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2. Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order.
- Author
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Pavlotsky, I. P. and Strianese, M.
- Subjects
MATHEMATICS ,INTEGRAL calculus ,INTEGRALS ,NUMERICAL analysis - Abstract
A second-order equation can have singular sets of first and second type, and (see the introduction), where the integral curve does not exist in the ordinary sense but where it can be extended by using the first integral [1--5]. Denote by the Cartesian axis . If the function has a derivative at a point of local extremum of this function, then this point belongs to . The extrema at which does not exist can be placed on . In [5--8], the stability and instability of extrema on under small perturbations of the equation were considered, and the stability of the mutual arrangement of the maxima and minima of on the singular set was studied (locally as a rule, i.e., in small neighborhoods of singular points). In the present paper, sufficient conditions for the preservation of type of a local extremum on the finite part of or are found for the case in which the perturbation on all of this part does not exceed some explicitly indicated quantity which is the same on the entire singular set. [ABSTRACT FROM AUTHOR]
- Published
- 2004
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3. How and how much to invest for fighting cheaters: from an ODE to a Cellular Automata model
- Author
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Luca Meacci
- Subjects
education.field_of_study ,Ordinary differential equation ,Population ,Ode ,Tax evasion ,Type (model theory) ,education ,Mathematical economics ,Cellular automaton ,Mathematics - Abstract
In this paper, we present a study which is the completion of the problem left open in the appendix of the paper Nuno et al. (Eur J Appl Math 21:459–478, 2010). The goal of the paper is to offer a deeper analysis of the type of the ODE system featured in the paper above concerning the evolution of the total wealth and cheater population and, mainly, the purpose is to generalize this one to a Cellular Automata model introducing spatial and local effects. The work presented shows the original interaction between an Ordinary Differential Equations model and a Cellular Automata model.
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4. The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE
- Author
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James S.W. Wong and Man Kam Kwong
- Subjects
Partial differential equation ,Algebra and Number Theory ,Differential equation ,Mathematical analysis ,Ode ,lcsh:QA299.6-433 ,lcsh:Analysis ,Shooting method ,Ordinary differential equation ,Boundary value problem ,Differential algebraic equation ,Analysis ,Mathematics ,Numerical partial differential equations - Abstract
In a recent paper, Sun et al. (2007) studied the existence of positive solutions of nonhomogeneous multipoint boundary value problems for a second-order differential equation. It is the purpose of this paper to show that the shooting method approach proposed in the recent paper by the first author can be extended to treat this more general problem.
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5. On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition
- Author
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Yulian An and Hua Luo
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Partial differential equation ,Algebra and Number Theory ,lcsh:Mathematics ,Applied Mathematics ,Mathematical analysis ,lcsh:QA1-939 ,Infimum and supremum ,Nonlinear system ,Bifurcation theory ,Ordinary differential equation ,Boundary value problem ,Value (mathematics) ,Analysis ,Mathematics ,Numerical partial differential equations - Abstract
By defining a new terminology, scatter degree, as the supremum of graininess functional value, this paper studies the existence of solutions for a nonlinear two-point dynamic boundary value problem on time scales. We do not need any growth restrictions on nonlinear term of dynamic equation besides a barrier strips condition. The main tool in this paper is the induction principle on time scales.
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6. Existence of solutions for a class of quasilinear Schrödinger equations on R ${\mathbb{R}}$
- Author
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Kuo Yang and Da-Bin Wang
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Partial differential equation ,Algebra and Number Theory ,Mathematical analysis ,Euler equations ,Nonlinear system ,symbols.namesake ,Method of characteristics ,Simultaneous equations ,Ordinary differential equation ,symbols ,Differential algebraic equation ,Analysis ,Mathematics ,Numerical partial differential equations - Abstract
In this paper, we study the existence of nontrivial solution for a class of quasilinear Schrodinger equations in ${\mathbb{R}}$ with the nonlinearity asymptotically linear and, furthermore, the potential indefinite in sign. The tool used in this paper is the direct variation method.
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7. A Ritz-Galerkin approximation to the solution of parabolic equation with moving boundaries
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Jianrong Zhou and Heng Li
- Subjects
Approximation solution ,Partial differential equation ,Algebra and Number Theory ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Boundary (topology) ,Mathematics::Numerical Analysis ,Ordinary differential equation ,Free boundary problem ,Uniqueness ,Boundary value problem ,Galerkin method ,Analysis ,Mathematics - Abstract
The present paper is devoted to the investigation of a parabolic equation with moving boundaries arising in ductal carcinoma in situ (DCIS) model. Approximation solution of this problem is implemented by Ritz-Galerkin, which is a first attempt at tackling such problem. In process of dealing with this moving boundary condition, we use a trick of introducing two transformations to convert moving boundary to nonclassical boundary that can be handled with Ritz-Galerkin method. Also, existence and uniqueness are proved. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper.
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8. Partial permanence and extinction on stochastic Lotka-Volterra competitive systems
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Yonghui Sun, Chunwei Dong, and Lei Liu
- Subjects
Stochastic partial differential equation ,Extinction ,Partial differential equation ,Algebra and Number Theory ,Ordinary differential equation ,Applied Mathematics ,Mathematical analysis ,Quantitative Biology::Populations and Evolution ,Itō's lemma ,Analysis ,Mathematics - Abstract
This paper discusses an autonomous competitive Lotka-Volterra model in random environments. The contributions of this paper are as follows. (a) Some sufficient conditions for partial permanence and extinction on this system are established; (b) By using some novel techniques, the conditions imposed on permanence and extinction of one-species are weakened. Finally, a numerical experiment is conducted to validate the theoretical findings.
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9. Exponential stability criteria for fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays and impulses
- Author
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Longxian Chu and Weina He
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Lyapunov function ,0209 industrial biotechnology ,Partial differential equation ,Algebra and Number Theory ,Artificial neural network ,Applied Mathematics ,02 engineering and technology ,Fuzzy logic ,symbols.namesake ,020901 industrial engineering & automation ,Exponential stability ,Control theory ,Ordinary differential equation ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,Bidirectional associative memory ,Differential inequalities ,Analysis ,Mathematics - Abstract
This paper is concerned with fuzzy bidirectional associative memory (BAM) Cohen-Grossberg neural networks with mixed delays and impulses. By constructing an appropriate Lyapunov function and a new differential inequality, we obtain some sufficient conditions which ensure the existence and global exponential stability of a periodic solution of the model. The results in this paper extend and complement the previous publications. An example is given to illustrate the effectiveness of our obtained results.
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10. A PD-type iterative learning control algorithm for singular discrete systems
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Qian Liu, Senping Tian, Xisheng Dai, and Jianxiang Zhang
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0209 industrial biotechnology ,Mathematical optimization ,Partial differential equation ,Algebra and Number Theory ,Applied Mathematics ,Iterative learning control ,02 engineering and technology ,State (functional analysis) ,020901 industrial engineering & automation ,Singular function ,Singular solution ,Ordinary differential equation ,Convergence (routing) ,0202 electrical engineering, electronic engineering, information engineering ,Decomposition (computer science) ,020201 artificial intelligence & image processing ,Algorithm ,Analysis ,Mathematics - Abstract
Based on a specific decomposition of discrete singular systems, in this paper, we study the problem of state tracking control by using PD-type algorithm of iterative learning control. The convergence conditions and theoretical analysis of the PD-type algorithm are presented in detail. An illustrative example supporting the theoretical results and the effectiveness of the PD-type iterative learning control algorithm for discrete singular systems is shown at the end of the paper.
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11. Iterative oscillation tests for differential equations with several non-monotone arguments
- Author
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George E. Chatzarakis, Elena Braverman, and Ioannis P. Stavroulakis
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Partial differential equation ,Algebra and Number Theory ,Differential equation ,Oscillation ,Applied Mathematics ,010102 general mathematics ,Dynamical Systems (math.DS) ,01 natural sciences ,34K11, 34K06 ,010101 applied mathematics ,Gronwall's inequality ,Ordinary differential equation ,Computer Science::Logic in Computer Science ,FOS: Mathematics ,Applied mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Non monotone ,Analysis ,Mathematics - Abstract
Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Gr\"{o}nwall inequality. Examples illustrating the significance of the results are also given., Comment: The paper was originally open access. The purpose of this publication is correct the original mistake that occurred in Theorems 6 and 10 (in particular, the variables under the integral in (2.11) and (2.27)) in the published paper
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12. Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem
- Author
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Dechu Chen and Binxiang Dai
- Subjects
Stochastic partial differential equation ,Equilibrium point ,Examples of differential equations ,Algebra and Number Theory ,Distributed parameter system ,Ordinary differential equation ,Mathematical analysis ,Mountain pass theorem ,Delay differential equation ,C0-semigroup ,Analysis ,Mathematics - Abstract
In this paper we use variational methods and generalized mountain pass theorem to investigate the existence of periodic solutions for some second-order delay differential systems with impulsive effects. To the authors’ knowledge, there is no paper about periodic solution of impulses delay differential systems via critical point theory. Our results are completely new.
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13. Global existence and blow-up for a class of nonlinear reaction diffusion problems
- Author
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Juntang Ding
- Subjects
Nonlinear system ,Class (set theory) ,Partial differential equation ,Algebra and Number Theory ,Global analysis ,Ordinary differential equation ,Reaction–diffusion system ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Finite time ,Upper and lower bounds ,Analysis ,Mathematics - Abstract
This paper deals with the global existence and blow-up of the solution for a class of nonlinear reaction diffusion problems. The purpose of this paper is to establish conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified. Finally, as applications of the obtained results, some examples are presented.
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14. The stability of evolutionary p ( x ) $p(x)$ -Laplacian equation
- Author
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Huashui Zhan
- Subjects
Partial differential equation ,Algebra and Number Theory ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,01 natural sciences ,Omega ,Stability (probability) ,010101 applied mathematics ,Ordinary differential equation ,Nabla symbol ,0101 mathematics ,Degeneracy (mathematics) ,Laplace operator ,Analysis ,Mathematics - Abstract
The paper studies the equation $$ {u_{t}}= \operatorname{div} \bigl(a(x)\vert {\nabla u} \vert ^{p(x) - 2}\nabla u \bigr), $$ with the boundary degeneracy coming from $a(x)\vert_{x\in \partial \Omega }=0$ . The paper introduces a new kind of weak solutions of the equation. One can study the stability of the new kind of weak solutions without any boundary value condition.
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15. Asymptotic behavior of the thermoelastic suspension bridge equation with linear memory
- Author
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Jum-Ran Kang
- Subjects
Partial differential equation ,Algebra and Number Theory ,Independent equation ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Bridge (interpersonal) ,Viscoelasticity ,010101 applied mathematics ,Condensed Matter::Soft Condensed Matter ,Thermoelastic damping ,global attractors ,suspension bridge equation ,viscoelasticity ,memory ,thermoelasticity ,asymptotically compact ,Ordinary differential equation ,Attractor ,0101 mathematics ,Suspension (vehicle) ,Analysis ,Mathematics - Abstract
This paper is concerned with a thermoelastic suspension bridge equations with memory effects. For the suspension bridge equations without memory, there are many classical results. However, the suspension bridge equations with both viscoelastic and thermal memories were not studied before. The object of the present paper is to provide a result on the global attractor to a thermoelastic suspension bridge equation with past history.
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16. The eigenvalue characterization for the constant sign Green’s functions of ( k , n − k ) $(k,n-k)$ problems
- Author
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Lorena Saavedra, Alberto Cabada, and Universidade de Santiago de Compostela. Departamento de Estatística, Análise Matemática e Optimización
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Green’s functions ,Spectral theory ,Algebra and Number Theory ,010102 general mathematics ,Mathematical analysis ,Function (mathematics) ,Disconjugation ,01 natural sciences ,010101 applied mathematics ,Maximum principles ,Operator (computer programming) ,nth order boundary value problem ,Ordinary differential equation ,Interval (graph theory) ,0101 mathematics ,Constant (mathematics) ,Eigenvalues and eigenvectors ,Analysis ,Mathematics ,Sign (mathematics) - Abstract
This paper is devoted to the study of the sign of the Green’s function related to a general linear nth-order operator, depending on a real parameter, $T_{n}[M]$ , coupled with the $(k,n-k)$ boundary value conditions. If the operator $T_{n}[\bar{M}]$ is disconjugate for a given M, we describe the interval of values on the real parameter M for which the Green’s function has constant sign. One of the extremes of the interval is given by the first eigenvalue of the operator $T_{n}[\bar{M}]$ satisfying $(k,n-k)$ conditions. The other extreme is related to the minimum (maximum) of the first eigenvalues of $(k-1,n-k+1)$ and $(k+1,n-k-1)$ problems. Moreover, if $n-k$ is even (odd) the Green’s function cannot be nonpositive (nonnegative). To illustrate the applicability of the obtained results, we calculate the parameter intervals of constant sign Green’s functions for particular operators. Our method avoids the necessity of calculating the expression of the Green’s function. We finalize the paper by presenting a particular equation in which it is shown that the disconjugation hypothesis on operator $T_{n}[\bar{M}]$ for a given M cannot be eliminated.
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17. Almost periodic solution of an impulsive multispecies logarithmic population model
- Author
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Li Wang and Hui Zhang
- Subjects
Partial differential equation ,Algebra and Number Theory ,Degree (graph theory) ,Logarithm ,Population model ,Differential equation ,Ordinary differential equation ,Applied Mathematics ,Mathematical analysis ,Coincidence ,Analysis ,Mathematics - Abstract
In this paper, some easily verifiable conditions are derived for the existence of almost periodic solution of an impulsive multispecies logarithmic population model in terms of the continuous theorem of coincidence degree theory, which is rarely applied to studying the existence of almost periodic solution of an impulsive differential equation. Our results generalize previous results by Alzabut, Stamov and Sermutlu. Besides, our technique used in this paper can be applied to study the existence of almost periodic solution of an impulsive differential equation with linear impulsive perturbations.
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18. Discreteness of the spectrum of vectorial Schrödinger operators with δ-interactions
- Author
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Xiaoyun Liu, Xiaojing Zhao, and Guoliang Shi
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Standard form ,Partial differential equation ,Algebra and Number Theory ,010102 general mathematics ,Mathematical analysis ,Spectrum (functional analysis) ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Matrix (mathematics) ,Operator (computer programming) ,Ordinary differential equation ,symbols ,Embedding ,0101 mathematics ,Schrödinger's cat ,Analysis ,Mathematics - Abstract
This paper deals with the vectorial Schrodinger operators with δ-interactions generated by $L_{X,A,Q}:=-\frac{d^{2}}{dx^{2}} +Q(x)+\sum_{k=1}^{\infty}A_{k}\delta(x-x_{k})$ , $x\in[ 0,+\infty)$ . First, we obtain an embedding inequality. Then using standard form methods, we prove that the operator $\mathbf{H}_{X,A,Q}$ given in this paper is self-adjoint. Finally, a sufficient condition and a necessary condition are given for the spectrum of the operator $\mathbf {H}_{X,A,Q}$ to be discrete. By giving additional restrictions on the symmetric potential matrix $Q(x)$ and $A_{k}$ , we also give a necessary and sufficient condition for a special case. The conditions are analogous to Molchanov’s discreteness criteria.
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19. Stepanov-like pseudo almost automorphic solution to a parabolic evolution equation
- Author
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Desheng Ji and Yueming Lu
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Pure mathematics ,Partial differential equation ,Algebra and Number Theory ,Mathematics::Number Theory ,Applied Mathematics ,Mathematical analysis ,Banach space ,Monotonic function ,Term (logic) ,Ordinary differential equation ,Evolution equation ,Component (group theory) ,Analysis ,Mathematics - Abstract
In this paper, the parabolic evolution equation $u'(t)+A(t)u(t)=f(t)$ in a reflexive real Banach space is considered. Assuming strong monotonicity, pseudo almost automorphy and other appropriate conditions of the operators $A(t)$ and Stepanov-like pseudo almost automorphy of the forced term $f(t)$ , we obtain the Stepanov-like pseudo almost automorphy of the solution to the evolution equation by using the almost automorphic component equation method. This paper extends a known result in the case where $A(\cdot)$ and f are almost automorphic in certain senses. Finally, a concrete example is given to illustrate our results.
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20. Existence of periodic solutions for nonautonomous second-order discrete Hamiltonian systems
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Wen Guan, Da-Bin Wang, and Hua-Fei Xie
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Class (set theory) ,Partial differential equation ,Algebra and Number Theory ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Hamiltonian system ,Saddle point theorem ,010101 applied mathematics ,Ordinary differential equation ,Order (group theory) ,0101 mathematics ,Time variable ,Analysis ,Mathematics - Abstract
In this paper, we consider the existence of periodic solutions for a class of nonautonomous second-order discrete Hamiltonian systems in case the sum on the time variable of potential is periodic. The tools used in our paper are the direct variational minimizing method and Rabinowitz’s saddle point theorem.
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21. Robust iterative learning control design for linear systems with time-varying delays and packet dropouts
- Author
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Yang Junqi, Bu Xuhui, Hou Zhanwei, and Hou Zhongsheng
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0209 industrial biotechnology ,Sequence ,Mathematical optimization ,Algebra and Number Theory ,Network packet ,Applied Mathematics ,Linear system ,Iterative learning control ,Stability (learning theory) ,02 engineering and technology ,Bernoulli's principle ,020901 industrial engineering & automation ,Exponential stability ,Control theory ,Ordinary differential equation ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Analysis ,Mathematics - Abstract
This paper proposes the robust iterative learning control (ILC) design for uncertain linear systems with time-varying delays and random packet dropouts. The packet dropout is modeled by an arbitrary stochastic sequence satisfying the Bernoulli binary distribution, which renders the ILC system to be stochastic instead of a deterministic one. The main idea of this paper is to transform the ILC design into robust stability for a two-dimensional (2D) stochastic system described by the Roesser model with a delay varying in a range. A delay-dependent stability condition, which can guarantee mean-square asymptotic stability of such a 2D stochastic system, is derived in terms of linear matrix inequalities (LMIs), and formulas can be given for the ILC law design. An example for the injection molding is given to demonstrate the effectiveness of the proposed ILC method.
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22. Nagumo theorems of third-order singular nonlinear boundary value problems
- Author
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Ming Cheng
- Subjects
Discrete mathematics ,Third order ,Nonlinear system ,Schauder fixed point theorem ,Partial differential equation ,Algebra and Number Theory ,Ordinary differential equation ,Mathematical analysis ,Topological degree theory ,Multiplicity (mathematics) ,Boundary value problem ,Analysis ,Mathematics - Abstract
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.
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23. Dynamical analysis of the permanent-magnet synchronous motor chaotic system
- Author
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Fuchen Zhang, Chunlai Mu, and Xiaofeng Liao
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Lyapunov stability ,0209 industrial biotechnology ,Partial differential equation ,Algebra and Number Theory ,Permanent magnet synchronous motor ,Applied Mathematics ,Chaotic ,02 engineering and technology ,01 natural sciences ,010305 fluids & plasmas ,Nonlinear Sciences::Chaotic Dynamics ,020901 industrial engineering & automation ,Control theory ,Ordinary differential equation ,0103 physical sciences ,System parameters ,Synchronous motor ,Analysis ,Numerical stability ,Mathematics - Abstract
This paper is concerned with some dynamics of the permanent-magnet synchronous motor chaotic system based on Lyapunov stability theory and optimization theory. The innovation of the paper lies in that we derive a family of mathematical expressions of globally exponentially attractive sets for this chaotic system with respect to system parameters. Numerical simulations confirm that theoretical analysis results are correct.
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24. Existence results for fractional differential equations of arbitrary order with nonlocal integral boundary conditions
- Author
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Bashir Ahmad, Mohammed M. Matar, and Ravi P. Agarwal
- Subjects
Stochastic partial differential equation ,Algebra and Number Theory ,Picard–Lindelöf theorem ,Ordinary differential equation ,Mathematical analysis ,Fixed-point theorem ,Boundary value problem ,C0-semigroup ,Analysis ,Mathematics ,Numerical partial differential equations ,Fractional calculus - Abstract
In this paper, we investigate the existence of solutions for fractional differential equations of arbitrary order with nonlocal integral boundary conditions. The existence results are obtained by applying Krasnoselskii’s fixed point theorem and Leray-Schauder degree theory, while the uniqueness of the solutions is established by means of Banach’s contraction mapping principle. The paper concludes with illustrative examples.
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25. Existence and uniqueness of wave fronts in neuronal network with nonlocal post-synaptic axonal and delayed nonlocal feedback connections
- Author
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Lijun Zhang
- Subjects
Partial differential equation ,Model equation ,Algebra and Number Theory ,Quantitative Biology::Neurons and Cognition ,Applied Mathematics ,Mathematical analysis ,Synaptic coupling ,Superposition principle ,Lateral inhibition ,Ordinary differential equation ,Biological neural network ,Uniqueness ,Analysis ,Mathematics - Abstract
An integral-differential model equation, arising from neuronal networks with both axonal and delayed nonlocal feedback connections, is considered in this paper. The kernel functions in the feedback channel we study here include not only pure excitations but also lateral inhibition. For the kernel functions in the synaptic coupling, pure excitations, lateral inhibition, the lateral excitations and more general synaptic couplings (e.g., oscillating kernel functions) are considered. The main goal of this paper is the study of the existence and uniqueness of the traveling wave front solutions. The main method we applied is the speed index functions and principle of linear superposition.
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26. Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays
- Author
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Changjin Xu, Lin Lu, and Yusen Wu
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Partial differential equation ,Algebra and Number Theory ,Ordinary differential equation ,Applied Mathematics ,Mathematical analysis ,Quantitative Biology::Populations and Evolution ,Stability (probability) ,Differential inequalities ,Analysis ,Mathematics - Abstract
In this paper, we deal with a discrete Lotka-Volterra predator-prey model with time-varying delays. For the general non-autonomous case, sufficient conditions which ensure the permanence and global stability of the system are obtained by using differential inequality theory. For the periodic case, sufficient conditions which guarantee the existence of a unique globally stable positive periodic solution are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.
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27. Discrete matrix delayed exponential for two delays and its property
- Author
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Josef Diblík and Blanka Morávková
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Property (philosophy) ,Partial differential equation ,Algebra and Number Theory ,Generalization ,Applied Mathematics ,Mathematical analysis ,Exponential integrator ,Discrete matrix delayed exponential for two delays ,Exponential function ,Matrix (mathematics) ,main property of exponential ,Ordinary differential equation ,Applied mathematics ,C0-semigroup ,Diskrétní zpožděná maticová exponenciála pro dvě zpoždění ,hlavní vlastnost exponenciály ,Analysis ,Mathematics ,Hardware_LOGICDESIGN - Abstract
In recent papers, a discrete matrix delayed exponential for a single delay was defined and its main property connected with the solution of linear discrete systems with a single delay was proved. In the present paper, a generalization of the concept of discrete matrix delayed exponential is designed for two delays and its main property is proved as well. V nedávném článku byla definována diskrétní zpožděná maticová exponenciála pro jedno zpoždění a byla dokázána její hlavní vlastnost související s řešením lineárních diskrétních systémů s jedním zpožděním. V aktuálním článku je zkonstruováno zobecnění pojmu diskrétní zpožděné maticové exponenciály pro dvě zpoždění a rovněž dokázána její základní vlastnost.
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28. Existence and exponential stability of periodic solutions for a class of Hamiltonian systems on time scales
- Author
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Li Yang, Yongkun Li, and Yongzhi Liao
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Class (set theory) ,Partial differential equation ,Algebra and Number Theory ,Exponential stability ,Ordinary differential equation ,Applied Mathematics ,Mathematical analysis ,Fixed-point theorem ,Scale (descriptive set theory) ,Exponential integrator ,Analysis ,Hamiltonian system ,Mathematics - Abstract
In this paper, by using a fixed point theorem and the theory of calculus on time scales, we obtain some sufficient conditions for the existence and exponential stability of periodic solutions for a class of Hamiltonian systems on time scales. We also present numerical examples to show the feasibility of our results. The results of this paper are completely new and complementary to the previously known results even if the time scale T = R or ℤ.
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29. The general meromorphic solutions of the Petviashvili equation
- Author
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Zifeng Huang, Jinchun Lai, and Wenjun Yuan
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Partial differential equation ,Exact solutions in general relativity ,Algebra and Number Theory ,Independent equation ,Differential equation ,Ordinary differential equation ,Applied Mathematics ,Mathematical analysis ,Elliptic function ,Exact differential equation ,Analysis ,Mathematics ,Meromorphic function - Abstract
In this paper, we employ the complex method to first obtain all meromorphic exact solutions of complex Petviashvili equation, and then find all exact solutions of Petviashvili equation. The idea introduced in this paper can be applied to other non-linear evolution equations. Our results show that the complex method is simpler than other methods. Finally, we give some computer simulations to illustrate our main results. MSC:30D35, 34A05.
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30. Existence of positive solutions for a kind of periodic boundary value problem at resonance
- Author
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Piotr Drygaś and Mirosława Zima
- Subjects
Partial differential equation ,Algebra and Number Theory ,Ordinary differential equation ,Mathematical analysis ,Free boundary problem ,Exact differential equation ,Initial value problem ,Cauchy boundary condition ,Mixed boundary condition ,Boundary value problem ,Analysis ,Mathematics - Abstract
In the paper we provide sufficient conditions for the existence of positive solutions for some second-order differential equation subject to periodic boundary conditions. Our method employs a Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. Two examples are given to illustrate the main result of the paper.
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31. Approximation of eigenvalues of discontinuous Sturm-Liouville problems with eigenparameter in all boundary conditions
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Abdulaziz Alofi, Ali H. Bhrawy, and Mohammed M Tharwat
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Discontinuity (linguistics) ,Partial differential equation ,Algebra and Number Theory ,Sinc function ,Ordinary differential equation ,Mathematical analysis ,Internal point ,Sturm–Liouville theory ,Boundary value problem ,Eigenvalues and eigenvectors ,Analysis ,Mathematics - Abstract
In this paper, we apply a sinc-Gaussian technique to compute approximate values of the eigenvalues of Sturm-Liouville problems which contain an eigenparameter appearing linearly in two boundary conditions, in addition to an internal point of discontinuity. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than that of the classical sinc method. Numerical worked examples with tables and illustrative figures are given at the end of the paper. MSC:34L16, 94A20, 65L15.
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32. Stability and estimate of solution to uncertain neutral delay systems
- Author
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Michael Gitman, Roman Shklyar, and Alexander Domoshnitsky
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Partial differential equation ,Algebra and Number Theory ,Exponential stability ,Ordinary differential equation ,Bounded function ,Mathematical analysis ,Cauchy distribution ,Uniqueness ,Boundary value problem ,Cauchy matrix ,Analysis ,Mathematics - Abstract
The coefficients and delays in models describing various processes are usually obtained as a results of measurements and can be obtained only approximately. We deal with the question of how to estimate the influence of ‘mistakes’ in coefficients and delays on solutions’ behavior of the delay differential neutral system , , . This topic is known in the literature as uncertain systems or systems with interval defined coefficients. The goal of this paper is to obtain stability of uncertain systems and to estimate the difference between solutions of a ‘real’ system with uncertain coefficients and/or delays and corresponding ‘model’ system. We develop the so-called Azbelev W-transform, which is a sort of the right regularization allowing researchers to reduce analysis of boundary value problems to study of systems of functional equations in the space of measurable essentially bounded functions. In corresponding cases estimates of norms of auxiliary linear operators (obtained as a result of W-transform) lead researchers to conclusions about existence, uniqueness, positivity and stability of solutions of given boundary value problems. This method works efficiently in the case when a ‘model’ used in W-transform is ‘close’ to a given ‘real’ system. In this paper we choose, as the ‘models’, systems for which we know estimates of the resolvent Cauchy operators. We demonstrate that systems with positive Cauchy matrices present a class of convenient ‘models’. We use the W-transform and other methods of the general theory of functional differential equations. Positivity of the Cauchy operators is studied and then used in the analysis of stability and estimates of solutions. Results: We propose results about exponential stability of the given system and obtain estimates of difference between the solution of this uncertain system and the ‘model’ system , , . New tests of stability and in the future of existence and uniqueness of boundary value problems for neutral delay systems can be obtained on the basis of this technique.
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33. On a Mixed Problem for a Constant Coefficient Second-Order System
- Author
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Rita Cavazzoni
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Cauchy problem ,Constant coefficients ,Algebra and Number Theory ,Mathematical analysis ,lcsh:QA299.6-433 ,lcsh:Analysis ,Differential operator ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Bounded function ,Ordinary differential equation ,Data_FILES ,Free boundary problem ,Initial value problem ,Boundary value problem ,Analysis ,Mathematics - Abstract
The paper is devoted to the study of an initial boundary value problem for a linear second-order differential system with constant coefficients. The first part of the paper is concerned with the existence of the solution to a boundary value problem for the second-order differential system, in the strip , where is a suitable positive number. The result is proved by means of the same procedure followed in a previous paper to study the related initial value problem. Subsequently, we consider a mixed problem for the second-order constant coefficient system, where the space variable varies in and the time-variable belongs to the bounded interval , with sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solution , by studying two related mixed problems, whose solutions exist due to the results proved for the Cauchy problem in a previous paper and for the boundary value problem in the first part of this paper.
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34. Almost periodic solutions for neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scales
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Li Yang, Yongkun Li, and Ling Li
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Algebra and Number Theory ,Artificial neural network ,Exponential dichotomy ,Applied Mathematics ,Mathematical analysis ,Fixed-point theorem ,Delay differential equation ,Term (time) ,Exponential stability ,Ordinary differential equation ,Applied mathematics ,C0-semigroup ,Analysis ,Mathematics - Abstract
In this paper, a class of neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scales is considered. By utilizing the exponential dichotomy of linear dynamic equations on time scales, Banach’s fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of neural networks. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and the discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new and complementary to the previously known results even when the time scale .
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35. Existence and multiplicity results for a class of fractional differential inclusions with boundary conditions
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Yanping Gong and Peng Zhang
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Partial differential equation ,Algebra and Number Theory ,Ordinary differential equation ,Multiplicity results ,Mathematical analysis ,Boundary value problem ,Minimax ,Critical point (mathematics) ,Analysis ,Numerical partial differential equations ,Mathematics ,Principle of least action - Abstract
In this paper, we study the existence and multiplicity results of solutions for some class of fractional differential inclusions with boundary conditions. Some existence and multiplicity results of solutions are given by using the least action principle and minmax methods in nonsmooth critical point theory. Recent results in the literature are generalized and improved. Some examples are given in the paper to illustrate our main results.
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36. Dynamics of a nonautonomous Lotka-Volterra predator-prey dispersal system with impulsive effects
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Wenquan Wu and Lijun Xu
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Comparison theorem ,Partial differential equation ,Algebra and Number Theory ,Differential equation ,Applied Mathematics ,Mathematical analysis ,Dynamics (mechanics) ,Stability (probability) ,Lyapunov functional ,Ordinary differential equation ,Biological dispersal ,Quantitative Biology::Populations and Evolution ,Analysis ,Mathematics - Abstract
By applying the comparison theorem, Lyapunov functional, and almost periodic functional hull theory of the impulsive differential equations, this paper gives some new sufficient conditions for the uniform persistence, global asymptotical stability, and almost periodic solution to a nonautonomous Lotka-Volterra predator-prey dispersal system with impulsive effects. The main results of this paper extend some corresponding results obtained in recent years. The method used in this paper provides a possible method to study the uniform persistence, global asymptotical stability, and almost periodic solution of the models with impulsive perturbations in biological populations. MSC:34K14, 34K20, 34K45, 92D25.
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37. Properties of q-shift difference-differential polynomials of meromorphic functions
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Hong-Yan Xu, Xin-Li Wang, and Tang-Sen Zhan
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Pure mathematics ,Polynomial ,Algebra and Number Theory ,Degree (graph theory) ,Entire function ,Applied Mathematics ,Function (mathematics) ,Algebra ,Difference polynomials ,Ordinary differential equation ,Uniqueness ,Analysis ,Mathematics ,Meromorphic function - Abstract
In this paper, we deal with the zeros of the q-shift difference-differential polynomials [ P ( f ) ∏ j = 1 d f ( q j z + c j ) s j ] ( k ) − α ( z ) and ( P ( f ) ∏ j = 1 d [ f ( q j z + c j ) − f ( z ) ] s j ) ( k ) − α ( z ) , where P ( f ) is a nonzero polynomial of degree n, q j , c j ∈ C ∖ { 0 } ( j = 1 , … , d ) are constants, n , d , s j ( j = 1 , … , d ) ∈ N + and α ( z ) is a small function of f. The results of this paper are an extension of the previous theorems given by Chen and Chen and Qi. We also investigate the value sharing for q-shift difference polynomials of entire functions and obtain some results which extend the recent theorem given by Liu, Liu and Cao. MSC:39A50, 30D35.
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38. A generalized groundwater flow equation using the concept of variable-order derivative
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Abdon Atangana and Joseph Francois Botha
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Partial differential equation ,Algebra and Number Theory ,Differential equation ,Independent equation ,Ordinary differential equation ,Functional equation ,Mathematical analysis ,Groundwater flow equation ,First-order partial differential equation ,Order of accuracy ,Analysis ,Mathematics - Abstract
In this paper, the groundwater flow equation is generalized using the concept of the variational order derivative. We present a numerical solution of the modified groundwater flow equation with the variational order derivative. We solve the generalized equation with the Crank-Nicholson technique. Numerical methods typically yield approximate solutions to the governing equation through the discretization of space and time and can relax the rigid idealized conditions of analytical models or lumped-parameter models. They can therefore be more realistic and flexible for simulating field conditions. Within the discredited problem domain, the variable internal properties, boundaries, and stresses of the system are approximated. We perform the stability and convergence analysis of the Crank-Nicholson method and complete the paper with some illustrative computational examples and their simulations.
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39. Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term
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Alexander Domoshnitsky
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Exponential stability ,Differential equation ,Oscillation ,Ordinary differential equation ,Applied Mathematics ,Mathematical analysis ,Discrete Mathematics and Combinatorics ,Delay differential equation ,Instability ,Differential algebraic equation ,Stability (probability) ,Analysis ,Mathematics - Abstract
Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation can be oscillating and asymptotically unstable, the delay equation , where , can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper refute this delusion. MSC:34K20.
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40. Lower and upper estimates of solutions to systems of delay dynamic equations on time scales
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Jiří Vítovec and Josef Diblík
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Partial differential equation ,Algebra and Number Theory ,Scale (ratio) ,delay ,asymptotic behavior of solution ,Mathematical analysis ,time scale ,Of the form ,Function (mathematics) ,Domain (mathematical analysis) ,retract ,Retract ,Ordinary differential equation ,dynamic system ,Graph (abstract data type) ,retraction ,Analysis ,Mathematics - Abstract
In this paper we study a system of delay dynamic equations on the time scale $\T$ of the form $$y^{\Delta}(t)=f(t,y_{\tau}(t)),$$ where $f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$, $y_\tau(t)=(y_1(\tau_1(t)),\ldots,y_n(\tau_n(t)))$ and $\tau_i\colon\T\rightarrow \T$, $i=1,\ldots,n$ are the delay functions. We are interested about the asymptotic behavior of solutions of mentioned system. In this paper we study a system of delay dynamic equations on the time scale $\T$ of the form $$y^{\Delta}(t)=f(t,y_{\tau}(t)),$$ where $f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$, $y_\tau(t)=(y_1(\tau_1(t)),\ldots,y_n(\tau_n(t)))$ and $\tau_i\colon\T\rightarrow \T$, $i=1,\ldots,n$ are the delay functions. We are interested about the asymptotic behavior of solutions of mentioned system. More precisely, we formulate conditions on a function $f$, which guarantee that the graph of at least one solution of above mentioned system stays in the prescribed domain. This result generalizes some previous results concerning the asymptotic behavior of solutions of non-delay systems of dynamic equations or of delay dynamic equations. A relevant example is considered.
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41. Second derivative of high-order accuracy methods for the numerical integration of stiff initial value problems
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D. G. Yakubu and S. Markus
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Backward differentiation formula ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Numerical methods for ordinary differential equations ,Function (mathematics) ,01 natural sciences ,Numerical integration ,010101 applied mathematics ,Ordinary differential equation ,Initial value problem ,0101 mathematics ,Mathematics ,Second derivative ,Linear multistep method - Abstract
In Mitsui (Sci Eng Rev Doshisha Univ Jpn 51(3):181–190, 2010) the author introduced a new class of discrete variable methods known as look-ahead linear multistep methods (LALMMs) which consist of a pair of predictor-corrector (PC), including a function value at one more step beyond the present step for the numerical solution of ordinary differential equations. Two-step family of Look-Ahead linear multistep methods of fourth-order pair were derived and shown to be A( $$\theta $$ )-stable in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181–188, 2011). The derived integration methods are of low orders and unfortunately cannot cope with stiff systems of ordinary differential equations. In this paper we extend the concept adopted in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181–188, 2011) to construct second-derivative of high-order accuracy methods with off-step points which behave essentially like one-step methods. The resulting integration methods are A-stable, convergent, with large regions of absolute stability, suitable for stiff systems of ordinary differential equations. Numerical comparisons of the new methods have been made and enormous gains in efficiency are achieved.
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42. A Note on the Periodicity of the Lyness Max Equation
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Ramazan Karatas, Ali Gelisken, Cengiz Cinar, and Selçuk Üniversitesi
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Discrete mathematics ,Rational number ,Partial differential equation ,Algebra and Number Theory ,Differential equation ,Open problem ,lcsh:Mathematics ,Applied Mathematics ,lcsh:QA1-939 ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Ordinary differential equation ,Functional equation ,Data_FILES ,Applied mathematics ,Analysis ,Mathematics - Abstract
WOS: 000255773300001, We investigate the periodic nature of solutions of a "max-type" difference equation sometimes referred to as the "Lyness max" equation. The equation we consider is x(n+1) = max{x(n), A}/x(n-1). where A is a positive real parameter, x - 1 = A(r-1), and x(0) = A(ro) such that r(-1) and r(0) are positive rational numbers. The results in this paper answer the Open Problem of Grove and Ladas (2005). Copyright (c) 2008 Ali Gelisken et al.
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43. Asymptotic boundary value problems for evolution inclusions
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Tomáš Fürst
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Partial differential equation ,Algebra and Number Theory ,Function space ,Uniform convergence ,Mathematical analysis ,lcsh:QA299.6-433 ,lcsh:Analysis ,Continuation ,Ordinary differential equation ,Partial derivative ,Boundary value problem ,Analysis ,Mathematics ,Numerical partial differential equations - Abstract
When solving boundary value problems on infinite intervals, it is possible to use continuation principles. Some of these principles take advantage of equipping the considered function spaces with topologies of uniform convergence on compact subintervals. This makes the representing solution operators compact (or condensing), but, on the other hand, spaces equipped with such topologies become more complicated. This paper shows interesting applications that use the strength of continuation principles and also presents a possible extension of such continuation principles to partial differential inclusions.
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44. Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular m-Point Boundary Value Problems
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Zengqin Zhao and Xinsheng Du
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Nonlinear system ,Partial differential equation ,Algebra and Number Theory ,Singular solution ,Ordinary differential equation ,Mathematical analysis ,Uniqueness ,Boundary value problem ,Singular point of a curve ,Analysis ,Mathematics ,Numerical partial differential equations - Abstract
This paper investigates the existence and uniqueness of smooth positive solutions to a class of singular m-point boundary value problems of second-order ordinary differential equations. A necessary and sufficient condition for the existence and uniqueness of smooth positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity may be singular at and/or .
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45. The Existence of Positive Solutions for Third-Order p-Laplacian m-Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales
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Fuyi Xu and Zhaowei Meng
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Partial differential equation ,Algebra and Number Theory ,lcsh:Mathematics ,Applied Mathematics ,Mathematical analysis ,lcsh:QA1-939 ,Term (time) ,Nonlinear system ,Ordinary differential equation ,Boundary value problem ,Value (mathematics) ,Laplace operator ,Analysis ,Mathematics ,Sign (mathematics) - Abstract
We study the following third-order -Laplacian -point boundary value problems on time scales , , , , , where is -Laplacian operator, that is, , , , . We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.
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46. An Approximation Approach to Eigenvalue Intervals for Singular Boundary Value Problems with Sign Changing and Superlinear Nonlinearities
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Donal O'Regan, Ravi P. Agarwal, and Haishen Lü
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Singular boundary value problems ,Partial differential equation ,Algebra and Number Theory ,Mathematical analysis ,lcsh:QA299.6-433 ,Interval (mathematics) ,lcsh:Analysis ,Window function ,Ordinary differential equation ,Boundary value problem ,Eigenvalues and eigenvectors ,Analysis ,Mathematics ,Sign (mathematics) - Abstract
This paper studies the eigenvalue interval for the singular boundary value problem , where may be singular at , , and may change sign and be superlinear at . The approach is based on an approximation method together with the theory of upper and lower solutions.
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47. Existence of Three Monotone Solutions of Nonhomogeneous Multipoint BVPs for Second-Order Differential Equations
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Xingyuan Liu
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Algebra and Number Theory ,Mathematical analysis ,lcsh:QA299.6-433 ,lcsh:Analysis ,Integrating factor ,Stochastic partial differential equation ,Examples of differential equations ,Collocation method ,Ordinary differential equation ,C0-semigroup ,Differential algebraic equation ,Analysis ,Mathematics ,Numerical partial differential equations - Abstract
This paper is concerned with nonhomogeneous multipoint boundary value problems of second-order differential equations with one-dimensional -Laplacian. Sufficient conditions to guarantee the existence of at least three solutions (may be not positive) of these BVPs are established.
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48. Existence and Location Results for Fully Nonlinear Boundary Value Problem of nth-Order Nonlinear System
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Mingru Zhou, Guangwa Wang, and Li Sun
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Algebra and Number Theory ,Mathematical analysis ,lcsh:QA299.6-433 ,Delay differential equation ,lcsh:Analysis ,Stochastic partial differential equation ,Nonlinear system ,Ordinary differential equation ,Free boundary problem ,Boundary value problem ,Differential algebraic equation ,Analysis ,Numerical partial differential equations ,Mathematics - Abstract
By appropriate bounding function pair and modified functions, using the theory of differential inequalities, this paper presents the existence and location criteria of solutions for the system of general nth-order differential equations with nonlinear boundary conditions. We give an example showing that the results are sharp. Our results extend many existing results.
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49. Positive Solutions of nth-Order Nonlinear Impulsive Differential Equation with Nonlocal Boundary Conditions
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Meiqiang Feng, Xuemei Zhang, and Xiaozhong Yang
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Algebra and Number Theory ,Mathematical analysis ,lcsh:QA299.6-433 ,lcsh:Analysis ,Mixed boundary condition ,Poincaré–Steklov operator ,symbols.namesake ,Ordinary differential equation ,Dirichlet boundary condition ,symbols ,Free boundary problem ,Cauchy boundary condition ,Boundary value problem ,Hyperbolic partial differential equation ,Analysis ,Mathematics - Abstract
This paper is devoted to study the existence, nonexistence, and multiplicity of positive solutions for the th-order nonlocal boundary value problem with impulse effects. The arguments are based upon fixed point theorems in a cone. An example is worked out to demonstrate the main results.
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50. Rough functions: p-Variation, calculus, and index estimation
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Rimas Norvaiša
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Mathematics(all) ,Property (philosophy) ,Quantum stochastic calculus ,Stochastic process ,General Mathematics ,Ordinary differential equation ,Multivariable calculus ,Calculus ,Time-scale calculus ,Function (mathematics) ,Product integral ,Mathematics - Abstract
In this paper, we give an overview of several dissipated results on the p-variation property of a function presented in a suitable way. More specifically, we attempt to show: (1) usefulness of this property in a calculus of rough functions; (2) a relatively thorough knowledge of the p-variation property of a sample function of basic stochastic processes; and (3) an almost unexplored area of statistical analysis seeking to estimate the p-variation index.
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