A new analytical method for the linearization of dynamic equation on measure chains

In this paper, by introducing the concept of topological equivalence on measure chain, we investigate the relationship between the linear system xΔ=A(t)x and the nonlinear system xΔ=A(t)x+f(t,x). Some sufficient conditions are obtained to guarantee the existence of a equivalent function H(t,x) sending the (c,d)-quasibounded solutions of nonlinear system xΔ=A(t)x+f(t,x) onto those of linear system xΔ=A(t)x. Our results generalize the Palmerʹs linearization theorem in [K.J. Palmer, A generalization of Hartmanʹs linearization theorem, J. Math. Anal. Appl. 41 (1973) 753–758] to dynamic equation measure chains. In the present paper, we give a new analytical method to study the topological equivalence problem on measure chains. As we will see, due to the completely different method to investigate the topological equivalence problem, we have a considerably different result from that in the pioneering work of Hilger [S. Hilger, Generalized theorem of Hartman–Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157–191]. Moreover, we prove that equivalent function H(t,x) is also ω-periodic when the systems are ω-periodic. Hilger [S. Hilger, Generalized theorem of Hartman–Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157–191] never considered this important property of the equivalent function H(t,x).