On Volterra{Fredholm Equations with Partial Integrals

Abstract

t, τ ∈ [0, a], s, σ ∈ [0, b]; l, m, n, and f are given measurable functions, and integrals are treated in the Lebesgue sense. Numerous problems of continuous mechanics [1–3] and a number of other problems can be reduced to such equations. The properties of Eqs. (1) and (2) are substantially different. In particular, Eq. (1) is a Fredholm equation of index zero under natural constraints imposed on the kernels; however, Eq. (2) is not so even in the general case of bounded kernels. Let X be a complex Banach space. For a bounded linear operator R in X, we denote the spectral radius by r(R), the dimension of the kernel by n(R), and that of the cokernel by d(R). The operator R is said to be Fredholm if its range R(X) is closed and both n(R) and d(R) are finite. A Fredholm operator of index zero is defined as a Fredholm operator such that n(R) = d(R). These two properties of the operator R are equivalent to the corresponding properties of the adjoint operator R∗ [4, p. 295 of the Russian translation] and are preserved under compact perturbations. Such perturbations also preserve the index ind(R) = n(R)−d(R) of a Fredholm operator [4, p. 300]. The equation x−Rx = f is referred to as a Fredholm equation (respectively, a Fredholm equation of index zero) if I − R is a Fredholm operator (respectively, a Fredholm operator of index zero), where I is the identity operator in X.