Extensions on a local convergence result by Dennis and Schnabel for Newton's method with applications

Theorem 1.Let $D\ne \varnothing$ an open convex subset of a Banach space $E_1$ with correspondence in a Banach space $E_2$, the nonlinear operator H is $m=3,4,5,\dots$ times continuously differentiable operator in the sense of Fréchet from D into $E_2$. Suppose that ${\varphi }_{\star }$ is a solution of $H(\varphi )=0$ such that $B({\varphi }_{\star },r)\subseteq D$ and $\Vert {[H^{\prime }({\varphi }_{\star })]}^{-1}{H}^{(i)}({\varphi }_{\star })\Vert \le {\alpha }_i$ (for $2\le i\le m-1$) with $r,{\alpha }_i>0$ and the operator ${[H^{\prime }({\varphi }_{\star })]}^{-1}$ exists. Consider that conditions (4) and (6), ${\omega }_0(t)\le \bar{\omega }(t)$ for each $t\in [0,r)$, are satisfied and for the equation

Proof.Let $ϵ=\min \{r,{\bar{R}}_0\}$. First, we prove, for all $\varphi \in B({\varphi }_{\star },ϵ)$, that there exists ${[H^{\prime }(\varphi )]}^{-1}$ and $\Vert {[H^{\prime }(\varphi )]}^{-1}{H}^{\prime }({\varphi }_{\star })\Vert \le \frac{1}{1-{\omega }_0({\bar{R}}_0)}$. For this, we use (4) to obtain

It follows from (10) and the Banach lemma on invertible operators,¹⁵ that ${[H^{\prime }(\varphi )]}^{-1}$ exists and $\Vert {[H^{\prime }(\varphi )]}^{-1}{H}^{\prime }({\varphi }_{\star })\Vert \le \frac{1}{1-{\omega }_0(\Vert \varphi -{\varphi }_{\star }\Vert )}\le \frac{1}{1-{\omega }_0({\bar{R}}_0)}$.

As ${\varphi }_0\in B({\varphi }_{\star },ϵ)$, then the operator ${\mathrm{\Lambda }}_0={[H^{\prime }({\varphi }_0)]}^{-1}$ exists,

Thus, we obtain that

Hence, we conclude ${\varphi }_n\in B({\varphi }_{\star },ϵ)$ and ${\lim }_{n\to +\infty }{\varphi }_n={\varphi }_{\star }$. On the other hand, (9) follows from (8) and

Remark 1. 

• (a)

  The results hold with ${\omega }_0$ replacing $\bar{\omega }$ if $\bar{\omega }(t)\le {\omega }_0(t)$ for each $t\in [0,r)$.

• (b)

  As we can see in (9) the Q-order of convergence is at least two⁵ for Newton's method. If $ϵ<1$, then we obtain

  $$\Vert {\varphi }_n-{\varphi }_{\star }\Vert <\frac{1}{ϵ}\Vert {\varphi }_{n-1}-{\varphi }_{\star }{\Vert }^2\le {\left(\frac{1}{ϵ}\right)}^{1+2+\dots +2^{n-1}}\Vert {\varphi }_0-{\varphi }_{\star }{\Vert }^{2^n}<{\left(\sqrt{ϵ}\right)}^{2^n}\sqrt{ϵ}.$$
  Hence, the R-order of convergence is at least two for Newton's method.⁵

• (c)

  The corresponding equations in Reference 14 are obtained from (8) by setting $\bar{\omega }(t)=\omega (t)={\omega }_0(t)$ and $\bar{\omega }(t)=\omega (t)$. Denote by $r_0$, $R_0$, the corresponding solutions, respectively. Then, we have

  $$r_0\le R_0\le {\bar{R}}_0.$$ ()
  Moreover, the corresponding upper error bounds on $\Vert {\varphi }_n-{\varphi }_{\star }\Vert$ are
  $$c_n=\frac{d_n^2}{1-\omega (\Vert {\varphi }_n-{\varphi }_{\star }\Vert )},$$
  $$b_n=\frac{d_n^2}{1-{\omega }_0(\Vert {\varphi }_n-{\varphi }_{\star }\Vert )},$$
  $$a_n=\frac{d_n^1}{1-{\omega }_0(\Vert {\varphi }_n-{\varphi }_{\star }\Vert )},$$
  where
  $$d_n^2=\frac{m-1}{m}\left({\displaystyle \sum _{i=1}^{m-2}}\frac{{\alpha }_{i+1}}{i!}\Vert {\varphi }_n-{\varphi }_{\star }{\Vert }^{i-1}+\frac{\Vert {\varphi }_n-{\varphi }_{\star }{\Vert }^{m-2}}{(m-1)!}\omega (\Vert {\varphi }_{\star }\Vert +\Vert {\varphi }_n-{\varphi }_{\star }\Vert )\right)$$
  and
  $$d_n^1=\frac{m-1}{m}\left({\displaystyle \sum _{i=1}^{m-2}}\frac{{\alpha }_{i+1}}{i!}\Vert {\varphi }_n-{\varphi }_{\star }{\Vert }^{i-1}+\frac{\Vert {\varphi }_n-{\varphi }_{\star }{\Vert }^{m-2}}{(m-1)!}\bar{\omega }(\Vert {\varphi }_{\star }\Vert +\Vert {\varphi }_n-{\varphi }_{\star }\Vert )\right),$$
  so
  $$a_n\le b_n\le c_n.$$ ()
  Hence, the new radii of convergence are larger and the error bounds are tighter under our new approach (see also the numerical examples).

Remark 2. 

• (a)

  Using only (6), we have that

  $$\begin{array}{ll}\Vert I-{[H^{\prime }({\varphi }_{\star })]}^{-1}{H}^{\prime }(\varphi )\Vert & \le ‖{[H^{\prime }({\varphi }_{\star })]}^{-1}{\int }_0^1{H}^{\prime \prime }(\varphi +\zeta ({\varphi }_{\star }-\varphi ))d\zeta ({\varphi }_{\star }-\varphi )‖\\ & \le \left[‖{[H^{\prime }({\varphi }_{\star })]}^{-1}{\int }_0^1\left({\displaystyle \sum _{i=2}^{m-1}}\frac{{(\zeta -1)}^{i-2}}{(i-2)!}{H}^{(i)}({\varphi }_{\star }){({\varphi }_{\star }-\varphi )}^{i-2}d\zeta \right)({\varphi }_{\star }-\varphi )‖\right.\\ & +\frac{1}{(m-3)!}{\int }_0^1{\int }_0^1\Vert {[H^{\prime }({\varphi }_{\star })]}^{-1}{H}^{(m)}\left({\varphi }_{\star }+s(\zeta -1)({\varphi }_{\star }-\varphi )\right)\Vert \\ & \times {(1-s)}^{m-3}{(1-\zeta )}^{m-2}\Vert {\varphi }_{\star }-\varphi {\Vert }^{m-1}dsd\zeta ]\\ & \le [{\displaystyle \sum _{i=2}^{m-1}}\frac{1}{(i-1)!}{\alpha }_i\Vert {\varphi }_{\star }-\varphi {\Vert }^{i-1}+\frac{1}{(m-3)!}\bar{\omega }(\Vert {\varphi }_{\star }\Vert +ϵ)\left({\int }_0^1{(1-s)}^{m-3}ds\right)\\ & \times \left({\int }_0^1{(1-\zeta )}^{m-2}d\zeta \right)\Vert {\varphi }_{\star }-\varphi {\Vert }^{m-1}]\\ & \le [{\displaystyle \sum _{i=1}^{m-2}}\frac{1}{i!}{\alpha }_{i+1}{ϵ}^{i-1}+\frac{1}{(m-1)!}\bar{\omega }(\Vert {\varphi }_{\star }\Vert +ϵ){ϵ}^{m-2}]ϵ,\end{array}$$
  since
  $$\begin{array}{ll}{H}^{\prime \prime }\left(\varphi +\zeta ({\varphi }_{\star }-\varphi )\right)& ={\displaystyle \sum _{i=2}^{m-1}}\frac{{(\zeta -1)}^{i-2}}{(i-2)!}{H}^{(i)}({\varphi }_{\star }){({\varphi }_{\star }-\varphi )}^{i-2}\\ & +\frac{1}{(m-3)!}{\int }_0^1{H}^{(m)}\left({\varphi }_{\star }+s(\zeta -1)({\varphi }_{\star }-\varphi )\right){(1-s)}^{m-3}{(\zeta -1)}^{m-2}{({\varphi }_{\star }-\varphi )}^{m-2}ds,\end{array}$$
  $$\begin{array}{ll}‖{\int }_0^1({\displaystyle \sum _{i=2}^{m-1}}\frac{{(\zeta -1)}^{i-2}}{(i-2)!}{H}^{(i)}({\varphi }_{\star }){({\varphi }_{\star }-\varphi )}^{i-2}d\zeta ({\varphi }_{\star }-\varphi )‖& \le {\displaystyle \sum _{i=2}^{m-1}}\frac{1}{(i-1)!}{\alpha }_i\Vert {\varphi }_{\star }-\varphi {\Vert }^{i-1},\\ ‖H^{(m)}\left({\varphi }_{\star }+s(\zeta -1)({\varphi }_{\star }-\varphi )\right)‖& \le \bar{\omega }(\Vert {\varphi }_{\star }\Vert +ϵ),\end{array}$$
  so
  $$\Vert I-{[H^{\prime }({\varphi }_{\star })]}^{-1}{H}^{\prime }(\varphi )\Vert \le [{\displaystyle \sum _{i=1}^{m-2}}\frac{{\alpha }_{i+1}}{i!}{r}_0^{i-1}+\frac{r_0^{m-2}}{(m-1)!}\bar{\omega }(\Vert {\varphi }_{\star }\Vert +{\bar{r}}_0)]{\bar{r}}_0=\delta <1,$$ ()
  provided that $\delta \in \left(0,\frac{m}{2m-1}\right)$. Equation (13) is now presented by
  $$\left({\displaystyle \sum _{i=1}^{m-2}}\frac{{\alpha }_{i+1}}{i!}{t}^{i-1}+\frac{t^{m-2}}{(m-1)!}\bar{\omega }(\Vert {\varphi }_{\star }\Vert +t)\right)t-\delta =0,$$ ()
  provided that Equation (14) has a smallest positive solution denoted by ${\bar{r}}_0$. Then, the error estimate is
  $$\begin{array}{ll}\Vert {\varphi }_{\star }-{\varphi }_n\Vert & \le \frac{1}{1-\delta }({\displaystyle \sum _{i=2}^{m-1}}\frac{i-1}{i!}{\alpha }_i{r}_0^{i-2}+\frac{m-1}{m!}\bar{\omega }(\Vert {\varphi }_{\star }\Vert +{\bar{r}}_0))\Vert {\varphi }_{\star }-{\varphi }_n{\Vert }^2\\ & \le \frac{\delta (m-1)}{(1-\delta )m}\Vert {\varphi }_{\star }-{\varphi }_{n-1}\Vert :=b_{n,\delta }.\end{array}$$ ()

• (b)

  The corresponding to (14) equation in Reference 6 is given for ${\delta }_0\in (0,\frac{m}{2m-1})$ by

  $$\left({\displaystyle \sum _{i=1}^{m-2}}\frac{{\alpha }_{i+1}}{i!}{t}^{i-1}+\frac{t^{m-2}}{(m-1)!}\omega (\Vert {\varphi }_{\star }\Vert +t)\right)t-{\delta }_0=0$$ ()
  and the error estimates
  $$\begin{array}{ll}\Vert {\varphi }_{\star }-{\varphi }_n\Vert & \le \frac{1}{1-{\delta }_0}\left({\displaystyle \sum _{i=2}^{m-1}}\frac{i-1}{i!}{\alpha }_i{r}_0^{i-2}+\frac{m-1}{m!}\omega (\Vert {\varphi }_{\star }\Vert +{\overline{\stackrel{‾}{r}}}_0)\right)\Vert {\varphi }_{\star }-{\varphi }_n{\Vert }^2\\ & \le \frac{{\delta }_0(m-1)}{(1-{\delta }_0)m}\Vert {\varphi }_{\star }-{\varphi }_{n-1}\Vert :=b_{n,{\delta }_0},\end{array}$$ ()
  where ${\overline{\stackrel{‾}{r}}}_0$ is the least positive solution of (16) and ${\delta }_0$ is as $\delta$ in (13) but ${\bar{r}}_0$, $\omega$ are ${\overline{\stackrel{‾}{r}}}_0$ and $\omega$, respectively.

Then, it follows from (3), (4), (6), (7), and the above definitions that the new error bounds are at least as tight whereas the new convergence radii at least as large as the ones in Reference 14 or Reference 6 (see also the numerical examples). We can certainly set $\delta ={\delta }_0={\lambda }_0$. In particular, we have

Remark 3. 

• (a)

  We can develop another approach as follows: there exist two nondecreasing continuous functions ${\omega }_1:[0,+\infty )\to ℝ$ and ${\bar{\omega }}_1(t):[0,\rho )\to [0,+\infty )$ with ${\omega }_1(0)\ge 0$ and ${\bar{\omega }}_1(0)\ge 0$ such that

  $$\Vert {[H^{\prime }({\varphi }_{\star })]}^{-1}(H^{\prime }(\psi )-H^{\prime }(\varphi ))\Vert \le {\omega }_1(\Vert \psi -\varphi \Vert ),\text{for each}\varphi \text{and}\psi \in D$$ ()
  and
  $$\Vert {[H^{\prime }({\varphi }_{\star })]}^{-1}(H^{\prime }(\psi )-H^{\prime }(\varphi ))\Vert \le {\bar{\omega }}_1(\Vert \psi -\varphi \Vert ),\text{for each}\varphi \text{and}\psi \in D_0.$$ ()
  Note that in general
  $${\omega }_0(t)\le {\omega }_1(t)\text{and}{\bar{\omega }}_1(t)\le {\omega }_1(t)\text{for each}t\in [0,\rho ]$$ ()
  hold and $\frac{{\omega }_0}{{\omega }_1}$ can be arbitrarily small.^{3, 8}

  We get the estimate using (4) and (22)

  $$\begin{array}{ll}\Vert {\varphi }_{n+1}-{\varphi }_{\star }\Vert & =\Vert {\varphi }_n-{\varphi }_{\star }-H^{\prime }{({\varphi }_n)}^{-1}H({\varphi }_n)\Vert \\ & \le \Vert H^{\prime }{({\varphi }_n)}^{-1}{H}^{\prime }({\varphi }_{\star })\Vert {\int }_0^1{H}^{\prime }{({\varphi }_{\star })}^{-1}(H^{\prime }({\varphi }_{\star }+\theta ({\varphi }_n-{\varphi }_{\star }))\\ & -H^{\prime }({\varphi }_n))d\theta ({\varphi }_n-{\varphi }_{\star })\Vert \\ & \le \frac{{\int }_0^1{\omega }_1((1-\theta )\Vert {\varphi }_n-{\varphi }_{\star }\Vert )d\theta }{1-{\omega }_0(\Vert {\varphi }_n-{\varphi }_{\star }\Vert )}\Vert {\varphi }_n-{\varphi }_{\star }\Vert \\ & =e_n^2\Vert {\varphi }_n-{\varphi }_{\star }\Vert ,\end{array}$$
  where
  $$e_n^2=\frac{{\int }_0^1{\omega }_1((1-\theta )\Vert {\varphi }_n-{\varphi }_{\star }\Vert )d\theta }{1-{\omega }_0(\Vert {\varphi }_n-{\varphi }_{\star }\Vert )}$$ ()
  and ${\rho }_2$ is a radius of convergence provided by the smallest positive solution of equation
  $${\int }_0^1{\omega }_1((1-\theta )t)dt+{\omega }_0(t)-1=0.$$ ()

  But if we use (4) and (21), we get

  $$\begin{array}{ll}\Vert {\varphi }_{n+1}-{\varphi }_{\star }\Vert & \le \frac{{\int }_0^1{\bar{\omega }}_1((1-\theta )t)d\theta }{1-{\omega }_0(t)}\Vert {\varphi }_n-{\varphi }_{\star }\Vert \\ & =e_n^1\Vert {\varphi }_n-{\varphi }_{\star }\Vert ,\end{array}$$
  where
  $$e_n^1=\frac{{\int }_0^1{\bar{\omega }}_1((1-\theta )t)d\theta }{1-{\omega }_0(t)},$$ ()
  and ${\rho }_1$ denotes the least positive solution of
  $${\int }_0^1{\bar{\omega }}_1((1-\theta )t)dt+{\omega }_0(t)-1=0.$$ ()

  If we only use (20), we get

  $$\begin{array}{ll}\Vert {\varphi }_{n+1}-{\varphi }_{\star }\Vert & \le \frac{{\int }_0^1{\omega }_1((1-\theta )t)d\theta }{1-{\omega }_1(t)}\Vert {\varphi }_n-{\varphi }_{\star }\Vert \\ & =e_n^3\Vert {\varphi }_n-{\varphi }_{\star }\Vert ,\end{array}$$
  where
  $$e_n^3=\frac{{\int }_0^1{\omega }_1((1-\theta )t)d\theta }{1-{\omega }_1(t)},$$ ()
  and ${\rho }_3$ denotes the least positive solution of
  $${\int }_0^1{\omega }_1((1-\theta )t)dt+{\omega }_1(t)-1=0.$$ ()

  Then, we have

  $${\rho }_3\le {\rho }_2\le {\rho }_1$$ ()
  and
  $$e_n^1\le e_n^2\le e_n^3.$$ ()

• (b)

  Uniqueness of the solution results can be found in References 3-5, 7, 8, 11, and 14.

Proposition 1.Suppose: equation $H(\varphi )=0$ has a simple solution ${\varphi }_{\star }\in D$ such that (4) holds; and there exists $a\ge 0$ such that

Then, ${\varphi }_{\star }$ is the only solution of equation $H(\varphi )=0$ in $\mathrm{\Omega }=\bar{B}({\varphi }_{\star },a)\cap D$.

Proof.Set $S={\int }_0^1{H}^{\prime }({\varphi }_{\star \star }+\theta ({\varphi }_{\star }-{\varphi }_{\star \star }))d\theta$, where ${\varphi }_{\star \star }\in \mathrm{\Omega }$ with $H({\varphi }_{\star \star })=0$. Then, by (4) and (31), we get in turn that

Example 1.Let $E_1=E_2={ℝ}^3$, $D=B(0,1)$, and ${\varphi }_{\star }={(0,0,0)}^T$. Consider the operator H from D to $E_2$ by