In this paper, we have obtained upper bounds for the total chromatic number of some classes of Cayley graphs, odd graphs and mock threshold graphs.

In this paper, we find all the solutions of Diophantine equation written in our title, where $n$, $m$, $x$ and $y$ are positive integers and $F_k$ denotes the $k$th term of the Fibonacci sequence.

We investigate Bergman type operators on the complex unit ball, which are singular integral operators induced by the modified Bergman kernel. We consider the $L^p$-$L^q$ boundedness and compactness of Bergman type operators. The results of boundedness can be viewed as the Hardy–Littlewood–Sobolev (HLS) type theorem in the case unit ball. We also give some sharp norm estimates of Bergman type operators which in fact gives the upper bounds of the optimal constants in the HLS type inequality on the unit ball. Moreover, a trace formula is given.

In this paper, we study an initial boundary value problem for a $p$-Laplacian hyperbolic equation with logarithmic nonlinearity. By combining the modified potential well method with the Galerkin method, the existence of the global weak solution is studied, and the polynomial and exponential decay estimation under certain conditions are also given. Moreover, by using the concavity method and other techniques, we obtain the blow up results at finite time.

In this paper we are concerned with the existence of invariant curves of quasi-periodic reversible mappings with higher order degeneracy of the twist condition under the Brjuno–Rüssmann's non-resonant condition. In the proof we use a new variant of the KAM theory, containing an artificial parameter $q$, $0 \lt q \lt 1$, which makes the steps of the KAM iteration infinitely small in the speed of function $q^n \varepsilon$, rather than super exponential function.

In this paper we will consider various kinds of chaotic points of multifunctions and show their application to the theory of infinite topological games.

We prove Sobolev-type inequalities for modified Riesz potentials of functions in Morrey spaces of an integral form over non-doubling metric measure spaces. Our results are new even for the doubling metric measure setting. In particular, our results extend the previous results in Morrey spaces of an integral form in the Euclidean case.

A class of conformable fractional differential equations \[ D_x^{\alpha} D_x^{\alpha} u(x) + \omega(x) f(u(x)) = 0 \quad \textrm{on $(0,1)$} \] is considered. We first give a sufficient condition for the existence of sign-changing solutions with the prescribed number of zeros to this problem. On the basis of this result, we turn to a specific case of the above problem and give a uniqueness theorem related to the function $\omega$. Essentially, the main methods using in this work are properties of conformable fractional calculus, the scaling argument and Prüfer-type substitutions.

We are concerned with divergences on the Cartan–Hadamard Riemannian manifold of symmetric cones, self-dual homogeneous cones in Euclidean spaces, and related optimization problems. We introduce a parameterized version of fidelity on symmetric cones, namely sandwiched quasi-relative entropies, and construct a one-parameter family of divergences based on these entropies. We consider the median minimization problem of finite points over these divergences and establish existence and uniqueness of minimizer. The global linear rate convergence of a gradient projection algorithm for solving the median minimization problem is analyzed based on the derived upper bound of the condition number of the Hessian function.