Some new extensions of Edelstein-Suzuki-type fixed point theorem to G-metric and G-cone metric spaces

In this paper, we prove some fixed point theorems for generalized contractions in the setting of G-metric spaces. Our results extend a result of Edelstein [M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79] and a result of Suzuki [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317]. We prove, also, a fixed point theorem in the setting of G-cone metric spaces.     

Common fixed point theorems for three maps in discontinuous Gb metric spaces

In [Aghajani A, Abbas M, Roshan JR. Common fixed point of generalized weak contractive mappings in partially ordered G_b-metric spaces. Filomat, 2013, in press], using the concepts of G-metric and b-metric Aghajani et al. defined a new type of metric which is called generalized b-metric or G_b-metric. In this paper, we prove a common fixed point theorem for three mappings in G_b-metric space which is not continuous. An example is presented to verify the effectiveness and applicability of our main result.     

Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces

In this paper we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally G-convex uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally G-convex uniform spaces. These theorems are new and generalize some known results in the literature.     

A result of suzuki type in partial G-metric spaces

Recently, Suzuki [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008), 1861-1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and characterizes the metric completeness. Paesano and Vetro [D. Paesano and P. Vetro, Suzuki's type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl., 159 (2012), 911-920] proved an analogous fixed point result for a self-mapping on a partial metric space that characterizes the partial metric 0-completeness. In this article, we introduce the notion of partial G-metric spaces and prove a result of Suzuki type in the setting of partial G-metric spaces. We deduce also a result of common fixed point.     

Generalized G-KKM Theorems in Generalized Convex Spaces and Their Applications

Some new generalized G-KKM and generalized S-KKM theorems are proved under the noncompact setting of generalized convex spaces. As applications, some new minimax inequalities, saddle point theorems, a coincidence theorem, and a fixed point theorem are given in generalized convex spaces. These theorems improve and generalize many important known results in recent literature.     

A note on iterative process for G-multi degree reduction of Bézier curves

In the paper [A. Rababah, S. Mann, Iterative process for G²-multi degree reduction of Bézier curves, Applied Mathematics and Computation 217 (2011) 8126-8133], Rababah and Mann proposed an iterative method for multi-degree reduction of Bézier curves with C¹ and G²-continuity at the endpoints. In this paper, we provide a theoretical proof for the existence of the unique solution in the first step of the iterative process, while the proof in their paper applies only in some special cases. Also, we give a complete convergence proof for the iterative method. We solve the problem by using convex quadratic optimization.     

The matching theorems in G-convex spaces with applications

The purpose of this paper is to establish some new matching theorems in G-convex spaces and, as applications, to obtain some new fixed point theorems, section theorems and a minimax theorem in G-convex spaces. The results presented in this paper improve and generalize the corresponding results in [1], [2], [3], [4], [5], [7], [8], [9], [10], [11] and [12].     

Duality of dams via mountain processes

We consider a G/M/1-type dam having finite capacity and a general release rule, and construct a ‘dual’ M/G/1-type dam with state-dependent jump sizes and without dry periods whose content process has the same stationary density (up to some transformation). For the dual dam the stationary distribution can be computed in closed form.     

Convergence theorem for the common solution for a finite family of ?-strongly accretive operator equations

The purpose of this paper is to study a strong convergence of multi-step iterative scheme to a common solution for a finite family of uniformly continuous ?-strongly accretive operator equations in an arbitrary Banach space. As a consequence, the strong convergence theorem for the multi-step iterative sequence to a common fixed point for finite family of ?-strongly pseudocontractive mappings is also obtained. The results presented in this paper thus improve and extend the corresponding results of Inchan [6], Kang [8] and [9] and many others.     

Some relations among term rank, clique number and list chromatic number of a graph

Let G be a graph with a nonempty edge set, we denote the rank of the adjacency matrix of G and term rank of G, by rk(G) and Rk(G), respectively. van Nuffelen conjectured that for any graph G, χ(G)?rk(G). The first counterexample to this conjecture was obtained by Alon and Seymour. In 2002, Fishkind and Kotlov proved that for any graph G, χ(G)?Rk(G). Here we improve this upper bound and show that χ_l(G)?(rk(G)+Rk(G))/2, where χ_l(G) is the list chromatic number of G.     

On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v₁,…,v_kof G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K₄ and K_3,3. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K₄ and K_3,3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.     

Bandwidth theorem for random graphs

A graph G is said to have bandwidth at most b, if there exists a labeling of the vertices by 1,2,…,n, so that |i−j|?b whenever {i,j} is an edge of G. Recently, Böttcher, Schacht, and Taraz verified a conjecture of Bollobás and Komlós which says that for every positive r, Δ, γ, there exists β such that if H is an n-vertex r-chromatic graph with maximum degree at most Δ which has bandwidth at most βn, then any graph G on n vertices with minimum degree at least (1−1/r+γ)n contains a copy of H for large enough n. In this paper, we extend this theorem to dense random graphs. For bipartite H, this answers an open question of Böttcher, Kohayakawa, and Taraz. It appears that for non-bipartite H the direct extension is not possible, and one needs in addition that some vertices of H have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed r-chromatic graph H₀ which one can find in a spanning subgraph of G(n,p) with minimum degree (1−1/r+γ)np.     

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

Let G be a mixed graph and let L(G) be the Laplacian matrix of the graph G. The first eigenvalue and the first eigenvectors of G are respectively referred to the least nonzero eigenvalue and the corresponding eigenvectors of L(G). In this paper we focus on the properties of the first eigenvalue and the first eigenvectors of a nonsingular unicyclic mixed graph (abbreviated to a NUM graph). We introduce the notion of characteristic set associated with the first eigenvectors, and then obtain some results on the sign structure of the first eigenvectors. By these results we determine the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order and fixed girth, and the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order.     

Design and G connection of developable surfaces through Bézier geodesics

For potential application in shoemaking and garment manufacture industries, the G¹ connection of (1, k) developable surfaces with abutting geodesic is important. In this paper, we discuss the developable surface which contains a given 3D Bézier curve as geodesic and prove the corresponding conclusions in detail. Primarily we study G¹ connection of developable surfaces through abutting cubic Bézier geodesics and give some examples.     

Compact group actions on equivariant absolute neighborhood retracts and their orbit spaces

We apply equivariant joins to give a new and more transparent proof of the following result: if G is a compact Hausdorff group and X a G-ANR (respectively, a G-AR), then for every closed normal subgroup H of G, the H-orbit space X/H is a G/H-ANR (respectively, a G/H-AR). In particular, X/G is an ANR (respectively, an AR).     

The nucleus of a point determining graph

A graph is point determining if distinct points have distinct neighborhoods. In this paper we investigate the nucleus G⁰={υ?G:G–υ is point determining} of a point determining graph G. In particular, we characterize those graphs that are the nucleus of some connected point determining graph.     

A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G+e is k-factor-critical for every missing edge e∉E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for 1?k?n-1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but G+e is k-extendable for every missing edge e∉E(G) . A connected bipartite graph G with a bipartitioning set (X,Y) such that |X|=|Y|=n is maximal non-k-extendable bipartite if G is not k-extendable but G+xy is k-extendable for any edge xy∉E(G) with x∈X and y∈Y. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.     

A surjection theorem and a fixed point theorem for a class of positive operators

This paper is concerned with α-convex operators on ordered Banach spaces. A surjection theorem for 1-convex operators in order intervals is established by means of the properties of cone and monotone iterative technique. It is assumed that 1-convex operator A is increasing and satisfies Ay−Ax?M(y−x) for θ?x?y?v₀, where θ denotes the zero element and v₀ is a constant. Moreover, we prove a fixed point theorem for -convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations.     

Graph powers and k-ordered Hamiltonicity

It is known that if G is a connected simple graph, then G³ is Hamiltonian (in fact, Hamilton-connected). A simple graph is k-ordered Hamiltonian if for any sequence v₁, v₂,…,v_k of k vertices there is a Hamiltonian cycle containing these vertices in the given order. In this paper, we prove that if k?4, then G^⌊3k/2⌋-2 is k-ordered Hamiltonian for every connected graph G on at least k vertices. By considering the case of the path graph P_n, we show that this result is sharp. We also give a lower bound on the power of the cycle C_n that guarantees k-ordered Hamiltonicity.     

Construction for bicritical graphs and k-extendable bipartite graphs

A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.