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In this note, we point out several gaps in Gurudwan and Sharma (Appl. Math. Comput. 217(15):6748-6754, 2011) and Yang (Appl. Math. Comput. 218(21):10367-10369, 2012) and give the main results under weaker conditions. MSC: 47H10; 47H09; 46B20

The mathematical analysis of the shape of chemical reactors is studied in this paper through the research of the optimization of its effectiveness η such as introduced by R. Aris around 1960. Although our main motivation is the consideration of reactors specially designed for the treatment of wastewaters our results are relevant also in more general frameworks. We simplify the modeling by assuming a single chemical reaction with a monotone kinetics leading to a parabolic equation with a non-necessarily differentiable function. In fact we consider here the case of a single, non-reversible catalysis reaction of chemical order q,00). We assume the chemical reactor of cylindrical shape Ω=G×(0,H) with G and open regular set of R2 not necessarily symmetric. We show that among all the sections G with prescribed area the ball is the set of lowest effectiveness η(t,G). The proof uses the notions of Steiner rearrangement. Finally, we show that if the height H is small enough then the effectiveness can be made as close to 1 as desired. 888 The mathematical analysis of the shape of chemical reactors is studied in this paper through the research of the optimization of its effectiveness (Formula presented.) such as introduced by R. Aris around 1960. Although our main motivation is the consideration of reactors specially designed for the treatment of wastewaters our results are relevant also in more general frameworks. We simplify the modeling by assuming a single chemical reaction with a monotone kinetics leading to a parabolic equation with a non-necessarily differentiable function. In fact we consider here the case of a single, non-reversible catalysis reaction of chemical order (Formula presented.) (i.e., the kinetics is given by (Formula presented.) for some (Formula presented.)). We assume the chemical reactor of cylindrical shape (Formula presented.) with G and open regular set of (Formula presented.) not necessarily symmetric. We show that among all the sections G with prescribed area the ball is the set of lowest effectiveness (Formula presented.). The proof uses the notions of Steiner rearrangement. Finally, we show that if the height H is small enough then the effectiveness can be made as close to 1 as desired.

The algebraic approach by E.W. Dijkstra and C.S. Scholten to formal logic is a proof calculus, where the notion of proof is a sequence of equivalences proved – mainly – by using substitution of ‘equals for equals’. This paper presents Set , a first-order logic axiomatization for set theory using the approach of Dijkstra and Scholten. What is novel about the approach presented in this paper is that symbolic manipulation of formulas is an effective tool for teaching an axiomatic set theory course to sophomore-year undergraduate students in mathematics. This paper contains many examples on how argumentative proofs can be easily expressed in Set and points out how the rigorous approach of Set can enrich the learning experience of students. The results presented in this paper are part of a larger effort to formally study and mechanize topics in mathematics and computer science with the algebraic approach of Dijkstra and Scholten., El enfoque algebraico de E.W. Dijkstra y C.S. Scholten a la lógica formal es un cálculo de prueba, donde la noción de prueba es una secuencia de equivalencias probadas, principalmente, mediante la sustitución de "iguales por iguales". Este artículo presenta Set, una axiomatización lógica de primer orden para la teoría de conjuntos utilizando el enfoque de Dijkstra y Scholten. Lo novedoso del enfoque presentado en este artículo es que la manipulación simbólica de fórmulas es una herramienta eficaz para enseñar un curso de teoría axiomática de conjuntos a estudiantes de segundo año de pregrado en matemáticas. Este artículo contiene muchos ejemplos sobre cómo las pruebas argumentativas se pueden expresar fácilmente en Set y señala cómo el enfoque riguroso de Set puede enriquecer la experiencia de aprendizaje de los estudiantes. Los resultados presentados en este artículo son parte de un esfuerzo mayor para estudiar y mecanizar formalmente temas en matemáticas e informática con el enfoque algebraico de Dijkstra y Scholten., Colombian Conference on Computing

This paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family. The research of the first author is supported by ANID-Fondecyt 1190012, Proyecto/Grant PIA AFB-170001, MICIU of Spain and Universidad de Alicante (Contract Beatriz Galindo BEA- GAL 18/00205). The second author is supported by the Research Project PGC2018-097960-B-C21 from MICINN, Spain, and by the Australian ARC—Discovery Projects DP 180100602.

We analyse the performance of well-known evolutionary algorithms (1+1)EA and (1+$\lambda$)EA in the prior noise model, where in each fitness evaluation the search point is altered before evaluation with probability $p$. We present refined results for the expected optimisation time of the (1+1)EA and the (1+$\lambda$)EA on the function LeadingOnes, where bits have to be optimised in sequence. Previous work showed that the (1+1)EA on LeadingOnes runs in polynomial expected time if $p = O((\log n)/n^2)$ and needs superpolynomial expected time if $p = \omega((\log n)/n)$, leaving a huge gap for which no results were known. We close this gap by showing that the expected optimisation time is $\Theta(n^2) \cdot \exp(\Theta(\min\{pn^2, n\}))$ for all $p \le 1/2$, allowing for the first time to locate the threshold between polynomial and superpolynomial expected times at $p = \Theta((\log n)/n^2)$. Hence the (1+1)EA on LeadingOnes is much more sensitive to noise than previously thought. We also show that offspring populations of size $\lambda \ge 3.42\log n$ can effectively deal with much higher noise than known before. Finally, we present an example of a rugged landscape where prior noise can help to escape from local optima by blurring the landscape and allowing a hill climber to see the underlying gradient. We prove that in this particular setting noise can have a highly beneficial effect on performance., Comment: This is an extended version of a paper that appeared in the Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2018), https://doi.org/10.1145/3205455.3205595

The purpose of the paper is to study the uniqueness of meromorphic functions sharing a small function with weight. The results of the paper improve and extend some recent results due to Banerjee and Sahoo (Sarajevo J Math 20:69–89, 2012), which in turn radically improve, extend and supplement some results of Dyavanal (J Math Anal Appl 372(1):252–261, 2010; 374(1):334, 2011; 374(1):345–355, 2011).

For any commutative semigroup S and any positive integer m, the power function f:S→S defined by f(x)=xm is an endomorphism of S. In this paper we characterize finite cyclic semigroups as those finite commutative semigroups whose endomorphisms are power functions. We also prove that if S is a finite commutative semigroup with 1≠0, then every endomorphism of S preserving 1 and 0 is equal to a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. Immediate consequences of the results are, on the one hand, a characterization of commutative rings whose multiplicative endomorphisms are power functions given by Greg Oman in the paper (Semigroup Forum, 86 (2013), 272–278), and on the other hand, a partial solution of Problem 1 posed by Oman in the same paper.

The purpose of this paper is to introduce and study the strong convergence problem of the explicit iteration process for a Lipschitzian and demicontraction semigroups in arbitrary Banach spaces. The main results presented in this paper not only extend and improve some recent results announced by many authors, but also give an affirmative answer for the open questions raised by Suzuki (2003) and Xu (2005).

A k-weighting w of a graph is an assignment of an integer weight $$w(e)\in \{1,...k\}$$w(e)ź{1,...k} to each edge e. Such an edge weighting induces a vertex coloring c defined by $$c(v)=\mathop {\displaystyle {\prod }}\limits _{v\in e}w(e).$$c(v)=źvźew(e). A k-weighting of a graph G is multiplicative vertex-coloring if the induced coloring c is proper, i.e., $$c(u)\ne c(v)$$c(u)źc(v) for any edge $$uv\in E(G).$$uvźE(G). This paper studies the parameter $$\mu (G),$$μ(G), which is the minimum k for which G has a multiplicative vertex-coloring k-weighting. Chang, Lu, Wu, Yu investigated graphs with additive vertex-coloring 2-weightings (they considered sums instead of products of incident edge weights). In particular, they proved that 3-connected bipartite graphs, bipartite graphs with the minimum degree 1, and r-regular bipartite graphs with $$r\ge 3$$rź3 permit an additive vertex-coloring 2-weighting. In this paper, the multiplicative version of the problem is considered. It was shown in Skowronek-Kaziow (Inf Process Lett 112:191---194, 2012) that $$\mu (G)\le 4$$μ(G)≤4 for every graph G. It was also proved that every 3-colorable graph admits a multiplicative vertex-coloring 3 -weighting. A natural problem to consider is whether every 2-colorable graph (i.e., a bipartite graph) has a multiplicative vertex-coloring 2-weighting. But the answer is no, since the cycle $$C_{6}$$C6 and the path $$P_{6}$$P6 do not admit a multiplicative vertex-coloring 2-weighting. The paper presents several classes of 2-colorable graphs for which $$\mu (G)=2$$μ(G)=2, including trees with at least two adjacent leaf edges, bipartite graphs with the minimum degree 3 and bipartite graphs $$G=(A,B,E)$$G=(A,B,E) with even $$\left| A\right| $$A or $$\left| B\right| $$B.

Let M be a shift invariant subspace in the vector-valued Hardy space \({H_{E}^{2}(\mathbb{D})}\). The Beurling–Lax–Halmos theorem says that M can be completely characterized by \({\mathcal{B}(E)}\)-valued inner function \({\Theta}\). When \({E = H^{2}(\mathbb{D}),\,H_{E}^{2}(\mathbb{D})}\) is the Hardy space on the bidisk \({H^{2}(\mathbb{D}^2)}\). Recently, Qin and Yang (Proc Am Math Soc, 2013) determines the operator valued inner function \({\Theta(z)}\) for two well-known invariant subspaces in \({H^{2}(\mathbb{D}^{2})}\). This paper generalizes the \({\Theta(z)}\) by Qin and Yang (Proc Am Math Soc, 2013) and deal with the structure of \({M = {\Theta}(z)H^{2}(\mathbb{D}^{2})}\) when M is an invariant subspace in \({H^{2}(\mathbb{D}^{2})}\). Unitary equivalence, spectrum of the compression operator and core operator are studied in this paper.

We study common fixed point theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.

In this survey paper we present some recent results in the iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embedddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in $${{\mathbb{R}}^n}$$ space; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers–Ulam stability of the translation equation. Most of the results presented here was obtained by the means of functional equations. We indicate the relations between the iteration theory and functional equations.

The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping and a finite family of nonexpansive mappings , respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others.

This paper investigates properties of convergence of distances of $$p$$ -cyclic $$\alpha$$ - $$\psi$$ -type contractions on the union of the $$p$$ subsets of a space $$X$$ defining probabilistic metric spaces and Menger spaces. The paper also investigates the characterization of both Cauchy and $$G$$ -Cauchy sequences which are convergent, in particular, to best proximity points. On the other hand, the existence and uniqueness of fixed points and best proximity points of $$p$$ -cyclic $$\alpha$$ - $$\psi$$ -type contractions are also investigated. The fixed points of the $$p$$ -composite self-mappings, which are obtained from the $$p$$ -cyclic self-mapping restricted to each of the $$p$$ subsets in the cyclic disposal, are also investigated while a generalization and some illustrative examples are also given.

We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now clas- sic result of Skvortsova for the Medvedev lattice. 1. Introduction Among the structures arising from computability theory, the lattices intro- duced by Medvedev and Muchnik stand out for several distinguished features and a broad range of applications. In particular these lattices have additional structure that makes them suitable as models of certain propositional cal- culi. The structure of the Medvedev lattice as a Brouwer algebra, and thus as a model for propositional logics, has been extensively studied in several papers, see e.g. (10), (15), (17), (20), (22). Originally motivated in (10) as a formalization of Kolmogorov's calculus of problems (7), the Medvedev lattice fails to provide an exact interpretation of the intuitionistic propositional cal- culus IPC; however, as shown by Skvortsova (15), there are initial segments of the Medvedev lattice that model exactly IPC. On the other hand, little is known about the structure of the Muchnik lattice, and of its dual, as Brouwer algebras. The goal of this paper is to show that there are initial segments (equivalently: factors obtained dividing the lattice by principal filters) of the Muchnik lattice, in which the set of valid propositional sentences coincides with IPC. This shows that the analogue of Skvortsova's theorem also holds for the Muchnik lattice. From this, it readily follows that the propositional sentences that are valid in the Muchnik lattice are exactly the sentences of the so-called logic of the weak law of the excluded middle ((17)). Similar results (as announced, with outlined proofs, in (18)) hold of the dual of the Muchnik lattice: detailed proofs are provided in Section 5. For all unexplained notions from computability theory, the reader is re- ferred to Rogers (14); our main source for Brouwer algebras and the algebraic semantics of propositional calculi is Rasiowa-Sikorski (13). A comprehensive survey on the Medvedev and Muchnik lattices, and their mutual relation- ships, can be found in (19). Throughout the paper we use the symbols + and × to denote the join and meet operations, respectively, in any lattice. 1.1. The Medvedev and the Muchnik lattices. Although our main ob- ject of study is the Muchnik lattice, reference to the Medvedev lattice will be sometimes useful. Therefore, we start by reviewing some basic definitions and facts concerning both lattices. Following Medvedev (10), a mass problem is a set of functions from the set of natural numbers ω to ω. There are two natural ways to extend Turing reducibility to mass problems: following (10)

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly -Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend the corresponding results announced by Chang (2001), Cho et al. (2005), Ofoedu (2006), Schu (1991) and Zeng (2003 and 2005), and many others.

In this paper, by asymptotic center techniques, we shown that the set of fixed points of a uniformly k-lipschitzian semigroup (one-parameter or left reversible semi-topological) in a uniformly convex Banach space is a retract of the domain if k is close to 1. The results presented in this paper includes (among others, in the discrete situation) many known results as special cases.

This paper has been withdrawn by the author because there are some typos in proofs., Comment: This paper has been withdrawn by the author because there are some typos in proofs

In the previous paper by the first and the third authors, we present six algorithms for determining whether a given symmetric matrix is strictly copositive, copositive (but not strictly), or not copositive. The algorithms for matrices of order are not guaranteed to produce an answer. It also shows that for 1000 symmetric random matrices of order 8, 9, and 10 with unit diagonal and with positive entries all being less than or equal to 1 and negative entries all being greater than or equal to , there are 8, 6, and 2 matrices remaing undetermined, respectively. In this paper we give two more algorithms for and our experiment shows that no such matrix of order 8 or 9 remains undetermined; and almost always no such matrix of order 10 remains undetermined. We also do some discussion based on our experimental results.

In this paper, we investigate the structure of reachable sets from a given point q 0 for a class of analytic control affine systems characterized, among other things, by possessing two singular trajectories initiating at q 0. The aim of the paper is to establish the connection between the minimal number of analytic functions needed for describing reachable sets and the number of geometrically optimal singular trajectories. The paper is written in a language of the sub-Lorentzian geometry. Also, the sub-Lorentzian geometry methods are used to prove theorems.

This paper strengthens the low-error PCP characterization of NP, coming closer to the upper limit of the BGLR conjecture. Consider the task of verifying a written proof for the membership of a given input in an NP language. In this paper, this is achieved by making a constant number of accesses to the proof, obtaining error probability that is exponentially small in the total number of bits that are read. We show that the number of bits that are read in each access to the proof can be made as high as log β n , for any constant β < 1, where n is the length of the proof. The BGLR conjecture asserts the same for any constant β, for β smaller or equal to 1. Our results are in fact stronger, implying that the Gap-Quadratic-Solvability problem with a constant number of variables in each equation is NP-hard. That is, given a system of n quadratic equations over a field $${\mathcal{F}}$$ of size up to $$2^{\log^\beta n}$$, where each equation depends on a constant number of variables, it is NP-hard to distinguish between the case where there is a common solution to all of the equations and the case where any assignment satisfies at most a $${2 / |\mathcal{F}|}$$ fraction of them. At the same time, our proof presents a direct construction of a low-degree test whose error-probability is exponentially small in the number of bits accessed. Such a result was previously known only relying on recursive applications of the entire PCP theorem.

The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi- -asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

In this paper we study extensions between Cohen–Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine \(\mathrm{Ext}^i(L_I, L_J)\) for arbitrary rank 1 modules \(L_I\) and \(L_J\). An explicit combinatorial algorithm is given for the computation of \(\mathrm{Ext}^i(L_I, L_J)\) when i is odd, and when i even, we show that \(\mathrm{Ext}^i(L_I, L_J)\) is cyclic over the centre, and we give an explicit formula for its computation. At the end of the paper we give a vanishing condition of \(\mathrm{Ext}^i(L_I, L_J)\) for any \(i>0\).

In this paper, we prove that the sequence {xn} generated by modified Krasnoselskii–Mann iterative algorithm introduced by Yao et al. [J Appl Math Comput 29:383–389, 2009] converges strongly to a fixed point of a nonexpansive mapping T in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. Furthermore, we present an example that illustrates our result in the setting of a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. The results of this paper extend and improve several results presented in the literature in the recent past. Open image in new window

Low-rank structured matrices have attracted much attention in the last decades, since they arise in many applications and all share the fundamental property that can be represented by O(n)$\mathcal {O}(n)$ parameters, where n×n is the size of the matrix. This property has allowed the development of fast algorithms for solving numerically many problems involving low-rank structured matrices by performing operations on the parameters describing the matrices, instead of directly on the matrix entries. Among these problems, the solution of linear systems of equations is probably the most basic and relevant one. Therefore, it is important to measure, via structured computable condition numbers, the relative sensitivity of the solutions of linear systems with low-rank structured coefficient matrices with respect to relative perturbations of the parameters representing such matrices, since this sensitivity determines the maximum accuracy attainable by fast algorithms and allows us to decide which set of parameters is the most convenient from the point of view of accuracy. To develop and analyze such condition numbers is the main goal of this paper. To this purpose, a general expression is obtained for the condition number of the solution of a linear system of equations whose coefficient matrix is any differentiable function of a vector of parameters with respect to perturbations of such parameters. Since there are many different classes of low-rank structured matrices and many different types of parameters describing them, it is not possible to cover all of them in a single work. Therefore, the general expression of the condition number is particularized to the important case of {1,1}-quasiseparable matrices and to the quasiseparable and the Givens-vector representations, in order to obtain explicit expressions of the corresponding two condition numbers that can be estimated in O(n) operations. In addition, detailed theoretical and numerical comparisons of these two condition numbers between themselves, and with respect to unstructured condition numbers, are provided, which show that there are situations in which the unstructured condition number is much larger than the structured ones, but that the opposite never happens. The approach presented in this manuscript can be generalized to other classes of low-rank structured matrices and parameterizations.

In the paper by Olaleru and Akewe (2007), the authors tried to generalize Gregus fixed point theorem. In this paper we give a counterexample on their main statement.

We consider the minimum affine separating committee (MASC) combinatorial optimization problem, which is related to ensemble machine learning techniques on the class of linear weak classifiers combined by the rule of simple majority. Actually, the MASC problem is a mathematical formalization of the famous Vapnik---Chervonenkis principle of structural risk minimization in the mentioned class of classifiers. According to this principle, it is required to construct a best performance ensemble classifier belonging to a family of the least possible VC-dimension. It is known that the MASC problem is NP-hard and remains intractable in spaces of any fixed dimension $$n>1$$n>1 even under an additional constraint on the separated sets to be in general position. This special case of the MASC problem called MASC-GP(n) is the main subject of interest of the present paper. To design polynomial-time approximation algorithms for a class of combinatorial optimization problems containing the MASC problem, we propose a new framework, adjusting the well-known Multiple Weights Update method. Following this approach, we construct polynomial-time approximation algorithms with state-of-the-art approximation guarantee for the MASC-GP(n) problem. The results obtained provide a theoretical framework for learning a high-performance ensembles of affine classifiers.

In this paper, we propose an iteration method for finding a split common fixed point of asymptotically nonexpansive semigroups in the setting of two Banach spaces, and we obtain some weak and strong convergence theorems of the iteration scheme proposed. The results presented in the paper are new and improve and extend some recent corresponding results.

In this paper, our aim is to introduce the concept of G-g-contraction mapping and prove some coupled coincidence and coupled common fixed point theorems for nonlinear contraction mappings in the new set up of partially ordered complete metric spaces endowed with a directed graph. As an application, we apply our results to present an existence theorem for solution of some particular integral equations. Our paper is inspired by the work of Chifu and Petrusel (Fixed Point Theory Appl. 2014:151, 2014); the authors introduced the concept of a coupled fixed point. In the current paper, however, we have established the results by introducing the new notion of a coupled coincidence fixed point instead of the coupled fixed point in the setting of a partially ordered complete metric space with graph.

In this paper, we prove Browder’s convergence theorem for G-nonexpansive mappings in a Hilbert space with a directed graph. Moreover, we also prove strong convergence of the Halpern iteration process to a fixed point of G-nonexpansive mappings in a Hilbert space endowed with a directed graph. The main results obtained in this paper extend and generalize many well-known results in the literature.

In the present paper, we propose three kinds of new algorithms for a finite family of quasi-asymptotically pseudocontractive mappings in real Hilbert spaces. By using some new analysis techniques, we prove the strong convergence of the proposed algorithms. Some numerical examples are also included to illustrate the effectiveness of the proposed algorithms. The results presented in this paper are interesting extensions of those well-known results.

The purpose of this paper is to present a new iterative scheme for finding a common solution to a variational inclusion problem with a finite family of accretive operators and a modified system of variational inequalities in infinite-dimensional Banach spaces. Under mild conditions, a strong convergence theorem for approximating this common solution is proved. The methods in the paper are novel and different from those in the early and recent literature.

In this paper, we introduce a new class of multivalued contractions with respect to b-generalized pseudodistances and prove a best proximity point theorem for this class of non-self mappings. In this way, we improve and extend several existing results in the literature. Examples are given to support our main results. This work is a continuation of studies on the use of a new type of distances in the fixed point theory. The pioneering effort in direction of defining distance is inter alia paper of O. Kada, T. Suzuki and W. Takahashi.

After the appearance of Nieto and Rodriguez-Lopez’s theorem, the branch of fixed point theory devoted to the setting of partially ordered metric spaces have attracted much attention in the last years, especially when coupled, tripled, quadrupled and, in general, multidimensional fixed points are studied. Almost all papers in this direction have been forced to present two results assuming two different hypotheses: the involved mapping should be continuous or the metric framework should be regular. Both conditions seem to be different in nature because one of them refers to the mapping and the other one is assumed on the ambient space. In this paper, we unify such different conditions in a unique one. By introducing the notion of continuity of a mapping from a metric space into itself depending on a function α, which is the case that covers the partially ordered setting, we extend some very recent theorems involving control functions that only must be lower/upper semi-continuous from the right. Finally, we use metric spaces endowed with transitive binary relations rather than partial orders.

In this paper, some multifunctions on partial metric space are defined and common fixed points of such multifunctions are discussed. The results presented in the paper generalize some of the existing results in the literature. Several conclusions of the main results are given.

In this paper, a new iterative scheme for a finite family of $I_{i}$ -generalized asymptotically nonexpansive nonself-mappings $\{T_{i}\} _{i=1}^{r}$ is constructed in a uniformly convex hyperbolic space. We establish strong convergence theorems of this iterative scheme to a common fixed point of $\{T_{i}\}_{i=1}^{r}$ and $\{ I_{i}\}_{i=1}^{r}$ under certain conditions. Our results of this paper extend some results in the literature.

In this paper, we introduce and analyze a new general hybrid iterative algorithm for finding a common element of the set of common zeros of two families of finite maximal monotone mappings, the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping in a real Hilbert space. Our algorithm is based on four well-known methods: Mann’s iteration method, composite method, outer-approximation method and extragradient method. We prove the strong convergence theorem for the proposed algorithm. The results presented in this paper extend and improve the corresponding results of Wei and Tan (Fixed Point Theory Appl. 2014:77, 2014). Some special cases are also discussed.

In this paper we establish some best proximity point results using generalized weak contractions with discontinuous control functions. The theorems are established in metric spaces with a partial order. We view the main problem in the paper as a problem of finding an optimal approximate solution of a fixed point equation. We also discuss several corollaries and give an illustrative example. We apply our result to obtain some coupled best proximity point results.

In this paper, the concept of ordered contractive mappings is introduced on cone metric spaces over Banach algebras, and some existence theorems of fixed points for such mappings are obtained. As an application of a result, an example is given at the end of the paper.

Based on the notions of both contractibility and local contractibility, many works were done in fixed point theory. The present paper concerns a relation between digital contractibility and the existence of fixed points of digitally continuous maps. In this paper, establishing a new digital homotopy named by a K-homotopy in the category of Khalimsky topological spaces, we prove that in digital topology, whereas contractibility implies local contractibility, the converse does not hold. Furthermore, we address the following problem, which remains open. Let X be a Khalimsky (K- for short) topological space with K-contractibility. Then we may pose the following question: does the space X have the fixed point property (FPP)? In this paper, we prove that not every K-topological space with K-contractibility has the FPP.

In this paper, we apply order-theoretic fixed point theorems and isotone selection theorems to study quasi-equilibrium problems. Some existence theorems of solutions to quasi-equilibrium problems are obtained on Hilbert lattices, chain-complete lattices and chain-complete posets, respectively. In contrast to many papers on equilibrium problems, our approach is order-theoretic and all results obtained in this paper do not involve any topological continuity with respect to the considered mappings.

In this paper, we establish the Nagumo theorems for boundary value problems associated with a class of third-order singular nonlinear equations: $(p(t)x')''=f(t,x,p(t)x',(p(t)x')')$ , $\forall t\in(0,1)$ by the method of upper and lower solutions and the Schauder fixed point theorem. We also consider the multiplicity of the solutions by using topological degree theory. There are some examples to illustrate how the results of this paper can be applied.

In this paper, we introduce the notions of soft Meir-Keeler contractive mappings and weaker ϕ-Meir-Keeler contractive mappings, and the purpose of this paper is to prove two theorems which assures the existence of fixed points for these two soft Meir-Keeler type contractive mappings on a soft metric space. Our results generalize and improve many recent fixed point results in the literature.

This paper compares different approaches to the multivariate analysis of interval data, focusing on discriminant analysis. Three fundamental approaches are considered. The first approach assumes an uniform distribution in each observed interval, derives the corresponding measures of dispersion and association, and appropriately defines linear combinations of interval variables that maximize the usual discriminant criterion. The second approach expands the original data set into the set of all interval description vertices, and proceeds with a classical analysis of the expanded set. Finally, a third approach replaces each interval by a midpoint and range representation. Resulting representations, using intervals or single points, are discussed and distance based allocation rules are proposed. The three approaches are illustrated on a real data set.

We are pleased to present this special issue of Algorithmica, which contains the extended journal versions of selected papers previously presented at the 9th International Symposium on Parameterized and Exact Computation, IPEC 2014, September 10–12, Wroclaw, Poland. The symposium is an established annual meeting of the multivariate and exact algorithms communities. This issue consists of nine papers, reviewed thoroughly according to the usual, high standard of the journal. In The Parameterized Complexity of Geometric Graph Isomorphism, V. Arvind and Gaurav Rattan present improved FPT algorithm for Geometric Graph Isomorphism problem, where one is to decide whether there is a distance preserving bijection between two sets of points in k-dimensional euclidian space. Igor Razgon in On the read-once property of branching programs and CNFs of bounded treewidthproves a space lower bound for non-deterministic read-oncebranching programs on functions expressible as CNFswith treewidth at most k of their primal graphs. In Finding Shortest Paths between Graph Colourings, Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, and Daniel Paulusma give a complete picture of the parameterized complexity of the k-colouring reconfiguration problem, where the goal is to modify one proper colouring into another one, by changing the colour of one vertex at a time. Given a system of linear equations Ax = b over the binary field one can ask whether there is a solution of weight at most t , exactly t or at least t . In Solving Linear

The purpose of this paper is to introduce a new hybrid projection method based on modified Mann iterative scheme by the generalized f-projection operator for a countable family of relatively quasi-nonexpansive mappings and the solutions of the system of generalized mixed equilibrium problems. Furthermore, we prove the strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniform smooth Banach space. Finally, we also apply our results to the problem of finding zeros of B-monotone mappings and maximal monotone operators. The results presented in this paper generalize and improve some well-known results in the literature. 2000 Mathematics Subject Classification: 47H05; 47H09; 47H10.

In this paper, we introduce the modified iterations of Mann’s type for nonexpansive mapping and asymptotically nonexpansive mapping to have the strong and weak convergence in a uniformly convex Banach space. We also proved strong convergence theorems of our modified Mann’s iteration processes for nonexpansive semigroups and asymptotically nonexpansive semigroups. The results presented in the paper give a partially affirmative answer to the open question raised by Kim and Xu (Nonlinear Anal. 64:1140-1152, 2006). Applications to the accretive operators are also included.