Integral Type Contraction in the Setting of \(\omega_t\)-Distance Mappings with an Application

Abstract

The purpose of this paper is to introduce a new family of contractive conditions of integral type formulated through an \(\omega_t\)-distance mapping defined on a \(b\)-metric space. To this end we employ two auxiliary classes of functions: the class \(\Phi\) of nonnegative Lebesgue integrable mappings whose integral over every nondegenerate interval is positive and finite, and the class \(\Gamma\) of continuous non-decreasing comparison mappings whose iterates converge pointwise to zero. Coupling these two classes with the class \(\Theta\) of altering distance functions, we define the notion of an \((\omega_t,\Theta)\) integral contraction and we prove that every such self-mapping of a complete \(b\)-metric space possesses a unique fixed point, provided that either the mapping is continuous or a Kada type infimum condition is fulfilled. Two corollaries, corresponding to the linear comparison function \(\gamma(a)=\eta a\) with \(\eta\in[0,1)\) and to the power type comparison function \(\gamma(a)=(1+a)^{r}-1\) with \(r\in(0,1)\), are deduced. Since neither the symmetry of the \(\omega_t\)-distance nor the continuity of the \(b\)-metric is required, our results extend and unify several known theorems obtained in metric, \(b\)-metric and \(\Omega\)-distance settings. A nontrivial example illustrating the sharpness of the hypotheses is supplied, and the abstract theory is applied to establish the existence and uniqueness of the solution of a class of Volterra integral equations.