Integral Equations on Function Spaces and Dichotomy on the Real Line

Abstract

The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy of evolution families on the real line. We consider two general classes denoted T (R) and H(R) and we prove that if V, W are Banach function spaces with V ∈ T (R) and W ∈ H(R), then the admissibility of the pair (W (R, X), V (R, X)) for an evolution family U = {U (t, s)} t≥s implies the uniform dichotomy of U. In addition, we consider a subclass W(R) ⊂ H(R) and we prove that if W ∈ W(R), then the admissibility of the pair (W (R, X), V (R, X)) implies the uniform exponential dichotomy of the family U. This condition becomes necessary if V ⊂ W . Finally, we present some applications of the main results. Mathematics Subject Classification (2000). Primary 34D09; Secondary 34D05.

Nonuniform Exponential Dichotomy for Evolution Families on the Real Line

“…The dichotomy on R + was studied by van Minh et al [19] and Huy [8][9][10][11][12]. Similar results for the dichotomy on the real line were obtained by Sasu and Sasu [26][27][28] and Sasu [29][30][31].…”

Pairs of Function Spaces and Exponential Dichotomy on the Real Line

“…The uniqueness of f is immediate see, e.g., 16, the Necessity part of Theorem 5.3 . In conclusion, the pair W R, X , V R, X is admissible for the evolution family U.…”

“…Taking into account the results obtained in 11,16 , an interesting open question is whether in the study of exponential dichotomy, the output space may belong to the general class H R . To answer this question, the first purpose of this paper is to prove the following theorem.…”

“…If one analyzes the dichotomous properties of differential equations, then it is easily seen that there are some main technical differences between the case of evolution families 2 Advances in Difference Equations on the half-line see 5, 9, 10 and the case of evolution families on the real line see [11][12][13][14][15][16] , which require a distinct analysis for each case. For instance, when one determines sufficient conditions for the existence of exponential dichotomy on the half-line, an important hypothesis is that the initial stable subspace is closed and complemented see, e.g., 5, Theorem 4.3 or 9, Theorem 3.3 .…”

Nonuniform exponential stability for evolution families on the real line

“…All the above results are obtained for t 0 ∈ R + . In [15] and [16] are considered systems described by evolution families with exponential growth on the real line. It can also be mentioned the results obtained by M. Marin and O. Florea in [10] as well as K. Sharma and M. Marin in [17].…”