On Linear Singular Functional-differential Equations in One Functional Space

The case where K and S are continuous on Lp[0,1] operators is well studied (see, e.g., [1] and the references therein). Here we suppose that the functions K(t,s) and B(t) may be nonintegrable at t = 0. More precisely, we will formulate conditions on operators K and S in Sections 2 and 3. Under such conditions, those operators are not bounded on L[0,1] and one has to choose other functional spaces for studying (1.1). We propose a space of integrable functions on [0,1] and show that it may be useful in such a case. We call ∆ p ψ space the space of all measurable functions y : [0,1] → Rn, for which