Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator
Abstract The authors consider the impulsive differential equation with Monge-Ampère operator in the form of { ( ( u ′ ( t ) ) n ) ′ = λ n t n − 1 f ( − u ( t ) ) , t ∈ ( 0 , 1 ) , t ≠ t k , k = 1 , 2 , … , m , Δ ( u ′ ) n | t = t k = λ I k ( − u ( t k ) ) , k = 1 , 2 , … , m , u ′ ( 0 ) = 0 , u ( 1...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-11-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13661-017-0904-8 |
| _version_ | 1811211175863517184 |
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| author | Xuemei Zhang Meiqiang Feng |
| author_facet | Xuemei Zhang Meiqiang Feng |
| author_sort | Xuemei Zhang |
| collection | DOAJ |
| description | Abstract The authors consider the impulsive differential equation with Monge-Ampère operator in the form of { ( ( u ′ ( t ) ) n ) ′ = λ n t n − 1 f ( − u ( t ) ) , t ∈ ( 0 , 1 ) , t ≠ t k , k = 1 , 2 , … , m , Δ ( u ′ ) n | t = t k = λ I k ( − u ( t k ) ) , k = 1 , 2 , … , m , u ′ ( 0 ) = 0 , u ( 1 ) = 0 , $$\textstyle\begin{cases} ( (u'(t) )^{n} )'=\lambda nt^{n-1}f (-u(t) ), \quad t\in(0,1), t\neq t_{k}, k=1, 2, \ldots, m, \\ \Delta (u' )^{n}|_{t=t_{k}}=\lambda I_{k} (-u(t_{k}) ), \quad k=1, 2, \ldots , m, \\ u'(0)=0, \quad\quad u(1)=0, \end{cases} $$ where λ is a nonnegative parameter and n ≥ 1 $n\geq1$ . We show the existence, uniqueness, and continuity results. Our approach is largely based on the eigenvalue theory and the theory of α-concave operators. The nonexistence result of a nontrivial convex solution is also studied by taking advantage of the internal geometric properties related to the problem. |
| first_indexed | 2024-04-12T05:08:21Z |
| format | Article |
| id | doaj.art-64db8af0fda8414aa75ce62de9ee256a |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-04-12T05:08:21Z |
| publishDate | 2017-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-64db8af0fda8414aa75ce62de9ee256a2022-12-22T03:46:50ZengSpringerOpenBoundary Value Problems1687-27702017-11-012017111710.1186/s13661-017-0904-8Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operatorXuemei Zhang0Meiqiang Feng1Department of Mathematics and Physics, North China Electric Power UniversitySchool of Applied Science, Beijing Information Science & Technology UniversityAbstract The authors consider the impulsive differential equation with Monge-Ampère operator in the form of { ( ( u ′ ( t ) ) n ) ′ = λ n t n − 1 f ( − u ( t ) ) , t ∈ ( 0 , 1 ) , t ≠ t k , k = 1 , 2 , … , m , Δ ( u ′ ) n | t = t k = λ I k ( − u ( t k ) ) , k = 1 , 2 , … , m , u ′ ( 0 ) = 0 , u ( 1 ) = 0 , $$\textstyle\begin{cases} ( (u'(t) )^{n} )'=\lambda nt^{n-1}f (-u(t) ), \quad t\in(0,1), t\neq t_{k}, k=1, 2, \ldots, m, \\ \Delta (u' )^{n}|_{t=t_{k}}=\lambda I_{k} (-u(t_{k}) ), \quad k=1, 2, \ldots , m, \\ u'(0)=0, \quad\quad u(1)=0, \end{cases} $$ where λ is a nonnegative parameter and n ≥ 1 $n\geq1$ . We show the existence, uniqueness, and continuity results. Our approach is largely based on the eigenvalue theory and the theory of α-concave operators. The nonexistence result of a nontrivial convex solution is also studied by taking advantage of the internal geometric properties related to the problem.http://link.springer.com/article/10.1186/s13661-017-0904-8continuity on a parameterexistence of nontrivial convex solutionsMonge-Ampère operatorimpulsive differential equationgeometric properties |
| spellingShingle | Xuemei Zhang Meiqiang Feng Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator Boundary Value Problems continuity on a parameter existence of nontrivial convex solutions Monge-Ampère operator impulsive differential equation geometric properties |
| title | Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator |
| title_full | Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator |
| title_fullStr | Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator |
| title_full_unstemmed | Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator |
| title_short | Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Ampère operator |
| title_sort | nontrivial convex solutions on a parameter of impulsive differential equation with monge ampere operator |
| topic | continuity on a parameter existence of nontrivial convex solutions Monge-Ampère operator impulsive differential equation geometric properties |
| url | http://link.springer.com/article/10.1186/s13661-017-0904-8 |
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