Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side
The paper brings multiplicity results for a Dirichlet problem in one-dimensional billiard space with right-hand side depending on the velocity of the ball, i.e. a problem in the form x'' = f(t, x, x') if x(t) ∈ int K, x'(t+) = -x'(t-) if x(t) ∈ ∂K, ...
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| Language: | English |
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Vilnius University Press
2024-12-01
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| Series: | Lietuvos Matematikos Rinkinys |
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| Online Access: | https://ojs.test/index.php/LMR/article/view/37775 |
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| author | Vĕra Krajščáková Jan Tomeček |
| author_facet | Vĕra Krajščáková Jan Tomeček |
| author_sort | Vĕra Krajščáková |
| collection | DOAJ |
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The paper brings multiplicity results for a Dirichlet problem in one-dimensional billiard space with right-hand side depending on the velocity of the ball, i.e. a problem in the form
x'' = f(t, x, x') if x(t) ∈ int K, x'(t+) = -x'(t-) if x(t) ∈ ∂K,
x(0) = A, x(T) = B,
where T > 0, K = [0, R], R > 0, f is a Carathéodory function on [0, T] × K × ℝ, A, B ∈ int K. Sufficient conditions ensuring the existence of at least two solutions having prescribed number of impacts with the boundary of the billiard table K are obtained. In particular, if the right-hand side has at most sublinear growth in the last variable, there exist infinitely many solutions of the problem.
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| format | Article |
| id | doaj-art-82e725de7cf84f42aed953d939b16238 |
| institution | DOAJ |
| issn | 0132-2818 2335-898X |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Lietuvos Matematikos Rinkinys |
| spelling | doaj-art-82e725de7cf84f42aed953d939b162382025-08-20T02:58:54ZengVilnius University PressLietuvos Matematikos Rinkinys0132-28182335-898X2024-12-0165A10.15388/LMD.2024.37775Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand sideVĕra Krajščáková0Jan Tomeček1Palacký UniversityPalacký University The paper brings multiplicity results for a Dirichlet problem in one-dimensional billiard space with right-hand side depending on the velocity of the ball, i.e. a problem in the form x'' = f(t, x, x') if x(t) ∈ int K, x'(t+) = -x'(t-) if x(t) ∈ ∂K, x(0) = A, x(T) = B, where T > 0, K = [0, R], R > 0, f is a Carathéodory function on [0, T] × K × ℝ, A, B ∈ int K. Sufficient conditions ensuring the existence of at least two solutions having prescribed number of impacts with the boundary of the billiard table K are obtained. In particular, if the right-hand side has at most sublinear growth in the last variable, there exist infinitely many solutions of the problem. https://ojs.test/index.php/LMR/article/view/37775billiard problemDirichlet problemmultiplicity resultsublinear growthlinear growth |
| spellingShingle | Vĕra Krajščáková Jan Tomeček Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side Lietuvos Matematikos Rinkinys billiard problem Dirichlet problem multiplicity result sublinear growth linear growth |
| title | Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side |
| title_full | Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side |
| title_fullStr | Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side |
| title_full_unstemmed | Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side |
| title_short | Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side |
| title_sort | dirichlet problem in one dimensional billiard space with velocity dependent right hand side |
| topic | billiard problem Dirichlet problem multiplicity result sublinear growth linear growth |
| url | https://ojs.test/index.php/LMR/article/view/37775 |
| work_keys_str_mv | AT verakrajscakova dirichletprobleminonedimensionalbilliardspacewithvelocitydependentrighthandside AT jantomecek dirichletprobleminonedimensionalbilliardspacewithvelocitydependentrighthandside |