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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.

MSC:

45G05 Singular nonlinear integral equations
45M20 Positive solutions of integral equations
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