Document Type : Research Paper

Author

• Babak Mohammadi

Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran

Abstract

‎‎Many authors such as Amini-Harandi‎, ‎Rezapour ‎et al., ‎Kadelburg ‎et al.‎‎, ‎have tried to find at least one fixed point for quasi-contractions when $\alpha \in [\frac{1}{2}‎,‎1)$ but no clear answer exists right now and many of them either have failed or changed to a lighter version‎. In this paper‎, ‎we introduce some new strict fixed point results in the set of multi-valued '{C}iri'{c}-generalized weak quasi-contraction mappings of integral type‎. ‎We consider a necessary and sufficient condition on such mappings which guarantees the existence of unique strict fixed point of such mappings. Our result is a partial positive answer for the mentioned problem ‎ which has remained open for many years‎. Also, we give an strict fixed point result of ‎$‎\alpha ‎$-‎$‎\psi ‎$‎-quasicontractive multi-valued mappings of integral type. Our results generalize and improve many existing results on multi-valued mappings in literature. ‎Moreover‎, ‎some examples are presented to support our new class of multi-valued contractions.

Keywords

• ‎strict fixed point
• '{C}iri'{c}-generalized weak quasi-contraction
• multi-valued mappings
• integral type