A multiparameter family of solutions to the Volterra linear integral equation of the first kind (Q6638442)

The author studies the following Volterra equation of the first kind with a singularity and a sufficiently smooth kernel in some Banach space with weight: \N\[\N\int_0^x K(x,t)\varphi(t)\, dt =f(x), \quad 0\leq x\leq \delta. \N\]\NThe kernel \(K(x,t)\) is a given function with values in \(L(E)\), the space of linear bounded operators on the Banach space \(E\) and in addition to differentiability assumptions it is assumed that for some \(q\geq 1\) and non-negative integer \(n\) the limits \(\lim_{x\to +0} x^{-nq}K(x,x)=C_0\), \(\lim_{x\to +0} x^{-(n-1)q}(-1)K^{(1)}_t(x,x)=C_1\), \(\ldots\), \(\lim_{x\to +0} x^0(-1)^nK^{(n)}_{t\ldots t}(x,x)=C_n\) exist, that \(C_0\) is invertible and that the eigenvalues, eigenvectors and adjoined vectors of the operator pencil \(B_\nu = \sum_{p=0}^{n-1} C_{n-1-p} (-1)^p \nu^p -C_n \frac 1\nu\) satisfy certain assumptions. The function \(f\) is assumed to belong to the space \(M_{q,\nu}^{1,0}\). \N\NIt is shown there is a multi-parameter family of solutions in the space \( M_{q,\nu}^{0,-(n+1)q}\) where \(\nu <0\) and \(M_{q,\nu}^{k,\alpha}= \{ \varphi(x) : \varphi^{(i)}(x)= x^{\alpha-qi} \exp(\nu\int_x^\delta t^{-q}\,dt)\omega_i(x), \; 0\leq x\leq \delta, \; \sup_{x\in [0,\delta]} \|\omega_i(x)\|_E <\infty, 1\leq i\leq k)\}\).\N\NThe proof is based on reducing the equation to an integro-differential equation using integration by parts.