The paper studies a class of nonlinear integral equations on the semiaxis with a non-compact Hammerstein operator. It is assumed that the kernel of the equation decreases exponentially on the positive part of the number axis. Equations of this kind arise in various fields of natural science. In particular, such equations are found in the theory of radiation transfer in spectral lines, in the mathematical theory of the space-time propagation of an epidemic, in the kinetic theory of gases. A distinctive feature of the considered class of equations is the non-compactness of the corresponding nonlinear Hammerstein integral operator in the space of essentially bounded functions on the positive part of the number line and the criticality condition (i.e., the presence of a trivial zero solution). Under certain restrictions on nonlinearity, the existence of a positive bounded and summable solution is proved. The asymptotic behavior of the solution at infinity is also investigated. The proof of the existence theorem is constructive. First, solving a certain simple characteristic equation the zero approximation in the considered iterations is costructed. Then a special auxiliary nonlinear integral equation is studied, the solution of which is constructed using simple successive approximations. After that, it is proved that the iterations introduced by us increase monotonically and are bounded from above by the solution of the aforementioned auxiliary integral equation. Further, using the appropriate conditions for nonlinearity and for the kernel, it is proved that the limit of these iterations is a solution to the original equation and exponentially decreases at infinity. Under an additional constraint on nonlinearity, the uniqueness of the constructed solution is established in a certain class of measurable functions. At the end of the work, specific examples of the kernel and nonlinearity of an applied nature are given for which all the conditions of the theorem proved are automatically satisfied.