A priori bounds of solutions of the nonlinear integral convolution type equation and their applications
In connection with applications in the theories of filtration, shock waves, heat exchange, and others an equation of the form \[ u^ \alpha(x)= \int^ x_ 0 k(x- t) u(t)+ f(x),\qquad \alpha> 1,\;x> 0,\tag{1} \] has been considered in the literature in the class of non- negative \(\Gamma\)-functions continuous on \([0, \infty)\) and under the rather restrictive condition \(f(0)= 0\). Since equations of the form (1) are exactly solvable only in a few particular cases, we see that both for the theory and applications the development of approximate methods for equations of this type as well as the stability of solutions under changes of given functions \(k\), \(f\) are of great importance. In the present paper we obtain new a priori bounds for solutions of equation (1) that make the corresponding bounds of the authors more precise. Using these bounds enables us to prove the existence and uniqueness of a solution of equation (1) for \(f\) not necessarily vanishing at \(x= 0\) and to define the precise boundaries of the solution. For \(\alpha\geq 2\) we show that the solution may be obtained by the method of sequential approximations for which the error bound and that of the degree of their convergence to the precise solution are given, as well as the stability of the solution relative to changes of \(k\) and \(f\) in the same metric is also proved.
nonlinear integral convolution type equation
scientific article; zbMATH DE number 795594
scientific article