Root bounds and $C^1$ dependence for the perturbed scalar recurrence $v^2-a-u-w+e(u,v,w)=0$

Bounds for the defect: on the cube, [F1] gives $\mathrm{\mid }V(v,w)-w\mathrm{\mid }\le \epsilon ,\mathrm{\mid }V(u,v)-v\mathrm{\mid }\le \epsilon ,\mid {\mathrm{\partial }}_u(V(u,v)-v)\mid \le \frac{\epsilon }{1-\epsilon },\mid {\mathrm{\partial }}_v(V(u,v)-v)\mid \le \frac{\epsilon }{1-\epsilon },$ and the corresponding partials of $V(v,w)-w$ in $(v,w)$ are bounded by $\epsilon \mathrm{/}(1-\epsilon )$; [F2] gives $\mathrm{\mid }B(u,V(u,v))-(V(u,v{)}^2-a-u)\mathrm{\mid }\le \epsilon$ and partial bounds $\epsilon$ plus Lipschitz factors $2\mathrm{\mid }V\mathrm{\mid }+1\le 2(R+1)+1$ from the derivative bounds of $B$ and [F1]. Since $\mathrm{\mid }2v\mathrm{\mid }\le 2R$, every term of $e$ and every first partial derivative of $e$ (computed by the chain rule and product rule) is bounded by a sum of finitely many terms each $O(\epsilon )$, so there is $C=C(a,R)$ with $\mathrm{\parallel }e{\mathrm{\parallel }}_{C^1}\le C\epsilon$; the factors $1\mathrm{/}(1-\epsilon )$ are bounded by $6\mathrm{/}5<2$ under the stated hypothesis. With $\eta :=C\epsilon \le 1$ clause 1 follows.

Endpoint values of $P$: fix $u,w\in [-R,R]$. By step 1.1, $\mathrm{\mid }e(u,v,w)\mathrm{\mid }\le \eta$ for every $v$. Using [F4], $P_{u,w}(l)=l^2-a-u-w+e(u,l,w)\le -(4\sqrt{a}-15)+\eta <0,P_{u,w}(R)=R^2-a-u-w+e(u,R,w)\ge (4\sqrt{a}+3)-\eta >0,$ because $\eta \le 1\le 25\le 4\sqrt{a}-15$ and $\eta \le 1<43\le 4\sqrt{a}+3$. Similarly $P_{u,w}(-R)\ge (4\sqrt{a}+3)-\eta >0$ and $P_{u,w}(-l)\le -(4\sqrt{a}-15)+\eta <0$.

Monotonicity: for $v\in [l,R]$, $P_{u,w}^{\mathrm{\prime }}(v)=2v+{\mathrm{\partial }}_{v}e(u,v,w)\ge 2l-\eta \ge 14-1>0,$ and for $v\in [-R,-l]$, $P_{u,w}^{\mathrm{\prime }}(v)=2v+{\mathrm{\partial }}_{v}e\le -2l+\eta <0$; hence $P_{u,w}$ is strictly increasing on $[l,R]$ and strictly decreasing on $[-R,-l]$.

Orbit form of the root equation: fix $(u,v,w)\in [-R,R{]}^3$ and put ${\delta }_B:=B(u,V(u,v))-(V(u,v{)}^2-a-u)$, so that $\mathrm{\mid }{\delta }_B\mathrm{\mid }\le \epsilon$ by [F2]. Expanding the two inverse-branch values, $V(u,v{)}^2=v^2+2v(V(u,v)-v)+(V(u,v)-v{)}^2,V(v,w)=w+(V(v,w)-w),$ one computes $B(u,V(u,v))-V(v,w)=v^2-a-u-w+[-(V(v,w)-w)+2v(V(u,v)-v)+(V(u,v)-v{)}^2+{\delta }_B]=P_{u,w}(v),$ because the bracket is exactly $e(u,v,w)$. Hence $P_{u,w}(v)=0$ if and only if $B(u,V(u,v))=V(v,w)$, which is clause 4.

Existence, uniqueness and margins of the root: by steps 2.1 and 2.2 and the intermediate value theorem [F5], $P_{u,w}$ has exactly one zero $M_{+}(u,w)$ in $[l,R]$ and exactly one zero $M_{-}(u,w)$ in $[-R,-l]$. For the margins, on $[l,R]$ one has $\mathrm{\mid }{P}_{u,w}^{\mathrm{\prime }}\mathrm{\mid }\le 2R+\eta \le 2\sqrt{a}+7$ by step 1.1, while $P_{u,w}(l)\le -(4\sqrt{a}-15)+\eta \le -(4\sqrt{a}-16)$ and $P_{u,w}(R)\ge (4\sqrt{a}+3)-\eta \ge 4\sqrt{a}+2$ by step 2.1. The mean value theorem applied on $[l,M_{+}]$ and on $[M_{+},R]$ therefore gives $M_{+}-l\ge \frac{4\sqrt{a}-16}{2\sqrt{a}+7},R-M_{+}\ge \frac{4\sqrt{a}+2}{2\sqrt{a}+7},$ and both right-hand sides are at least $3\mathrm{/}4$ because $\sqrt{a}\ge 10$: the first inequality is equivalent to $10\sqrt{a}\ge 85$ and the second to $10\sqrt{a}\ge 13$. At $a=100$ the two bounds read $24\mathrm{/}27$ and $42\mathrm{/}27$. The same computation on the interval $[-R,-l]$, where $P_{u,w}$ is decreasing and $P_{u,w}(-l)\le -(4\sqrt{a}-16)$, $P_{u,w}(-R)\ge 4\sqrt{a}+2$, gives $\mathrm{\mid }{M}_{-}\mathrm{\mid }-l\ge \frac{3}{4}$ and $R-\mathrm{\mid }{M}_{-}\mathrm{\mid }\ge \frac{3}{4}$; thus $s{M}_s-l\ge \frac{3}{4}$ and $R-s{M}_s\ge \frac{3}{4}$ for each sign $s$.

Regularity and derivative bounds: at a zero $v$ of $P_{u,w}$ with $sv\in [l,R]$, step 2.2 gives ${\mathrm{\partial }}_v{P}_{u,w}(v)\ge 2l-\eta >0$ (respectively $\le -(2l-\eta )$), so the implicit function theorem [F6], applied to the $C^1$ function $(u,w,v)\mapsto P_{u,w}(v)$ and combined with the uniqueness of step 3.1, makes $M_s$ of class $C^1$ with ${\mathrm{\partial }}_u{M}_s=-\frac{{\mathrm{\partial }}_{u}P}{{\mathrm{\partial }}_{v}P}=\frac{1-{\mathrm{\partial }}_{u}e}{2v+{\mathrm{\partial }}_{v}e},{\mathrm{\partial }}_w{M}_s=-\frac{{\mathrm{\partial }}_{w}P}{{\mathrm{\partial }}_{v}P}=\frac{1-{\mathrm{\partial }}_{w}e}{2v+{\mathrm{\partial }}_{v}e}\mathrm{.}$ Both numerators are bounded by $1+\eta$ in absolute value, and the denominator by $2l-\eta$ from below in absolute value; hence $\mathrm{\mid }{\mathrm{\partial }}_u{M}_s\mathrm{\mid },\mathrm{\mid }{\mathrm{\partial }}_w{M}_s\mathrm{\mid }\le (1+\eta )\mathrm{/}(2l-\eta )=d$, which is clause 2.

Open-neighborhood $C^2$ regularity for later applications: the “Regularity needed by the half-tail construction” paragraph in the proof of A $C^2$ map $C^1$-close to $F$ has a unique inverse branch for its first component, with $O(\epsilon )$ derivative bounds extends $V$ as a $C^2$ map to an open rectangle containing $[-R,R{]}^2$. Since $B$ is $C^2$ on $U$, the formula for $e$, hence $P$, is $C^2$ on an open neighborhood of the cube after shrinking the rectangle so that $(u,V(u,v))\in U$. The endpoint signs in step 2.1 and the derivative lower bound in step 2.2 have uniform strictly positive margins on the compact parameter square and the two root intervals. Continuity and compactness therefore preserve those signs and derivative bounds on a slightly enlarged open parameter rectangle. On that rectangle the intermediate value theorem and strict monotonicity give unique roots in the same two intervals; the implicit function theorem makes them $C^1$. Their first partials are the quotients in step 4.1 of $C^1$ functions composed with these $C^1$ roots, with nonzero denominator, so the partials are $C^1$ and the roots are $C^2$. This supplies $C^2$ extensions of $M_s$ on an open neighborhood of $[-R,R{]}^2$, as needed by the weighted half-tail lemma; the original derivative bounds on the closed square are unchanged.

Numerical inequalities: since $l\ge 7$ and $\eta \le 1$, $d\le (1+1)\mathrm{/}(14-1)=2\mathrm{/}13$; therefore $2d\le 4\mathrm{/}13<1$; with $\theta =1\mathrm{/}2$, $d(\theta +{\theta }^{-1})\le (2\mathrm{/}13)(5\mathrm{/}2)=5\mathrm{/}13<1$; and $d\mathrm{/}(1-d\theta )\le (2\mathrm{/}13)\mathrm{/}(1-1\mathrm{/}13)=2\mathrm{/}12=1\mathrm{/}6<2\mathrm{/}3$.

Conclusion: step 1.1 proves clause 1; steps 2.1, 2.2, 3.1 and 4.1 prove clause 2; step 5.2 proves clause 3; step 2.3 proves clause 4.