Solutions existence for two types of mixed variational relation problems

Abstract

In the last decade, more and more authors have attempted to treat in a unified manner, by means of some binary or ternary relations, various equilibrium or quasi-equilibrium problems encountered in literature. Several types of variational relation problems have been investigated in many recent papers in which, as for other mathematical models, the main focus has been on sufficient conditions for the existence of solution. The existence of solution has been also studied for several types of systems of two variational relation problems called either mixed variational relation problems or simultaneous variational relation problems. In this paper, we deal with the existence of solutions for two types of mixed variational relation problems in which the involved relations (one binary and the other ternary) are linked by a certain condition of compatibility.

Keywords:

• variational relation problem
• set-valued mapping
• fixed point
• - relation

AMS Subject Classifications:

• 49J53
• 49J40

Acknowledgements

The author is extremely grateful to Professor Dinh The Luc for useful advice and suggestions. The author wish also to thank the two anonymous referees for the careful reviews and valuable comments which helped him to improve the presentation form of the paper.