## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 3, Number 1 (2009), 52-60.

### A functional method applied to operator equations

Abderrezak Chaoui and Assia Guezane-Lakoud

#### Abstract

We consider second order hyperbolic equations with unbounded operator's coefficients possessing time dependent domain of definition in a Hilbert space. Existence and uniqueness of the strong generalized solution are studied. The proofs rely on a generalization of the well known energy integral method. First, we derive a priori estimates for the strong generalized solutions with the help of Yosida operator approximation. Then, using previous results, we show that the range of the operators generated by the posed problem is dense.

#### Article information

**Source**

Banach J. Math. Anal., Volume 3, Number 1 (2009), 52-60.

**Dates**

First available in Project Euclid: 21 April 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1240336423

**Digital Object Identifier**

doi:10.15352/bjma/1240336423

**Mathematical Reviews number (MathSciNet)**

MR2461746

**Zentralblatt MATH identifier**

1172.35458

**Subjects**

Primary: 35B45: A priori estimates

Secondary: 35D05 35L90: Abstract hyperbolic equations 35L10: Second-order hyperbolic equations 35L20: Initial-boundary value problems for second-order hyperbolic equations

**Keywords**

evolution equation a priori estimate strong generalized solution hyperbolic equation boundary value problem

#### Citation

Guezane-Lakoud, Assia; Chaoui, Abderrezak. A functional method applied to operator equations. Banach J. Math. Anal. 3 (2009), no. 1, 52--60. doi:10.15352/bjma/1240336423. https://projecteuclid.org/euclid.bjma/1240336423