Short-interval real Volterra contraction

The space $C(J;E)$ with supremum norm is complete: a uniform Cauchy sequence has a pointwise limit by completeness of $E$, converges uniformly to it by its Cauchy estimate, and the epsilon/three argument using one continuous sequence member proves continuity of the limit. Thus it is a nonempty complete metric space. Norm-continuity of $P$ and continuity of $v$ imply continuity of $s\mapsto P(s)v(s)$. By [F1] the affine map $\mathrm{\Phi }(v)=W(\cdot -r)x+D_r(Pv+f)$ maps this whole space to itself.

For $v,w\in C(J;E)$, the norm bound gives $\mathrm{\parallel }\mathrm{\Phi }(v)-\mathrm{\Phi }(w){\mathrm{\parallel }}_C\le MKh\mathrm{\parallel }v-w{\mathrm{\parallel }}_C=q\mathrm{\parallel }v-w{\mathrm{\parallel }}_C$. Therefore [F2] gives one and only one fixed point. This estimate uses the length of the unoriented interval and consequently is identical in the backward direction.

At the fixed point, $\mathrm{\parallel }u{\mathrm{\parallel }}_C\le M\mathrm{\parallel }x\mathrm{\parallel }+M\mathrm{\parallel }f{\mathrm{\parallel }}_1+q\mathrm{\parallel }u{\mathrm{\parallel }}_C$, which yields the displayed estimate. Real linear combinations of solutions satisfy the same equation with the corresponding real linear combinations of their data, by real linearity of all operators and integration. Uniqueness identifies them with the constructed solutions; applying the bound to differences proves bounded dependence.