Existence and uniqueness for a neutral differential problem with unbounded delay via fixed point results

Abstract

Jleli and Samet (2018) introduced a new metric space and named it as $F$-space. In this paper we consider the notion of $\alpha$-$\psi$-contraction in the setting of $F$-metric spaces. We present some fixed point and coupled fixed point results in the generalized setting. Moreover, our purpose in this paper is to concerned with the solution of nonlinear neutral differential equation $x^{\prime }\left(t\right)=-a\left(t\right)x\left(t\right)+b\left(t\right)g\left(x(t-r\left(t\right))\right)+c\left(t\right)x^{\prime }(t-r\left(t\right))$with unbounded delay using fixed point theory in $F$-metric space.