On The Stability of The Quasi-Linear Implicit Equations in Hilbert Spaces

Document Type : Original Article

Authors

Mehdi Benabdallah ¹

Mohamed Hariri ²

¹ Department of Mathematics, Faculty of Math and Computer, USTOran, 31000, Algeria

² Department of Mathematics, University of Djillali Liabes, Sidi Bel-Abbes, 22000, Algeria.

10.22034/kjm.2019.81222

Abstract

We use the generalized theorem of Liapounov to obtain some necessary and sufficient conditions for the stability of the stationary implicit equation $$Ax'(t)=Bx(t) ,\quad t\geq 0 ,$$ where $A$ and $B$ are bounded operators in Hilbert spaces. The achieved results can be applied to the stability for the quasi-linear implicit equation $$Ax'(t)=Bx(t)+\theta(t,x(t)),\quad t\geq 0 .$$

Keywords

Exponential stability

operator theory

implicit equations

Subjects

• 34 Ordinary differential equations
• 47 Operator theory

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