Volume 72 , issue 4 ( 2020 ) | back |

GENERALIZED CONTRACTIONS AND FIXED POINT THEOREMS OVER BIPOLAR CONE$_{tvs}$ $b$-METRIC SPACES WITH AN APPLICATION TO HOMOTOPY THEORY | 281$-$296 |

**Abstract**

In this paper, we introduce the concept of bipolar cone$_{tvs}$ $b$-metric space and prove some generalized fixed point theorems on it. These theorems extend and generalize some recent results obtained by other authors for mappings on a bipolar metric space. Also, a brief study on topological properties of this newly introduced space has been made and in support of our theorems, we give some examples. Moreover, our fixed point result is applied to homotopy theory on such spaces.

**Keywords:** Fixed point; bipolar cone$_{tvs}$ $b$-metric space; covariant and contravariant mappings; contravariant $A$-contraction mappings; homotopic mappings.

**MSC:** 47H10, 54H25

A NOTE ON THE MINIMAL DISPLACEMENT FUNCTION | 297$-$302 |

**Abstract**

Let $(X,d)$ be a metric space and ${Iso}(X,d)$ the associated isometry group. We study the subadditivity of the minimal displacement function $f:{Iso}(X,d)\to {R}$ for different metric spaces. When $(X,d)$ is ultrametric, we prove that the minimal displacement function is subadditive. We show, by a simple algebraic argument, that subadditivity does not hold for the direct isometry group of the hyperbolic plane. The same argument can be used for other metric spaces.

**Keywords:** Minimal displacement function; metric space; subadditivity.

**MSC:** 51F99, 51K05

SOME CHEBYSHEV TYPE INEQUALITIES INVOLVING THE HADAMARD PRODUCT OF HILBERT SPACE OPERATORS | 303$-$313 |

**Abstract**

In this paper, we prove that if ${A}$ is a Banach $*$-subalgebra of $B(H)$, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$¦ and $\alpha:T\rightarrow [0,\infty)$ is a integrable function and $(A_t), (B_t)$ are appropriate integrable fields of operators in ${A}$ having the almost synchronous property for the Hadamard product, then $$ \int_T\!\alpha(s)d\mu(s)\int_T\!\alpha(t)\big(A_t\circ B_t\big) d\mu(t) \geq \int_T\!\alpha(t)A_td\mu(t)\circ\int_T\!\alpha(t)B_td\mu(t). $$ We also introduce a semi-inner product for square integrable fields of operators in a Hilbert space and using it, we prove the Schwarz and Chebyshev type inequalities dealing with the Hadamard product and the trace of operators.

**Keywords:** Grüss inequality; Chebyshev inequality; operator inequality.

**MSC:** 26D10, 26D15, 46C50, 46G12

THE ZARIOUH'S PROPERTY $(gaz)$ THROUGH LOCALIZED SVEP | 314$-$326 |

**Abstract**

In this paper we study the property $(gaz)$ for a bounded linear operator $T\in L(X)$ on a Banach space $X$, introduced by Zariouh in [\emph{Property $(gz)$ for bounded linear operators}, Mat.\ Vesnik, {\bf 65(1)}(2013), 94--103], through the methods of local spectral theory. This property is a stronger variant of generalized $a$-Browder's theorem. In particular, we shall give several characterizations of property $(gaz)$, by using the localized SVEP.

**Keywords:** Property $(gaz)$; localized SVEP; Browder type theorems.

**MSC:** 47A10, 47A11, 47A53, 47A55

ON GENERALIZED DISTANCE SPECTRAL RADIUS OF A BIPARTITE GRAPH | 327$-$336 |

**Abstract**

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)$ and $D^{Q}(G)$ respectively be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix and the distance signless Laplacian matrix of a graph $G$. The convex linear combination $D_{\alpha}(G)$ of $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$, this matrix reduces to merging the distance spectral, signless distance Laplacian spectral theories. In this paper, we study the spectral radius of the generalized distance matrix $D_{\alpha}(G)$ of a graph $G$. We obtain bounds for the generalized distance spectral radius of a bipartite graph in terms of various parameters associated with the structure of the graph and characterize the extremal graphs. For $\alpha=0$, our results improve some previously known bounds.

**Keywords:** Distance matrix (spectrum); Distance signless Laplacian matrix (spectrum); generalized distance matrix; spectral radius.

**MSC:** 05C30, 05C50

INVARIANTS, SOLUTIONS AND INVOLUTION OF HIGHER ORDER DIFFERENTIAL SYSTEMS | 337$-$350 |

**Abstract**

The paper is concerned with the interpretation of the fixed points of an involution as invariant solutions under certain Lie algebra of symmetries of a given equation. Our aim is to study the involutivity in terms of the symmetries of an equation. We prove that if $\pi:E\to M$ is a fiber bundle and $\nabla:T^*M\to J^1T^*M$ is a linear connection on the base space, then there exists a unique involutive linear automorphism, $\alpha_{_{\nabla}}$ in $J^1J^1E$, that commutes with the projections $\pi_{11}$ and $J^1\pi_{1,0}$. Moreover, we prove that the space $J^k(\pi)$ is the quotient space of the iterated sesqui-holonomics jets $\^{J}^1J^{k-1}(\pi)$ relative to the subgroup of symmetries determined by some involution $\alpha_{g}$.

**Keywords:** Geometric structures on manifolds; differential systems; contact theory; co-contact of higher order.

**MSC:** 53C15, 53B25

A NOTE ON THE EIGENVALUES OF $\boldsymbol{n}$-CAYLEY GRAPHS | 351$-$357 |

**Abstract**

A graph $\Gamma$ is called an $n$-Cayley graph over a group $G$ if its automorphism contains a semi-regular subgroup isomorphic to $G$ with $n$ orbits. Every $n$-Cayley graph over a group $G$ is completely determined by $n^2$ suitable subsets of $G$. If each of these subsets is a union of conjugacy classes of $G$, then it is called a quasi-abelian $n$-Cayley graph over $G$. In this paper, we determine the characteristic polynomial of quasi-abelian $n$-Cayley graphs. Then we exactly determine the eigenvalues and the number of closed walks of quasi-abelian semi-Cayley graphs. Furthermore, we construct some integral graphs.

**Keywords:** Semi-Cayley graph; $n$-Cayley graph; quasi-abelian; eigenvalue.

**MSC:** 05C50, 05C25, 05C31

THE EXISTENCE OF ONE WEAK SOLUTION FOR A SECOND-ORDER IMPULSIVE HAMILTONIAN SYSTEM | 358$-$367 |

**Abstract**

In this work, we are concerned with the existence of at least one non-trivial weak solution for a second-order impulsive Hamiltonian system. The proof of the main result is based on the critical point theory.

**Keywords:** Weak solution; impulsive Hamiltonian system; critical point theory; variational methods.

**MSC:** 34B15, 47J10

INTERPOLATIVE HARDY-ROGERS AND REICH-RUS-ĆIRIĆ TYPE CONTRACTIONS IN $\boldsymbol b$-METRIC SPACES AND RECTANGULAR $\boldsymbol b$-METRIC SPACES | 368$-$374 |

**Abstract**

In this paper, we investigate two fixed point theorems in the framework of $b$-metric spaces and rectangular $b$-metric spaces, using interpolative approach. One is Hardy-Rogers and the other is Reich-Rus-{Ć}iri{ć} type contraction. Examples are provided in support of the results.

**Keywords:** Fixed point; $b$-metric space; rectangular $b$-metric space; contraction map.

**MSC:** 47H10, 54H25, 54E50