Some characterizations for the uniform exponential stability of linear skew-product semiflows
YUE Tian1, SONG Xiaoqiu2
1. School of Science, Hubei University of Automotive Technology, Shiyan 442002, Hubei Province, China;
2. School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu Province, China
Abstract：Based on the definition of linear skew-product semiflows, a uniform exponential stability concept of dynamical systems is presented in Banach spaces. The main purpose of this paper is to give several characterizations for the uniform exponential stability of linear skew-product semiflows by means of Datko's approach in stability theory. Some necessary and sufficient conditions concerning the uniform exponential stability of linear skew-product semiflows are given. The obtained conclusions are generalizations of the well-known results about the exponential stability and exponential instability.
岳田, 宋晓秋. 线性斜积半流的一致指数稳定性的若干刻画[J]. 浙江大学学报（理学版）, 2018, 45(5): 545-548.
YUE Tian, SONG Xiaoqiu. Some characterizations for the uniform exponential stability of linear skew-product semiflows. Journal of ZheJIang University(Science Edition), 2018, 45(5): 545-548.
[1] DATKO R. Uniform asymptotic stability of evolutionary processes in a Banach space[J]. SIAM J Math Anal, 1973, 3(3):428-445.
[2] ROLEWICZ S. On uniform N-equistability[J]. J Math Anal Appl, 1986, 115(2):434-441.
[3] MEGAN M,SASU A L, SASU B. Perron conditions for uniform exponential expansiveness of linear skew-product flows[J]. Monatsh Math, 2003, 138(2):145-157.
[4] MEGAN M,SASU A L, SASU B. Exponential stability and exponential instability for linear skew-product flows[J]. Math Bohem, 2004, 129(3):225-243.
[5] MEGAN M,SASU A L, SASU B. Exponential instability of linear skew-product semiflows in terms of Banach function spaces[J]. Results Math, 2004, 45(3):309-318.
[6] MEGAN M, BULIGA L. Functionals on normed function spaces and exponential instability of linear skew-product semiflows[J]. Bull Belg Math Soc Simon Stevin, 2007, 14(2):355-362.
[7] PREDA C,PREDA P, BǎTǎRAN F. An extension of a theorem of R. Datko to the case of (non)uniform exponential stability of linear skew-product semiflows[J]. J Math Anal Appl, 2015, 425(2):1148-1154.
[8] BUSE C, NICULESCU C P. An ergodic characterization of uniformly exponentially stable evolution families[J]. Bull Math Soc Sci Math Moumanie, 2009, 52(1):33-40.
[9] LUPA N, MEGAN M, POPA I L. On weak exponential stability of evolution operators in Banach spaces[J]. Nolinear Anal, 2010, 73(8):2445-2450.
[10] STOICA C, MEGAN M. On uniform exponential stability for skew-evolution semiflows on Banach spaces[J]. Nolinear Anal, 2010, 72(3/4):1305-1313.
[11] HAI P V. Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows[J]. Nolinear Anal, 2010, 72(12):4390-4396.
[12] YUE T, SONG X Q, LI D Q. On weak exponential expansiveness of skew-evolution semiflows in Banach spaces[J]. J Inequal Appl, 2014, 2014:165.
[13] 岳田, 雷国梁, 宋晓秋. 线性斜演化半流一致指数膨胀性的若干刻画[J]. 数学进展, 2016, 45(3):433-442. YUE T, LEI G L, SONG X Q. Some characterizations for the uniform exponential expansiveness of linear skew-evolution semiflows[J]. Advances in Mathematics, 2016, 45(3):433-442.
[14] 岳田, 宋晓秋. 巴拿赫空间上斜演化半流的非一致指数不稳定性的存在条件[J]. 浙江大学学报(理学版), 2016, 43(2):181-183. YUE T, SONG X Q. Criteria for the existence of nonuniform exponential instability of skew-evolution semiflows in Banach spaces[J]. Journal of Zhejiang University (Science Edition), 2016, 43(2):181-183.
2018, Vol. 45 
