A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities
Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X⁎. Let T:X⊇DT→2X⁎ be maximal monotone of type Γdϕ (i.e., there exist d≥0 and a nondecreasing function ϕ:0,∞→0,∞ with ϕ(0)=0 such that 〈v⁎,x-y〉≥-dx-ϕy for all x∈DT, v⁎∈Tx, and y∈X),L:X⊃D(L)→X⁎ be...
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Wiley
2016-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2016/3970621 |
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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 reflexive locally uniformly convex Banach space with locally uniformly convex dual space X⁎. Let T:X⊇DT→2X⁎ be maximal monotone of type Γdϕ (i.e., there exist d≥0 and a nondecreasing function ϕ:0,∞→0,∞ with ϕ(0)=0 such that 〈v⁎,x-y〉≥-dx-ϕy for all x∈DT, v⁎∈Tx, and y∈X),L:X⊃D(L)→X⁎ be linear, surjective, and closed such that L-1:X⁎→X is compact, and C:X→X⁎ be a bounded demicontinuous operator. A new degree theory is developed for operators of the type L+T+C. The surjectivity of L can be omitted provided that RL 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+C, where C 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=L20,T;H01Ω of the nonlinear equation given by ∂u/∂t-∑i=1N(∂/∂xi)Aix,u,∇u+Hλx,u,∇u=fx,t, x,t∈QT; ux,t=0, x,t∈∂QT; and ux,0=ux,T, x∈Ω, where λ>0, QT=Ω×0,T, ∂QT=∂Ω×0,T, Aix,u,∇u=∂/∂xiρx,u,∇u+aix,u,∇u (i=1,2,…,N), Hλx,u,∇u=-λΔu+gx,u,∇u, Ω is a nonempty, bounded, and open subset of RN with smooth boundary, and ρ,ai,g:Ω¯×R×RN→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. |
| format | Article |
| id | doaj-art-a4cf1825f8894b589b78eeb148dae5d9 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
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| series | Journal of Function Spaces |
| spelling | doaj-art-a4cf1825f8894b589b78eeb148dae5d92025-02-03T01:30:06ZengWileyJournal of Function Spaces2314-88962314-88882016-01-01201610.1155/2016/39706213970621A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone NonlinearitiesTeffera M. Asfaw0Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USALet X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X⁎. Let T:X⊇DT→2X⁎ be maximal monotone of type Γdϕ (i.e., there exist d≥0 and a nondecreasing function ϕ:0,∞→0,∞ with ϕ(0)=0 such that 〈v⁎,x-y〉≥-dx-ϕy for all x∈DT, v⁎∈Tx, and y∈X),L:X⊃D(L)→X⁎ be linear, surjective, and closed such that L-1:X⁎→X is compact, and C:X→X⁎ be a bounded demicontinuous operator. A new degree theory is developed for operators of the type L+T+C. The surjectivity of L can be omitted provided that RL 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+C, where C 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=L20,T;H01Ω of the nonlinear equation given by ∂u/∂t-∑i=1N(∂/∂xi)Aix,u,∇u+Hλx,u,∇u=fx,t, x,t∈QT; ux,t=0, x,t∈∂QT; and ux,0=ux,T, x∈Ω, where λ>0, QT=Ω×0,T, ∂QT=∂Ω×0,T, Aix,u,∇u=∂/∂xiρx,u,∇u+aix,u,∇u (i=1,2,…,N), Hλx,u,∇u=-λΔu+gx,u,∇u, Ω is a nonempty, bounded, and open subset of RN with smooth boundary, and ρ,ai,g:Ω¯×R×RN→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.http://dx.doi.org/10.1155/2016/3970621 |
| spellingShingle | Teffera M. Asfaw A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities Journal of Function Spaces |
| title | A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities |
| title_full | A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities |
| title_fullStr | A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities |
| title_full_unstemmed | A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities |
| title_short | A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities |
| title_sort | new topological degree theory for perturbations of demicontinuous operators and applications to nonlinear equations with nonmonotone nonlinearities |
| url | http://dx.doi.org/10.1155/2016/3970621 |
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