Proyecciones. Journal of Mathematics Universidad Católica del Norte. en-US Proyecciones. Journal of Mathematics 0716-0917 On semi-open sets and Feebly open sets in generalized topological spaces <p>In this paper, we introduce the notion of semi-open sets and feebly open sets in generalized topological spaces. Several properties of these notions are discussed. Also this paper considers (semi and feebly)-separation axioms for generalized topological spaces. We further investigate (semi-continuous, feebly-continuous, almost open)-functions in generalized topological spaces.</p> B. K. Tyagi Harsh V. S. Chauhan Copyright (c) 2019 B. K. Tyagi, Harsh V. S. Chauhan 2019-12-15 2019-12-15 38 5 875 896 10.22199/issn.0717-6279-2019-05-0057 Left and right generalized Drazin invertible operators and local spectral theory <p>In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property (C), and property (β), to its generalized Drazin inverse.</p> Mohammed Benharrat Kouider Miloud Hocine Bekkai Messirdi Copyright (c) 2019 Mohammed Benharrat, Kouider Miloud Hocine, Bekkai Messirdi 2019-12-15 2019-12-15 38 5 897 919 10.22199/issn.0717-6279-2019-05-0058 θω−Connectedness and ω−R1 properties <p><em>We use the theta omega closure operator to define theta omega connectedness as a property which is weaker than connectedness and stronger than θ-connectedness. We give several sufficient conditions for the equivalence between θ<sub>ω</sub>-connectedness and connectedness, and between θω-connectedness and θ-connectedness. We give two results regarding the union of θω-connected sets and also we show that the weakly θ<sub>ω</sub>-continuous image of a connected set is θ<sub>ω</sub>-connected. We define and investigate V -θ<sub>ω</sub>-connectedness as a strong form of V - θ-connectedness, and we show that the θ<sub>ω</sub>-connectedness and V -θ<sub>ω</sub>-connectedness are independent. We continue the study of R<sub>1</sub> as a known topological property by giving several results regarding it. We introduce ω-R<sub>1</sub> (I), ω-R<sub>1</sub> (II), ω-R<sub>1</sub> (III) and weakly ω-R<sub>1</sub> as four weaker forms of R<sub>1</sub> by utilizing ω-open sets, we give several relationships regarding them and we raise two open questions.</em></p> Samer Al Ghour Salma El-Issa Copyright (c) 2019 Samer Al Ghour, Salma El-Issa 2019-12-15 2019-12-15 38 5 921 942 10.22199/issn.0717-6279-2019-05-0059 A new generalization of Wilson’s functional equation <p><em>Let G be a group, let σ : G → G be an involutive automorphism and let χ<sub>1</sub>, χ<sub>2</sub> : G → <strong>C</strong></em><em><sup>∗</sup></em><em> be two characters of G such that χ<sub>2</sub>(xσ(x)) = 1 for all x </em><em>∈</em><em> <strong>G.</strong> The aim of this paper is to describe the solutions f, g : G → </em>C<em> of the functional equation </em></p> <p><em>χ<sub>1</sub>(y)f (xy) + χ<sub>2</sub>(y)f (σ(y)x) = 2f (x)g(y), x,y </em><em>∈</em><em> G, </em></p> <p><em>in terms of characters and additive functions.</em></p> Hajira Dimou Abdellatif Chahbi Samir Kabbaj Copyright (c) 2019 Hajira Dimou, Abdellatif Chahbi, Samir Kabbaj 2019-12-15 2019-12-15 38 5 943 954 10.22199/issn.0717-6279-2019-05-0060 On a class of a boundary value problems involving the p(x)-Biharmonic operator <p>Our aim is to establish the existence of weak solution for a class of Robin problems involving fourth order operator. The nonlinearity is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz condition.<br>The proof is made with and without variational structure.</p> Anass Ourraoui Copyright (c) 2019 Anass Ourraoui 2019-12-16 2019-12-16 38 5 955 967 10.22199/issn.0717-6279-2019-05-0061 The b-radical of generalized alternative b-algebras II <p><em>We prove that if (U, ω) is a finite dimensional generalized alternative b-algebra II over a field F of characteristic different from 2 and 3, then rad(U) = R(U) </em>⋂<em> (bar(U))<sup>3</sup>.</em></p> B. L. M. Ferreira Copyright (c) 2019 B. L. M. Ferreira 2019-12-16 2019-12-16 38 5 969 979 10.22199/issn.0717-6279-2019-05-0062 On approximation of signals in the generalized Zygmund class via (E, 1) (N̅, pn) summability means of conjugate Fourier series <p><em>Approximation of functions of different classes have been considered by various researchers under different summability means. In the present paper, presumably a new theorem has been established under (E, 1)(N̅ p<sub>n</sub>) -product summability mean of conjugate Fourier series of a function of Zr(⍵) r (r ≥ 1) -class (generalized Zygmund class). Moreover, the result obtained here is a generalization of several known theorems.</em></p> T. Pradhan S. K. Paikray A. A. Das Hemen Dutta Copyright (c) 2019 T. Pradhan, S. K. Paikray, A. A. Das, Hemen Dutta 2019-12-16 2019-12-16 38 5 981 998 10.22199/issn.0717-6279-2019-05-0063 Zero forcing in Benzenoid network <p><em>A set S of vertices in a graph G is called a dominating set of G if every vertex in V (G)\S is adjacent to some vertex in S. A set S is said to be a power dominating set of G if every vertex in the system is monitored by the set S following a set of rules for power system monitoring. The power domination number of G is the minimum cardinality of a power dominating set of G. A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set (zero forcing set) of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number of G, denoted Z(G), is the minimum cardinality of a zero forcing set of G. In this paper, we obtain the zero forcing number for certain benzenoid networks.</em></p> <p><strong><em>&nbsp;</em></strong></p> J. Anitha Indra Rajasingh Copyright (c) 2019 J. Anitha, Indra Rajasingh 2019-12-18 2019-12-18 38 5 999 1010 10.22199/issn.0717-6279-2019-05-0064 I-statistical limit points and I-statistical cluster points <p><em>In this paper we have extended the notion of statistical limit point as introduced by Fridy</em>[8]<em> to I-statistical limit point of sequences of real numbers and studied some basic properties of the set of all Istatistical limit points and I-statistical cluster points of real sequences including their interrelationship. Also introducing additive property of I-asymptotic density zero sets we establish I-statistical analogue of some completeness theorems of </em><strong>R</strong><em>.</em></p> Prasanta Malik Argha Ghosh Samiran Das Copyright (c) 2019 Prasanta Malik, Argha Ghosh, Samiran Das 2019-12-17 2019-12-17 38 5 1011 1026 10.22199/issn.0717-6279-2019-05-0065 Zero-sum flow number of octagonal grid and generalized prism <p><em>A zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k-flow if the absolute values of edges are less than k. We recall the zero-sum flow number of G as the least integer k for which G admitting a zero sum k-flow. In this paper we gave complete zero-sum flow and zero sum numbers for Octagonal Grid and Generalized Prism. </em></p> Muhammad Naeem Muhammad Imran Sarfraz Ahmad Muhammad Kamran Siddiqui Copyright (c) 2019 Muhammad Naeem, Muhammad Imran, Sarfraz Ahmad, Muhammad Kamran Siddiqui 2019-12-17 2019-12-17 38 5 1027 1038 10.22199/issn.0717-6279-2019-05-0066 µ-Statistically convergent function sequences in probabilistic normed linear spaces <p><em>In this article, we introduce the concept of µ-statistical convergence and µ-density convergence of sequences of functions defined on a compact subset D of the probabilistic normed space (X, N, </em><em>∗</em><em>), where µ is a finitely additive two valued measure. In particular, we introduce the notions of µ-statistical uniform convergence as well as µ-statistical point-wise convergence of sequences of functions in probabilistic normed space (in short PN-space) and we give some characterization results on these two convergences of sequences of functions in PN-space. We have also observed that µ-statistical uniform convergence of sequences of functions in PN-spaces inherits the basic properties of uniform convergence. </em></p> Mausumi Sen Rupam Haloi Binod Chandra Tripathy Copyright (c) 2019 Mausumi Sen, Rupan Haloi, Binod Chandra Tripathy 2019-12-17 2019-12-17 38 5 1039 1056 10.22199/issn.0717-6279-2019-05-0067 Nonlocal quantum stochastic differential equations with impulsive effects <p><em>The aim of this work is to establish further existence of solution results for quantum stochastic differential equations with unbounded coefficients, which are also stochastic processes. Existence of solutions for nonlocal quantum stochastic differential equations with impulsive effects is established. The nonlocal condition extends the traditional initial value condition and cases of Lipschitz and non-Lipschitz continuous conditions were established. The quantum stochastic differential equation considered are driven by noises on Boson Fock spaces and measure of non compactness was employed to prove the main result. </em></p> M. O. Ogundiran Copyright (c) 2019 M. O. Ogundiran 2019-12-18 2019-12-18 38 5 1057 1069 10.22199/issn.0717-6279-2019-05-0068 On star coloring of degree splitting of join graphs <p><em>A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χ<sub>s</sub> (G) of G is the least number of colors needed to star color G. In this paper, we have generalized the star chromatic number of degree splitting of join of any two graph G and H denoted by G + H, where G is a path graph and H is any simple graph. Also, we determine the star chromatic number for degree splitting of join of path graph G of order m with path P<sub>n</sub>, complete graph K<sub>n</sub> and cyclevgraph C<sub>n</sub>.</em></p> S. Ulagammal Vernold Vivin J. Copyright (c) 2019 S. Ulagammal, Vernold Vivin J. 2019-12-18 2019-12-18 38 5 1071 1080 10.22199/issn.0717-6279-2019-05-0069 Computing the Schultz polynomials and indices for ladder related graphs <p><em>Distance is an important graph invariant that has wide applications in computing science and other fields of sciences. A topological index is a genuine number connected with compound constitution indicating for relationship of compound structure with different physical properties, synthetic reactivity or natural action. The Schultz and modified Schultz polynomials and their corresponding indices are used in synthetic graph theory as in light of vertex degrees. In this paper, the Schultz and modified Schultz polynomials and their corresponding indices for Mongolian tent graph, diamond graph and double fan are determined.</em></p> Ali Ahmad Copyright (c) 2019 Ali Ahmad 2019-12-18 2019-12-18 38 5 1081 1092 10.22199/issn.0717-6279-2019-05-0070 (p, q)-Lucas polynomials and their applications to bi-univalent functions <p><em>In the present paper, by using the L<sub>p,q,n</sub>(x) functions, our methodology intertwine to yield the Theory of Geometric Functions and that of Special Functions, which are usually considered as very different fields. Thus, we aim at introducing a new class of bi-univalent functions defined through the (p, q)-Lucas polynomials. Furthermore, we derive coefficient inequalities and obtain Fekete-Szegö problem for this new function class.</em></p> Şahsene Altınkaya Sibel Yalçın Copyright (c) 2019 Şahsene Altınkaya, Sibel Yalçın 2019-12-26 2019-12-26 38 5 1093 1105 10.22199/issn.0717-6279-2019-05-0071