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The sufficient conditions for existence and uniqueness of continuous solutions of the Volterra operator equations of the first kind with piecewise continuous kernel are derived. The asymptotic approximation of the parametric family of solutions are constructed in case of non-unique solution. The algorithm for the solution's improvement is proposed using the successive approximations method. Keywords: Volterra operator equations of the first kind, asymptotic, discontinuous kernel, successive approximations, Fredholm's point, regularization. n 1 D i . Let us introduce biparametric family of linear continuous operator-functions K i (t, s), defined for t, s ∈ D i , i = 1, n, which are differentiable wrt t and acting from Banach space E 1 into Banach space E 2 . Therefore K i (t, s) ∈ L(E 1 → E 2 ) and ∂K i (t,s) ∂t ∈ L(E 1 → E 2 ) for t, s ∈ D i , i = 1, n. Let the space of continuous functions x(t) defined on [0, T ] with ranges on E 1 be denoted as C ([0,T ];E 1 ) . Let us introduce the integral operatorwith piecewise kernel

The branches of a solution of the nonlinear integral equation u(x) = b a K(x, s)q(s, u(s), λ) ds,where q (s, u, λ∞ k=1 q ik (s)u i λ k and λ is a parameter, are constructed by successive approximations. Under consideration is the case when unity is a characteristic number of the kernel K(x, s) of rank n ≥ 1, and λ = 0 is a bifurcation point. The principal term of the asymptotic expansion constructed is used as an initial approximation. The uniform convergence is established in some neighborhood about the bifurcation point on using the implicit function theorem and the Schmidt lemma.We examine the equationwhere K(x, s) and g (s, u, λIf unity is not a characteristic number of K(x, s) then (1) has the unique solution such that u → 0 for λ → 0 and a broad class of methods can be used for its construction [1]. Assume that unity is the characteristic number of K(x, s) of rank n, while {φ i } n 1 are the corresponding eigenfunctions and {ψ i } n 1 are the eigenfunctions of the adjoint kernel K(s, x). We need to construct a solution in some neighborhood about the bifurcation point λ = 0 such that u λ (x) → 0 as λ → 0.To construct branching solutions of (1), we can employ the classical Trenogin results of analytic bifurcation theory [2][3][4].Alongside asymptotic methods, of great importance in solving (1) is the development of the techniques of successive approximations in a neighborhood of a bifurcation point which were not considered in [2].The latter methods can be constructed on the base of explicit and implicit parametrizations, in particular, under the condition of group symmetry of (1) (see [5,6]). For example, [5][6][7][8][9] contain the schemes of successive approximations with implicit parametrization of branches of solutions to operator equations in Banach spaces. The implicit parametrization was used for solving particular problems also in [10,11].

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