Exact Derivation of All Periodic Solutions for Impact Oscillator (Proposition of Initial Value Correction Periodize Method)

It is shown that all solutions of ẍ + 2x3 = p(t) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p.The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almost-periodic solutions and, for each positive integer m, an infinity of periodic solutions of least period 2mπ (2π being taken as the least period of p ).It is suggested that for a large class of equations the same attack would show all solutions of ẍ + g(x) = p(t) bounded. However, in order to show the method clearly, no generalisation is attempted here.

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On periodic solutions to a class of delay differential variational inequalities

Evolution Equations and Control Theory ◽

10.3934/eect.2021045 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Nguyen Thi Van Anh

Keyword(s):

Variational Inequalities ◽

Periodic Solutions ◽

Delay Differential Equations ◽

Conditional Stability ◽

Sufficient Condition ◽

Initial Value ◽

Periodic Problems ◽

Existence Of Periodic Solutions ◽

Delay Differential ◽

Differential Variational Inequalities

<p style='text-indent:20px;'>In this paper, we introduce and study a class of delay differential variational inequalities comprising delay differential equations and variational inequalities. We establish a sufficient condition for the existence of periodic solutions to delay differential variational inequalities. Based on some fixed point arguments, in both single-valued and multivalued cases, the solvability of initial value and periodic problems are proved. Furthermore, we study the conditional stability of periodic solutions to this systems.</p>

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A MONOTONE-ITERATIVE METHOD FOR FINDING PERIODIC SOLUTION OF AN IMPULSIVE DIFFERENTIAL EQUATIONS AND APPLICATION

International Journal of Biomathematics ◽

10.1142/s1793524508000205 ◽

2008 ◽

Vol 01 (02) ◽

pp. 247-256

Author(s):

JIAWEI DOU

Keyword(s):

Differential Equations ◽

Periodic Solutions ◽

Upper And Lower Solutions ◽

Impulsive Differential Equations ◽

Iteration Process ◽

Impulsive System ◽

Monotone Iterative Method ◽

Initial Value ◽

Monotone Iterative ◽

Monotone Iterations

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

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