Show simple item record

dc.contributor.authorAsfaw, Teffera M.
dc.date.accessioned2017-03-28T18:59:08Z
dc.date.available2017-03-28T18:59:08Z
dc.date.issued2016-10-18
dc.identifier.urihttp://hdl.handle.net/10919/76720
dc.description.abstractLet 𝑋 be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space π‘‹βˆ—. Let 𝑇 : 𝑋 βŠ‡ 𝐷(𝑇)β†’ 2π‘‹βˆ— be maximal monotone of type πšͺπœ™ 𝑑 (i.e., there exist 𝑑 β‰₯ 0 and a nondecreasing function πœ™ : [0,∞) β†’ [0,∞) with πœ™(0) = 0 such that ⟨Vβˆ—, π‘₯ βˆ’ π‘¦βŸ© β‰₯ βˆ’π‘‘β€–π‘₯β€– βˆ’ πœ™(‖𝑦‖) for all π‘₯ ∈ 𝐷(𝑇), Vβˆ— ∈ 𝑇π‘₯, and𝑦 ∈ 𝑋),𝐿 : 𝑋 βŠƒ 𝐷(𝐿) β†’ π‘‹βˆ— be linear, surjective, and closed such that 𝐿⁻¹ : π‘‹βˆ— β†’ 𝑋 is compact, and 𝐢 : 𝑋 β†’ π‘‹βˆ— be a bounded demicontinuous operator. A new degree theory is developed for operators of the type 𝐿+𝑇+𝐢.The surjectivity of 𝐿 can be omitted provided that 𝑅(𝐿) is closed, 𝐿 is densely defined and self-adjoint, and 𝑋 = 𝐻, a real Hilbert space.The theory improves the degree theory of Berkovits and Mustonen for 𝐿+𝐢, where 𝐢 is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when 𝐿 is monotone, a maximality result is included for 𝐿 and 𝐿+𝑇.The theory is applied to prove existence of weak solutions in 𝑋 = 𝐿₂(0, 𝑇;𝐻¹₀ (Ξ©)) of the nonlinear equation given by πœ•π‘’/πœ•π‘‘βˆ’Ξ£π‘ 𝑖=1((πœ•/πœ•π‘₯𝑖)𝐴𝑖(π‘₯, 𝑒, βˆ‡π‘’))+π»πœ†(π‘₯, 𝑒, βˆ‡π‘’) = 𝑓(π‘₯, 𝑑), (π‘₯, 𝑑) ∈ Q𝑇; 𝑒(π‘₯, 𝑑) = 0, (π‘₯, 𝑑) ∈ πœ•Q𝑇; and𝑒(π‘₯, 0) = 𝑒(π‘₯, 𝑇), π‘₯ ∈ Ξ©, whereπœ† > 0, 𝑄𝑇 = Ω×(0,𝑇), πœ•π‘„π‘‡ = πœ•Ξ©Γ—(0,𝑇), 𝐴𝑖(π‘₯, 𝑒, βˆ‡π‘’) = (πœ•/πœ•π‘₯𝑖)𝜌(π‘₯, 𝑒, βˆ‡π‘’)+π‘Žπ‘–(π‘₯, 𝑒, βˆ‡π‘’) (𝑖 = 1, 2, . . . , 𝑁),π»πœ†(π‘₯, 𝑒, βˆ‡π‘’) = βˆ’πœ†Ξ”π‘’ + 𝑔(π‘₯, 𝑒, βˆ‡π‘’), Ξ© is a nonempty, bounded, and open subset of ℝ𝑁 with smooth boundary, and 𝜌, π‘Žπ‘–, 𝑔 : Ξ© Γ— ℝ Γ— ℝ𝑁 β†’ ℝ satisfy suitable growth conditions. In addition, a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity.en_US
dc.format.mimetypeapplication/pdfen_US
dc.language.isoen_USen_US
dc.publisherHindawien_US
dc.titleA New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearitiesen_US
dc.typeArticle - Refereeden_US
dc.title.serialJournal of Function Spacesen_US
dc.identifier.doihttps://doi.org/10.1155/2016/3970621
dc.identifier.volume2016en_US
dc.type.dcmitypeTexten_US
ο»Ώ

Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record