Abstract and Applied Analysis http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. On the Boundary of Self-Affine Sets Tue, 24 Feb 2015 09:26:49 +0000 http://www.hindawi.com/journals/aaa/2015/573604/ This paper is devoted to studying the boundary behavior of self-affine sets. We prove that the boundary of an integral self-affine set has Lebesgue measure zero. In addition, we consider the variety of the boundary of a self-affine set when some other contractive maps are added. We show that the complexity of the boundary of the new self-affine set may be the same, more complex, or simpler; any one of the three cases is possible. Qi-Rong Deng and Xiang-Yang Wang Copyright © 2015 Qi-Rong Deng and Xiang-Yang Wang. All rights reserved. Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods Mon, 23 Feb 2015 09:09:04 +0000 http://www.hindawi.com/journals/aaa/2015/797594/ A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs. Krzysztof Gdawiec, Wiesław Kotarski, and Agnieszka Lisowska Copyright © 2015 Krzysztof Gdawiec et al. All rights reserved. Approximate Derivations with the Radical Ranges of Noncommutative Banach Algebras Sun, 22 Feb 2015 11:51:15 +0000 http://www.hindawi.com/journals/aaa/2015/594075/ We consider the derivations on noncommutative Banach algebras, and we will first study the conditions for a derivation on noncommutative Banach algebra. Then, we examine the stability of functional inequalities with a derivation. Finally, we take the derivations with the radical ranges on noncommutative Banach algebras. Jaiok Roh and Ick-Soon Chang Copyright © 2015 Jaiok Roh and Ick-Soon Chang. All rights reserved. -Multipliers on Banach Algebras and Topological Modules Sun, 22 Feb 2015 11:10:45 +0000 http://www.hindawi.com/journals/aaa/2015/951021/ We prove some results concerning Arens regularity and amenability of the Banach algebra of all -multipliers on a given Banach algebra . We also consider -multipliers in the general topological module setting and investigate some of their properties. We discuss the -strict and -uniform topologies on . A characterization of -multipliers on -module , where is a compact group, is given. Marjan Adib Copyright © 2015 Marjan Adib. All rights reserved. Qualitative Theory of Functional Differential and Integral Equations Sun, 22 Feb 2015 07:22:48 +0000 http://www.hindawi.com/journals/aaa/2015/515162/ Cemil Tunç, Mouffak Benchohra, Bingwen Liu, Muhammad N. Islam, and Samir H. Saker Copyright © 2015 Cemil Tunç et al. All rights reserved. Recent Trends in Boundary Value Problems 2014 Sun, 22 Feb 2015 06:54:54 +0000 http://www.hindawi.com/journals/aaa/2015/261754/ Bashir Ahmad, Juan J. Nieto, Donal O’Regan, and Agacik Zafer Copyright © 2015 Bashir Ahmad et al. All rights reserved. Outer Synchronization of Complex Networks with Nondelayed and Time-Varying Delayed Couplings via Pinning Control or Impulsive Control Thu, 19 Feb 2015 11:18:06 +0000 http://www.hindawi.com/journals/aaa/2015/414596/ The outer synchronization problem between two complex networks with nondelayed and time-varying delayed couplings via two different control schemes, namely, pinning control and impulsive control, is considered. Firstly, by applying pinning control to a fraction of the network nodes and using a suitable Lyapunov function, we obtain some new and useful synchronization criteria, which guarantee the outer synchronization between two complex networks. Secondly, impulsive control is added to the nodes of corresponding response network. Based on the generalized inequality about time-varying delayed different equation, the sufficient conditions for outer synchronization are derived. Finally, some examples are presented to demonstrate the effectiveness and feasibility of the results obtained in this paper. Jianwen Feng, Sa Sheng, Ze Tang, and Yi Zhao Copyright © 2015 Jianwen Feng et al. All rights reserved. Organically Structured Control of Large-Scale Systems with Expanding Construction Based on State Observation Thu, 19 Feb 2015 09:47:14 +0000 http://www.hindawi.com/journals/aaa/2015/807202/ This paper presents a new method on the problem of organically structured control based on state observation for a class of large-scale systems with expanding construction. This problem is to design a local state feedback controller and an observer for a new subsystem which is added to a large-scale system without changing the decentralized state feedback control laws of the original construction, so that both the new subsystem and the resulting expanded system are robustly connectively stable. Firstly, based on state observers, the mathematical model of a large-scale system with expanding construction is reestablished and analyzed. In addition, the sufficient condition for robust decentralized connective stabilization of the expanded construction of large-scale systems is deduced by taking an LMI approach, which is further relaxed by removing the square matrix condition on the output matrix. This problem is transformed into solving an LMI problem. The new design method of an organically structured controller and observer for the expanded construction is also given. Finally, the simulation examples show the effectiveness of the proposed method. Li Xiaohua, Liu Xiaoping, Liu Yang, and Gao Jiawang Copyright © 2015 Li Xiaohua et al. All rights reserved. Trajectory Tracking and Stabilization of a Quadrotor Using Model Predictive Control of Laguerre Functions Thu, 19 Feb 2015 09:32:57 +0000 http://www.hindawi.com/journals/aaa/2015/916864/ This paper presents a solution to stability and trajectory tracking of a quadrotor system using a model predictive controller designed using a type of orthonormal functions called Laguerre functions. A linear model of the quadrotor is derived and used. To check the performance of the controller we compare it with a linear quadratic regulator and a more traditional linear state space MPC. Simulations for trajectory tracking and stability are performed in MATLAB and results provided in this paper. Mapopa Chipofya, Deok Jin Lee, and Kil To Chong Copyright © 2015 Mapopa Chipofya et al. All rights reserved. Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces Wed, 18 Feb 2015 07:59:54 +0000 http://www.hindawi.com/journals/aaa/2015/942814/ In uniform spaces with symmetric structures determined by the -families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the -families of generalized pseudodistances on are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by -families. Results are new also in locally convex and metric spaces. Examples are provided. Kazimierz Włodarczyk and Robert Plebaniak Copyright © 2015 Kazimierz Włodarczyk and Robert Plebaniak. All rights reserved. JPD-Coloring of the Monohedral Tiling for the Plane Wed, 18 Feb 2015 07:57:25 +0000 http://www.hindawi.com/journals/aaa/2015/258436/ We introduce a definition of coloring by using joint probability distribution “JPD-coloring” for the plane which is equipped by tiling . We investigate the JPD-coloring of the r-monohedral tiling for the plane by mutually congruent regular convex polygons which are equilateral triangles at r = 3 or squares at r = 4 or regular hexagons at r = 6. Moreover we present some computations for determining the corresponding probability values which are used to color in the three studied cases by MAPLE-Package. S. A. El-Shehawy and M. Basher Copyright © 2015 S. A. El-Shehawy and M. Basher. All rights reserved. -Stably Limit Shadowing Diffeomorphisms Tue, 17 Feb 2015 13:15:31 +0000 http://www.hindawi.com/journals/aaa/2015/751769/ Let be a diffeomorphism on a closed surface. In this paper, we show that if has the -stably limit shadowing property, then we have the following: (i) satisfies the Kupka-Smale condition; (ii) if is dense in the nonwandering set () and if there is a dominated splitting on (), then satisfies both Axiom and the strong transversality condition. Manseob Lee and Junmi Park Copyright © 2015 Manseob Lee and Junmi Park. All rights reserved. Pontryagin’s Maximum Principle for the Optimal Control Problems with Multipoint Boundary Conditions Sun, 15 Feb 2015 12:46:11 +0000 http://www.hindawi.com/journals/aaa/2015/428042/ Optimal control problem with multipoint boundary conditions is considered. Sufficient conditions for the existence and uniqueness of the solution of boundary value problem for every fixed admissible control are obtained. First order increment formula for the functional is derived. Pontryagin’s maximum principle is proved by using the variations of admissible control. M. J. Mardanov and Y. A. Sharifov Copyright © 2015 M. J. Mardanov and Y. A. Sharifov. All rights reserved. On Convergence in -Valued Fuzzy Topological Spaces Wed, 11 Feb 2015 12:25:20 +0000 http://www.hindawi.com/journals/aaa/2015/730940/ We introduce the concept of L-fuzzy neighborhood systems using complete MV-algebras and present important links with the theory of L-fuzzy topological spaces. We investigate the relationships among the degrees of L-fuzzy r-adherent points (r-convergent, r-cluster, and r-limit, resp.) in an L-fuzzy topological spaces. Also, we investigate the concept of LF-continuous functions and their properties. A. A. Ramadan, M. El-Dardery, and Hu Zhao Copyright © 2015 A. A. Ramadan et al. All rights reserved. Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus Thu, 05 Feb 2015 10:03:40 +0000 http://www.hindawi.com/journals/aaa/2015/594685/ Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions. Uğur Kadak and Muharrem Özlük Copyright © 2015 Uğur Kadak and Muharrem Özlük. All rights reserved. A Weak Solution of a Stochastic Nonlinear Problem Thu, 05 Feb 2015 08:47:36 +0000 http://www.hindawi.com/journals/aaa/2015/482410/ We consider a problem modeling a porous medium with a random perturbation. This model occurs in many applications such as biology, medical sciences, oil exploitation, and chemical engineering. Many authors focused their study mostly on the deterministic case. The more classical one was due to Biot in the 50s, where he suggested to ignore everything that happens at the microscopic level, to apply the principles of the continuum mechanics at the macroscopic level. Here we consider a stochastic problem, that is, a problem with a random perturbation. First we prove a result on the existence and uniqueness of the solution, by making use of the weak formulation. Furthermore, we use a numerical scheme based on finite differences to present numerical results. M. L. Hadji Copyright © 2015 M. L. Hadji. All rights reserved. Regularization Methods for Phase Retrieval and Phase Contrast Tomography Wed, 04 Feb 2015 13:46:48 +0000 http://www.hindawi.com/journals/aaa/2015/943501/ Phase contrast tomography is a high sensitivity medical imaging technique. Several regularization methods have been used in the literature to obtain stable solutions for the phase retrieval or the phase contrast tomography problems. Yet, the functional framework and the convergence properties of the methods have not been studied in detail. In this work, the convergence properties of regularization approaches for phase retrieval and phase contrast tomography are investigated. Bruno Sixou Copyright © 2015 Bruno Sixou. All rights reserved. On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators Wed, 04 Feb 2015 10:52:28 +0000 http://www.hindawi.com/journals/aaa/2015/434020/ We prove that there is no bijective map between the set of all positive definite operators and the set of all self-adjoint operators on a Hilbert space with dimension greater than 1 which preserves the usual order (the one coming from the concept of positive semidefiniteness) in both directions. We conjecture that a similar assertion is true for general noncommutative -algebras and present a proof in the finite dimensional case. Lajos Molnár Copyright © 2015 Lajos Molnár. All rights reserved. Global Stability Results in a SVIR Epidemic Model with Immunity Loss Rate Depending on the Vaccine-Age Sun, 01 Feb 2015 08:17:20 +0000 http://www.hindawi.com/journals/aaa/2015/341854/ We formulate a susceptible-vaccinated-infected-recovered (SVIR) model by incorporating the vaccination of newborns, vaccine-age, and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states. Raúl Peralta, Cruz Vargas-De-León, and Pedro Miramontes Copyright © 2015 Raúl Peralta et al. All rights reserved. Semi-Iterative Method for Computing the Generalized Inverse Thu, 29 Jan 2015 12:56:53 +0000 http://www.hindawi.com/journals/aaa/2015/849487/ The main aim of this paper is to compute the generalized inverse over Banach spaces by using semi-iterative method and to present the error bounds of the semi-iterative method for approximating . Xiaoji Liu and Caijing Jiang Copyright © 2015 Xiaoji Liu and Caijing Jiang. All rights reserved. Asymptotic Behavior of Nonlinear Evolution Equations Thu, 29 Jan 2015 11:49:11 +0000 http://www.hindawi.com/journals/aaa/2015/430238/ Hui-Sheng Ding, Carlos Lizama, Gaston M. N’Guérékata, and Claudio Cuevas Copyright © 2015 Hui-Sheng Ding et al. All rights reserved. Function Spaces and Applications Thu, 29 Jan 2015 10:01:47 +0000 http://www.hindawi.com/journals/aaa/2015/531282/ Henryk Hudzik, Vakhtang M. Kokilashvili, Baoxiang Wang, and Marek Wisła Copyright © 2015 Henryk Hudzik et al. All rights reserved. Global Exponential Stability of Learning-Based Fuzzy Networks on Time Scales Thu, 29 Jan 2015 07:47:24 +0000 http://www.hindawi.com/journals/aaa/2015/283519/ We investigate a class of fuzzy neural networks with Hebbian-type unsupervised learning on time scales. By using Lyapunov functional method, some new sufficient conditions are derived to ensure learning dynamics and exponential stability of fuzzy networks on time scales. Our results are general and can include continuous-time learning-based fuzzy networks and corresponding discrete-time analogues. Moreover, our results reveal some new learning behavior of fuzzy synapses on time scales which are seldom discussed in the literature. Juan Chen, Zhenkun Huang, and Jinxiang Cai Copyright © 2015 Juan Chen et al. All rights reserved. First-Order Regular and Degenerate Identification Differential Problems Wed, 28 Jan 2015 11:46:13 +0000 http://www.hindawi.com/journals/aaa/2015/393624/ We are concerned with both regular and degenerate first-order identification problems related to systems of differential equations of weakly parabolic type in Banach spaces. Several applications to partial differential equations and systems will be given in a subsequent paper to show the fullness of our abstract results. A. Favini, A. Lorenzi, and H. Tanabe Copyright © 2015 A. Favini et al. All rights reserved. Applications of Multivalued Contractions on Graphs to Graph-Directed Iterated Function Systems Wed, 28 Jan 2015 07:02:56 +0000 http://www.hindawi.com/journals/aaa/2015/345856/ We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graph and a suitable -contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system. T. Dinevari and M. Frigon Copyright © 2015 T. Dinevari and M. Frigon. All rights reserved. Windowed Fourier Frames to Approximate Two-Point Boundary Value Problems Mon, 26 Jan 2015 09:20:57 +0000 http://www.hindawi.com/journals/aaa/2015/153010/ Boundary value problems arise while modeling various physical and engineering reality. In this communication we investigate windowed Fourier frames focusing two-point BVPs. We approximate BVPs using windowed Fourier frames. We present some numerical results to demonstrate the efficiency of such frame functions to approximate BVPs. Abdullah Aljouiee and Samir Kumar Bhowmik Copyright © 2015 Abdullah Aljouiee and Samir Kumar Bhowmik. All rights reserved. Eigenvalues of Vectorial Sturm-Liouville Problems with Parameter Dependent Boundary Conditions Mon, 26 Jan 2015 07:34:01 +0000 http://www.hindawi.com/journals/aaa/2015/796086/ We generalize the regularized sampling method introduced in 2005 by the author to compute the eigenvalues of scalar Sturm-Liouville problems (SLPs) to the case of vectorial SLP with parameter dependent boundary conditions. A few problems are worked out to illustrate the effectiveness of the method and show by the same token that we have indeed a general method capable of handling with ease very broad classes of SLPs, whether scalar or vectorial. Bilal Chanane Copyright © 2015 Bilal Chanane. All rights reserved. Analysis of a Fractal Boundary: The Graph of the Knopp Function Thu, 22 Jan 2015 13:07:45 +0000 http://www.hindawi.com/journals/aaa/2015/587347/ A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or local regularity exponents (the so-called p-exponent). These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function which is defined for as , where and . The Knopp function itself has everywhere the same p-exponent . Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute the p-exponent of the characteristic function of the domain under the graph of F at each point and show that p-exponents, weak and strong accessibility exponents, change from point to point. Furthermore we will derive a characterization of the local extrema of the function according to the values of these exponents. Mourad Ben Slimane and Clothilde Mélot Copyright © 2015 Mourad Ben Slimane and Clothilde Mélot. All rights reserved. Variational Analysis, Optimization, and Fixed Point Theory 2014 Mon, 19 Jan 2015 12:45:34 +0000 http://www.hindawi.com/journals/aaa/2015/312823/ Jen-Chih Yao, Abdul Latif, Chong Li, and Adrian Petrusel Copyright © 2015 Jen-Chih Yao et al. All rights reserved. -Solvability of the Dirichlet Problem for Elliptic Equations with Singular Data Mon, 19 Jan 2015 12:43:45 +0000 http://www.hindawi.com/journals/aaa/2015/635035/ We give an overview on some results concerning the unique solvability of the Dirichlet problem in , , for second-order linear elliptic partial differential equations in nondivergence form and with singular data in weighted Sobolev spaces. We also extend such results to the planar case. Loredana Caso, Roberta D’Ambrosio, and Maria Transirico Copyright © 2015 Loredana Caso et al. All rights reserved.