omega}\frac{|\eta|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\eta-|\eta|^{\overline{p}(x)}\eta\right)\\ \; \; \; = \lambda F_{\eta}(x, \eta, \xi)+\mu G_{\eta}(x, \eta, \xi), \; \; x \in \Omega, \\ -M_{2}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\xi(x)-\xi(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}} \ \ \ \ \ dxdy +\int_{\Omega}\frac{|\xi|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\xi-|\xi|^{\overline{p}(x)}\xi\right)\\ \; \; \; = \lambda F_{\xi}(x, \eta, \xi)+\mu G_{\xi}(x, \eta, \xi), \; \; x \in \Omega, \\ \; \eta = \xi = 0, \; \; x \in \mathbb{R}^{N}\backslash \Omega, \end{array} \right. \end{equation*} $\end{document} where $ M_{1}(t), M_{2}(t) $ are the models of Kirchhoff coefficient, $ \Omega $ is a bounded smooth domain in $ \mathbb R^{N} $, $ (-\Delta)_{p(\cdot)}^{s(\cdot)} $ is a fractional Laplace operator, $ \lambda, \mu $ are two real parameters, $ F, G $ are continuous differentiable functions, whose partial derivatives are $ F_{\eta}, F_{\xi}, G_{\eta}, G_{\xi} $. With the help of direct variational methods, we study the existence of solutions for nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order, and obtain at least two and three weak solutions based on Bonanno's and Ricceri's critical points theorem. The outstanding feature is the case that the Palais-Smale condition is not requested. The major difficulties and innovations are nonlocal Kirchhoff functions with the presence of the Laplace operator involving two variable parameters.

This paper is devoted to derive optimality conditions and duality theorems for interval-valued optimization problems based on gH-symmetrically derivative. Further, the concepts of symmetric pseudo-convexity and symmetric quasi-convexity for interval-valued functions are proposed to extend above optimization conditions. Examples are also presented to illustrate corresponding results.

In this paper, we present the gH-symmetrical derivative of interval-valued functions andits properties. In application, we apply this new derivative to investigate the Karush–Kuhn–Tucker(KKT) conditions of interval-valued optimization problems. Meanwhile, some examples are workedout to illuminate the obtained results.