Abstract: New graph invariant, which is called mixed metric dimension, has been recently introduced. In this paper, exact results of mixed metric dimension on two special classes of graphs are found: flower snarks $J_n$ and wheels $W_n$. It is proved that mixed metric dimension for $J_5$ is equal to 5, while for higher dimensions it is constant and equal to 4. For $W_n$, its mixed metric dimension is not constant, but it is equal to $n$ when $n\geq 4$, while it is equal to 4, for $n=3$.

Abstract: More than 20 years after Fickett attempted to prove the Hyers-Ulam stability of isometries defined on bounded subsets of $\mathbb{R}^n$ in 1981, Vaisala improved Fickett's result significantly. In this paper, we will improve Fickett's theorem by proving the Hyers-Ulam stability of isometries defined on bounded subsets of $\mathbb{R}^n$ using a more intuitive method different from that used by Vaisala.

Abstract: Abstract In this study, we first obtain a new identity for generalized fractional integrals which contains some parameters. Then by this equality, we establish some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals. Moreover, we show that the results proved in the main section reduce to several Simpson-, trapezoid- and midpoint-type inequalities for various values of parameters.

Abstract: We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(\phi(| abla u|) abla u\big)+V(x)\phi (|u|)u=f(u)\qquad u\in W^{1,\Phi}(\mathbb{R}^{N}), \] where $\phi$ and $f$ satisfy suitable conditions. The positive potential $V\in C(\mathbb{R}^{N})$ exhibits a finite or infinite potential well in the sense that $V(x)$ tends to its supremum $V_{\infty}\le+\infty$ as $|x|\to\infty$. Nontrivial solutions are obtained by variational methods. When $V_{\infty }=+\infty$, a compact embedding from a suitable subspace of $W^{1,\Phi }(\mathbb{R}^{N})$ into $L^{\Phi}(\mathbb{R}^{N})$ is established, which enables us to get infinitely many solutions for the case that $f$ is odd. For the case that $V(x)=\lambda a(x) + 1$ exhibits a steep potential well controlled by a positive parameter $\lambda$, we get nontrivial solutions for large $\lambda$.