## Abstract and Applied Analysis

### A Characterization of Semilinear Dense Range Operators and Applications

#### Abstract

We characterize a broad class of semilinear dense range operators ${G}_{H}:W\to Z$ given by the following formula, ${G}_{H}w=Gw+H\left(w\right),w\in W$, where $Z$, $W$ are Hilbert spaces, $G\in L\left(W,Z\right)$, and $H:W\to Z$ is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator $G$ to have dense range. Second, under some condition on the nonlinear term $H$, we prove the following statement: If $\overline{\text{R}\text{a}\text{n}\text{g}\left(G\right)}=Z$, then $\overline{\text{R}\text{a}\text{n}\text{g}\left({G}_{H}\right)}=Z$ and for all $z\in Z$ there exists a sequence $\left\{{w}_{\alpha }\in Z:0<\alpha \le 1\right\}$ given by ${w}_{\alpha }={G}^{*}\left(\alpha I+G{G}^{*}{\right)}^{-1}\left(z-H\left({w}_{\alpha }\right)\right)$, such that $\mathrm{ }\text{l}\text{i}\text{m}\alpha \to {0}^{+}\left\{G{u}_{\alpha }+H\left({u}_{\alpha }\right)\right\}=z$. Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: ${z}^{\prime }=Az+Bu\left(t\right)+F\left(t,z,u\left(t\right)\right),z\in Z,u\in U,t>0$, where $Z$, $U$ are Hilbert spaces, $A:D\left(A\right)\subset Z\to Z$ is the infinitesimal generator of strongly continuous compact semigroup $\left\{T\left(t\right){\right\}}_{t\ge 0}$ in $Z,B\in L\left(U,Z\right)$, the control function $u$ belongs to ${L}^{2}\left(0,\tau ;U\right)$, and $F:\left[0,\tau \right]×Z×U\to Z$ is a suitable function. As a particular case we consider the controlled semilinear heat equation.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 729093, 11 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511777

Digital Object Identifier
doi:10.1155/2013/729093

Mathematical Reviews number (MathSciNet)
MR3035310

Zentralblatt MATH identifier
1287.47054

#### Citation

Leiva, H.; Merentes, N.; Sanchez, J. A Characterization of Semilinear Dense Range Operators and Applications. Abstr. Appl. Anal. 2013 (2013), Article ID 729093, 11 pages. doi:10.1155/2013/729093. https://projecteuclid.org/euclid.aaa/1393511777