On the theory of partial integral operators (Q1182516)

The authors study operators of the form \[ K:=C+L+M+N,\tag{+} \] where \[ Cx(t,s):=c(t,s)x(t,s), \] \[ Lx(t,s):=\int_ Sl(t,s,\sigma)x(t,\sigma)d\nu(\sigma), \] \[ Mx(t,s):=\int_ Tm(t,s,\tau)x(\tau,s)d\mu(\tau) \] and \[ Nx(t,s):=\int_ T\int_ Sn(t,s,\tau,\sigma)\times(\tau\sigma)d(\mu\times\nu)(\tau\sigma). \] Here \(T\) and \(S\) are arbitrary nonempty sets, \(\mu\) and \(\nu\) separable measures, \(\mu\times\nu\) the product measure and \(c(t,s),\) \(l(t,s,\sigma)\), \(m(t,s,\tau)\) and \(n(t,s,\tau,\sigma)\) are measurable functions. Operators of the form (+) are called partial integral operators (PIO). Several theorems on the continuity, the regularity, the duality theory and the algebra of PIO's are studied. The existence and the uniqueness of a solution of linear equations of Volterra type \((I-L-M-N)x=f\) are proved. An application to a linear integral equation, occuring in the mechanics of continuous media, is given.