Criteria for the existence of nonuniform exponential instability of skew-evolution semiflows in Banach spaces
YUE Tian1, SONG Xiaoqiu2
1. School of Science, Hubei University of Automotive Technology, Shiyan 442002, Hubei Province, China;
2. College of Science, China University of Mining and Technology, Xuzhou 221116, Jiangsu Province, China
Abstract：Based on the definition of uniform exponential instability, a skew-evolution semiflow with nonuniform exponential instability is presented in Banach spaces. An illustrating example is used to clarify the relationship between the two concepts. Exponential stability technique is applied to study the features of nonuniform exponential instability of skew-evolution semilflows. Two necessary and sufficient conditions concerning the nonuniform exponential instability of skew-evolution semiflows are given. The obtained conclusions are generalizations of the well-known results about the exponential stability and uniform exponential instability.
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 Tian, SONG Xiaoqiu. Criteria for the existence of nonuniform exponential instability of skew-evolution semiflows in Banach spaces. Journal of ZheJIang University(Science Edition), 2016, 43(2): 181-183.
MEGAN M, STOICA C. Exponential instability of skew-evolution semiflows in Banach spaces[J]. Stud Univ Babes-Bolyai Math,2008,53(1):17-24.
[3]
HAI P H. Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows[J]. Nolinear Anal,2010,72(12):4390-4396.
[4]
HAI P H. Discrete and continuous versions of Barbashin-type theorems of linear skew-evolution semiflows[J]. Appl Anal,2011,90(12):1897-1907.
[5]
STOICA C, MEGAN M. On nonuniform exponential stability for skew-evolution semiflows in Banach spaces[J]. Carpathian J Math,2013,29(2):259-266.
[8]
YUE T, SONG X Q, LI D Q. On weak exponential expansiveness of skew-evolution semiflows in Banach spaces[J]. J Inequal Appl,2014(1):1-11.
[2]
STOICA C, MEGAN M. On uniform exponential stability for skew-evolution semiflows on Banach spaces[J]. Nolinear Analysis,2010,72(3/4):1305-1313.
[6]
STOICA C, BORLEA D. Exponential stability versus polynomial stability for skew-evolution semiflows in infinite dimensional spaces[J]. Theory Appl Math Comput Sci,2014,4(2):221-229.
[7]
DATKO R. Uniform asymptotic stability of evolutionary processes in Banach spaces[J]. SIAM J Math Anal,1972,3(3):428-445.
[9]
YUE T, LEI G L, SONG X Q. Some characterizations for the uniform exponential expansiveness of linear skew-evolution semiflows[J]. Advances in Mathematics (China),2015,45,doi:10.11845/sxjz.2014173b.
2016, Vol. 43 
