Gronwall-Bellman type integral inequalities in measure spaces (Q1922989)

Let \((X,{\mathcal A},\mu)\) be a measure space. In the present paper, the author has given explicit bounds on the solutions of an inequality of the form \[ \phi(x)\leq f(x)+ g(x)\int_{S(x)}\phi d\mu,\quad x\in X, \] where \(\phi\), \(f\), and \(g\) are real-valued functions on \(X\) and \(S(x)\in {\mathcal A}\) for every \(x\in X\). The unique solvability of certain integral equations is also obtained.