This work deals with the study of a class of nonlocal Navier boundary value problems involving the degenerate $p(x)$-biharmonic operator with a potential term \mbox{$q(x)$-Hardy} \begin{align*} \begin{cases} \Delta( \omega (|\Delta u |^{p(x)}) |\Delta u |^{p(x)- 2}\Delta u ) - \lambda \frac{ |u|^{q(x)- 2}u}{{\delta(x)}^{2q(x)}}= \mu \vartheta(x)|u|^{q(x)- 2}u\bigg(\int_{\Omega}\frac{\vartheta(x)}{q(x)}| u|^{q(x)}dx\bigg)^{r} & \mbox{in}\ \Omega, \\ u = \Delta u =0, & \mbox{on}\ \partial \Omega. \end{cases} \end{align*} In this new setting, our objectif is to extend the results obtained in the paper [M. Laghzal, A. El Khalil, M. D. Morchid Alaoui, A. Touzani, Eigencurves of the $p(\cdot)$-biharmonic operator with a Hardy-type term potential, Moroccan J. Pure Appl. Anal., 6(2) (2020), 198--209] for the nonhomogeneous case $p(x) \neq q(x),$ where $ \vartheta$ is a weight function. The main results are established by using the variational method and min-max arguments based on Ljusternik-Schnirelmann theory on $C^1$ manifoleds [A. Szulkin, Schnirelmann theory on $C^1$-manifolds, Ann. Inst. Henri Poincaré C, Anal. Non Linéaire, 5(2) (1988), 119--139]. A direct characterization of the principal curve (first one) is provided.
An open-locating-dominating (OLD) set in a graph is a subset of vertices that dominates the graph and distinguishes all pairs of vertices via their open neighborhoods within the set. In this paper, we determine the exact OLD number $\gamma_{\mathrm{OLD}}(G)$ for three families of structured graphs. We first establish that for helm graphs $H_n$ and cycle coronae $C_n \circ K_1$, the parameter equals $n$ for all $n \ge 3$, except for $n=4$ where $\gamma_{\mathrm{OLD}}(H_4)=6$. We then focus on gear graphs $G_n$, where standard degree-based bounds prove insufficient. By modeling the problem via binary sequences and analyzing forbidden local patterns, we prove that for $n \ge 4$, $\gamma_{\mathrm{OLD}}(G_n) = 2\lceil 2n/3\rceil$, and $\gamma_{\mathrm{OLD}}(G_3)=5$.
In this paper, we investigate various dynamical properties of multiparameter strongly continuous semigroups of operators induced by semiflows. We also provide many illustrative examples and useful remarks concerning issues under our consideration.
The purpose of this communication is to define and illustrate the notions of $\mathcal{J}(S_\sigma: \mathcal{G})$-summable, $\mathcal{J}(S_\sigma: \vartheta: \mathcal{G})$-summable and $\mathcal{J}(N_\sigma: \vartheta: \mathcal{G})$-summable sequences in gradual normed linear spaces (briefly abbreviated as $GNLP$) and to demonstrate some connections between their corresponding sequence spaces.
For a Tychonoff space $X$, we consider the space $I_{f}(X)$ that is a subspace of the space of all idempotent probability measures on $X$. Using a cover $\gamma_{X}$ of the origin space $X$ we build an open cover $\gamma_{0}$ of $I_{f}(X)$. Then we show that if $\gamma_{X}$ is a finite-component family, then $\gamma_{0}$ is also a finite-component family. We obtain that the space $I_{f}(\beta\, X)$ of idempotent probability measures on the Stone-\v{C}ech compactification $\beta\,X$ of $X$ is a perfect compactification of $I_{f}(X)$. Finally, applying the superparacompactness criterion, we establish that $I_{f}(X)$ is superparacompact if and only if the given Tychonoff space $X$ is superparacompact.
