On the solvability of a class of Volterra operator equations of the first kind with piecewise continuous kernels

Abstract

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

Solvability of Nonlinear Equations in Case of Branching Solutions

“…Results of this paper enable applications of the existence theorems for bifurcation points of nonlinear BVP problems and make it possible to construct an appropriate solutions. Our method has been also applied for solution of degenerate operator-differential and integral equations [17,18,21,23,32], [37].…”

“…Proof. Let λ = λ(2(θ − 1)δ) in branching system (21) for arbitrary small δ > 0 and θ ∈ [0, 1] and consider the continuous vector field…”

“…, n + 1 are defined iterations (27) converges as m → ∞ to desired small solution x(ε) → 0, λ(ε) → 0 as ε → 0. Right hand sides of formulae (21), (22) used in iterations (27) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 April 2020 doi:10.20944/preprints202004.0288.v1…”

“…Trenogin [4] the basement of the analytical theory of branching solutions in Banach spaces with applications is given. These works contributed to the modern functional analysis development with many applications [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Volterra Models in Load Leveling Problem

“…Studies of linear Volterra integral equations of the first kind with piecewise continuous kernels (1) were initiated in [20,21] and pursued in [19], [22][23][24][25] followed by monograph [15]. In general, VIE of the first kind can be solved by reduction of the equations of the second kind.…”