scholarly journals Reducibility for a Class of Almost-Periodic Differential Equations with Degenerate Equilibrium Point under Small Almost-Periodic Perturbations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Wenhua Qiu ◽  
Jianguo Si
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Equilibrium Point ◽  
Almost Periodic Solution ◽  
Time Dependent ◽  
Almost Periodic ◽  
Degenerate Equilibrium ◽  
Periodic Transformation ◽  
Perturbed Equation ◽  
Kam Method

This paper focuses on almost-periodic time-dependent perturbations of an almost-periodic differential equation near the degenerate equilibrium point. Using the KAM method, the perturbed equation can be reduced to a suitable normal form with zero as equilibrium point by an affine almost-periodic transformation. Hence, for the equation we can obtain a small almost-periodic solution.

scholarly journals On annth-order infinitesimal generator and time-dependent operator differential equation with a strongly almost periodic solution

2002 ◽  
Vol 32 (9) ◽  
pp. 573-578 ◽  
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Infinitesimal Generator ◽  
Almost Periodic Solution ◽  
Time Dependent ◽  
Operator Differential Equation ◽  
Almost Periodic ◽  
Periodic Forcing ◽  
Forcing Function ◽  
Weakly Almost Periodic

In a Banach space, ifuis a Stepanov almost periodic solution of a certainnth-order infinitesimal generator and time-dependent operator differential equation with a Stepanov almost periodic forcing function, thenu,u′,…,u (n−2)are all strongly almost periodic andu (n−1)is weakly almost periodic.

scholarly journals On the Stepanov-almost periodic solution of a second-order operator differential equation

1975 ◽  
Vol 19 (3) ◽  
pp. 261-263 ◽  
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Periodic Solution ◽  
Almost Periodic Solution ◽  
Second Order ◽  
Almost Periodicity ◽  
Operator Differential Equation ◽  
Almost Periodic ◽  
Order Operator ◽  
Image Position

Suppose X is a Banach space and J is the interval −∞<t<∞. For 1 ≦ p<∞, a function is said to be Stepanov-bounded or Sp-bounded on J if(for the definitions of almost periodicity and Sp-almost periodicity, see Amerio-Prouse (1, pp. 3 and 77).

scholarly journals On the Spectrum of almost periodic solutions of an abstract differential equation

1974 ◽  
Vol 18 (4) ◽  
pp. 385-387
Author(s):  
Aribindi Satyanarayan Rao ◽  
Walter Hengartner
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Periodic Solution ◽  
Linear Operator ◽  
Periodic Function ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Abstract Differential Equation

AbstractIf a linear operator A in a Banach space satisfies certain conditions, then the spectrum of any almost periodic solution of the differential equation u′ = Au + f is shown to be identical with the spectrum of f, where f is a Stepanov almost periodic function.

scholarly journals On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Necessary And Sufficient

For the differential equation , on a Hilbert space , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

On the almost periodic solution of an abstract diffferential equation

1974 ◽  
Vol 18 (2) ◽  
pp. 252-256
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Function ◽  
Bounded Solution ◽  
Almost Periodic Solution ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Abstract Differential Equation ◽  
Weakly Almost Periodic

Abstract: Under certain suitable conditions, the Stepanov-bounded solution of an abstract differential equation corresponding to a Stepanov almost periodic function is strongly (weakly) almost periodic.

scholarly journals On the spectrum of weakly almost periodic solutions of certain abstract differential equations

1985 ◽  
Vol 8 (1) ◽  
pp. 109-112 ◽  
Author(s):  
Aribindi Satyanarayan Rao ◽  
L. S. Dube
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Periodic Solution ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic Function ◽  
Dual Operator ◽  
Almost Periodic ◽  
Abstract Differential Equations ◽  
Weakly Almost Periodic

In a sequentially weakly complete Banach space, if the dual operator of a linear operatorAsatisfies certain conditions, then the spectrum of any weakly almost periodic solution of the differential equationu′=Au+fis identical with the spectrum offexcept at the origin, wherefis a weakly almost periodic function.

scholarly journals Positive Almost Periodic Solution on a Nonlinear Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Ni Hua ◽  
Tian Li-xin ◽  
Liu Xun
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Nonlinear Equation ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic ◽  
Positive Almost Periodic Solution ◽  
The Stability

We study the following nonlinear equationdx(t)/dt=x(t)[a(t)-b(t)xα(t)-f(t,x(t))]+g(t), by using fixed point theorem, the sufficient conditions of the existence of a unique positive almost periodic solution for above system are obtained, by using the theories of stability, the sufficient conditions which guarantee the stability of the unique positive almost periodic solution are derived.

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