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Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

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Abstract

The following system considered in this paper:

$$x' = -\,e(t)x + f(t)\phi_{p^*}(y), \qquad y'= -\,(p-1)g(t)\phi_p(x) - (p-1)h(t)y,$$

where \({p > 1, p^* > 1 (1/p + 1/p^* = 1)}\) and \({\phi_q(z) = |z|^{q-2}z}\) for q = p or q = p *. This system is referred to as a half-linear system. The coefficient f(t) is assumed to be bounded, but the coefficients e(t), g(t) and h(t) are not necessarily bounded. Sufficient conditions are obtained for global asymptotic stability of the zero solution. Our results can be applied to not only the case that the signs of f(t) and g(t) change like the periodic function but also the case that f(t) and g(t) irregularly have zeros. Some suitable examples are included to illustrate our results.

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  1. Department of Mathematics and Computer Science, Shimane University, Matsue, 690-8504, Japan

    Jitsuro Sugie & Saori Hata

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  1. Jitsuro Sugie
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  2. Saori Hata
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Correspondence to Jitsuro Sugie.

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Communicated by Ansgar Jüngel.

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Sugie, J., Hata, S. Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients. Monatsh Math 166, 255–280 (2012). https://doi.org/10.1007/s00605-011-0297-1

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