Communications in Mathematical Analysis

New Developments on Nirenberg's Problem for Compact Perturbations of Quasimonotone Expansive Mappings in Reflexive Banach Spaces

Teffera M. Asfaw

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Abstract

Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^*$. Let $T:X\to X^*$ be demicontinuous, quasimonotone and $\alpha$-expansive, and $C: X\to X^*$ be compact such that either (i) $\langle Tx+Cx, x\rangle \geq -d\|x\|$ for all $x\in X$ or (ii) $\langle Tx+Cx, x\rangle \geq-d\|x\|^2$ for all $x\in X$ and some suitable positive constants $\alpha$ and $d.$ New surjectivity results are given for the operator $T+C.$ The results are new even for $C=\{0\}$, which gives a partial positive answer for Nirenberg's problem for demicontinuous, quasimonotone and $\alpha$-expansive mapping. Existence result on the surjectivity of quasimonotone perturbations of multivalued maximal monotone operator is included. The theory is applied to prove existence of generalized solution in $H^{1}_{0}(\Omega)$ of nonlinear elliptic equation of the type \begin{align*} \begin{split} \left\{\begin{array}{cc} -\sum\limits_{i=1}^{N}{\frac{\partial}{\partial x_i} a_i(x, u(x), \nabla u(x))})+G_{\lambda}(x, u(x))=f(x) &\textrm{in $\Omega$}\\ u(x)=0&\textrm{$x\in\partial \Omega$},\\ \end{array}\right. \end{split} \end{align*} where $f\in L^{2}(\Omega)$, $\Omega$ is a nonempty, bounded and open subset of $\mathbb{R}^{N}$ with smooth boundary, $\lambda>0$, $ G_{\lambda}(x, u)=-div (\beta (\nabla u(x)))+\lambda u(x)+a_0(x, u(x), \nabla u(x))+g(x, u(x))$, $\beta: \mathbb{R}^{N}\to\mathbb{R}^{N}$, $a_i: \Omega\times \mathbb{R}\times \mathbb{R}^{N}\to\mathbb{R}$ ($i=0, 1, 2, ..., N$) and $g:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}$ satisfy certain conditions.

Article information

Source
Commun. Math. Anal., Volume 18, Number 2 (2015), 54-75.

Dates
First available in Project Euclid: 30 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.cma/1446210173

Mathematical Reviews number (MathSciNet)
MR3411396

Zentralblatt MATH identifier
1323.47055

Subjects
Primary: 47H04: Set-valued operators [See also 28B20, 54C60, 58C06] 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
Expansive mappings Nirenberg's problem Quasimonotone Compact perturbations Elliptic equations Eigenvalue problem

Citation

Asfaw, Teffera M. New Developments on Nirenberg's Problem for Compact Perturbations of Quasimonotone Expansive Mappings in Reflexive Banach Spaces. Commun. Math. Anal. 18 (2015), no. 2, 54--75. https://projecteuclid.org/euclid.cma/1446210173


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