Differential and Integral Equations

Existence and uniqueness of solutions of some abstract degenerate nonlinear equations

Angelo Favini and Anatoliy Rutkas

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Abstract

The nonlinear abstract differential equation $$ \frac{d}{dt}(Ay)+By(t)=F(t,Ky),\quad 0\le t\le\tau, $$ where $A,B,K$ are linear closed operators from a complex Banach space $Y$ into a Banach space $X$ is considered. The main assumption reads that the point $\xi =0$ is a polar singularity of the resolvent $(T-\xi I)^{-1}$, where $T=A(\lambda A+B)^{-1}$, $\lambda$ being a regular point of the operator pencil $\lambda A+B$. Mainly the case of a simple pole and of a second order pole are considered. Some examples of application to concrete partial differential equations are given. In particular, we show that the results work for mathematical models of nonlinear electrical networks.

Article information

Source
Differential Integral Equations, Volume 12, Number 3 (1999), 373-394.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367265217

Mathematical Reviews number (MathSciNet)
MR1674394

Zentralblatt MATH identifier
1014.35047

Subjects
Primary: 34G20: Nonlinear equations [See also 47Hxx, 47Jxx]
Secondary: 34A09: Implicit equations, differential-algebraic equations [See also 65L80] 35K99: None of the above, but in this section 47J05: Equations involving nonlinear operators (general) [See also 47H10, 47J25] 47N20: Applications to differential and integral equations

Citation

Favini, Angelo; Rutkas, Anatoliy. Existence and uniqueness of solutions of some abstract degenerate nonlinear equations. Differential Integral Equations 12 (1999), no. 3, 373--394. https://projecteuclid.org/euclid.die/1367265217


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