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We study optimization 1. Introduction. Problems involving optimal control of processes occurring in solid media, seeking an optimal structure shape and the like often arise in practice (see [1-5, 8, 15, 18]). The mathematical formulation of a large class of problems of this type consists of minimizing (or maximizing) a certain functional, called the objectfi, e functional, on solutions of boundary-value or initial/boundary-value problems for partial differential equations, on solutions of Cauchy problems for ordinary differential equations, and so forth.Numerous works have been devoted to various aspects of the study of problems of this type (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]). But finding effective necessary conditions for optimality, for example, generalization of the Pontryagin maximum principle for these problems, remains one of the most important problems of this theory. In the present paper we devote our attention to obtaining necessary optimality conditions, for example, in the form of the Pontryagin maximum principle. It is not our purpose to state the most general necessary optimality condition or to compile a list of all the necessary conditions characterizing the optimal control in any particular minimization problem. In this paper we describe schemes for obtaining necessary optimality conditions on solutions of general operator equations defined in Banach spaces, and the scheme we discuss does not require that there be no global functional constraints on the controlling parameters. In specific Banach spaces suitable optimality conditions can be obtained by the introduction of special variations (for example, the Pontryagin-McShane variations). One such example is given at the end of the present paper. We note also that no specific physical (or other) formulations of problems are given in this paper, but the physical meaning of these examples is clear.