On the structure of Levinson center of monotone almost periodic systems
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2023-03-06 11:06
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CHEBAN, David. On the structure of Levinson center of monotone almost periodic systems. In: Mathematics and IT: Research and Education, 1-3 iulie 2021, Chişinău. Chișinău, Republica Moldova: 2021, p. 22.
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Mathematics and IT: Research and Education 2021
Conferința "Mathematics and IT: Research and Education "
Chişinău, Moldova, 1-3 iulie 2021

On the structure of Levinson center of monotone almost periodic systems


Pag. 22-22

Cheban David
 
Moldova State University
 
Disponibil în IBN: 30 iunie 2021


Rezumat

The aim of this talk is studying the problem of existence of Levitan/Bohr almost periodic solutions for dissipative differential equationformulawhen the second right hand side is monotone with respect to spacial variable. The existence of at least two quasi periodic (respectively, Bohr/Levitan almost periodic) solutions of (1) is proved under the condition that every solution of equation (1) is positively uniformly Lyapunov stable. These results we establish in the framework of general non-autonomous (cocycle) dynamical systems. Along with the equation (1) we consider its H-class, i.e., the family of the equationsformulawhere g 2 H(f) = ff¿ : ¿ 2 Rg and f¿ (t; u) = f(t+¿; u), where the bar indicates the closure in the compact-open topology. This study is a continuation of the author’s work, which gives a positive answer to the I. U. Bronshtein’s conjecture for monotone systems. I. U. Bronshtein’s conjecture [1]. If an equation (1) with right hand side (Bohr) almost periodic in t satisfies the conditions of uniform positive stability and positive dissipativity, then it has at least one (Bohr) almost periodic solution. Below we will use the following conditions. Condition (A1). The function f 2 C(R £W;Rd) is Bohr/Levitan almost periodic [2,3] in t 2 R uniformly in u on every compact subset K ½ W and the equation (1) is monotone. Condition (A2). Equation (1) with regular right hand side f admits a compact global attractor (Levinson center) and every solution of equation (2) is positively uniformly stable. Theorem. Under conditions (A1)¡(A2) if the function f 2 C(R£Rn;Rn) is quasi-periodic (respectively, Bohr/Levitan almost periodic) in t 2 R uniformly with respect to u on every compact subset from Rn, then equation (1) has at least two (lower and upper) quasi-periodic (respectively, Bohr/Levitan almost periodic) solutions.