Abstract
Recently, Jleli and Samet [J. Inequal. Appl. (2014), 2014:38] introduced and studied a new contraction to prove a generalization of the Banach contraction principle. In this paper, we introduce the concept of --contraction with respect to a general family of functions H and we establish Jleli-Samet-type fixed point results in metric and ordered metric spaces. As an application of our results we deduce Suzuki-type fixed point results for -contractions. We also derive certain fixed and periodic point results for orbitally continuous generalized -contractions. Moreover, we present an illustrative example to highlight the obtained improvements.