Integral equations of convolution type with power nonlinearity.

*(English)*Zbl 0738.45003This survey paper is written on the basis of some works of the author, the reviewer and a brief paper by the author, the reviewer and A. Ya. Yakubov. He gives full proofs for the results from these papers concerning the existence and uniqueness of the solution of the nonlinear integral equation of the form \(u^{\alpha}=k*u+f\), \(x>0\), \(\alpha>1\). This equation in the case \(k(0)>0\) is connected with the Boussinesq equation and was considered by W. Okrasiński; his investigation essentially depends on \(s/k(0)\).

In this paper the problem of solvability in the case \(k(0)=0\) is considered. Moreover, the stability of the solutions with respect to perturbations of \(k\), \(\alpha\), \(f\) is studied. In the case \(0<\alpha<1\) the method of monotone operators is applied.

It should be noted that an extensive survey of these and some other results can be found in the paper by the author, the reviewer, and A. Ya. Yakubov [Nonlinear Wiener-Hopf equations. Chechen-Inguch State University, Grosnyi (1988), 144 p. Deposited at VINITI 25.11.88 N 8341, RZ Mat 1989 35 523]. For some new results see the author, the reviewer, and A. Ya. Yakubov [Dokl. Akad. Nauk SSSR 311, No. 5, 1035-1039 (1990; Zbl 0726.45006)].

In this paper the problem of solvability in the case \(k(0)=0\) is considered. Moreover, the stability of the solutions with respect to perturbations of \(k\), \(\alpha\), \(f\) is studied. In the case \(0<\alpha<1\) the method of monotone operators is applied.

It should be noted that an extensive survey of these and some other results can be found in the paper by the author, the reviewer, and A. Ya. Yakubov [Nonlinear Wiener-Hopf equations. Chechen-Inguch State University, Grosnyi (1988), 144 p. Deposited at VINITI 25.11.88 N 8341, RZ Mat 1989 35 523]. For some new results see the author, the reviewer, and A. Ya. Yakubov [Dokl. Akad. Nauk SSSR 311, No. 5, 1035-1039 (1990; Zbl 0726.45006)].

Reviewer: N.K.Karapetyants (Rostov-na-Donu)