Fredholm Property of Linear Equations with Partial Integrals in the Space of Integrable Functions
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
FREDHOLM PROPERTY OF LINEAR EQUATIONS WITH PARTIAL INTEGRALS IN THE SPACE OF INTEGRABLE FUNCTIONS V. A. Kalitvin Lipetsk State Pedagogical University 42, Lenina St., Lipetsk 398020, Russia [email protected]
UDC 517.968
We study the Fredholm property of linear equations with partial integrals and kernels in some classes. We show that a linear operator with partial integrals acts in the space of integrable functions and in the space of continuous functions defined in a square. We give an example of a non-Fredholm homogeneous linear equation of the second kind with partial integrals and continuous kernels. We obtain Fredholm criteria for linear operators and equations with partial integrals in the space of integrable functions. Bibliography: 6 titles.
1
Introduction
Linear equations with partial integrals have numerous applications, for example, in continuum mechanics, the theory of elastic shells, the theory of differential and integro-differential equations with partial derivatives. Solutions to such equations are understood in different ways. Therefore, the choice of function spaces depends on the definition of a solution [1]–[4]. For example, some evolution type mixed problems are reduced to the equations [1] 1
1 m(s − σ)x(t, σ)dσ+
l(t, τ )x(τ, s)dτ +
x(t, s)+ 0
t 1
−1
n(t, τ )m(s − σ)x(τ, σ)dτ dσ = g(t, s), 0 −1
whereas some quasistatic mixed problems and problems of elasticity, viscoelasticity, and fluid mechanics are reduced to the Volterra–Fredholm equation with partial integrals [1] t
1 m(s − σ)x(t, σ)dσ = g(t, s).
l(t, τ )x(τ, s)dτ +
λx(t, s) + 0
−1
The integral equation with partial integrals and difference kernels Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 137-142. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0967
967
t
t
l(t − τ )x(τ, s)dτ −
x(t, s) = 0
m(s − σ)x(t, σ)dσ+
n(t − τ, s − σ)x(τ, σ)dτ dσ + f (t, s) 0
Γ
Γ
is a generalization of an equation obtained from the initial-boundary value problems for the equation of gravity gyroscopic waves in the Boussinesq approximation [1]. In the above examples, solutions to equations can be found by the method of successive approximations. The important properties of linear integral equations are connected with the Fredholm property. As known, linear equations of the second kind with partial integrals are not Fredholm even in the general case of continuous kernels [1, 2]. A Fredholm criterion for the integral equation with partial integrals (1.1) x(t, s) = l(t, τ )x(τ, s)dτ + m(s, σ)x(t, σ)dσ + f (t, s) T
S
in the space Lp = Lp (T × S) (1 p < ∞) was established in [3, 4]. Here, T and S are compact sets in Rn , the space Rm is equipped with the continuous Lebesgue measure, and the kernels l(t, τ ) and m(s, σ) generate compact integral operators acting in Lp (T ) and Lp (S) respectively. This criterion means that Equation (1.1) is Fredholm in Lp (T × S) if and only if the Fredholm integral equations of the second ord
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