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Abstract Mathematical modeling of real life problems usually results in functional equations and the initial value problem of the dierential equations and integral equation. Many mathematical formulation of physical phenomena contain dierential equations, functional equations and functional dierential equations.The theory of integral operators, integral equations and dierential equations are impor-tant parts of nonlinear analysis. It is caused by the fact that this theories are frequently applicable in other branches of mathematical physics, biology as well in describing problem connected with real life[7, 8, 9, 11, 17].Our aim in this thesis is to consider the question of the existence and uniqueness of solutions of the initial value problem of the dierential equations with parameters. |