Semilinear systems with a multi-valued nonlinear term
Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of mono...
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| Format: | Article |
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De Gruyter
2017-05-01
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| Series: | Open Mathematics |
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| Online Access: | http://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0056/math-2017-0056.xml?format=INT |
| _version_ | 1818515699507658752 |
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| author | Kim In-Sook Hong Suk-Joon |
| author_facet | Kim In-Sook Hong Suk-Joon |
| author_sort | Kim In-Sook |
| collection | DOAJ |
| description | Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = (N1, N2), provided that L = (L1, L2) satisfies dim Ker L1 = ∞ and dim Ker L2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed. |
| first_indexed | 2024-12-11T00:32:11Z |
| format | Article |
| id | doaj.art-2f059e78ad5240d59437e57082879b7e |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-12-11T00:32:11Z |
| publishDate | 2017-05-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-2f059e78ad5240d59437e57082879b7e2022-12-22T01:27:17ZengDe GruyterOpen Mathematics2391-54552017-05-0115162864410.1515/math-2017-0056math-2017-0056Semilinear systems with a multi-valued nonlinear termKim In-Sook0Hong Suk-Joon1Department of Mathematics, Sungkyunkwan University, Natural Science Campus, Seobu-ro 2066, Suwon 16419, Republic of KoreaDepartment of Mathematics, Sungkyunkwan University, Natural Science Campus, Seobu-ro 2066, Suwon 16419, Republic of KoreaIntroducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = (N1, N2), provided that L = (L1, L2) satisfies dim Ker L1 = ∞ and dim Ker L2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed.http://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0056/math-2017-0056.xml?format=INTsemilinear systemmulti-valued operatoroperators of monotone typedegree theory47h0447h0547h1135a1635b1035l71 |
| spellingShingle | Kim In-Sook Hong Suk-Joon Semilinear systems with a multi-valued nonlinear term Open Mathematics semilinear system multi-valued operator operators of monotone type degree theory 47h04 47h05 47h11 35a16 35b10 35l71 |
| title | Semilinear systems with a multi-valued nonlinear term |
| title_full | Semilinear systems with a multi-valued nonlinear term |
| title_fullStr | Semilinear systems with a multi-valued nonlinear term |
| title_full_unstemmed | Semilinear systems with a multi-valued nonlinear term |
| title_short | Semilinear systems with a multi-valued nonlinear term |
| title_sort | semilinear systems with a multi valued nonlinear term |
| topic | semilinear system multi-valued operator operators of monotone type degree theory 47h04 47h05 47h11 35a16 35b10 35l71 |
| url | http://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0056/math-2017-0056.xml?format=INT |
| work_keys_str_mv | AT kiminsook semilinearsystemswithamultivaluednonlinearterm AT hongsukjoon semilinearsystemswithamultivaluednonlinearterm |