Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems
Let X be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space X∗. Let T:X⊇D(T)→2X∗ be maximal monotone and S:X⊇D(S)→X∗ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space W⊂D(S), dense and continu...
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Wiley
2015-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2015/357934 |
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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 separable Banach space with locally uniformly convex dual space X∗. Let T:X⊇D(T)→2X∗ be maximal monotone and S:X⊇D(S)→X∗ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space W⊂D(S), dense and continuously embedded in X. Assume, further, that there exists d≥0 such that 〈v∗+Sx,x〉≥-dx2 for all x∈D(T)∩D(S) and v∗∈Tx. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type T+S. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator L:X⊇D(L)→X∗ is given as a result of surjectivity of L+S, where S is of type (M) with respect to L. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in X=Lp(0,T;W01,p(Ω)) of a nonlinear parabolic problem of the type ut-∑i=1n(∂/∂xi)ai(x,t,u,∇u)=f(x,t), (x,t)∈Q; u(x,t)=0, (x,t)∈∂Ω×(0,T); u(x,0)=0, x∈Ω, where p>1, Ω is a nonempty, bounded, and open subset of RN, ai:Ω×(0,T)×R×RN→R (i=1,2,…,n) satisfies certain growth conditions, and f∈Lp′(Q), Q=Ω×(0,T), and p′ is the conjugate exponent of p. |
| format | Article |
| id | doaj-art-de26658af67a470fa9dcad57eaa67642 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-de26658af67a470fa9dcad57eaa676422025-02-03T00:59:33ZengWileyAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/357934357934Noncoercive Perturbed Densely Defined Operators and Application to Parabolic ProblemsTeffera M. Asfaw0Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USALet X be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space X∗. Let T:X⊇D(T)→2X∗ be maximal monotone and S:X⊇D(S)→X∗ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space W⊂D(S), dense and continuously embedded in X. Assume, further, that there exists d≥0 such that 〈v∗+Sx,x〉≥-dx2 for all x∈D(T)∩D(S) and v∗∈Tx. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type T+S. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator L:X⊇D(L)→X∗ is given as a result of surjectivity of L+S, where S is of type (M) with respect to L. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in X=Lp(0,T;W01,p(Ω)) of a nonlinear parabolic problem of the type ut-∑i=1n(∂/∂xi)ai(x,t,u,∇u)=f(x,t), (x,t)∈Q; u(x,t)=0, (x,t)∈∂Ω×(0,T); u(x,0)=0, x∈Ω, where p>1, Ω is a nonempty, bounded, and open subset of RN, ai:Ω×(0,T)×R×RN→R (i=1,2,…,n) satisfies certain growth conditions, and f∈Lp′(Q), Q=Ω×(0,T), and p′ is the conjugate exponent of p.http://dx.doi.org/10.1155/2015/357934 |
| spellingShingle | Teffera M. Asfaw Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems Abstract and Applied Analysis |
| title | Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems |
| title_full | Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems |
| title_fullStr | Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems |
| title_full_unstemmed | Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems |
| title_short | Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems |
| title_sort | noncoercive perturbed densely defined operators and application to parabolic problems |
| url | http://dx.doi.org/10.1155/2015/357934 |
| work_keys_str_mv | AT tefferamasfaw noncoerciveperturbeddenselydefinedoperatorsandapplicationtoparabolicproblems |