ISRN Mathematical Analysis

On the Honesty in Nonlocal and Discrete Fragmentation Dynamics in Size and Random Position

S. C. Oukouomi Noutchie and E. F. Doungmo Goufo — Thu, 06 Jun 2013 18:53:10 +0000

A discrete initial-value problem describing multiple fragmentation processes, where the fragmentation rate is size and position dependent and where new particles are spatially randomly distributed according to some probabilistic law, is investigated by means of parameter-dependent operators together with the theory of substochastic semigroups with a parameter. The existence of semigroups is established for each parameter and “glued” together so as to obtain a semigroup to the full space. Under certain conditions on each fragmentation rate, we used Kato’s Theorem in to show the existence of the generator and we provide sufficient conditions for honesty.

Global Existence and Blowup for a Reaction-Diffusion System with Nonlocal Boundary Condition

Jun Zhou — Sun, 02 Jun 2013 09:05:56 +0000

This paper considers the singularity properties of positive solutions for a reaction-diffusion system with nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the blow-up rate estimate for the blow-up solution.

Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

Yongqin Xie, Zhufang He, Chen Xi, and Zheng Jun — Tue, 21 May 2013 14:50:55 +0000

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in . Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor which is bounded in , where the nonlinear term satisfies a critical exponential growth condition.

Optimization Problems of Excess-of-Loss Reinsurance and Investment under the CEV Model

Qicai Li and Mengdi Gu — Sun, 19 May 2013 08:35:37 +0000

We consider that the insurer purchases excess-of-loss reinsurance and invests its wealth in the constant elasticity of variance (CEV) stock market. We model risk process by Brownian motion with drift and study the optimization problem of maximizing the exponential utility of terminal wealth under the controls of excess-of-loss reinsurance and investment. Using stochastic control theory and power transformation technique, we obtain explicit expressions for the optimal polices and value function. We also show that the optimal excess-of-loss reinsurance is always better than optimal proportional reinsurance. Some numerical examples are given.

On the Continuity of Hausdorff Dimension of Julia Sets Concerning Potts Models

Gang Liu and Junyang Gao — Thu, 18 Apr 2013 10:39:21 +0000

Considering the Julia sets of a family of rational maps concerning two-dimensional diamond hierarchical Potts models in statistical mechanics, we show the continuity of their Hausdorff dimension.

On Global Existence of Solutions of the Neumann Problem for Spherically Symmetric Nonlinear Viscoelasticity in a Ball

Jerzy A. Gawinecki and Wojciech M. Zajączkowski — Tue, 12 Mar 2013 16:18:22 +0000

We examine spherically symmetric solutions to the viscoelasticity system in a ball with the Neumann boundary conditions. Imposing some growth restrictions on the nonlinear part of the stress tensor, we prove the existence of global regular solutions for large data in the weighted Sobolev spaces, where the weight is a power function of the distance to the centre of the ball. First, we prove a global a priori estimate. Then existence is proved by the method of successive approximations and appropriate time extension.

Time-Delayed Interactions in Networks of Self-Adapting Hopf Oscillators

Julio Rodriguez, Max-Olivier Hongler, and Philippe Blanchard — Thu, 07 Feb 2013 15:31:49 +0000

A network of coupled limit cycle oscillators with delayed interactions is considered. The parameters characterizing the oscillator’s frequency and limit cycle are allowed to self-adapt. Adaptation is due to time-delayed state variables that mutually interact via a network. The self-adaptive mechanisms ultimately drive all coupled oscillators to a consensual cyclostationary state, where the values of the parameters are identical for all local systems. They are analytically expressible. The interplay between the spectral properties of the coupling matrix and the time delays determines the conditions for which convergence towards a consensual state takes place. Once reached, this consensual state subsists even if interactions are removed. In our class of models, the consensual values of the parameters depend neither on the delays nor on the network’s topologies.

A Comparison Principle for Some Types of Elliptic Equations

Maria Emilia Amendola — Sun, 02 Dec 2012 14:57:27 +0000

In this paper a comparison principle between a continuous viscosity supersolution and a continuous viscosity subsolution is presented. The operator of interest is a fully nonlinear uniformly elliptic one with a gradient term which could be noncontinuous and grow like some BMO functions, as shown in the last section.

Exponential Stability for a Class of Switched Nonlinear Systems with Mixed Time-Varying Delays via an Average Dwell-Time Method

N. Yotha, T. Botmart, and T. Mouktonglang — Sat, 29 Sep 2012 10:39:04 +0000

The problem of exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays is studied. The constraint on the derivative of the time-varying delay is not required which allows the time delay to be a fast time-varying function. We study the stability properties of switched nonlinear systems consisting of both stable and unstable subsystems. Average dwell-time approached and improved piecewise Lyapunov functional combined with Leibniz-Newton are formulated. New delay-dependent sufficient conditions for the exponential stabilization of the switched systems are first established in terms of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.

Pseudo Almost Automorphic Solutions for Differential Equations Involving Reflection of the Argument

Elhadi Ait Dads, Samir Fatajou, and Lahcen Khachimi — Thu, 20 Sep 2012 15:04:41 +0000

By means of the fixed point methods and the properties of the pseudo almost automorphic functions, the existence and uniqueness of pseudo almost automorphic solutions are obtained for differential equations involving reflection of the argument. For the nonscalar, case we use the exponential dichotomy properties.

Regularity Criteria for Hyperbolic Navier-Stokes and Related System

Jishan Fan and Tohru Ozawa — Thu, 06 Sep 2012 09:19:24 +0000

We prove a regularity criterion for strong solutions to the hyperbolic Navier-Stokes and related equations in Besov space.

Comparing Numerical Methods for Solving Time-Fractional Reaction-Diffusion Equations

Veyis Turut and Nuran Güzel — Wed, 29 Aug 2012 13:19:55 +0000

Multivariate Padé approximation (MPA) is applied to numerically approximate the solutions of time-fractional reaction-diffusion equations, and the numerical results are compared with solutions obtained by the generalized differential transform method (GDTM). The fractional derivatives are described in the Caputo sense. Two illustrative examples are given to demonstrate the effectiveness of the multivariate Padé approximation (MPA). The results reveal that the multivariate Padé approximation (MPA) is very effective and convenient for solving time-fractional reaction-diffusion equations.

A Class of Integral Operators Preserving Subordination and Superordination for Analytic Functions

H. A. Al-Kharsani, N. M. Al-Areefi, and Janusz Sokół — Thu, 16 Aug 2012 13:39:52 +0000

The purpose of the paper is to investigate several subordination- and superordination-preserving properties of a class of integral operators, which are defined on the space of analytic functions in the open unit disk. The sandwich-type theorem for these integral operators is also presented. Moreover, we consider an application of the subordination and superordination theorem to the Gauss hypergeometric function.

Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation

Yaning Tang and Pengpeng Su — Tue, 07 Aug 2012 13:36:42 +0000

Based on the Hirota bilinear method and Wronskian technique, two different classes of sufficient conditions consisting of linear partial differential equations system are presented, which guarantee that the Wronskian determinant is a solution to the corresponding Hirota bilinear equation of a (3+1)-dimensional generalized shallow water equation. Our results show that the nonlinear equation possesses rich and diverse exact solutions such as rational solutions, solitons, negatons, and positons.

Differential Transcendency in the Theory of Linear Differential Systems with Constant Coefficients

Thu, 12 Jul 2012 13:46:45 +0000

We consider a reduction of a nonhomogeneous linear system of first-order operator equations to a totally reduced system. Obtained results are applied to Cauchy problem for linear differential systems with constant coefficients and to the question of differential transcendency.

Odd-Ary Approximating Subdivision Schemes and RS Strategy for Irregular Dense Initial Data

Muhammad Aslam and W. P. Abeysinghe — Mon, 11 Jun 2012 11:22:45 +0000

We investigate the implementation of approximating subdivision schemes on noisy or irregular initial control data. Presence of noise in the initial data generates oscillatory curves by subdivision schemes. To reduce or completely eliminate these oscillations, we combine subdivision schemes with other noise removal schemes such as variational regularization method. This setup will allow us to produce the limit curve with less oscillations and still stay as close as possible to the initial data points.

Subclasses of Analytic Functions Associated with Generalised Multiplier Transformations

Rashidah Omar and Suzeini Abdul Halim — Thu, 24 May 2012 12:16:36 +0000

New subclasses of analytic functions in the open unit disc are introduced which are defined using generalised multiplier transformations. Inclusion theorems are investigated for functions to be in the classes. Furthermore, generalised Bernardi-Libera-Livington integral operator is shown to be preserved for these classes.

Bifurcation of Sign-Changing Solutions for 𝑚-Point Boundary Value Problems

Yulian An and Maoan Han — Tue, 22 May 2012 10:46:51 +0000

With the help of bifurcation techniques, some multiplicity results and global structure for sign-changing solutions of some 𝑚-point boundary value problems are obtained when the nonlinear term is sublinear at 0.

Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Hee Sun Jung, Gou Nakamura, Ryozi Sakai, and Noriaki Suzuki — Wed, 16 May 2012 10:55:17 +0000

Let ℝ=(−∞,∞), and let 𝑤𝜌(𝑥)=|𝑥|𝜌𝑒−𝑄(𝑥), where 𝜌>−1/2 and 𝑄∈𝐶1(ℝ)∶ℝ→ℝ+=[0,∞) is an even function. Then we can construct the orthonormal polynomials 𝑝𝑛(𝑤2𝜌;𝑥) of degree 𝑛 for 𝑤2𝜌(𝑥). In this paper for an even integer 𝜈≥2 we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros {𝑥𝑘,𝑛,𝜌}𝑛𝑘=1 of 𝑝𝑛(𝑤2𝜌;𝑥). Moreover, for an odd integer 𝜈≥1, we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros {𝑥𝑘,𝑛,𝜌}𝑛𝑘=1 of 𝑝𝑛(𝑤2𝜌;𝑥).

Existence of Alternate Steady States in a Phosphorous Cycling Model

Dagny Butler, Sarath Sasi, and R. Shivaji — Mon, 14 May 2012 18:39:03 +0000

We analyze the positive solutions to the steady-state reaction diffusion equation with Dirichlet boundary conditions of the form: −Δ𝑢=𝜆[𝐾−𝑢+𝑐(𝑢4/(1+𝑢4))],𝑥∈Ω,𝑢=0,𝑥∈𝜕Ω. Here, Δ𝑢=div(∇𝑢) is the Laplacian of 𝑢, 1/𝜆 is the diffusion coefficient, 𝐾 and 𝑐 are positive constants, and Ω⊂ℝ𝑁 is a smooth bounded region with 𝜕Ω in 𝐶2. This model describes the steady states of phosphorus cycling in stratified lakes. Also, it describes the colonization of barren soils in drylands by vegetation. In this paper, we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of subsuper solutions.

On Certain Subclasses of Meromorphic Functions with Positive and Fixed Second Coefficients Involving the Liu-Srivastava Linear Operator

N. Magesh, N. B. Gatti, and S. Mayilvaganan — Tue, 08 May 2012 10:53:29 +0000

We introduce and study a subclass Σ𝑃(𝛾,𝑘,𝜆,𝑐) of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class Σ𝑃(𝛾,𝑘,𝜆,𝑐) by fixing the second coefficient. Further, it is shown that the class Σ𝑃(𝛾,𝑘,𝜆) is closed under convex linear combination.

Boundedness and Compactness of the Mean Operator Matrix on Weighted Hardy Spaces

Bahmann Yousefi and Ebrahim Pazouki — Tue, 08 May 2012 10:49:59 +0000

We investigate the boundedness and the compactness of the mean operator matrix acting on the weighted Hardy spaces.

The Theory for 𝐽-Hermitian Subspaces in a Product Space

Huaqing Sun and Jiangang Qi — Tue, 24 Apr 2012 11:08:37 +0000

This paper is concerned with the theory for 𝐽-Hermitian subspaces. The defect index of a 𝐽-Hermitian subspace is defined, and a formula for the defect index is established; the result that every 𝐽-Hermitian subspace has a 𝐽-self-adjoint subspace extension is obtained; all the 𝐽-self-adjoint subspace extensions of a 𝐽-Hermitian subspace are characterized. This theory will provide a fundamental basis for characterizations of 𝐽-self-adjoint extensions for linear nonsymmetric expressions on general time scales in terms of boundary conditions, including both differential and difference cases.

Some New Double-Sequence Spaces in 2-Normed Spaces Defined by Ideal Convergence and an Orlicz Function

Orhan Tuğ, Mutlay Doğan, and Abdullah Kurudirek — Sun, 22 Apr 2012 11:08:04 +0000

We generalize some sequence spaces from single to double, we study some topological properties of these double sequence spaces by using ideal convergence, difference sequence spaces, and an Orlicz function in 2-normed spaces, and we give some results related to these sequence spaces.

Some Dense Linear Subspaces of Extended Little Lipschitz Algebras

Davood Alimohammadi and Sirous Moradi — Thu, 19 Apr 2012 15:31:11 +0000

Let (𝑋,𝑑) be a compact metric space. In 1987, Bade, Curtis, and Dales obtained a sufficient condition for density of a subspace 𝑃 of little Lipschitz algebra lip(𝑋,𝛼) in this algebra and in particular showed that Lip(𝑋,1) is dense in lip(𝑋,𝛼), whenever 0<𝛼<1. Let 𝐾 be a compact subset of 𝑋. We define new classes of Lipchitz algebras Lip(𝑋,𝐾,𝛼) for 𝛼∈(0,1] and lip(𝑋,𝐾,𝛼) for 𝛼∈(0,1), consisting of those continuous complex-valued functions 𝑓 on 𝑋 such that 𝑓|𝐾∈Lip(𝐾,𝛼) and 𝑓|𝐾∈lip(𝐾,𝛼), respectively. In this paper we obtain a sufficient condition for density of a linear subspace 𝑃 of extended little Lipschitz algebra lip(𝑋,𝐾,𝛼) in this algebra and in particular show that Lip(𝑋,𝐾,1) is dense in lip(𝑋,𝐾,𝛼), whenever 0<𝛼<1.

On Locally Uniformly Differentiable Functions on a Complete Non-Archimedean Ordered Field Extension of the Real Numbers

Khodr Shamseddine and Todd Sierens — Tue, 17 Apr 2012 11:13:03 +0000

We study the properties of locally uniformly differentiable functions on 𝒩, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In particular, we show that locally uniformly differentiable functions are 𝐶1, they include all polynomial functions, and they are closed under addition, multiplication, and composition. Then we formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions.

Regularity Criterion for the 3D Nematic Liquid Crystal Flows

Jishan Fan and Tohru Ozawa — Wed, 11 Apr 2012 18:20:10 +0000

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.

Inclusion Relationships for Certain Subclasses of Meromorphic Functions Defined by Using the Extended Multiplier Transformations

R. M. El-Ashwah — Tue, 10 Apr 2012 13:19:12 +0000

Let ∑ denote the class of analytic functions in the punctured unit disc 𝑈∗={𝑧∶0<|𝑧|<1}. Set 𝜑𝑚𝜆,ℓ∑(𝑧)=1/𝑧+∞𝑘=0[ℓ+𝜆(𝑘+1)/ℓ]𝑚𝑧𝑘(𝑚∈ℕ0;ℓ>0;𝜆≥0;𝑧∈𝑈∗), and define 𝜑𝑚,𝜇𝜆,ℓ in terms of the Hadamard product by 𝜑𝑚𝜆,ℓ(𝑧)∗𝜑𝑚,𝜇𝜆,ℓ(𝑧)=1/𝑧(1−𝑧)𝜇(𝜇>0;𝑧∈𝑈∗). In this paper, we introduce several new subclasses of analytic functions defined by means of the operator 𝐼𝑚𝜇(𝜆,ℓ)𝑓(𝑧)=𝜑𝑚,𝜇𝜆,ℓ∑(𝑧)∗𝑓(𝑧)(𝑓∈;𝑚∈ℕ0;ℓ>0;𝜆≥0;𝜇>0).Inclusion properties of these classes and some applications involving integral operator are also considered.

Continuation Criterion for the 2D Liquid Crystal Flows

Jishan Fan and Tohru Ozawa — Sun, 08 Apr 2012 09:01:10 +0000

We consider the 2D liquid crystal systems, which consists of Navier-Stokes system coupled with wave maps or biharmonic wave maps, respectively. By logarithmic Sobolev inequalities, we obtain a blow-up criterion ∇𝑑,𝜕𝑡𝑑∈𝐿1̇𝐵(0,𝑇;0∞,∞(ℝ2)) for the case with wave maps, and we prove the existence of a global-in-time strong solutions for the case with biharmonic wave maps.

Positive Solutions to Periodic Boundary Value Problems for Four-Order Differential Equations

Huantao Zhu and Zhiguo Luo — Wed, 04 Apr 2012 13:31:42 +0000

We apply fixed point theorem in a cone to obtain sufficient conditions for the existence of single and multiple positive solutions of periodic boundary value problems for a class of four-order differential equations.