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2015 Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems
Teffera M. Asfaw
Abstr. Appl. Anal. 2015: 1-11 (2015). DOI: 10.1155/2015/357934


Let X be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space X. Let T:XD(T)2X be maximal monotone and S:XD(S)X quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space WD(S), dense and continuously embedded in X. Assume, further, that there exists d0 such that v+Sx,x-dx2 for all xD(T)D(S) and vTx. 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:XD(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, xΩ, where p>1, Ω is a nonempty, bounded, and open subset of RN, ai:Ω×(0,T)×R×RNR (i=1,2,…,n) satisfies certain growth conditions, and fLp(Q), Q=Ω×(0,T), and p is the conjugate exponent of p.


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Teffera M. Asfaw. "Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems." Abstr. Appl. Anal. 2015 1 - 11, 2015.


Published: 2015
First available in Project Euclid: 13 October 2015

zbMATH: 06929097
MathSciNet: MR3399540
Digital Object Identifier: 10.1155/2015/357934

Rights: Copyright © 2015 Hindawi


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