In this paper, we generalize several important inequalities and investigate the random solutions of a class of random nonlinear operator equations by means of the random fixed point index. Meanwhile, the main results generalize the famous Altman’s theorem and Petryshyn’s theorem, etc. Finally, an application to random nonlinear differential equation and random integral equation is given to illustrate the usability of the obtained results.

random fixed point index, random semi-closed 1-set-contract operator, real Banach space.