Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed‐point principles |
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| Authors: | Josef Diblík Mária Kúdelčíková |
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| Affiliation: | 1. Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno, Czech Republic;2. Department of Structural Mechanics and Applied Mathematics, Faculty of Civil Engineering, University of ?ilina, ?ilina, Slovak Republic |
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| Abstract: | The paper considers a system of advanced‐type functional differential equations ![]() where F is a given functional,  , r > 0 and xt(θ) = x(t + θ), θ∈[0,r]. Two different results on the existence of solutions, with coordinates bounded above and below by the coordinates of the given vector functions if t→∞, are proved using two different fixed‐point principles. It is illustrated by examples that, applying both results simultaneously to the same equation yields two positive solutions asymptotically different for t→∞. The equation ![]() where a,τ∈(0,∞), a < 1/(τe),  are constants can serve as a linear example. The existence of a pair of positive solutions asymptotically different for t→∞ is proved and their asymptotic behavior is investigated. The results are also illustrated by a nonlinear equation. Copyright © 2016 John Wiley & Sons, Ltd. |
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| Keywords: | advanced differential equation monotone iterative method Schauder– Tychonoff theorem positive solution asymptotic behavior of solutions nonlinear system |
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, r > 0 and xt(θ) = x(t + θ), θ∈[0,r]. Two different results on the existence of solutions, with coordinates bounded above and below by the coordinates of the given vector functions if t→∞, are proved using two different fixed‐point principles. It is illustrated by examples that, applying both results simultaneously to the same equation yields two positive solutions asymptotically different for t→∞. The equation 
are constants can serve as a linear example. The existence of a pair of positive solutions asymptotically different for t→∞ is proved and their asymptotic behavior is investigated. The results are also illustrated by a nonlinear equation. Copyright © 2016 John Wiley & Sons, Ltd.