On the solvability of one infinite system of integral equations with power nonlinearity on the semi-axis (Q6637619)

The following system of nonlinear integral equations on the positive half-line is studied:\N\[\Nf_i(x)=\sum\limits_{j=0}^{\infty}\int\limits_{0}^{\infty}K_{i-j}(x,t)f_{j}^{\alpha}(t)dt, \quad i\in\mathbb{Z}^+=\{0,1,2,\ldots\}, \qquad x\in\mathbb{R}^+=[0,\infty),\N\]\Nwith respect to an unknown infinite vector function \(f(x)=\left(f_0(x),\ldots,f_n(x),\ldots\right)^T\).\N\NThe existence, uniqueness, and asymptotic behavior of its solutions at \(+\infty\) are discussed.\NIn particular, the authors prove the existence of a component-wise positive solution of this system in the class of infinite vector functions. Considering an additional constraint on the matrix kernel, the integral asymptotic of the constructed solution is obtained.\NUnder an additional condition of kernel symmetry the uniqueness of the solution in a certain subclass of the class of infinite vector functions is proved. Some examples of matrix kernels satisfying all conditions of the proved theorems are also suggested.