An Application of Fixed Point Theory to a Nonlinear Differential Equation

Abstract

We introduce a new family of mappings on $[\mathrm{0},+\infty )$ by relaxing the nondecreasing condition on the mappings and by using the properties of this new family we present some fixed point theorems for $\alpha$-$\psi$-contractive-type mappings in the setting of complete metric spaces. By applying our obtained results, we also assure the fixed point theorems in partially ordered complete metric spaces and as an application of the main results we provide an existence theorem for a nonlinear differential equation.