Integral Equations with Multidimensional Partial Integrals
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
INTEGRAL EQUATIONS WITH MULTIDIMENSIONAL PARTIAL INTEGRALS A. S. Kalitvin Lipetsk State Pedagogical University 42, Lenina St., Lipetsk 398020, Russia [email protected]
A. I. Inozemtsev Lipetsk State Pedagogical University 42, Lenina St., Lipetsk 398020, Russia [email protected]
V. A. Kalitvin ∗ Lipetsk State Pedagogical University 42, Lenina St., Lipetsk 398020, Russia [email protected]
UDC 517.968
We obtain Fredholm criteria for linear equations with multidimensional partial integrals in the space of continuous functions of three variables provided that the kernels are continuous vector-valued functions on a rectangle with the values in spaces of integrable functions. We establish the Fredholm criterion for equations with degenerate kernels and describe the scheme of studying the Fredholm properties of linear equations with partial integrals in spaces of continuous functions of at least four variables. Bibliography: 8 titles.
1
Introduction
Various problems for integro-differential equations with partial derivatives [1]–[3] are connected with integral equations with multidimensional partial integrals. We mention two such cases: the Kolmogorov–Feller equations for a purely discontinuous random process [2] ∂ϕ(t, x; τ, y) = ϕ(t, x; τ, y) ∂t
+∞ +∞ L(t, x; z)dz − L(t, x; z)ϕ(t, z; τ, y)dz,
−∞ +∞
∂ϕ(t, x; τ, y) = −ϕ(t, x; τ, y) ∂τ
L(τ, y; z)dz +
−∞
∗
−∞ +∞
L(τ, z; y)ϕ(t, x; τ, z)dz −∞
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 125-136. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0954
954
and continual analogues of the system of evolution first order partial differential equations in the case of large dimension [3] ∂v(x, y, t) = iξd(y)v(x, y, t) + ∂x ∂v(x, y, t) = ∂t
+∞ N (x, y, z, t)v(x, z, t)dz, −∞
+∞ Q(x, y, z, t)v(x, z, t)dz, −∞
where the kernel N (x, y, z, t) satisfies the nonlinear integro-differential equation ∂N (x, y, z, t) ∂N (x, y, z, t) = a(x, y) + ∂t ∂z
+∞ [a(y, z ) − a(z , z)]N (x, y, z , t)N (x, z , z, t)dz −∞
and a(y, z) = a(z, y) [3]. Here, the partial derivatives of unknown functions are taken with respect to some variables, whereas the integrals are taken with respect to the remaining variables. The above equations are referred to as Barbashin integro-differential equations [4]. Under additional conditions at t = t0 and τ = τ0 , integrating the Kolmogorov–Feller equations, one can obtain a system of linear Volterra integral equations with partial integrals. The integro-differential equation for the continual analogues of the system of evolution first order partial differential equations in large dimension can be also reduced to a system of linear Volterra equations with partial integrals, whereas the nonlinear integro-differential equation with the kernel N (x, y, z, t) is reduced to a nonlinear Volterra integral equation with partial integrals. Thus, the above integro-differential equations are reduce
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