Common fixed points of $$\alpha _{*}$$-$$\psi$$-contractive without order closed multi-valued mappings

Recently Samet et al. introduced the notion of $$\alpha$$ - $$\psi$$ -contractive type mappings. They have been establish some fixed point theorems for the mappings in complete metric spaces. In this paper, we introduce the notion of $$\alpha _{*}$$ - $$\psi$$ -contractive and weakly increasing without order closed multi-valued mappings on ordered metric spaces with application to initial value problem.