Karapetyants, N. K.; Yakubov, A. Ya. A convolution equation with power nonlinearity of negative order. (English. Russian original) Zbl 0770.45009 Sov. Math., Dokl. 44, No. 2, 517-520 (1992); translation from Dokl. Akad. Nauk SSSR 320, No. 4, 777-780 (1991). This paper discusses the convolution equation with power nonlinearity of negative order of the form (1) \(u^{-\alpha}(x)=(u^*k)(x)\), \(x>0\), \(\alpha>1\). Let \(\Gamma\) denote the cone of continuous functions that are positive for \(x>0\) and possibly have an integrable singularity at \(x=0\).The author proves that equation (1) has a unique solution in the class of almost increasing functions in the cone \(\Gamma\) under the assumption that \(k(x)\in L^{loc}_ 1(R_ +')\) is nonnegative and nondecreasing. In the case of the special form of the kernel \(k(x)=t^{-\gamma}\), \(0<\gamma<1\), the uniqueness result for the nonlinear Abel equation (1) in the class of nondecreasing functions in \(\Gamma\) is obtained. Reviewer: Zeng Yuesheng (Huaihua) Cited in 8 Documents MSC: 45G05 Singular nonlinear integral equations 45M20 Positive solutions of integral equations Keywords:almost increasing; convolution equation; power nonlinearity; cone of continuous functions; Abel equation PDFBibTeX XMLCite \textit{N. K. Karapetyants} and \textit{A. Ya. Yakubov}, Sov. Math., Dokl. 44, No. 2, 517--520 (1991; Zbl 0770.45009); translation from Dokl. Akad. Nauk SSSR 320, No. 4, 777--780 (1991)