Existence and global attractivity of solutions of a nonlinear functional integral equation

In this paper we prove a result on the existence and global attractivity of solutions of a nonlinear functional integral equations with deviating arguments. The investigations are placed in the Banach space of real functions defined, continuous and bounded on the real half-axis. The main tool used in consideration is a fixed point theorem of Krasnosel’skii type. A few examples illustrating the obtained results are also included.     

The global attractivity and asymptotic stability of solution of a nonlinear integral equation

In this paper, we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation. Some new results are given.     

On global existence and attractivity results for nonlinear functional integral equations

In this paper two existence results concerning the global attractivity and global asymptotic attractivity for a certain functional nonlinear integral equation are proved. Our existence results include several existence as well as attractivity results obtained earlier by Banas and Dhage (2008) [1], Hu and Yan (2006) [3], Dhage (2009) [15] and Banas and Rzepka (2003) [7] as special cases under some weaker Lipschitz conditions. A measure theoretic fixed point theorem of Dhage (2008) [6] is used in formulating our main results and the characterizations of solutions are obtained in the space of functions defined, continuous and bounded on unbounded intervals.     

Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation in N variables

Motivated by recent known results about the solvability of nonlinear functional integral equations in one, two or N variables, this paper proves the existence of asymptotically stable solutions for a mixed functional integral equation in N variables with values in a general Banach space via a fixed point theorem of Krasnosels'ki?  type. In order to illustrate the results obtained here, an example is given.     

一类非线性差分方程的持续生存与全局吸引

研究了非线性差分方程xn 1=x^snf(xn,xn-k,…,xn-kr)s∈{1,2,…}得到该系统永久持续生存和全局吸引的充分条件。     

On local attractivity of solutions of a functional integral equation of fractional order with deviating arguments

In this paper, we study the existence of solutions of a nonlinear functional integral equation of fractional order with deviating arguments. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive. An example is provided to illustrate the theory.     

Existence and global attractivity of positive periodic solutions of functional differential equations with feedback control

Sufficient conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with feedback control

The method involves the application of Krasnoselskii's fixed point theorem and estimates of uniform upper and lower bounds of solutions. When these results are applied to some special delay population models with multiple delays, some new results are obtained and some known results are generalized.     

Integrable solutions of a nonlinear functional integral equation on an unbounded interval

In this paper we prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020]. An example which shows the importance and the applicability of our result is also included.     

EXISTENCE OF POSITIVE SOLUTIONS AND 1／c-GLOBAL ATTRACTIVITY OF A NONLINEAR DELAY DIFFERENCE EQUATION WITH VARIABLE COEFFICIENTS

Abstract. The existence of positive solutions and the global attractivity of the difference equa-tion     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

Hopfield神经网络的周期解存在性及其全局吸引性

利用重合度理论和微分不等式分析等技巧，给出了Hopfield神经网络周期解存在性及其全局吸引性的判别准则。     

Existence and global attractivity of positive periodic solutions of competition systems

In this paper, we study the existence and global attractivity of periodic solutions of a competition system. We obtain sufficient conditions for the existence and global attractivity of positive periodic solutions by Krasnoselskii’s fixed point theorem and the construction of Lyapunov functions.     

Existence of solutions to a global eikonal equation

We derive a geometric necessary and sufficient condition for the existence of solutions to a global eikonal equation. We also study the existence of a minimal solution to this equation, and its relation with the well-known minimal time function.     

Existence and nonexistence of global solutions for nonlinear evolution equation of fourth order

Abstract. The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation ua-a1uxx-a2uxxl-a3uxxu=ψ(ux)x are proved     

On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation

The existence of solutions of a nonlinear quadratic Volterra integral equation is studied. In our considerations we apply the technique of measures of noncompactness in conjunction with the classical Schauder fixed point principle. Such an approach allows us to obtain a result on the existence of solutions of an equation in question which are uniformly locally attractive or asymptotically stable.     

Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation

The paper studies the existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation arising in the model in variational form for the neo–Hookean elastomer rod where k₁, k₂>0 are real numbers, g(s) is a given nonlinear function. When g(s)=sⁿ (where n?2 is an integer), by using the Fourier transform method we prove that for any T>0, the Cauchy problem admits a unique global smooth solution u∈C^∞((0, T]; H^∞( R ))∩C([0, T]; H³( R ))∩C¹([0, T]; H^?1( R )) as long as initial data u₀∈W^{4, 1}( R )∩H³( R ), u₁∈L¹( R )∩H^?1( R ). Moreover, when (u₀, u₁)∈H²( R ) × L²( R ), g∈C²( R ) satisfy certain conditions, the Cauchy problem has no global solution in space C([0, T]; H²( R ))∩C¹([0, T]; L²( R ))∩H¹(0, T; H²( R )). Copyright © 2009 John Wiley & Sons, Ltd.     

Existence of global attractors for a nonlinear wave equation

Using a new method developed in [Q. Ma, S. Wang, C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Math. J. 5 (6) (2002) 1542–1558], we prove the existence of the global attractor for a nonlinear wave equation.     

Existence and global attractivity of a positive periodic solution of a class of delay differential equation

The existence and the global attractivity of a positive periodic solution of the delay differential equationy(t)=y(t) F[t, y](t-τ ₁ (t)),⋯,y(t−τ _n (t))] are studied by using some techniques of the Mawhin coincidence degree theory and the constructing suitable Liapunov functionals. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved. Project partially supported by the National Natural Science Foundation of China (Grant No. 10572057) and the Applied Basic Research Foundation of Yunnan Province.