A theorem on existence of nonnegative and nontrivial solutions of the nonlinear convolution equation \(\int^{+\infty}_{-\infty}k(t- s)u(s)ds=u^{\alpha}(t)\), \(t\in R\), is proved. The assumptions: 1) \(\alpha\in (0,1)\), \(\alpha =const.\), 2) \(k: R\to R\) is nonnegative, \(k\in L^{\infty}(R)\cap L^ p(R)\) where \(1<p<(1+\alpha)/2\alpha\), 3) k is a.e. symmetric on R and a.e. nonincreasing on \(R_+\). Applications to differential equations are indicated.