On the Constructive Solvability of one Class of the Hammerstein–Volterra type Nonlinear Multidimensional Integral Equations

The questions of the existence and uniqueness of a non-negative (non-trivial) bounded and continuous solution of one class of multidimensional integral equations of the Hammerstein–Volterra type with monotonic and concave nonlinearity are studied. Under some conditions on the kernel and nonlinearity, a constructive theorem for existence of a non-negative bounded and continuous solution is proved. Moreover, special successive approximations are introduced for the equation under study and a uniform estimate is established for the difference of neighboring iterations, the right side of which decreases as a geometric progression. In particular, this estimate implies the uniform convergence of successive approximations to a continuous and bounded solution of the integral equation. In a sufficiently wide subclass of nonnegative nontrivial and bounded functions, it is also proved the uniqueness theorem for the solution. The results obtained are used to study the issue of global solvability of one Cauchy problem for the nonlinear heat equation. The developed methods also make it possible to study questions of the existence and uniqueness of a solution to a nonlinear integral equation arising in the mathematical theory of the spatial-temporal spread of epidemic diseases within the framework of the Diekmann–Kaper model. At the end of the work, specific applied examples of the kernel and nonlinearity of the equation under study are given that satisfy the conditions of the proven theorems.