Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

Asfaw, Teffera M.

Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

dc.contributor.author	Asfaw, Teffera M.	en
dc.contributor.department	Mathematics	en
dc.date.accessioned	2017-03-07T20:48:42Z	en
dc.date.available	2017-03-07T20:48:42Z	en
dc.date.issued	2015-08-09	en
dc.description.abstract	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⟩ ≥ −d‖𝑥‖² 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− Σ𝑛𝑖=1(𝜕/𝜕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.	en
dc.description.version	Published version	en
dc.format.mimetype	application/pdf	en
dc.identifier.citation	Teffera M. Asfaw, "Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems", Abstract and Applied Analysis, vol. 2015, Article ID 357934, 11 pages, 2015. https://doi.org/10.1155/2015/357934	en
dc.identifier.doi	https://doi.org/10.1155/2015/357934	en
dc.identifier.uri	http://hdl.handle.net/10919/75304	en
dc.identifier.volume	2015	en
dc.language.iso	en	en
dc.publisher	Hindawi	en
dc.relation.uri	https://doi.org/10.1155/2017/2739102	en
dc.rights	Creative Commons Attribution 4.0 International	en
dc.rights.uri	http://creativecommons.org/licenses/by/4.0/	en
dc.title	Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems	en
dc.title.serial	Abstract and Applied Analysis	en
dc.type	Article - Refereed	en
dc.type.dcmitype	Text	en