1 Introduction and main result

1.1 Background and main problem

Let \(F:D(F)\subset B_1\rightarrow B_2\) be a nonlinear operator between Banach spaces \(B_1\) and \(B_2.\) Here D(F) represents the domain of F. In this paper, we are concerned about the solution of following equation:

$$\begin{aligned} F(u)=v,\quad u\in D(F),\ v\in {B_2}. \end{aligned}$$
(1.1)

For practical applications, it is known that v is never available. Instead some perturbed data \(v^{\delta }\) fulfilling \(\Vert v^{\delta }-v\Vert \le \delta \) is available, where \(\delta >0.\) Consequently, (1.1) is ill-posed due to no continuous dependence between the solution and data. We assume that (1.1) has a solution. Let it be represented by \(u^{\dagger }.\) To approximately solve (1.1), a number of regularization methods are known in Hilbert as well as Banach spaces (cf. [2,3,4,5,6, 8, 10, 17]). A well known and classical regularization method is the Landweber iteration [7, 17]:

$$\begin{aligned} J_p(u_{r+1})=J_p(u_{r})-\mu F'(u_r)^*j_p(F(u_{r})-v),\quad u_{r+1}=J_q^*(J_p(u_{r+1})),\ r\ge 0.\nonumber \\ \end{aligned}$$
(1.2)

Here \(F'(u_r)\) is the Fréchet derivative of F at \(u_r,\) \(F'(u_r)^*\) is the adjoint of \(F'(u_r),\) \(u_0\) is an initial guess of the exact solution \(u^{\dagger },\) \(p>1,\) p and q are conjugate exponents, and \(J_p : B_1 \rightarrow 2^{B_1^*}\) defined as \(J_p(u) := \{ u^* \in B_1^*\ | \ \langle u, u^*\rangle = \Vert u\Vert ^p,\Vert u^*\Vert = \Vert u\Vert ^{p-1}\}\) is the duality mapping of \(B_1\) with the gauge function \(s \rightarrow s^{p-1}.\) For a gauge function \(s\rightarrow s^{q-1},\) the corresponding duality mapping \(J_{q}^*:B_1^*\rightarrow B_1\) is the inverse of \(J_{p}.\) The convergence analysis of (1.2) is well studied by utilizing a tangential cone condition [4] in Hilbert as well as Banach spaces [10, 17]. In addition, the convergence rates for this method have been obtained by incorporating the source conditions and variational inequalities [10, 17]. Recently, de Hoop et al. [7] studied the convergence analysis of (1.2) by utilizing the following Hölder-type stability:

$$\begin{aligned} \Delta _p(u, {\tilde{u}})\le {\mathfrak {A}}^p\Vert F(u)-F({\tilde{u}})\Vert ^{\frac{p(1+\epsilon )}{2}},\quad \forall u,\ {\tilde{u}}\in {\mathcal {B}}_{\rho }(u^{\dagger }), \end{aligned}$$
(1.3)

where \(\epsilon \in [0, 1],\) \(p>1,\) \({\mathfrak {A}}>0,\) \(\Delta _p(u, {\bar{u}})\) is the Bregman distance of \({\bar{u}}\) from u and it is given by

$$\begin{aligned} \Delta _p(u, {\bar{u}}):= p^{-1}\Vert {\bar{u}}\Vert ^p-p^{-1}\Vert u\Vert ^p-\langle J_p(u), {\bar{u}}-u\rangle . \end{aligned}$$

Here, we assume that \({\mathcal {B}}_{\rho }(u^{\dagger }):=\{{\bar{u}}\in B_1:\Delta _p({\bar{u}}, u^{\dagger })\le \rho \}\subset D(F)\) for some \(\rho >0.\) However, the convergence analysis of the perturbed version of (1.2), i.e.,

$$\begin{aligned} J_p(u^{\delta }_{r+1})=J_p(u^{\delta }_{r})-\mu F'(u_r^{\delta })^*j_p(F(u_{r}^{\delta })-v^{\delta }),\quad u_{r+1}^{\delta }=J_q^*(J_p(u_{r+1}^{\delta })),\ r\ge 0,\nonumber \\ \end{aligned}$$
(1.4)

is not yet studied in the literature via the stability estimates (1.3). In this paper, we fill this important gap in the literature. The importance of studying the convergence analysis of an iterative method via stability estimates is that these provide the convergence rates without the requirement of any additional smoothness condition. This is in contrary to the standard analysis. We refer to [11,12,13,14,15] for studying the convergence analysis of several other regularization methods through stability estimates. Very recently, Jin [9] studied the convergence rates for the method (1.4) in Banach spaces for linear ill-posed problems with perturbed data.

To this end, we recall some basic definitions and known results related to our work (see [11, 17] for more details). For \(u\in B_1\) and \(\zeta \in B_1^*,\) we write \(\langle \zeta , u\rangle =\zeta (u)\) for the duality pairing. In this paper, in order to ensure the well-definedness of the method (1.4), we require the following definitions of the modulus of convexity \(\delta _{B_1}(\cdot )\) and the modulus of smoothness \(\rho _{B_1}(\cdot )\):

$$\begin{aligned} \delta _{B_1}(\epsilon ):= & {} \inf _{0\le \epsilon \le 2} \big \{(2-\Vert u+{\bar{u}}\Vert ): \ u, {\bar{u}} \in {\mathbb {S}}, \Vert u-{\bar{u}}\Vert \ge \epsilon \big \},\\ \rho _{B_1}(\tau ):= & {} \sup _{\tau \ge 0} \big \{(\Vert u+\tau {\tilde{u}}\Vert + \Vert u-\tau {\tilde{u}}\Vert - 2): \ u, {\tilde{u}} \in {\mathbb {S}}\big \}. \end{aligned}$$

Here \({\mathbb {S}}\) denotes the boundary of the unit sphere in \(B_1.\) We say that \(B_1\) is p-convex if \(\delta _{B_1}(\epsilon ) \ge {\mathcal {C}}_1 \epsilon ^p\) for all \(\epsilon \in [0, 2],\) where \(p\ge 0\) and \({\mathcal {C}}_1>0.\) Further, we say that \(B_1\) is q-smooth if \(\rho _{B_1}(\tau ) \le {\mathcal {C}}_2 \tau ^q\) for all \(\tau \ge 0,\) where \(q>1\) and \({\mathcal {C}}_2>0.\) Also, \(B_1\) is uniformly convex if for any \(\epsilon \in (0, 2],\) \(\delta _{B_1}(\epsilon )>0\) and it is uniformly smooth if \(\lim _{\tau \rightarrow 0} \rho _{B_1}(\tau )\tau ^{-1}=0.\) We note that \(B_1\) is uniformly convex if and only if \(B_1^*\) is uniformly smooth. Moreover, any uniformly convex or uniformly smooth Banach space is reflexive. We emphasize that uniform smoothness of \(B_1\) guarantees that \(J_p(u)\) is single valued for all \(u\in B_1,\) i.e., the method (1.4) becomes well-defined.

Finally, we recall a known result that will be utilized in our work.

Lemma 1

([7, 16]). Let \(B_1\) be a uniformly convex and uniformly smooth Banach space. Then,  for all \(u, {\bar{u}} \in B_1\) and \(u^*, {\bar{u}}^*\in B_1^*,\) we have : 

  1. (1)

    \(\Delta _p(u, {\bar{u}}) \ge 0\) and \(\Delta _p(u, {\bar{u}}) = 0 \iff u = {\bar{u}}.\)

  2. (2)

    If \(B_1\) is p-convex,  then \(\Delta _p(u, {\bar{u}}) \ge {\mathcal {C}}_3 p^{-1}\Vert u-{\bar{u}}\Vert ^p,\) where \({\mathcal {C}}_3 > 0\) is a constant.

  3. (3)

    If \(B_1^*\) is q-smooth,  then \(\Delta _q(u^*, {\bar{u}}^*) \le {\mathcal {C}}_4 q^{-1}\Vert u^*-{\bar{u}}^*\Vert ^q,\) where \({\mathcal {C}}_4 > 0\) is a constant.

  4. (4)

    The following are equivalent :  (a) \(\lim _{r\rightarrow \infty }\Vert u_r-u\Vert =0.\) (b) \(\lim _{r\rightarrow \infty }\) \(\Delta _p(u_r, u)=0.\) (c) \(\lim _{r\rightarrow \infty }\Vert u_r\Vert =\Vert u\Vert \) and \(\lim _{r\rightarrow \infty }\langle J_p(u_r), u\rangle =\langle J_p(u), u\rangle .\)

  5. (5)

    \(\Delta _p(u, {\tilde{u}}) =p^{-1}\Vert {\tilde{u}}\Vert ^p+q^{-1}\Vert u\Vert ^{p}-\langle J_p(u), {\tilde{u}}\rangle = p^{-1}\Vert {\tilde{u}}\Vert ^p-p^{-1}\Vert u\Vert ^{p}-\langle J_p(u), {\tilde{u}}\rangle +\Vert u\Vert ^p.\)

1.2 Main result

In order to formulate our main result, we discuss certain assumptions. With the gauge function \(s\rightarrow s^{p-1},\) we assume that \(j_p\) denotes the single valued selection of the duality mapping. For the method (1.4), we engage the following well known discrepancy criterion:

$$\begin{aligned} \Vert v^{\delta }-F(u_{r_*}^{\delta })\Vert \le \tau \delta< \Vert v^{\delta }-F(u_{r}^{\delta })\Vert ,\quad 0\le r<r_*, \end{aligned}$$
(1.5)

where \(\tau >1\) satisfies

$$\begin{aligned} 1 -\frac{2}{\tau }-\frac{1}{2\tau ^p}>0 \end{aligned}$$
(1.6)

and \(r_*=r_*(\delta , v^{\delta })\) is the stopping index. The utilization of (1.5) yields \(r_*\) and \(u_{r_*}^{\delta }\) which is the required approximate solution. Our main result is as follows:

Theorem 1

Let \(B_1\) be p-convex and q-smooth with conjugate exponents \(1<p,q<\infty \) and let \(B_2\) be an arbitrary Banach space. Moreover,  we assume:

  1. (1)

    For all \(u, {\bar{u}}\in {\mathcal {B}}_{\rho }(u^{\dagger }),\) it holds that

    $$\begin{aligned} \Vert F'(u)-F'({\bar{u}})\Vert \le {\mathcal {C}}_5\Vert u-{\bar{u}}\Vert ,\quad \text {where}\ {\mathcal {C}}_5>0. \end{aligned}$$
    (1.7)
  2. (2)

    For all \(u\in {\mathcal {B}}_{\rho }(u^{\dagger }),\) it holds that \(\Vert F'(u)\Vert \le {\mathcal {C}}_6,\) where \({\mathcal {C}}_6>0.\)

  3. (3)

    \(u^{\dagger }\) is a solution of (1.1) such that \( \Delta _p(u_0, u^{\dagger })\le \rho \) for

    $$\begin{aligned} \rho ^{\frac{1}{p}}=2^{-p\epsilon }{\mathcal {C}}_6^{-1} ({\mathcal {C}}_3{\mathfrak {A}}^2)^{-\frac{1}{\epsilon }} (p^{-1}{\mathcal {C}}_3)^{(1+\frac{2}{\epsilon })\frac{1}{p}}. \end{aligned}$$
    (1.8)
  4. (4)

    \(\mu \) in (1.4) is such that

    $$\begin{aligned} \mu<\bigg (\frac{q}{2{\mathcal {C}}_4{\mathcal {C}}_6}\bigg )^{\frac{1}{q-1}}\ \text {and}\ 4{\mathcal {C}}_4 {\mathcal {C}}_6^qq^{-1} \mu ^{q-1} <1. \end{aligned}$$
    (1.9)
  5. (5)

    The Hölder stability estimate (1.3) and (1.5) hold with \(\tau \) the same as in (1.6).

Further,  assume that

$$\begin{aligned} {\mathfrak {R}}:=1 -\frac{2}{\tau }-\frac{1}{2\tau ^p}. \end{aligned}$$
(1.10)

Then,  we have : 

  1. (a)

    For \(0\le r<r_*,\) \( \Delta _p(u_{r+1}^{\delta }, u^{\dagger })\le \Delta _p(u_{r}^{\delta }, u^{\dagger }).\)

  2. (b)

    The stopping index \(r_*\) is finite.

  3. (c)

    Moreover,  for a given \(\delta >0,\) if \(\rho >0\) is such that \(\rho \le {\mathcal {C}}_7\delta ^p\) for some \({\mathcal {C}}_7>0,\) then the following convergence rates can be derived : 

    $$\begin{aligned} \Delta _p(u_{r_*}^{\delta }, u^{\dagger })\le {\mathcal {C}}_8\delta ^p, \end{aligned}$$

    where \({\mathcal {C}}_8={\mathcal {C}}_7-r_*\frac{\mu {\mathfrak {R}}}{2}.\)

Proof

We engage the fundamental theorem of the Fréchet derivative along with (1.7) to deduce that

$$\begin{aligned} \Vert F(u_{r}^{\delta })-v^{\delta }-F'(u_{r+1}^{\delta })(u_{r+1}^{\delta }-u^{\dagger })\Vert \le \delta +\frac{{\mathcal {C}}_5}{2}\Vert u_{r+1}^{\delta }-u^{\dagger }\Vert ^2. \end{aligned}$$
(1.11)

It is known that \(\Delta _p(u_0, u^{\dagger })\le \rho .\) Suppose by the induction principle that

$$\begin{aligned} \Delta _p(u_s^{\delta }, u^{\dagger })\le \rho \quad \text {for}\ s=0, 1, \ldots , r. \end{aligned}$$

We claim that \(\Delta _p(u_{r+1}^{\delta }, u^{\dagger })\le \rho .\) Using induction, the mean value inequality, (2) of Theorem 1 and (2) of Lemma 1, we obtain

$$\begin{aligned} \Vert F(u_{s}^{\delta })-v\Vert \le {\mathcal {C}}_6(p{\mathcal {C}}_3^{-1})^{\frac{1}{p}}\Delta _p(u_{s}^{\delta }, u^{\dagger })^{\frac{1}{p}}\le {\mathcal {C}}_6(p{\mathcal {C}}_3^{-1})^{\frac{1}{p}}\rho ^{\frac{1}{p}}, \end{aligned}$$
(1.12)

where \(s=0, 1, \ldots , r.\) Next, by taking \({\bar{u}}^*=J_p(u_{r+1}^{\delta })\) and \(u^*=J_p(u_{r}^{\delta })\) in (3) of Lemma 1, we derive that

$$\begin{aligned} \Delta _q(J_p(u_{r}^{\delta }), J_p(u_{r+1}^{\delta })) \le {\mathcal {C}}_4 q^{-1}\Vert J_p(u_{r+1}^{\delta })-J_p(u_{r}^{\delta })\Vert ^q. \end{aligned}$$
(1.13)

After applying (5) of Lemma 1 and the result that \(J_p^{-1}(u^*)=J_{q}^*(u^*)\) along with the definition of the duality mapping, we note that

$$\begin{aligned}{} & {} \Delta _q(J_p(u_{r}^{\delta }), J_p(u_{r+1}^{\delta }))\\{} & {} \quad = q^{-1}\Vert J_p(u_{r+1}^{\delta })\Vert ^{q}-q^{-1}\Vert J_p(u_{r}^{\delta })\Vert ^q- \langle J_q^*(J_p(u_{r}^{\delta })), J_p(u_{r+1}^{\delta })\rangle +\Vert J_p(u_{r}^{\delta })\Vert ^q\\{} & {} \quad = q^{-1}\Vert J_p(u_{r+1}^{\delta })\Vert ^{q}-q^{-1}\Vert J_p(u_{r}^{\delta })\Vert ^q- \langle J_q^*(J_p(u_{r}^{\delta })), J_p(u_{r+1}^{\delta })\rangle +\Vert u_{r}^{\delta }\Vert ^p\\{} & {} \quad = q^{-1}\Vert J_p(u_{r+1}^{\delta })\Vert ^{q}-q^{-1}\Vert J_p(u_{r}^{\delta })\Vert ^q- \langle J_p(u_{r+1}^{\delta })-J_p(u_{r}^{\delta }), u_{r}^{\delta }\rangle . \end{aligned}$$

Plugging (1.13) in the last estimate we deduce that

$$\begin{aligned}{} & {} q^{-1}(\Vert J_p(u_{r+1}^{\delta })\Vert ^{q}-\Vert J_p(u_{r}^{\delta })\Vert ^q) \nonumber \\{} & {} \quad \le \Delta _q(J_p(u_{r}^{\delta }), J_p(u_{r+1}^{\delta }))+ \langle J_p(u_{r+1}^{\delta })- J_p(u_{r}^{\delta }), u_{r}^{\delta }\rangle \nonumber \\{} & {} \quad \le {\mathcal {C}}_4 q^{-1}\Vert J_p(u_{r+1}^{\delta })-J_p(u_{r}^{\delta })\Vert ^q+\langle J_p(u_{r+1}^{\delta })- J_p(u_{r}^{\delta }), u_{r}^{\delta }\rangle . \end{aligned}$$
(1.14)

Again we note from (5) of Lemma 1 that

$$\begin{aligned} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })-\Delta _p(u_{r}^{\delta }, u^{\dagger })= & {} q^{-1}(\Vert J_p(u_{r+1}^{\delta })\Vert ^{q}-\Vert J_p(u_{r}^{\delta })\Vert ^q)\nonumber \\{} & {} -\langle J_p(u_{r+1}^{\delta })-J_p(u_{r}^{\delta }), u^{\dagger }\rangle . \end{aligned}$$
(1.15)

By combining (1.14) and (1.15), we note that

$$\begin{aligned} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })-\Delta _p(u_{r}^{\delta }, u^{\dagger })\le & {} {\mathcal {C}}_4 q^{-1}\Vert J_p(u_{r+1}^{\delta })-J_p(u_{r}^{\delta })\Vert ^q\\{} & {} +\langle J_p(u_{r+1}^{\delta })- J_p(u_{r}^{\delta }), u_{r}^{\delta }-u^{\dagger }\rangle . \end{aligned}$$

We incorporate (1.4) and assumption (2) of Theorem 1 in the last inequality to derive that

$$\begin{aligned}{} & {} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })-\Delta _p(u_{r}^{\delta }, u^{\dagger })\le {\mathcal {C}}_4 ({\mathcal {C}}_6\mu )^qq^{-1}\Vert {\mathcal {A}}_r^{\delta }\Vert ^p-\mu \langle j_p({\mathcal {A}}_r^{\delta }), F'(u_r^{\delta })(u_{r}^{\delta }-u^{\dagger })\rangle \nonumber \\{} & {} \quad = {\mathcal {C}}_4 ({\mathcal {C}}_6\mu )^qq^{-1}\Vert {\mathcal {A}}_r^{\delta }\Vert ^p-\mu \langle j_p({\mathcal {A}}_r^{\delta }), {\mathcal {A}}_r^{\delta }\rangle +\mu \langle j_p({\mathcal {A}}_r^{\delta }), {\mathcal {A}}_r^{\delta }- F'(u_r^{\delta })(u_{r}^{\delta }-u^{\dagger })\rangle \nonumber \\{} & {} \quad \le {\mathcal {C}}_4 ({\mathcal {C}}_6\mu )^qq^{-1}\Vert {\mathcal {A}}_r^{\delta }\Vert ^p-\mu \Vert {\mathcal {A}}_r^{\delta }\Vert ^p +\mu \Vert {\mathcal {A}}_r^{\delta }\Vert ^{p-1} \Vert {\mathcal {A}}_r^{\delta }- F'(u_r^{\delta })(u_{r}^{\delta }-u^{\dagger })\Vert \nonumber \\{} & {} \quad \le {\mathcal {C}}_4 ({\mathcal {C}}_6\mu )^qq^{-1}\Vert {\mathcal {A}}_r^{\delta }\Vert ^p-\mu \Vert {\mathcal {A}}_r^{\delta }\Vert ^p +\mu \Vert {\mathcal {A}}_r^{\delta }\Vert ^{p-1} \bigg (\delta +\frac{{\mathcal {C}}_5}{2}\Vert u_{r+1}^{\delta }-u^{\dagger }\Vert ^2\bigg ), \end{aligned}$$
(1.16)

where \({\mathcal {A}}_r^{\delta }=F(u_{r}^{\delta })-v^{\delta }\) and the last inequality holds due to (1.11) and the definition of the duality mapping. We plug the Hölder stability estimate (1.3) and (2) of Lemma 1 in (1.16) to further write it as

$$\begin{aligned}{} & {} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })-\Delta _p(u_{r}^{\delta }, u^{\dagger })\le \frac{1}{2}(2{\mathcal {C}}_4 ({\mathcal {C}}_6\mu )^qq^{-1} -\mu )\Vert {\mathcal {A}}_r^{\delta }\Vert ^p-\frac{\mu }{2}\Vert {\mathcal {A}}_r^{\delta }\Vert ^p \\{} & {} \quad +\mu \delta \Vert {\mathcal {A}}_r^{\delta }\Vert ^{p-1}+\frac{\mu }{2}{\mathcal {C}}_5{\mathfrak {A}}^2 (p{\mathcal {C}}_3^{-1})^{\frac{2}{p}}\Vert F(u_{r}^{\delta })-v\Vert ^{p+\epsilon }. \end{aligned}$$

Inserting (1.12) in the last estimate, we derive that

$$\begin{aligned}{} & {} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })-\Delta _p(u_{r}^{\delta }, u^{\dagger })\le \frac{1}{2}(2{\mathcal {C}}_4 ({\mathcal {C}}_6\mu )^qq^{-1} -\mu )\Vert {\mathcal {A}}_r^{\delta }\Vert ^p-\frac{\mu }{2}\Vert {\mathcal {A}}_r^{\delta }\Vert ^p \nonumber \\{} & {} \quad +\mu \delta \Vert {\mathcal {A}}_r^{\delta }\Vert ^{p-1} +\frac{\mu }{2}{\mathcal {C}}_5{\mathfrak {A}}^2(p{\mathcal {C}}_3^{-1})^{\frac{2}{p}} ({\mathcal {C}}_6(p{\mathcal {C}}_3^{-1})^{\frac{1}{p}}\rho ^{\frac{1}{p}})^{\epsilon } \Vert F(u_{r}^{\delta })-v\Vert ^{p}. \end{aligned}$$
(1.17)

Using (1.9) and the estimate

$$\begin{aligned} (\alpha _1+\alpha _2)^p\le 2^{p-1}(\alpha _1^p+\alpha _2^p)\quad \text {for}\ \alpha _1, \alpha _2\ge 0, \ p\ge 1 \end{aligned}$$
(1.18)

in (1.17), we get

$$\begin{aligned} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })-\Delta _p(u_{r}^{\delta }, u^{\dagger })\le & {} -\frac{\mu }{2}\Vert {\mathcal {A}}_r^{\delta }\Vert ^p +\mu \delta \Vert {\mathcal {A}}_r^{\delta }\Vert ^{p-1} +\frac{\mu }{4} \delta ^p\nonumber \\{} & {} +\frac{1}{2}\bigg (2{\mathcal {C}}_4 ({\mathcal {C}}_6\mu )^qq^{-1} -\mu +\frac{\mu }{2}\bigg )\Vert {\mathcal {A}}_r^{\delta }\Vert ^p. \end{aligned}$$
(1.19)

By incorporating the discrepancy principle (1.5) and (1.9) in (1.19), we obtain

$$\begin{aligned} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })-\Delta _p(u_{r}^{\delta }, u^{\dagger })\le -\frac{1}{2}\bigg (1 -\frac{2}{\tau }-\frac{1}{2\tau ^p} \bigg )\mu \Vert {\mathcal {A}}_r^{\delta }\Vert ^p, \end{aligned}$$
(1.20)

where \(r+1\le r_{*}.\) This and the choice of \(\tau \) mentioned in (1.6) together with the induction hypothesis guarantee that \( \Delta _p(u_{r+1}^{\delta }, u^{\dagger })<\rho .\) Therefore, our claim holds which completes the proof of assertion (a).

Next, we show that the stopping index \(r_*<\infty .\) For this, we incorporate (1.9) and (1.20) to write

$$\begin{aligned} \frac{{\mathfrak {R}}}{2}\mu \Vert {\mathcal {A}}_r^{\delta }\Vert ^{p}\le \Delta _p(u_{r}^{\delta }, u^{\dagger })-\Delta _p(u_{r+1}^{\delta }, u^{\dagger }). \end{aligned}$$

Summing this from \(r=0\) to \(r^*-1\), we deduce that

$$\begin{aligned} \sum _{r=0}^{r^*-1}\Vert {\mathcal {A}}_r^{\delta }\Vert ^{p} \le \frac{2}{\mu {\mathfrak {R}}}\Delta _p(u_{0}, u^{\dagger }). \end{aligned}$$

This, the choice of \(u_0,\) and (1.5) yield

$$\begin{aligned} r_*(\tau \delta )^p\le \sum _{r=0}^{r^*-1}\Vert {\mathcal {A}}_r^{\delta }\Vert ^{p}\le \frac{2}{\mu {\mathfrak {R}}}\Delta _p(u_{0}, u^{\dagger })\le \frac{2\rho }{\mu {\mathfrak {R}}}. \end{aligned}$$

We note that as \(\frac{2\rho }{\mu {\mathfrak {R}}}<\infty \) and both \(\tau , \delta \) are positive quantities, \(r_*\) can never be infinite. This proves assertion (b).

To this end, we deduce the convergence rates for the method (1.4). It follows from the Hölder stability estimate (1.3) that

$$\begin{aligned} \Delta _p(u_{r}^{\delta }, u^{\dagger })\le {\mathfrak {A}}^p \Vert F(u_{r}^{\delta })-F(u^{\dagger })\Vert ^{\frac{p(1+\epsilon )}{2}}. \end{aligned}$$

This with a slightly modified version of (1.18) (i.e., \((\alpha _1+\alpha _2)^p\le 2^{p}(\alpha _1^p+\alpha _2^p)\) for \(\alpha _1, \alpha _2\ge 0, \ p\ge 0\)) implies that

$$\begin{aligned} \Delta _p(u_{r}^{\delta }, u^{\dagger })\le 2^{p_1}{\mathfrak {A}}^p(\Vert {\mathcal {A}}_r^{\delta }\Vert ^{p_1}+\delta ^{p_1})\implies -\Vert {\mathcal {A}}_r^{\delta }\Vert ^{p_1}\le -\frac{\Delta _p(u_{r}^{\delta }, u^{\dagger })}{2^{p_1}{\mathfrak {A}}^{p}}+\delta ^{p_1}, \end{aligned}$$

where \(p_1=\frac{p(1+\epsilon )}{2}.\) Inserting the last estimate in (1.20), we obtain

$$\begin{aligned} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })-\Delta _p(u_{r}^{\delta }, u^{\dagger })\le -\frac{1}{2} \mu {\mathfrak {R}}\left( \frac{\Delta _p(u_{r}^{\delta }, u^{\dagger })}{2^{p_1}{\mathfrak {A}}^{p}}+\delta ^{p_1}\right) ^{\frac{2}{1+\epsilon }}. \end{aligned}$$
(1.21)

With some minor rearrangements, (1.21) leads to

$$\begin{aligned} \Delta _p(u_{r+1}^{\delta }, u^{\dagger })\le \Delta _p(u_{r}^{\delta }, u^{\dagger }) -\frac{\mu {\mathfrak {R}}}{2} \delta ^p. \end{aligned}$$

Consequently, by the induction hypothesis, we derive that

$$\begin{aligned} \Delta _p(u_{r_*}^{\delta }, u^{\dagger })\le \Delta _p(u_{0}^{\delta }, u^{\dagger })- \frac{\mu {\mathfrak {R}}}{2}r_*\delta ^p. \end{aligned}$$

From the last inequality, we can deduce the convergence rates in assertion (c) which completes the proof. \(\square \)

Remark 1

The assumptions considered in our work are standard and similar to [7]. Consequently, our results are applicable on a severely ill-posed inverse conductivity problem related to electrical impedance tomography (EIT) [1]. de Hoop et al. [7] showed that the inverse conductivity problem fulfills a Hölder stability estimate (1.3) for \(p=2\) and \(\epsilon =1.\) In addition to this, it is also known that the operator associated with this inverse conductivity problem fulfills (1) and (2) of Theorem 1. Therefore, by carefully choosing the other parameters such as \(\tau ,\) \(\mu \) etc., one can apply our results on the inverse conductivity problem.

2 Conclusion and future scope

In this paper, we have shown that one can obtain the convergence rates of the Landweber iteration method through stability estimates in the presence of perturbed data without the utilization of any additional smoothness concept. This paper fills an important gap in the literature. With this paper, the study of convergence analysis of the Landweber method for perturbed as well as unperturbed data via stability estimates is complete. One of the most important future tasks in the direction of studying the convergence analysis via stability estimates is to derive the optimal convergence rates. In this direction, the optimality conditions discussed in [4, 17] can be used as a reference.