The article considers a problem for a linear integro-functional equation with a given value of the required function outside the main interval and constraints (additional conditions) imposed on the required function. These restrictions are integral. The main and auxiliary tasks are formulated. Step-by-step considerations have been made on the relationship between these tasks. For the values included in the given problem, it is required that they meet a number of necessary conditions. It is shown that under these conditions the initial problem will be equivalent to some integral Fredholm equation of the second kind with a completely continuous operator and additional conditions for the desired solution. In addition to the main problem, the auxiliary problem is also considered - the problem with control, when in case of compatibility an additional, correcting value is introduced. The conditions of compatibility of the initial problem are formulated and substantiated.
The article also presents and substantiates the iterative, namely the method of successive approximations and collocation-iterative methods of constructing approximate solutions of the initial problem with constraints. The algorithms of these methods and sufficient conditions for their convergence are indicated. In this case, we use the fact that the initial problem under certain conditions is equivalent to an integral equation with constraints. In particular, the method will be convergent for some fixed n if the unit is not a point of the spectrum of the integral operator T. The collocation nodes are chosen depending on the system of basis functions. These methods of constructing approximate solutions of the integro-functional equation with additional conditions can be successfully implemented on computers by creating appropriate programs. It should be noted that the proposed methods for constructing approximate solutions of the integro-functional equation with additional conditions are quite effective. In the future, we can transfer the study of this nature to the boundary value problem for a differential equation with a deviation of the argument of the neutral type.

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