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Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.

We show that the main results in the recent paper [4] by Sadiq Basha (2012) are not real generalizations but particular cases of existing fixed point theorems in the literature.

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant M 1 and for each k > 1 there are cones with normal constant M > k . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results.

In this paper we extend the notion of stability of each aggregation function as presented in a previous paper of the authors, where an aggregation function was denominated Family of Aggregation Operators (FAO) in order to stress that aggregation operators within an aggregation function should be consistent. We will show that the previous definition presents certain problems when dealing with a FAO that is not idempotent or continuous. Meanwhile the previous definition was based on the study of fixed points, without taking into account the environment of each point, as many applications demand, the new approach we propose now is more flexible, and at the end allows a more appropriate definition of the consistency of a FAO. It will be shown that under certain conditions our new proposal is equivalent to the previous definition of strict stability, but the new definition covers families of aggregation operators that are not strictly stable, but intuitively stable, and vice versa. In short, we will see that our new proposal fits much better the intuition of stability of a FAO. This reformulation of stability is based upon the concept of penalty function as a measure of proximity between a value and an array of values.

In this paper, we focus on synchronization issue of coupled multistable memristive neural networks (CMMNNs) with time delay under multiple stable equilibrium states. First, we build delayed CMMNNs consisting of one master subnetwork without controller and N−1 identical slave subnetworks with controllers, and every subnetwork has n nodes. Moreover, this paper investigates multistability of delayed CMMNNs with continuous nonmonotonic piecewise linear activation function (PLAF) owning 2 r + 2 corner points. By using the theorems of differential inclusion and fixed point, sufficient conditions are derived such that master subnetwork of CMMNNs can acquire ( r + 2 ) n exponentially stable equilibrium points, stable periodic orbits or hybrid stable equilibrium states. Then, this paper proposes hybrid multisynchronization of delayed CMMNNs related with various external inputs under multiple stable equilibrium states for the first time. There exist ( r + 2 ) n hybrid multisynchronization manifolds in CMMNNs with different initial conditions and external inputs. Finally, two numerical simulations are given to illustrate the effectiveness of the obtained results.

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove an equidistribution result (out of 6 equidistribution results we prove in total), and conjecture 6 more equidistributions. Finally, we find seemingly unknown distribution of the well known permutation statistic ``strict fixed point'', which plays a key role in many of our enumerative results. This paper is the first systematic study of distributions of mesh patterns. Our techniques to obtain the results include, but are not limited to, obtaining functional relations for generating functions, and finding recurrence relations and bijections., Comment: Advances in Applied Mathematics 110 (2019) 1-32; there is a mistake in the proof of Theorem 5.1 in version 1 of the paper; its statement is now a conjecture

The paper investigates the problem of defining a specificity relation in Pyt'ev possibility theory. That branch of possibility theory developed by prof. Yu.P. Pyt'ev in the late 1990s defines possibility as a monotone measure Π which numerical values except 0 and 1 are meaningless. The following information and its consequent implications are meaningful: Π ( A ) = 0 (A is impossible), or Π ( A ) > Π ( B ) (A is more possible than B), or Π ( A ) = Π ( B ) (A and B are equally possible). All results obtained with Pyt'ev possibility theory are invariant to any strictly increasing lower semi-continuous transformation of all possibility values with the fixed points: 0 and 1. This fact allows anyone to assign numerical values to possibilities using an individual subjective scale of possibility values. Unlike Zadeh possibility theory based on fuzzy sets, where specificity of possibility distributions is evaluated by means of fuzzy set inclusion and distribution π 1 is more specific than π 2 iff π 1 ( ⋅ ) ⩽ π 2 ( ⋅ ) pointwise, in Pyt'ev possibility theory such an approach can not be applied if numerical values of π 1 and π 2 are assigned through different subjective scales. A thesis we put forward in this paper states that the specificity relation definition has to be consistent with the possibilistic decision-making approaches. The following is intended to be true: the more specific an underlying possibility and a corresponding possibility distribution are, the narrower (more specific) a set of the optimal decisions is (a decision is optimal if it minimizes the possibility of error). A specificity relation satisfying such a requirement is defined and studied.

In this paper, some new p-stability criteria and boundedness results of reaction–diffusion BAM neural networks are derived by way of fixed point theorem, Laplacian semigroup theory and L∞-estimate technique, which are novel against those of the previous related literature. Since the T–S fuzzy impulsive reaction–diffusion neural networks was investigated by previous literature, the main difficulty of this paper is to find out a novel method to give simpler conclusions than existing results. Finally, a numerical example is presented to illustrate the effectiveness and feasibility of the proposed methods.

Jleli and Samet (2018) introduced a new metric space and named it as F -space. In this paper we consider the notion of α - ψ -contraction in the setting of F -metric spaces. We present some fixed point and coupled fixed point results in the generalized setting. Moreover, our purpose in this paper is to concerned with the solution of nonlinear neutral differential equation x ′ ( t ) = − a ( t ) x ( t ) + b ( t ) g ( x ( t − r ( t ) ) ) + c ( t ) x ′ ( t − r ( t ) ) with unbounded delay using fixed point theory in F -metric space.

As one of the main factors affecting the dynamic behaviors of the ultra-large spatial structure, the non-sphere perturbation effects have been considered in the dynamic analysis of the rigid spatial structures recently. In this paper, employing the structure-preserving method developed in our previous job, the non-sphere perturbation effects on the orbit dynamic behavior, the attitude dynamic behavior as well as the transverse vibration of the spatial flexible damping beam are investigated in detail. The remarkable effects of the non-sphere perturbation on the dynamic behaviors of the beam obtained numerically in this paper include: The non-sphere perturbation effects on the evolution of the orbit radius should not be neglected when the attitude angle is far away from the stable attitude state; The non-sphere perturbation accelerates the dissipation of the transverse vibration of the spatial flexible damping beam; It is more important that, the consideration of the non-sphere perturbation results in a new stable attitude of the beam when the initial attitude angle is close to π / 2 and the initial attitude angle velocity is small enough. The above novel findings give some good advice on the attitude adjustment scheme design and the vibrational control strategy design for the ultra-large spatial structures.

This paper presents NAR-RM, a method for learning robot reaching motions from a set of demonstrations using Nonlinear AutoRegressive (NAR) polynomial models. Reaching motions are modeled as solutions to autonomous discrete-time nonlinear dynamical systems, so that the movements started near the data of the demonstrations follow the trained trajectories and always reach and stop at the target. Since NAR models obtained using standard system identification techniques do not always adequately model the reaching motions, in this paper we present a method that uses a least-squares estimator with constraints to impose the location of fixed points in the model. With the imposition of new fixed points it is possible to change the location of the original fixed points of the model, thus allowing the learning of stable reaching motions. We evaluate our method using a library of human handwriting motions, a mobile robot and an industrial manipulator.

Let Π be a finite lattice of integer points in a box of R n and f an increasing mapping in terms of the componentwise ordering from Π to itself. The well-known Tarski's fixed point theorem asserts that f has a fixed point in Π. A simple expansion of f from Π to a larger lattice C of integer points in a box of R n yields that the smallest point in C is always a fixed point of f (an expanded Tarski's fixed point problem). By introducing an integer labeling rule and applying a cubic triangulation of the Euclidean space, we prove in this paper that the expanded Tarski's fixed point problem is in the class PPA when f is given as an oracle. It is shown in this paper that Nash equilibria of a bimatrix game can be reformulated as fixed points different from the smallest point in C of an increasing mapping from C to itself. This implies that the expanded Tarski's fixed point problem has at least the same complexity as that of the Nash equilibrium problem. As a byproduct, we also present a homotopy-like simplicial method to compute a Tarski fixed point of f. The method starts from an arbitrary lattice point and follows a finite simplicial path to a fixed point of f.

Let M m be a closed smooth manifold equipped with a smooth involution having fixed point set of the form F n ∪ F 3 , where F n and F 3 are submanifolds with dimensions n and 3, respectively, where 3 n m and with the normal bundles over F n and F 3 being nonbounding. The authors of this paper, together with Patricia E. Desideri, previously showed that, when n is even, then m ≤ n + 4 , which we call a small codimension phenomenon. Further, they showed that this small bound is best possible. In this paper we study this problem for n odd, which is much more complicated, requiring more sophisticated techniques involving characteristic numbers. We show in this case that m ≤ M ( n − 3 ) + 6 , where M ( n ) is the Stong–Pergher number (see the definition of M ( n ) in Section 1). Further, we show that this bound is almost best possible, in the sense that there exists an example with m = M ( n − 3 ) + 5 , which means that for n odd the small codimension phenomenon does not occur and the bound in question is meaningful. The existence of these bounds is guaranteed by the famous Five Halves Theorem of J. Boardman, which establishes that, under the above hypotheses, m ≤ 5 2 n .

In a paper by Biro et al. [3] , a novel twist on guarding in art galleries, motivated by geographical greedy routing in sensor networks, is introduced. A beacon is a fixed point that when activated induces a force of attraction that can move points within the environment. The effect of a beacon is similar to standard visibility with some additional properties. The effects of a beacon are asymmetric leading to separate algorithms to compute the “beacon kernel” and “inverse beacon kernel”. In Biro [2] O ( n 2 ) time algorithms are given to compute the beacon kernel and the inverse beacon kernel in simple polygons. In this paper we revisit the problem of computing the shortest watchtower to guard a 2D terrain, using the properties of beacons, and we present an O ( n log ⁡ n ) time algorithm that computes the shortest beacon watchtower. In doing this we introduce O ( n log ⁡ n ) time algorithms to compute the beacon kernel and the inverse beacon kernel in a monotone polygon. We also prove that Ω ( n log ⁡ n ) time is a lower bound for computing the beacon kernel of a monotone polygon.

This paper discusses the classifier ensemble problem with sparsity and diversity learning, which is a central issue in machine learning. The current approach for reducing the size and increasing the accuracy of a classifier ensemble is to formulate it as a convex quadratic programming problem, which is a relaxation problem, and then solve it by using the existing methods for convex quadratic programming or by computing closed-form solutions. This paper presents a novel computational approach for solving the classifier ensemble problem with sparsity and diversity learning without any recourse to relaxation problems and their associated methods. We first show that the classifier ensemble problem can be expressed as a minimization problem for the sum of certain convex functions over the intersection of fixed point sets of quasi-nonexpansive mappings. Next, we propose fixed point optimization algorithms for solving the minimization problem and show that the algorithms converge to the solution of the minimization problem. It is shown that the proposed algorithms can directly solve the classifier ensemble problem with sparsity and diversity learning. Finally, we compare the performance of the proposed sparsity and diversity learning methods against an existing method in classification experiments using data sets from the UCI machine learning repository and the LIBSVM. The experimental results show that the proposed methods have higher classification accuracies than the existing method.

The usability of advanced control methods of physiological processes have been several times demonstrated. Advanced (i.e. MPC) control approaches cope with practical difficulties of limited measurability of the state variables, model-imprecisions, significant inter-patient variability of the available model’s parameters and limitations in the sampling frequency of the variables that at least in principle can be directly measured. However, the lack of the necessary information prevents the use of state estimators. Compensation of the effects of the presence of model-imprecisions needs the application of robust control methods or adaptive techniques. The Proportional-Derivative (PD) control completed with Robust Fixed Point Transformation (RFPT)-based adaptive control was invented for tackling such difficulties. The current paper investigates the applicability of this technique in case of angiogenic growth of tumors using different scenarios of tumor volume measurement. Conclusions are drawn on the basis of numerical simulations.

Disruptions are prevalent phenomenons that prevent airline from operating as original scheduled. This paper adopts the iterative fixed-point method for integer programming proposed by Dang and Ye [1] to generate feasible flight routes that are used to construct an aircraft reassignment in response to the grounding of one aircraft. Two division methods are proposed with which the solution space can be divided into several independent segments and implemented a distributed computation. The second division method is emphasized in this paper for the good performance of partial feasible flight routes which are generated by this division approach. Comparison with CPLEX CP Optimizer [2] shows that less partial feasible flight routes which are generated by Dang’s algorithm [1] are required to find an aircraft reassignment when disruptions happen, and this division method is more promising when dealing with long haul airline disruption problem.

In this paper, we introduce an explicit parallel algorithm for finding common solutions to a system of variational inequalities over the set of common fixed points of a finite family of demi-contractive operators. Under suitable assumptions, we prove the strong convergence of this algorithm in the framework of a Hilbert space. The results obtained in this paper extend and improve the results of Tian and Jiang (2017), of Censor, Gibali and Reich (2012), and of many others.

This paper proposes a new general class of strategic games and develops an associated new existence result for pure-strategy Nash equilibrium. For a two-player game with scalar and compact action sets, existence entails that one reaction curve be increasing and continuous and the other quasi-increasing (i.e., not have any downward jumps). The latter property amounts to strategic quasi-complementarities. The paper provides a number of ancillary results of independent interest, including sufficient conditions for a quasi-increasing argmax (or non-monotone comparative statics), and new sufficient conditions for uniqueness of fixed points. For maximal accessibility of the results, the main results are presented in a Euclidean setting. We argue that all these results have broad and elementary applicability by providing simple illustrations with commonly used models in economic dynamics and industrial organization.

The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time scales and is often referred to as the diffusion coefficient a in u t − ∇ ( a ∇ u ) = f . In this paper we seek the unknown a assuming that a = a ( u ) depends only on the value of the solution at a given point. Such diffusion models are the basic of a wide range of physical phenomena such as nonlinear heat conduction, chemical mixing and population dynamics. We shall look at two types of overposed data in order to effect recovery of a ( u ) : the value of a time trace u ( x 0 , t ) for some fixed point x 0 on the boundary of the region Ω; or the value of u on an interior curve Σ lying within Ω. As examples, these might represent a temperature measurement on the boundary or a census of the population in some subset of Ω taken at a fixed time T > 0 . In the latter case we shall show a uniqueness result that leads to a constructive method for recovery of a. Indeed, for both types of measured data we shall show reconstructions based on the iterative algorithms developed in the paper.

This paper aims at applications of Lyapunov functions as tools for analyzing concrete dynamical systems with computer assistance, even for non-gradient-like systems. We want to know concrete form of Lyapunov functions around invariant sets and their domains of definition for applying Lyapunov functions to various analysis of both continuous and discrete dynamical systems. Although there are several abstract results for the existence of Lyapunov functions, they cannot induce a systematic and concrete procedure of Lyapunov functions with explicit forms. In this paper, we present a numerical verification method which can validate Lyapunov functions with explicit forms and their explicit domains of definition, which can be applied to arbitrary dynamical systems with (hyperbolic) equilibria or fixed points. The proposed procedure provides us with a powerful validation tool for analyzing asymptotic behavior of dynamical systems.

In this paper, we modify the general iterative method to approximate a common element of the set of solutions of split variational inclusion problem and the set of common fixed points of a finite family of k -strictly pseudo-contractive nonself mappings. Strong convergence theorem is established under some suitable conditions in a real Hilbert space, which also solves some variational inequality problems. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors. Finally, some examples to study the rate of convergence and some illustrative numerical examples are presented.

We extend in this paper an optimal family of three-step eighth-order methods developed by Duni etal. (2011) with higher-order weight functions employed in the second and third sub-steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics. Their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases. A wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development. In addition, this paper investigates the dynamics of selected existing optimal eighth-order iterative maps with the help of illustrative basins of attraction for various polynomials.

Multiple-zero finders with optimal quartic convergence for nonlinear equations are proposed in this paper with a weight function of the principal k th root of a derivative-to-derivative ratio. The optimality of the proposed multiple-zero finders is checked for their consistency based on Kung–Traub’s conjecture established in 1974. Through various test equations, relevant numerical experiments strongly support the claimed theory in this paper. Also investigated are extraneous fixed points of the iterative maps associated with the proposed methods. Their dynamics is explored along with illustrated basins of attraction for various polynomials.

In this paper, we consider infinite words that arise as fixed points of primitive substitutions on a finite alphabet and finite colorings of their factors. Any such infinite word exhibits a "hierarchal structure" that will allow us to define, under the additional condition of strong recognizability, certain remarkable finite colorings of its factors. In particular, we generalize two combinatorial results by Justin and Pirillo concerning arbitrarily large monochromatic $k$-powers occurring in infinite words, in view of a recent paper by de Luca, Pribavkina and Zamboni, we will give new examples of classes of infinite words $\mathbf{u}$ and finite colorings that do not allow infinite monochromatic factorizations $\mathbf{u}=\mathbf{u}_1\mathbf{u}_2\mathbf{u}_3\ldots$.

The implied volatility is the unique volatility value that makes the celebrated Black–Scholes formula yields a traded option’s price. Implied volatilities at varying strike prices and maturities form the implied volatility surface. Empirically, this surface is never flat. The local-volatility (LV) model is an option model that attempts to fit the implied volatility surface. It is popular because the preference freedom of the Black–Scholes model is retained. A tree consistent with the implied volatility surface generated by an LV model is known as implied tree. Past attempts to construct the implied tree, however, are prone to having invalid transition probabilities. In fact, this problem occurs even when the volatility surface is flat as in the Black–Scholes model. This paper unearths a potentially fundamental reason for that failure: the trees contain repelling fixed points. As efficient and valid trees for general LV models remain elusive despite decades of research, this paper turns to separable LV models. An efficient and valid binomial tree is then built for such models. Our novel tree is named the waterline tree because its upper part (the part that is above the water, so to speak) matches the moments of the price, whereas the lower part matches the moments of its logarithmic price. This break from traditional trees ensures that only attracting fixed points remain. Numerical results confirm the excellent performance of the waterline tree.

In this paper the optimal parameters of a dynamic vibration absorber (DVA) with negative stiffness is analytically studied. The analytical solution is obtained by Laplace transform method when the primary system is subjected to harmonic excitation. The research shows there are still two fixed points independent of the absorber damping in the amplitude-frequency curve of the primary system when the system contains negative stiffness. Then the optimum frequency ratio and optimum damping ratio are respectively obtained based on the fixed-point theory. A new strategy is proposed to obtain the optimum negative stiffness ratio and make the system remain stable at the same time. At last the control performance of the presented DVA is compared with those of three existing typical DVAs, which were presented by Den Hartog, Ren and Sims respectively. The comparison results in harmonic and random excitation show that the presented DVA in this paper could not only reduce the peak value of the amplitude-frequency curve of the primary system significantly, but also broaden the efficient frequency range of vibration mitigation.

The main goal of this paper is to investigate derivations in residuated lattices and characterize some special types of residuated lattices in terms of derivations. In the paper, we discuss related properties of some particular derivations and give some characterizations of (good) ideal derivations. Then we obtain that the fixed point set of good ideal derivations is still a residuated lattice. Also, we prove that the set of all ideal derivation filters on residuated lattices with good ideal derivations can form a complete Heyting algebra, which is isomorphic to the lattice of all filters in the fixed point set for good ideal derivations. Moreover, we study principal ideal derivations and their adjoint derivations. Finally, we get that the fixed point set of principal ideal derivations and that of their adjoint derivations are order isomorphism. In particular, using the fixed point set of principal ideal derivations, we give characterizations of Heyting algebras and linearly ordered Heyting algebras.

This paper proposes a novel variational model and a fast algorithm for its numerical approximation to remove multiplicative noise from digital images.By applying a maximum a posteriori (MAP), we obtained a strictly convex objective functional whose minimization leads to non-linear PDEs.To this end, we develop an efficient non-linear multi-grid algorithm with an improved smoother and also discuss a local Fourier analysis of the associated smoothers which leads to a new and more effective smoother.Experimental results using both synthetic and realistic images, illustrate advantages of our proposed model in visual improvement as well as an increase in the PSNR over comparing to related recent corresponding PDE methods.We compare numerical results of new multi-grid algorithm via modified smoother with traditional time marching schemes and with multi-grid method via (local and global) fixed point smoother as well. This paper proposes a novel variational model and a fast algorithm for its numerical approximation to remove multiplicative noise from digital images. By applying a maximum a posteriori (MAP), we obtained a strictly convex objective functional whose minimization leads to non-linear partial differential equations. As a result, developing a fast numerical scheme is difficult because of the high nonlinearity and stiffness of the associated Euler-Lagrange equation and standard unilevel iterative methods are not appropriate. To this end, we develop an efficient non-linear multi-grid algorithm with an improved smoother. We also discuss a local Fourier analysis of the associated smoothers which leads to a new and more effective smoother. Experimental results using both synthetic and realistic images, illustrate advantages of our proposed model in visual improvement as well as an increase in the peak signal-to-noise ratio over comparing to related recent corresponding PDE methods. We compare numerical results of new multigrid algorithm via modified smoother with traditional time marching schemes and with multigrid method via (local and global) fixed point smoother as well.

In this paper, by using a new fixed point alternative, we establish a new generalization of the Hyers-Ulam-Rassias stability concerning a Cauchy-Jensen functional equation in normed spaces which are not necessarily Banach spaces. Moreover, our paper consists of several non-trivial examples which signify the motivation of such investigations.

The planning and development of a hybrid energy source network is based on an optimal analysis which takes into account the connection costs and the production facilities. This paper aims at modeling a hybrid multisource system by employing the methods of the thermostatted kinetic theory. Specifically a network of energy sources is allocated into a geographical domain, which is divided into different regions each of them containing different energy sources. A hybrid kinetic theory framework is proposed where the energy source is identified by a quality parameter which can attain discrete values, whereas the energy supplied by the source is modeled by defining a continuous variable. The mathematical framework consists into a system of partial integro-differential equations with quadratic type nonlinearities whose parameters are interaction rates and probability density functions. The storage system is also modeled by introducing an external force field coupled with a mathematical thermostat for ensuring the conservation of the activation energy of the energy network. The existence and uniqueness of the solution is investigated by employing fixed point arguments and integration along the characteristic curves. A critical analysis and research perspectives conclude the paper showing that the proposed thermostatted kinetic theory frameworks can be considered as general paradigms for the derivation of specific models.

This paper is concerned with the stability and error estimates of the local discontinuous Galerkin (LDG) method coupled with semi-implicit spectral deferred correction (SDC) time-marching up to third order accuracy for the Allen–Cahn equation. Since the SDC method is based on the first order convex splitting scheme, the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step, which increases the difficulty of the theoretical analysis. For the LDG discretizations coupled with the second and third order SDC methods, we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimensional spaces. At the same time, the iteration and integral involved in the semi-implicit SDC scheme also increase the difficulty of the theoretical analysis. Comparing to the Runge–Kutta type semi-implicit schemes which exclude the left-most endpoint, the SDC scheme in this paper includes the left-most endpoint as a quadrature node. This makes the test functions of the SDC scheme more complicated and the energy equations are more difficult to construct. We provide two different ideas to overcome the difficulty of the nonlinear terms. By choosing the test functions carefully, the energy stability and error estimates are obtained in the sense that the time step Δ t only requires a positive upper bound and is independent of the mesh size h . Numerical examples are presented to illustrate our theoretical results.

In this paper, we present a new iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of common fixed points of a finite family of nonexpansive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters. The results presented in this paper improve and extend the corresponding results reported by some authors recently.

Let p be an analytic self-map of Bn , the unit ball in Cn , having 0 as the attracting fixed point, and having full-rank near 0. In the well-known case n = 1 (so Bn is the disk) Schroeder's equation asks if there is an analytic f and a number λ satisfying f c p = λf. Under the current assumption that p'(0) ≠ 0, Koenigs showed in 1884 that there is a solution f which is bijective near 0 precisely when λ = p'(0). In 2003, Cowen and MacCluer formulated an analogous functional equation in several variables, i.e. for a non-negative integer n, by considering F c p = AF, for an n × n matrix A and Cn -valued F. They showed that solving this equation with an F which has full rank near 0 is equivalent to solving F c p = p'(0)F with such an F, and F c p = p'(0)F shall be called Schroeder's equation in several variables. The main result of the 2003 Cowen and MacCluer paper provides necessary and sufficient conditions for an analytic solution, F, taking values in Cn and having full-rank near 0, under the additional assumption that p'(0) is diagonalizable. In 2007 Enoch provided many theorems giving formal power series solutions to Schroeder's equation in several variables. This thesis considers the more general equation, F c p = p'(0) kF with k a positive integer, and proves there is always a solution F with linearly independent component functions, but that such an F cannot have full rank except possibly when k = 1. Furthermore when k = 1, and we are considering Schroeder's equation, necessary and sufficient conditions on p are given to ensure F has has full rank near 0 without the added assumption of diagonalizability as needed in the 2003 Cowen and MacCluer paper. In response to Enoch's paper, it is proven in this thesis that any formal power series solution indeed represents an analytic function on the whole unit ball. How exactly resonance can lead to an obstruction of a full rank solution is discussed as well as some consequences of having solutions to Schroeder's equation.

This paper is concerned with the approximate controllability of the semilinear fractional evolution equations with nonlocal and impulsive conditions. Our main results are obtained by utilizing the technique of approximate solution and the theory of fixed point. In addition, the impulsive functions in this paper are supposed to be continuous and the nonlocal item is divided into two cases: Lipschitz continuous and only continuous, which generalizes the previous contributions. Finally two examples are worked out to illustrate our obtained results.

Due to the increasing scarcity of coal resources, and a drop in the quality of existing power plant coal, much research figure out best configuration of fluidized beds are required. Against this backdrop, this paper uses lignite coal to examine the heating delivery and spray dynamic characteristics of spouted bed. Unlike existing literature focusing on the research of signals of any fixed point under a fixed condition, this paper tests the signals under dynamic conditions that is, with the air temperature constantly rising. The dynamic temperature and pressure fluctuation signals of fixed points were measured and researched. The reason we chose to research dynamic conditions is that the temperature rising process are dynamic. This paper presents an effective quadric time–frequency representations of signal; namely, the Adaptive Optimal Kernel Time–Frequency Representation (AOK TFR) combined with spectral energy and entropy. The analysis focuses on the internal mechanism of pressure fluctuation signals corresponding to different temperature gradient distribution. We also did a regression analysis of the bed temperature and pressure drop. The results suggest that the AOK TFR method can reveal flow characteristics and spray mechanisms of high-temperature gas–solid two-phase flow from different analysis points, and that this method is an effective nonlinear time–frequency method to analyze other two-phase flow processes. There is a high correlation between drops in pressure and temperature when the operating temperature is between 140 °C and 350 °C.

In this paper, we investigate the factor properties and gap sequence of the Tribonacci sequence, the fixed point of the substitution σ( a , b , c ) = ( ab , ac , a ). Let ω p be the p-th occurrence of ω and G p ( ω ) be the gap between ω p and ω p +1 . We introduce a notion of kernel for each factor ω , and then give the decomposition of the factor ω with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper: for each factor ω , the gap sequence { G p ( ω )} p ≤1 is the Tribonacci sequence over the alphabet { G 1 ( ω ), G 2 ( ω ), G 4 ( ω )}, and the expressions of gaps are determined completely. As an application, for each factor ω and p ∈ ℕ , we determine the position of ω p . Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2015.08.039 Under the assumption of the known multiplicity of zeros of nonlinear equations, a class of two-point sextic-order multiple-zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double-Newton type of methods. Wit the introduction of a bivariate weight function dependent on function-to-function and derivative-to-derivative ratios, higher-order convergence is obtained. Additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction. National Research Foundation of Korea Ministry of Education, Science and Technology under the research grant (Project Number: 2015-R1D1A3A-01020808

We prove in this paper that policy iteration can be generally defined in finite domain of templates using Lagrange duality without any assumption on the templates. Such policy iteration algorithm always converges to a fixed point under a very simple technical condition. This fixed point furnishes a safe over-approximation of the set of reachable values taken by the variables of a program. The paper also discusses the choice of good templates and links these good templates to invariant algebraic relations. When templates are well chosen, the policy iteration algorithm developed in this paper can be easily initialised for one single loop programs.

To find new solutions through the information provided by previously found solution(s), this paper is first to develop a new singular transform to change the local basin-barrier structure of the original problem in an infinite-dimensional space and then to propose an augmented partial Newton method to solve the transformed problem on the solution set. Mathematical justifications of the new formulation and method are established. Details on algorithm implementation are presented. Examples for two very different variational PDE problems are carried out to illustrate the method. Their numerical results are displayed by profile-contour plots and error/convergence data. The formulation and method developed in this paper can also be applied to solve multiple fixed point problems.

In the paper a general and compact fixed point theorem for dislocated (or partial) metric spaces is proved in a simple way. It appears that the reasoning is sufficiently general to spread over the well known (and complicated) classical results. In consequence, the extensions of some celebrated fixed point theorems are obtained, while their proofs are short and consistent, which is the main aim of the paper. The more sophisticated results are contained in sections devoted to cyclic mappings, common fixed points, and dislocated quasi-metric spaces. Maybe the most exciting result is presented in the final corollary; this effective criterion for Cauchy sequences can make the life of many analysts bearable.

One of the known fixed point theorems for self-mappings of Banach spaces is the Kantorovich's fixed point theorem (see [1, Chapter XVIII, §1.2, Theorem 1] ). In the paper [2] , these results were extended to mappings between metric spaces. There we derived theorems on fixed and coincidence point existence for mappings between metric spaces. This work is a continuation of the paper [2] and is devoted to the issues of stability of fixed points and coincidence points of mappings between metric spaces under small perturbations of the corresponding mappings. The stability is proven under very general assumptions. Both single-valued and set-valued mappings are considered.

In this paper, we give a proper investigation of the problem of remanufacturing duopoly game. 1 A smooth piecewise map is introduced to describe such game. A rich analysis either analytical or numerical is presented in this paper in order to investigate the stability of the fixed points of the game’s model. Our obtained results show that those fixed points lose its stability due to flip and Neimark-Sacker border bifurcations. Furthermore, we study the stability of those points when there are periodic cycles with points jump from region to other region.

In 2008, T. Suzuki [33] defined one of most relevant extensions of the notion of nonexpansivity. This notion was extended in [15] defining the, so called, C λ -mappings. In the last years, some papers have appeared trying to extend the most important results about existence of fixed points for nonexpansive mappings to this wider class of mappings (see, for instance, [2] , [5] , [9] , [10] , [15] ). In this paper we continue this project proving that Banach spaces with extended unconditional bases satisfy the fixed point property for C λ -mappings if the unconditional constant of the basis is small enough. We also begin a new project on renorming with the fixed point property for C λ -mappings, proving that any separable Banach space can be renormed to satisfy the fixed point property for C λ -mappings in a stable sense, that is, the space with the new norm satisfies the fixed point property which, in addition, is inherited by those isomorphic spaces which are close to it in the sense of the Banach–Mazur distance.

This paper aims at the multichannel synthetic aperture radar (SAR) image speckle reduction. This paper proposes a novel energy minimized regularization model for multichannel image denoising, which is an extension of the non-local total variational model for gray-scale image. It contains two terms, namely the vectorial data fidelity term and the non-local vectorial total variation term. The latter is constructed by high-dimensional non-local gradient that contains the structure information of the multichannel image. The existence and the uniqueness of the solution of the model are proved. A fixed point iterative algorithm is designed to acquire the solution of this model. The convergence property of this algorithm is proved as well. This model is applied to the multi-polarimetric and multi-temporal RADARSAT-2 images despeckling. The result shows that this model performs better than the original vectorial total variational model on texture preserving.

Phyllotaxis, i.e. the arrangement of plant organs like leaves, florets, scales, bracts etc. around a shoot, stem, or cone, is often highly regular. Across the plant kingdom phyllotaxis shows not only qualitatively, but also quantitatively identical features, like the occurrence of divergence angles close to noble irrationals. In a previous study (Reick, 2012) a mechanism has been identified that explains the selection of these particular divergence angles on the basis of self-similarity and scaling, numerically found in the bifurcation diagrams of simple dynamical models of phyllataxis. In the present paper, by constructing a renormalization theory, the universality of this scaling is proved for a whole class of models, prototypically represented by Thornley’s model of phyllotaxis (Thornley, 1975). The renormalization is constructed from another self-similarity found numerically for the Fourier transform of the abstract potential governing the mutual inhibition of primordia. Surprisingly, the resulting renormalization transformation is already known from the treatment of the quasiperiodic transition to chaos but operates here on a different function space. It turns out that the fixed points of the renormalization transformation are characterized by divergences of the form Θ ( κ ) = 1 / τ ( κ ) , where, written as continued fraction, τ ( κ ) = [ κ ; κ , κ , … ] , κ ∈ N + . To show the universality of the scaling, it is demonstrated that the fixed points are unstable and that the associated scaling factors α ( κ ) = − ( τ ( κ ) ) 2 and β ( κ ) = τ ( κ ) are exactly those that were numerically found in (Reick, 2012) to rule the selfsimilarity of the bifurcation structure. Thereby, the present paper puts forward an explanation for the universal appearance of certain phyllotactic patterns that is independent of physiological detail of plant growth.

In this paper, we provide a key generalization of the supermarket model both from the impatient customers and from a doubly dynamic control, which may also be related to the size-based scheduling through the centered management of the customer resource as well as the total service ability. We first use an infinite-dimensional Markov process to express the states of this supermarket model, and set up an infinite-dimensional system of differential equations satisfied by the expected fraction vector. Then we use the operator semigroup to provide a mean-field limit for the sequence of infinite-dimensional Markov processes, which asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of limiting differential equations. Finally, we provide an effective and efficient algorithm for computing the fixed point of the infinite-dimensional system of limiting differential equations, and use the fixed point to give performance analysis of this supermarket model. Also, some numerical examples are given to demonstrate how the performance measures depend on some crucial parameters of this supermarket model. We believe that the mean-field method developed in this paper will be useful and effective for analyzing more complicated supermarket models in many practical areas.

Due to its possible applications, Fixed Point Theory has become one of the most useful branches of Nonlinear Analysis. In a very recent paper, Khojasteh et?al.?introduced the notion of simulation function in order to express different contractivity conditions in a unified way, and they obtained some fixed point results. In this paper, we slightly modify their notion of simulation function and we investigate the existence and uniqueness of coincidence points of two nonlinear operators using this kind of control functions.

Let X be a compact polyhedron that is homotopy equivalent to the figure-eight. This is a survey paper for the WYK algorithm which provides computations for the Nielsen number of self-maps of X. The WYK algorithm consists of three parts and each part was introduced separately by Wagner [11] , Yi and Kim [12] , and Kim [8] . In this paper, we present the whole algorithm so that people can more easily use the algorithm.