Volume 71 , issue 4 ( 2019 ) | back |

LOCAL PERSISTENCE OF GEOMETRIC STRUCTURES FOR BOUSSINESQ SYSTEM WITH ZERO VISCOSITY | 285$-$303 |

**Abstract**

The current paper deals with the local well-posedness problem for the two-dimensional partial viscous Boussinesq system when the initial vorticity belongs to the patch class. We prove in particular some results concerning the regularity persistence of the patch boundary and establish the convergence towards the inviscid limit when the molecular diffusivity goes to zero.

**Keywords:** Stratified system; vortex patches; local well-posedness; inviscid limit.

**MSC:** 35B65, 35Q35, 76D05

TAKENS-BOGDANOV NUMERICAL ANALYSIS IN PREDATOR-PREY MODEL WITH DELAY | 304$-$315 |

**Abstract**

The aim of this paper is to study the Bogdanov-Takens point in the case of a Predator-Prey model with delay.

**Keywords:** Bogdanov-Takens bifurcation; prey-predator model with delay; delay systems; Newton iteration.

**MSC:** 34K18, 65P30, 65L03

CAT$^1$-HYPERGROUPS AND PULLBACK CAT$^1$-HYPERGROUPS | 316$-$325 |

**Abstract**

Loday's 1-cat group definition plays very powerful role in making some new applications to crossed module due to Whitehead. There are many applications of cat$^1$-groups such as cat$^1$-polygroups and pullback cat$^1$-polygroups. The importance of hypergroups come from the properties of hypergroups such that hypergroups in the sense of Marty do not have identity element, inverse element and they are generalization of the well known groups. In this paper, we introduce the concept of cat$^1$-hypergroups, their examples and some related properties. Also, we investigate pullback cat$^1$-hypergroups and properties such as: every cat$^1$-group is a cat$^1$-hypergroup; construction of a cat$^1$-group from a crossed module of hypergroups and vice versa. Finally, we present the definition of pullback cat$^1$-hypergroups and some of their properties.

**Keywords:** Action; crossed module; hypergroup; crossed module of hypergroups; fundamental relation; cat$^1$-group; cat$^1$-hypergroup; pullback cat$^1$-hypergroup.

**MSC:** 20N20, 18D35

CHAIN TRANSITIVITY FOR MAPS ON $G$-SPACES | 326$-$337 |

**Abstract**

We define and study the notion of chain transitivity for maps on $G$-spaces. Through examples we justify that the notion of $G$-chain transitivity depends on the action of $G$. Further, we obtain characterization of $G$-chain transitivity in terms of chain transitivity. A relation between $G$-chain transitivity and $G$-chain recurrent points of a map is also obtained.

**Keywords:** Chain transitive; chain recurrent points; $G$-space.

**MSC:** 37C85, 37C50, 37C75, 54H20

STRONG CONVERGENCE OF AN INERTIAL-TYPE ALGORITHM TO A COMMON SOLUTION OF MINIMIZATION AND FIXED POINT PROBLEMS | 338$-$350 |

**Abstract**

In this paper, we introduce an inertial accelerated iterative algorithm for approximating a common solution of a minimization problem and a fixed point problem for quasi-pseudocontractive mapping in a real Hilbert space. Using the algorithm, we prove a strong convergence theorem for approximating a common solution of a minimization problem and a fixed point problem for quasi-pseudocontractive mapping. Furthermore, we give an application of our main result to solve convexly constrained linear inverse problems, and we also present a numerical example of our algorithm to illustrate its applicability.

**Keywords:** Minimization problem; quasi-pseudocontractive mappings; inertial iterative scheme; fixed point problem.

**MSC:** 47H06, 47H09, 47J05, 47J25

SPACES OF $\boldsymbol{u}\boldsymbol\tau$-DUNFORD-PETTIS AND $\boldsymbol{u}\boldsymbol\tau$-COMPACT OPERATORS ON LOCALLY SOLID VECTOR LATTICES | 351$-$358 |

**Abstract**

Suppose $X$ is a locally solid vector lattice. It is known that there are several non-equivalent spaces of bounded operators on $X$. In this paper, we consider some situations under which these classes of bounded operators form locally solid vector lattices. In addition, we generalize some notions of $uaw$-Dunford-Pettis operators and $uaw$-compact operators defined on a Banach lattice to general theme of locally solid vector lattices. With the aid of appropriate topologies, we investigate some relations between topological and lattice structures of these operators. In particular, we characterize those spaces for which these concepts of operators and the corresponding classes of bounded ones coincide.

**Keywords:** $u\tau$-convergence; $u\tau$-Dunford-Pettis operator; $u\tau$-compact operator; locally solid vector lattice.

**MSC:** 46A40, 54A20, 46B40, 46A03

QUANTITATIVE ESTIMATES FOR MODIFIED BETA OPERATORS | 359$-$367 |

**Abstract**

In this paper, we introduce a sequence of positive linear Beta type operators based on a function $\tau$ having certain properties. We firstly give some approximation properties of these operators. Next, we establish Voronovskaja type and Grüss-Voronovskaja type theorems in quantitative form with the help of the first order Ditzian-Totik modulus of smoothness.

**Keywords:** Beta operators; Voronovskaja type theorem; Grüss-Voronovskaja type theorem.

**MSC:** 41A36

A NEW APPROACH TO SOME FIXED POINT THEOREMS FOR MULTIVALUED NONLINEAR F-CONTRACTIVE MAPS | 369$-$377 |

**Abstract**

In this article, by introducing a new operator, we give a new generalized contraction condition for multivalued maps. Moreover, without assumption of lower semi-continuity, we prove some fixed point theorems in incomplete metric spaces. Our results are extension of the corresponding results of I. Altun et al.\ (Nonlinear Analysis: Modeling and control, 2016, Vol.\ 21, No.\ 2, 201--210). Also, we provide some examples to show that our main theorem is a generalization of some previous results.

**Keywords:** Fixed point; Hausdorff metric; multivalued contraction; $SO$-complete metric space; $SO$-lower semi-continuous.

**MSC:** 54H25, 47H10