Scielo RSS <![CDATA[Proyecciones (Antofagasta)]]> vol. 35 num. 3 lang. pt <![CDATA[SciELO Logo]]> <![CDATA[<strong>An alternative proof of a Tauberian theorem for Abel summability method</strong>]]> Using a corollary to Karamata’s main theorem [Math. Z. 32 (1930), 319-320], we prove that ifa slowly decreasing sequence ofreal numbers is Abel summable, then it is convergent in the ordinary sense. <![CDATA[<strong>Unicyclic graphs with equal domination and complementary tree domination numbers</strong>]]> Let G = (V, E) be a simple graph. A set <img src="http:/fbpe/img/proy/v35n3/art2_fig1.jpg" width="75" height="18"> is a dominating set if every vertex in V(G) \ D is adjacent to a vertex of D. A dominating set D of a graph G is a complementary tree dominating set if induced sub graph (V \ D) is a tree. The domination (complementary tree domination, respectively) number of G is the minimum cardinality of a dominating (complementary tree dominating, respectively) set of G. We characterize all unicyclic graphs with equal domination and complementary tree domination numbers. <![CDATA[<strong>Total edge irregularity strength of disjoint union of double wheel graphs</strong>]]> An edge irregular total k-labeling f : V ∪ E → {1, 2, 3,...,k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way thatfor any two different edges uv and u'v' their weights f (u) + f (uv) + f (v) and f (u') + f (u'v') + f (v') are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of disjoint union of p isomorphic double wheel graphs and disjoint union of p consecutive non-isomorphic double wheel graphs. <![CDATA[<strong>Asymptotic stability in delay nonlinear fractional differential equations</strong>]]> In this paper, we give sufficient conditions to guarantee the asymptotic stability of the zero solution to a kind of delay nonlinear fractional differential equations of order <img src="http:/fbpe/img/proy/v35n3/art4_fig1.jpg" width="85" height="11">. By using the Banach’s contraction mapping principle in a weighted Banach space, we establish new results on the asymptotic stability of the zero solution provided that g (t, 0) = f (t, 0, 0) = 0, which include and improve some related results in the literature. <![CDATA[<strong>One modulo three mean labeling of transformed</strong> <strong>trees</strong>]]> A graph G is said to be one modulo three mean graph if there is an injective function φ from the vertex set of G to the set {a|0 ≤ a ≤ 3q- 2 and either a ≡ 0(mod 3) or a ≡ 1(mod 3)} where q is the number of edges G and φ induces a bijection φ* from the edge set of G to {a|1 ≤ a ≤ 3q - 2 and either a ≡ 1(mod 3)} given by <img src="http:/fbpe/img/proy/v35n3/art5_fig1.jpg" width="242" height="77"> and the function φ is called one modulo three mean labeling of G. In this paper, we prove that the graphs T ° Kn, T ô K1,n, T ô Pn and T ô 2Pn are one modulo three mean graphs. <![CDATA[<strong>Weak forms of continuity and openness</strong>]]> Some new class of functions, called somewhat <img src="http:/fbpe/img/proy/v35n3/art6-fig1.jpg" width="10" height="17">-precontinuous, somewhat <img src="http:/fbpe/img/proy/v35n3/art6-fig1.jpg" width="10" height="17">-preopen and hardly <img src="http:/fbpe/img/proy/v35n3/art6-fig1.jpg" width="10" height="17">-preopenfunctions, have been defined and studied by utilizing <img src="http:/fbpe/img/proy/v35n3/art6-fig1.jpg" width="10" height="17">-preopen sets. Moreover, characterizations and properties of these functions are presented. <![CDATA[<strong>Non-linear maps preserving singular algebraic operators</strong>]]> Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space H. We prove that if Φ is a surjective map on B(H) such that Φ(I) = I + Φ(0) and for every pair T, S ∈ B(H), the operator T - S is singular algebraic if and only if Φ(T) - Φ(S) is singular algebraic, then Φ is either of the form Φ(T) = ATA-1 + Φ(0) or the form Φ(T) = AT*A-1 + Φ(0) where A : H → H is an invertible bounded linear, or conjugate linear, operator. <![CDATA[<strong>Stability and boundedness in differential systems of third order with variable delay</strong>]]> In this paper, we consider a non-linear system of differential equations ofthird order with variable delay. We discuss the globally asymptotic stability/uniformly stability, boundedness and uniformly boundedness ofsolutionsfor the considered system. The technique ofproofs involves defining an appropriate Lyapunov functional. The obtained results include and improve the results in literature. <![CDATA[<strong>Gliding Hump Properties in Abstract Duality Pairs with Projections</strong>]]> Let E, G be Hausdorff topological vector spaces and let F be a vector space. Assume there is a bilinear operator : E X F →G such that : E →G is continuous for every y £ F. The triple E, F, G is called an abstract duality pair with respect to G or an abstract triple and is denoted by (E,F : G). If {Pj} is a sequence of continuous projections on E, then (E,F : G) is called an abstract triple with projections. Under appropriate gliding hump assumptions, a uniform bounded principle is established for bounded subsets ofE and pointwise bounded subsets of F. Under additional gliding hump assumptions, uniform convergent results are established for series ∑ ∞ j=1 < Pj x,y&gt; when x varies over certain subsets of E and y varies over certain subsets of F. These results are used to establish uniform countable additivity results for bounded sets of indefinite vector valued integrals and bounded subsets of vector valued measures.