Volume 74 , issue 2 ( 2022 ) | back |

ON SINGULAR FIFTH-ORDER BOUNDARY VALUE PROBLEMS WITH DEFICIENCY INDICES $\boldsymbol{(5,5)}$ | 79$-$88 |

**Abstract**

This paper is devoted to introduce a way of construction of the well-defined boundary conditions for the solutions of a singular fifth-order equation with deficiency indices $(5,5)$. Imposing suitable separated and coupled boundary conditions some properties of the eigenvalues of the problems have been investigated.

**Keywords:** Fifth-order equation; eigenvalue problems; boundary conditions.

**MSC:** 34B05, 34B40, 34B09

ITERATIVE METHOD FOR FINDING ZEROS OF MONOTONE MAPPINGS AND FIXED POINT OF CERTAIN NONLINEAR MAPPING | 89$-$100 |

**Abstract**

In this article, an inertial Mann-type iterative algorithm is constructed using the so-called viscosity method of A. Moudafi, Viscosity approximation methods for fixed-point problems, J. Math. Anal. Appl. {241(1)} (2000), 46--55. A strong convergence theorem of mean ergodic-type is proved using the sequence of the iterative algorithm for finding zeros of monotone mappings and the fixed point of a strict pseudo nonspreading mapping in a real Hilbert space. Finally, we apply our result to solve some minimization problem.

**Keywords:** Monotone operators; strict-pseudo contractive mappings; strict-pseudo nonspreading mappings; resolvents; Hilbert space.

**MSC:** 47H04, 47H06, 47H15, 47H17, 47J25

APPROXIMATION OF GENERALIZED PALTANEA AND HEILMANN-TYPE OPERATORS | 101$-$109 |

**Abstract**

In this paper, we study the approximation on differences of two different positive linear operators (generalized Păltănea type operators and M. Heilmann type operators) with same basis functions. We estimates a quantitative difference of these operators in terms of modulus of continuity and Peetre's $K$-functional. We represent the rate of convergence, using modulus of continuity and Peetre's $K$-functional. Also, we represent Heilmann-type operators in terms of hypergeometric series.

**Keywords:** Difference operators; generalized Păltănea type operators; Heilmann type operators; modulus of continuity, rate of convergence.

**MSC:** 41A25, 26A15,41A30

$\boldsymbol{\mathcal I^{*}\text{-}\alpha}$ CONVERGENCE AND $\boldsymbol{\mathcal I^*}$-EXHAUSTIVENESS OF SEQUENCES OF METRIC FUNCTIONS | 110$-$118 |

**Abstract**

By a metric function, we mean a function from a metric space $(X,d)$ into a metric space $(Y,\rho)$. We introduce and study the notions of $\mathcal I^{*}\text{-}\alpha$ convergence and $\mathcal I^*$-exhaustiveness of sequences of metric functions, and we establish an inter-relationship between these two concepts. Moreover, we establish some relationship between our concepts with some well-established concepts such as $\mathcal I\text{-}\alpha$ convergence and $\mathcal I$-exhaustiveness of sequences of metric functions.

**Keywords:** $\mathcal I^{*}\text{-}\alpha$ convergence; $\mathcal I^*$-exhaustiveness; sequence of metric functions; ideal convergence.

**MSC:** 40A35, 26A03

BOUNDS FOR THE $\boldsymbol{A_{\alpha}}$-SPECTRAL RADIUS OF A DIGRAPH | 119$-$129 |

**Abstract**

Let $ D$ be a digraph of order $n$ and let $ A(D) $ be the adjacency matrix of $D$. Let $ Deg(D) $ be the diagonal matrix of vertex out-degrees of $ D$. For any real $ \alphaın [0,1], $ the generalized adjacency matrix $ A_{\alpha}(D) $ of the $D$ is defined as $ A_{\alpha}(D)=\alpha Deg(D)+(1-\alpha)A(D).$ The largest modulus of the eigenvalues of $ A_{\alpha}(D) $ is called the generalized adjacency spectral radius or the $ A_{\alpha} $-spectral radius of $ D$. In this paper, we obtain some new upper and lower bounds for the spectral radius of $ A_{\alpha}(D) $ in terms of the number of vertices $n$, the number of arcs, the vertex out-degrees, the average 2-out-degrees of the vertices of $ D $ and the parameter~$ \alpha $. We characterize the extremal digraphs attaining these bounds.

**Keywords:** Strongly connected digraphs; generalized adjacency matrix; generalized adjacency spectral radius; digraphs.

**MSC:** 05C50, 05C12, 15A18

FIXED POINTS OF ALMOST SUZUKI TYPE $\boldsymbol{\mathcal{Z}_s}$-CONTRACTIONS IN S-METRIC SPACES | 130$-$140 |

**Abstract**

In this paper, we introduce almost Suzuki type $\mathcal{Z}_s$-contractions and prove the existence and uniqueness of fixed points of such mappings in complete $S$-metric spaces. Our results generalize Theorem 1 from [N. Mlaiki, N. Yı lmaz Özgür, Nihal Ta\c s, Mathematics, 7 (583) 2019, 12 pages] and Theorem 3.1 from [S. Sedghi, N. Shobe, A. Aliouche, Mat. Vesnik, 64 (3) (2012), 258-266]. We give illustrative examples in support of our result.

**Keywords:** $S$-metric space; $\mathcal{Z}$-contraction; simulation function; $\mathcal{Z}_s$-contraction; almost Suzuki type $\mathcal{Z}_s$-contraction.

**MSC:** 47H10, 54H25

FIRST AND SECOND ORDER NONCONVEX SWEEPING PROCESS WITH PERTURBATION | 141$-$154 |

**Abstract**

We prove two existence results for functional differential inclusions governed by sweeping process. We consider the class of subsmooth moving sets. The perturbations depend on all the variables and their values are nonconvex.

**Keywords:** Differential inclusion; nonconvex sweeping process; subsmooth sets; set-valued map; normal cone.

**MSC:** 34A60, 34B15, 47H10