Abstract
The manuscript is an extension of the investigations concerning
θ
-contractions which have been newly proposed in (Jleli and Samet in J. Inequal. Appl. 2014:38,
2014
). The authors generalize the notion of
θ
-contractions to the case of nonlinear
θ
b
-contraction mappings and prove multi-valued fixed point theorems based on the
b
-Bianchini–Grandolfi gauge function in
b
-metric spaces. The manuscript consists of a tangible example which displays the motivation for such investigations. The work is compiled by the application of the proposed nonlinear
θ
b
-contractions to Liouville–Caputo fractional differential equations.