Bounded real-linear perturbations preserve solution estimates

Assume Countable Choice, with the real Banach and Bochner conventions of Continuous free convolution on a finite time interval. Let $E$ be a complex Banach space regarded as real, $0<T<\mathrm{\infty }$, $W(t)$ a strongly continuous bounded group, and $P:[0,T]\to {\mathcal{B}}_{\mathbb{R}}(E)$ a norm-continuous family of bounded real-linear operators. Write $M={\sup }_{\mathrm{\mid }t\mathrm{\mid }\le T}\mathrm{\parallel }W(t)\mathrm{\parallel }\ge 1$, $K={\sup }_{t\in [0,T]}\mathrm{\parallel }P(t)\mathrm{\parallel }$ and $C_0=M{e}^{MKT}$. The Volterra equation $S(t,r)x=W(t-r)x+{\int }_r^{t}W(t-\rho )P(\rho )S(\rho ,r)xd\rho$ has a unique evolution on $[0,T{]}^2$. It is constructed by A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point through Short-interval real Volterra contraction, is jointly strongly continuous and real-linear, satisfies $S(r,r)=I$, $S(t,q)S(q,r)=S(t,r)$ and $S(t,r{)}^{-1}=S(r,t)$, and obeys $\mathrm{\parallel }S(t,r)\mathrm{\parallel }\le M{e}^{MK\mathrm{\mid }t-r\mathrm{\mid }}$ by Global real evolution, cocycle and inverse and Gronwall's integral inequality with variable and constant coefficients.

For every strongly measurable $f\in L^1(0,T;E)$ the forced formula is $u(t)=S(t,r)x+{\int }_r^{t}S(t,\rho )f(\rho )d\rho$, with $\mathrm{\parallel }u{\mathrm{\parallel }}_{C([0,T];E)}\le C_0(\mathrm{\parallel }x\mathrm{\parallel }+\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1})$, as proved in Bochner variation of constants for real evolution. It uniquely solves $u(t)=W(t-r)x+{\int }_r^{t}W(t-\rho )(P(\rho )u(\rho )+f(\rho ))d\rho$. All integrals are Bochner integrals in $E$, oriented by ${\int }_r^t=-{\int }_t^r$ for $t<r$; their norm and real-linearity rules are Bochner integral norm inequality and Linearity of the Bochner integral.

If the actual free orbit map $H_{a}y(t)=W(t-a)y$ and retarded $L^1(I;E)$ forcing map $D_{a}g(t)={\int }_a^{t}W(t-\rho )g(\rho )d\rho$ are bounded into an additional compatible spacetime Banach norm $Y_I$ on a forward interval $I=[a,b]\subset [0,T]$, in exactly the common-ambient and map sense of Transfer of a compatible spacetime norm through the free forcing map, with bounds $A_I$ and $B_I$, then $v=u{\mathrm{\mid }}_I$ has that norm. Put $f_I=f{\mathrm{\mid }}_I$ and $C_I=M{e}^{MK(b-a)}$. The estimate in interval data is $\mathrm{\parallel }v{\mathrm{\parallel }}_{Y_I}\le A_I\mathrm{\parallel }u(a)\mathrm{\parallel }+B_I((b-a)K{C}_I(\mathrm{\parallel }u(a)\mathrm{\parallel }+\mathrm{\parallel }{f}_I{\mathrm{\parallel }}_{L^1(I;E)})+\mathrm{\parallel }{f}_I{\mathrm{\parallel }}_{L^1(I;E)})$. When $r=a$, one has $u(a)=x$, so this gives the estimate with initial datum $x$ and forcing restricted to $I$. For arbitrary $r$, one may use the global bound $\mathrm{\parallel }u(a)\mathrm{\parallel }\le C_0(\mathrm{\parallel }x\mathrm{\parallel }+\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(0,T;E)})$ on its right-hand side. For fixed free-map bounds the constant in interval data depends only on the interval, $M$ and $K$; uniform interval estimates require uniform free-map bounds. The corresponding backward assertion uses endpoint datum $u(b)$, forcing restricted to $I$, and the reflected free maps specified by the same supplier. One may not move multiplication by $i$ through a merely real-linear $S(t,r)$: a forcing $-if$ enters as ${\int }_r^{t}S(t,\rho )(-if(\rho ))d\rho$.