In this paper, the differential equation involving iterates of the unknown function,x＇（z）=[a^2-x^2（z）]x^[m]（z）with a complex parameter a, is investigated in the complex field C for the existence of analytic sol...In this paper, the differential equation involving iterates of the unknown function,x＇（z）=[a^2-x^2（z）]x^[m]（z）with a complex parameter a, is investigated in the complex field C for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the SchrSder transformation x（z） = y（αy^-1（z）） to the following another functional differential equation without iteration of the unknown functionαy＇（αz）=[a^2-y^2（αz）]y＇（z）y（α^mz）,which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y（αy^-1 （z）） for the original iterative differential equation are obtained. We discuss not only these α given in SchrSder transformation in the hyperbolic case 0 〈 ｜α｜ 〈 1 and resonance, i.e., at a root of the unity, but also those α near resonance （i.e., near a root of the unity） under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.展开更多
This paper is concerned with solutions of a functional differential equation. Using Krasnoselskii’s fixed point theorem, the solutions can be obtained from periodic solutions of a companion equation.
This paper is concerned with a nonlinear iterative equation with first order derivative. By construction a convergent power series solution, analytic solutions for the original equation are obtained.
基金supported by Natural Science Foundation of University of Ji'nan (Grant No. XKY0704)the second author is partially supported by National Natural Science Foundation of China (Grant No. 10871117)NSFSP (Grant No. Y2006A07)
文摘In this paper, the differential equation involving iterates of the unknown function,x＇（z）=[a^2-x^2（z）]x^[m]（z）with a complex parameter a, is investigated in the complex field C for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the SchrSder transformation x（z） = y（αy^-1（z）） to the following another functional differential equation without iteration of the unknown functionαy＇（αz）=[a^2-y^2（αz）]y＇（z）y（α^mz）,which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y（αy^-1 （z）） for the original iterative differential equation are obtained. We discuss not only these α given in SchrSder transformation in the hyperbolic case 0 〈 ｜α｜ 〈 1 and resonance, i.e., at a root of the unity, but also those α near resonance （i.e., near a root of the unity） under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.
基金The NSF(11326120 and 11501069)of Chinathe Foundation(KJ1400528 and KJ1600320)of Chongqing Municipal Education Commissionthe Foundation(02030307-00039)of Youth Talent of Chongqing Normal University
文摘This paper is concerned with solutions of a functional differential equation. Using Krasnoselskii’s fixed point theorem, the solutions can be obtained from periodic solutions of a companion equation.
文摘This paper is concerned with a nonlinear iterative equation with first order derivative. By construction a convergent power series solution, analytic solutions for the original equation are obtained.
