Unique solvability and exponential stability of differential hemivariational inequalities

ABSTRACT

In this paper, we study a differential hemivariational inequality (DHVI, for short) in the framework of reflexive Banach spaces. Our aim is three fold. The first one is to investigate the existence and the uniqueness of mild solution, by applying a general fixed-point principle. The second one is to study its exponential stability, by employing the formula for the variation of parameters and inequality techniques. Finally, the third aim is to illustrate an application of our abstract results in the study of an initial and boundary value problem which describes the contact of an elastic rod with an obstacle.

KEYWORDS:

• Differential hemivariational inequality
• $C_0$-semigroup of operators
• mild solution
• fixed point principle
• exponential stability
• elastic rod
• unilateral contact

AMS SUBJECT CLASSIFICATIONS:

• 34G20
• 47J20
• 49J40
• 74M15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The project was supported by NNSF of China [grant numbers 11671101, 11661001), NSF of Guangxi [grant number 2018GXNSFDA138002], Project of Guangxi Education Department [grant number KY2016YB417] and the funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie [grant number 823731 CONMECH].