Abstract

In this paper, we study the growth of higher order linear differential equations, where some coefficients are non-trivial solutions of certain second order linear differential equations. Some conditions guaranteeing that every non-trivial solution of the equation is of infinite order are obtained, in which the notion of accumulation rays of the zero sequence of an entire function is used.

Keywords: Linear differnetial equation; entire function; order of growth; asymptotic growth.

MSC: 34M10, 30D35

Abstract

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.

Keywords: Degenerate $p(x)$-biharmonic operator; variational methods; Ljusternik-Schnirelman; nonlinear eigenvalue problems; $q(x)$-Hardy's inequality.

MSC: 35J70, 35J35, 35J75, 58J05

Abstract

Let $b\ge 2$ be an integer. In this paper we study the base $b$ repdigits that can be expressed as sums or products of Fibonacci and Tribonacci numbers. As a corollary, it is shown that the numbers $1$ and $7$ are the only Mersenne numbers which can be expressed respectively as product and sum of Fibonacci and Tribonacci numbers. This is done using linear forms in logarithms of algebraic numbers (Baker's method) and the Baker-Davenport reduction method (the Dujella-Peth\H {o}'s version).

Keywords: Fibonacci numbers; tribonacci numbers; $b$ repdigits; logarithmic height; reduction method.

MSC: 35J70, 35J35, 35J75, 58J05

Abstract

For a compact orientable PL $4$-manifold $M$ with boundary, let $\hat{M}$ be the singular manifold obtained by capping of $\partial M$. In this article, we explore the class of compact orientable PL $4$-manifolds with empty or connected boundary, whose fundamental groups have rank $1$, and their corresponding singular manifolds admit weak semi-simple crystallizations. First, we show that if $M$ is a closed orientable PL $4$-manifold belonging to this class, then there exist complementary submanifolds $V$ and $V^\prime$ with a shared boundary such that $\mathcal{G}(V^\prime) \geq \mathcal{G}(M) \geq \mathcal{G}(V)$, where $\mathcal{G}(M)$, $\mathcal{G}(V)$, and $\mathcal{G}(V^\prime)$ denote the regular genera of $M$, $V$, and $V^\prime$, respectively. Next, we provide a handle decomposition for a compact orientable PL $4$-manifold $M$ with connected non-spherical boundary from this class. If the rank of the fundamental group of $\hat{M}$, $m^\prime$, is $1$, then $M$ admits a handle decomposition that takes one of the following forms:
1) one $0$-handle, one $1$-handle, $k$ $2$-handles and one $3$-handle, where $k=2+\beta_2(M)-\beta_1(M)-\beta_1(\hat{M})$, or
2) one $0$-handle, two $1$-handles, $k$ $2$-handles and one $3$-handle, where $k=3+\beta_2(M)-\beta_1(M)-\beta_1(\hat{M})$.
Further, if $m^\prime=0$, then $M$ admits a handle decomposition which consists of one $0$-handle, one $1$-handle and $\beta_2(M)$ $2$-handles. We further demonstrate that no manifold from this class with empty or connected spherical boundary can have a fundamental group isomorphic to a finite cyclic group. Finally, we provide a handle decomposition for such manifolds with empty or connected spherical boundary.

Keywords: PL-manifolds; crystallizations; regular genus; handle decomposition.

MSC: 54B15, 54C25, 57Q15, 57M15, 05C15

Abstract

In this paper, we investigate the value distribution of L-functions in the extended Selberg class. We show how two L-functions $L_1$ and $L_2$ satisfying certain condition are uniquely determined by the zero sharing between $P(L_1)$ and $P(L_2)$ for some polynomial $P$, or by a set sharing between $L_1$ and $L_2$. Considering the most general form of a polynomial in the set sharing we obtain some results which completely generalize and extend some recent results of [X. M. Li, X. R. Du, H. X. Yi, Dirichlet series satisfying a Riemann type functional equation and sharing one set, Complex Var. Elliptic Equ., 68(10) (2023), 1653--1677], which were actually proved as an answer of an analogous question of Gross [F. Gross, Factorization of meromorphic functions and some open problems, Complex Analysis (Proc. Conf. Univ. Kentucky, Lexington, Ky., 1976), pp. 51--69, Lect. Notes Math., Vol 599, Springer, Berlin, 1977] for L-functions. We also obtain uniqueness relation between two nonconstant L-functions (belonging to the extended Selberg class) by proving other two results, one concerning a prior result due to Yuan-Li-Yi [Q. Q. Yuan, X. M. Li, H. X. Yi, Value distribution of L-functions and uniqueness questions of F. Gross, Lithuanian Math. J., 58 (2018), 249--262] and another related to a result of Hao-Chen [W. J. Hao, J. F. Chen, Uniqueness theorems for L-functions in the extended Selberg class, Open Math., 16 (2018), 1291--1299].

Keywords: Dirichlet series; Selberg class; L-function; set sharing; uniqueness; meromorphic.

MSC: 30D35, 30D30, 11M06, 11M41

Abstract

The main result of this work is a simple characterization of the hyperstability of functional equations in a very general framework: for functions with the codomain endowed with a generalized homogeneous premetric. We also give some particularizations on Abelian groups equipped with generalized homogeneous norms, obtaining, among others, improvements of similar known results for Cauchy-type, Jensen-type equations, but also for equations having compound functions as solutions.

Keywords: Equation on restricted domain; hyperstability; Cauchy-type equation; Jensen-type equation.

MSC: 39B82, 39B52, 39A70, 39B62, 39A10

Abstract

In this paper, we prove a coupled fixed-point result for a hybrid mapping derived from generalized coupled Banach and Kannan-type contractions. We define the asymptotic regularity property for coupled maps and use it as a condition in our theorem. The results are derived in metric spaces with a preorder relation. The necessity of the hybrid contraction inequality is constrained by the preordering through the requirement that the inequality must be satisfied between points related by the preorder relation in a particular way. The main theorem is proven under several alternative additional conditions. There are several consequences of the main result, one of which is the relaxation of the contraction constant's range in the Kannan-type result. Two examples illustrate several features of the results presented herein. The paper concludes with an application to a system of integral equations.

Keywords: Asymptotic regularity; preorder; coupled $k$-continuity; coupled orbital continuity; coupled fixed point; Kannan-type contraction.

MSC: 47H10, 54H25, 54E50