On new generalized θ b $\theta _{b}$ -contractions and related fixed point theorems

Abstract

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 $\theta _{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 $\theta _{b}$ -contractions to Liouville–Caputo fractional differential equations.

Keywords

• Fixed points
• Multi-valued maps
• θ b $\theta _{b}$ -Contraction
• b-Bianchini–Grandolfi gauge function
• Liouville–Caputo fractional differential equations