Volume 75 , issue 3 ( 2023 ) | back |

CONSTRUCTION OF THE TYPE 2 DEGENERATE POLY-EULER POLYNOMIALS AND NUMBERS | 147$-$156 |

**Abstract**

In this paper, we introduce type 2 degenerate poly-Euler polynomials and numbers, briefly called degenerate poly-Euler polynomials and numbers, by using the modified degenerate polyexponential function and derive several properties on these polynomials and numbers. In the last section, we also consider the type 2 degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities of these polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

**Keywords:** Modified degenerate polyexponential functions; degenerate poly-Euler polynomials; degenerate unipoly functions; degenerate unipoly-Euler polynomials.

**MSC:** 11B73, 11B83, 05A19

EXISTENCE OF ONE WEAK SOLUTION FOR ELLIPTIC EQUATIONS INVOLVING A GENERAL OPERATOR IN DIVERGENCE FORM | 157$-$165 |

**Abstract**

In this article, we establish the existence of at least one non-trivial classical solution for a class of elliptic equations involving a general operator in divergence form, subject to Dirichlet boundary conditions in a smooth bounded domain in $\mathbb{R}^N$. A critical point result for differentiable functionals is discussed. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given.

**Keywords:** Existence result; weak solution; divergence type equations; variational methods; critical point theory.

**MSC:** 35J35, 35J60

GENERALIZED $\boldsymbol{V}$-$\boldsymbol{Ric}$ VECTOR FIELDS ON CONTACT PSEUDO-RIEMANNIAN MANIFOLDS | 166$-$174 |

**Abstract**

In this paper, we study contact pseudo-Riemannian manifold $M$ admitting generalized $V$-$Ric$ vector field. Firstly, for pseudo-Riemannian manifold, it is proved that $V$ is an infinitesimal harmonic transformation if $M$ admits $V$-$Ric$ vector field. Secondly, we prove that an $\eta$-Einstein $K$-contact pseudo-Riemannian manifold admitting a generalized $V$-$Ric$ vector field is either Einstein or has scalar curvature $r=\frac{2n\varepsilon(2n-1)}{4n-1}$. Finally, we consider a contact pseudo-Riemannian $(\kappa,\mu)$-manifold with a generalized $V$-$Ric$ vector field.

**Keywords:** Contact pseudo-Riemannian manifolds; genaralized $V$-$Ric$ vector field; $K$-contact pseudo-Riemannian manifold; Einstein manifold.

**MSC:** 53C50, 53C25, 53B30, 53C24

SLANT LIGHTLIKE SUBMANIFOLDS OF GOLDEN SEMI-RIEMANNIAN MANIFOLDS | 175$-$188 |

**Abstract**

In this paper, we introduce the notion of slant lightlike submanifold of a golden semi-Riemannian manifold and provide a characterization theorem with some non-trivial examples of such submanifolds. We find necessary and sufficient conditions for integrability of distributions. Finally, we study curvature properties of slant lightlike submanifolds of golden semi-Riemannian manifolds.

**Keywords:** Golden structure; golden semi-Riemannian manifolds; Gauss and Weingarten formulae; lightlike submanifolds; curvature-invariant.

**MSC:** 53C15, 53C40, 53C50

THREE SOLUTIONS FOR IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS WITH DIRICHLET BOUNDARY CONDITION | 189$-$203 |

**Abstract**

In this paper, we discuss the existence of at least three weak solutions for the following impulsive nonlinear fractional boundary value problem \begin{align*} {}_t D_T^{\alpha} \left({}_0^c D_t^{\alpha}u(t)\right) +a(t)u(t)&= \lambda f(t,u(t)), \quad t\neq t_j,\ \text{a.e. } t \in [0, T],\\ \Delta\left({}_t D_T^{\alpha-1} \left({}_0^c D_t^{\alpha}u\right)\right)(t_j)&= I_j(u(t_j)),\quad j=1,\ldots n,\\ u(0) = u(T) &= 0 \end{align*} where $\alpha \in (\frac{1}{2}, 1]$, $a \in C([0, T ])$ and $f : [0, T ]\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carathéodory function. Our technical approach is based on variational methods. An example is provided to illustrate the applicability of our results.

**Keywords:** Three solutions; fractional differential equation; impulsive effect; variational methods; critical point theory.

**MSC:** 26A33, 34B15, 35A15, 34B15, 34K45, 58E05

CONSTACYCLIC CODES OVER LIPSCHITZ INTEGERS | 204$-$215 |

**Abstract**

In this paper, the goal is to obtain constacyclic codes over Lipschitz integers in terms of Lipschitz metric. A decoding procedure is proposed for these codes, some of which have been shown to be perfect codes. Performance of constacyclic codes over Lipschitz integers is investigated over Additive White Gaussian Channel (AWGN) by means of symbol error rates and coding gain. According to the achieved results, these codes can be used in coded modulation schemes based on Quadrature Amplitude Modulation (QAM)-type constellations. Furthermore, it is shown that the Lipschitz metric is more suitable than Hamming metric and Lee metric for QAM type two dimensional constellations.

**Keywords:** Block code; QAM signal constellations; Lipschitz metric.

**MSC:** 94B05, 94B60

HOPF BIFURCATIONS IN DYNAMICAL SYSTEMS VIA ALGEBRAIC TOPOLOGICAL METHOD | 216$-$224 |

**Abstract**

A nonlinear phenomenon in nature is often modeled by a system of differential equations with parameters. The bifurcation occurs when a parameter varies in such systems, causing a qualitative change in its solution. In this paper, we study one of the most exciting bifurcations, which is Hopf bifurcation. We use tools from algebraic topology to analyze and reveal supercritical and subcritical Hopf bifurcations.

**Keywords:** Supercritical Hopf bifurcation; subcritical Hopf bifurcation; stable limit cycle; unstable limit cycle; homological Conley index; Morse sets.

**MSC:** 55-08, 37B30, 37G15, 37M20