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In this work, main results of the doctoral dissertation are presented, where linear differential and discrete problems with nonlocal conditions of a one variable are considered. Since many mathematical problems, modelling processes and phenomena of the real life or taken for theoretical purposes only, do not have unique solutions, we consider problems in the least squares sense, where the existence of a unique best approximate solution is possible. This function is often called a minimum norm least squares solution and nowadays is one of the most popular objects of investigation. Our aim is to describe the best approximate solution in a form related to the classical representation of the unique solution. Here the essential role is played by the concept of a Green's function. Indeed, if we know a Green's function, then a problem is considered as formally . Thus, developing this analogy to a unique best approximate solution, we focus our study on a generalized Green's function, which describes a minimum norm least squares solution and extends the classical meaning of an ordinary Green's function.

In this thesis, we investigate linear differential and discrete problems with nonlocal conditions of a one variable. Since many mathematical problems, modelling processes and phenomena of the real life or taken for theoretical purposes only, do not have unique solutions, we consider problems in the least squares sense, where the existence of a unique best approximate solution is possible. This function is often called a minimum norm least squares solution and nowadays is one of the most popular objects of investigation. Our aim is to describe the best approximate solution in a form related to the classical representation of the unique solution. Here the essential role is played by the concept of a Green's function. Indeed, if we know a Green's function, then a problem is considered as formally . Thus, developing this analogy to a unique best approximate solution, we focus our study on a generalized Green's function, which describes a minimum norm least squares solution and extends the classical meaning of an ordinary Green's function.

In this paper, we investigate the minimizer of the residual for the second order differential problem with one initial and other nonlocal Bitsadze-Samarskii condition. We obtain the representation of the minimizer and present an example.

In this paper, we obtain the representation of the nullspace for m-th order discrete linear problems with nonlocal conditions. An example is given.

In this paper, we investigate several second order discrete problems with nonlocal multipoint boundary conditions and present obtained results about their nullity.

In this paper we investigate the nullity of second order discrete problem with two nonlocal conditions. The classification of nullity with respect to rows and columns of discrete problem matrix is presented.