Volume 76 , issue 3 ( 2024 ) | back |

ON $(\alpha, \beta, \gamma)$-METRICS | 159$-$173 |

**Abstract**

In this paper, we introduce a new class of Finsler metrics that generalize the well-known $(\alpha, \beta)$-metrics. These metrics are defined by a Riemannian metric $\alpha$ and two 1-forms $\beta=b_i(x)y^i$ and $\gamma=\gamma_i(x)y^i$. This new class of metrics not only generalizes $(\alpha, \beta)$-metrics, but also includes other important Finsler metrics, such as all (generalized) $\gamma$-changes of generalized $(\alpha, \beta)$-metrics, $(\alpha, \beta)$-metrics, and spherically symmetric Finsler metrics in $\mathbb{R}^n$. We find a necessary and sufficient condition for this new class of metrics to be locally projectively flat. Furthermore, we prove the conditions under which these metrics are of Douglas type.

**Keywords:** Finsler geometry, $(\alpha, \beta, \gamma)$-metrics, Projectively flat, Douglas space.

**MSC:** 53B40, 53C60

CONSTRUCTION OF UNIVALENT HARMONIC MAPPINGS AND THEIR CONVOLUTIONS | 174$-$184 |

**Abstract**

In this article, we make use of convex analytic functions $H_a(z)=[1/(1-a)]\log[(1-az)/(1-z)]$, $a\in \mathbb{R}$, $|a|\leq 1$, $a\neq 1$ and starlike analytic functions $L_b(z)=z/[(1-bz)(1-z)]$, $b\in \mathbb{R}$, $|b|\leq 1$, to construct univalent harmonic functions by means of a transformation on some normalized univalent analytic functions. Besides exploring mapping properties of harmonic functions so constructed, we establish sufficient conditions for their harmonic convolutions or Hadamard products to be locally univalent and sense preserving, univalent and convex in some direction.

**Keywords:** Harmonic function; univalent function; convolution; convexity in one direction.

**MSC:** 30C45, 30C80

MULTIPLICITY OF SOLUTIONS FOR ANISOTROPIC DISCRETE BOUNDARY VALUE PROBLEMS | 185$-$195 |

**Abstract**

In this paper, we study the existence and multiplicity of nontrivial solutions for an anisotropic discrete nonlinear problem with variable exponent. The analysis makes use of variational methods and critical point theory.

**Keywords:** Discrete nonlinear boundary value problems; nontrivial solution; critical point theory;
variational methods.

**MSC:** 47A75, 34B15, 35B38, 65Q10

NEW MULTIPLE FIXED POINT THEOREMS FOR SUM OF TWO OPERATORS AND APPLICATION TO A SINGULAR GENERALIZED STURM-LIOUVILLE MULTIPOINT BVP | 196$-$209 |

**Abstract**

In this paper, we develop some new multiple fixed point theorems for the sum of two operators $T+S$ where $I-T$ is Lipschitz invertible and $S$ is a $k$-set contraction on translate of a cone in a Banach space. New existence criteria for multiple positive solutions of a singular generalized Sturm-Liouville multipoint boundary value problem are established. The article ends with an illustrative example.

**Keywords:** Fixed point; sum of operators; cone; Sturm-Liouville BVP; multiple positive solutions.

**MSC:** 47H10, 34B10, 34B24

ALMOST YAMABE SOLITON AND ALMOST RICCI-BOURGUIGNON SOLITON WITH GEODESIC VECTOR FIELDS | 210$-$217 |

**Abstract**

The aim of this paper is to prove some results about almost Yamabe soliton and almost Ricci-Bourguignon soliton with special soliton vector field. In fact, we prove that every compact non-trivial almost Ricci-Bourguignon soliton with constant scalar curvature is isometric to a Euclidean sphere. Then we show that every compact almost Ricci-Bourguignon soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. Finally, we prove that a complete almost Ricci-Bourguignon soliton $(M,g, V, \lambda, \rho)$ has $V$ as the contact vector field of a contact manifold $M$ with metric $g$ and its Reeb vector field is geodesic, then it becomes a Ricci-Bourguignon soliton and $g$ has constant scalar curvature. In particular, if $V$ is strict, then $g$ is a compact Sasakian Einstein.

**Keywords:** Almost Ricci-Bourguignon soliton; contact manifold; Einstein Sasakian.

**MSC:** 53C25, 53E20, 53C21

UNIQUENESS OF SOME DELAY-DIFFERENTIAL POLYNOMIALS SHARING A SMALL FUNCTION WITH FINITE WEIGHTS | 218$-$232 |

**Abstract**

In this paper, we study the uniqueness problems of $f^n(z)L(g)$ and $g^n(z)L(f)$ when they share a non-zero small function $\alpha(z)$ with finite weights, where $L(h)$ represents any one of $h^{(k)}(z),\; h(z+c), \;h(z+c)-h(z)$ and $h^{(k)}(z+c),$ $k\geq 1$ and $c$ is a non-zero constant. Here $f(z)$ and $g(z)$ are transcendental meromorphic (or entire) functions and $\alpha(z)$ is a small function with respect to both $f(z)$ and $g(z).$ Our results improve and supplement the recent results due to Gao and Liu [Bull. Korean Math. Soc. 59 (2022), 155-166].

**Keywords:** Uniqueness; Hayman conjecture; delay-differential polynomial; difference polynomial; weighted sharing.

**MSC:** 30D35, 39A05