$(F,\varphi‎ ,‎\alpha )_{s}$-contractions in ‎$‎b‎$‎-metric spaces and applications

Subject Areas : Fixed point theory

M. Sangurlu Sezen ¹

1 - Department of Mathematics, Faculty of Science and Arts, Giresun University, Turkey

Received: 2018-12-14 Accepted : 2019-04-09 Published : 2019-08-01

Keywords: $b$-metric space, &lrm, partial $b$-metric space&lrm, , ‎$(F, varphi‎, ‎alpha )_{s}$-contraction‎, ‎$varphi$-fixed point,

Abstract :

‎In this paper‎, ‎we introduce more general contractions called $\varphi $-fixed‎‎point point for $(F,\varphi‎ ,‎\alpha )_{s}$ and $(F,\varphi‎ ,‎\alpha )_{s}$-‎weak contractions‎. ‎We prove the existence and uniqueness of $\varphi $-‎fixed point point for $(F,\varphi‎ ,‎\alpha )_{s}$ and $(F,\varphi‎ ,‎\alpha‎‎)_{s}$-weak contractions in complete $b$-metric spaces‎. ‎Some examples are‎‎supplied to support the usability of our results‎. ‎As applications‎, ‎necessary‎‎conditions to ensure the existence of a unique solution for a nonlinear‎‎inequality problem are also discussed‎. ‎Also‎, ‎some new fixed point results in‎‎partial metric spaces are proved.

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