Integral conditions for nonuniform $μ$-dichotomy on the half-line

António J.G. Bento; Nicolae Lupa; Mihail Megan; César M. Silva

doi:10.3934/dcdsb.2017163

Article Contents

2017, Volume 22, Issue 8: 3063-3077. Doi: 10.3934/dcdsb.2017163

This issue Previous Article On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits Next Article An instability theorem for nonlinear fractional differential systems

Integral conditions for nonuniform $μ$-dichotomy on the half-line

1.
Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilh˜ã, Portugal
2.
Department of Mathematics, "Politehnica" University of Timişoara, Victoriei Square 2,300006 Timişoara, Romania
3.
Academy of Romanian Scientists, Independenţei 54,050094 Bucharest, Romania

Received: June 2016

Revised: January 2017

Published: October 2017

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Abstract

We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.

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Mathematics Subject Classification: 47D06, 34D09.

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