A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem
Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be maximal monotone, S:X→2X⁎ be bounded and of type (S+), and C:D(C)→X⁎ be compact with D(T)⊆D(C) such that C lies in Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2017/7236103 |
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| author | Teffera M. Asfaw |
| author_facet | Teffera M. Asfaw |
| author_sort | Teffera M. Asfaw |
| collection | DOAJ |
| description | Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be maximal monotone, S:X→2X⁎ be bounded and of type (S+), and C:D(C)→X⁎ be compact with D(T)⊆D(C) such that C lies in Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for all x∈D(C)). A new topological degree theory is developed for operators of the type T+S+C. The theory is essential because no degree theory and/or existence result is available to address solvability of operator inclusions involving operators of the type T+S+C, where C is not defined everywhere. Consequently, new existence theorems are provided. The existence theorem due to Asfaw and Kartsatos is improved. The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces. |
| format | Article |
| id | doaj-art-99b471ea51ba4ac680056ef3187f04ac |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-99b471ea51ba4ac680056ef3187f04ac2025-02-03T01:07:34ZengWileyAbstract and Applied Analysis1085-33751687-04092017-01-01201710.1155/2017/72361037236103A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic ProblemTeffera M. Asfaw0Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USALet X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be maximal monotone, S:X→2X⁎ be bounded and of type (S+), and C:D(C)→X⁎ be compact with D(T)⊆D(C) such that C lies in Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for all x∈D(C)). A new topological degree theory is developed for operators of the type T+S+C. The theory is essential because no degree theory and/or existence result is available to address solvability of operator inclusions involving operators of the type T+S+C, where C is not defined everywhere. Consequently, new existence theorems are provided. The existence theorem due to Asfaw and Kartsatos is improved. The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces.http://dx.doi.org/10.1155/2017/7236103 |
| spellingShingle | Teffera M. Asfaw A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem Abstract and Applied Analysis |
| title | A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem |
| title_full | A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem |
| title_fullStr | A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem |
| title_full_unstemmed | A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem |
| title_short | A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem |
| title_sort | degree theory for compact perturbations of monotone type operators and application to nonlinear parabolic problem |
| url | http://dx.doi.org/10.1155/2017/7236103 |
| work_keys_str_mv | AT tefferamasfaw adegreetheoryforcompactperturbationsofmonotonetypeoperatorsandapplicationtononlinearparabolicproblem AT tefferamasfaw degreetheoryforcompactperturbationsofmonotonetypeoperatorsandapplicationtononlinearparabolicproblem |