Demiclosedness Principles for Generalized Nonexpansive Mappings

Abstract

Demiclosedness principles are powerful tools in the study of convergence of iterative methods. For instance, a multi-operator demiclosedness principle for firmly nonexpansive mappings is useful in obtaining simple and transparent arguments for the weak convergence of the shadow sequence generated by the Douglas–Rachford algorithm. We provide extensions of this principle, which are compatible with the framework of more general families of mappings such as cocoercive and conically averaged mappings. As an application, we derive the weak convergence of the shadow sequence generated by the adaptive Douglas–Rachford algorithm.

1 Introduction

Demiclosedness principles play an important role in convergence analysis of fixed point algorithms. The concept of demiclosedness sheds light on topological properties of mappings, in particular, in the case where a weak topology is considered. More precisely, given a weakly sequentially closed subset D of a Hilbert space ${\mathcal {H}}$, the mapping $T:D\rightarrow {\mathcal {H}}$ is said to be demiclosed at $x\in D$, if for every sequence $(x_k)$ in D such that $(x_k)$ converges weakly to x and $T(x_k)$ converges strongly, say, to u, it follows that $T(x)=u$. By its definition, demiclosedness holds trivially whenever T is weakly sequentially continuous; however, it does not hold in general. Let ${\text {Id}}$ denote the identity mapping on ${\mathcal {H}}$. A fundamental result in the theory of nonexpansive mappings is Browder’s celebrated demiclosedness principle [1], which asserts that, if T is nonexpansive, then the mapping ${\text {Id}}-T$ is demiclosed at every point in D. Browder’s result holds in more general settings and, by now, has become a key tool in the study of asymptotic and ergodic properties of nonexpansive mappings; see [2,3,4,5], for example.

In [6], Browder’s demiclosedness principle was extended and a version for finitely many firmly nonexpansive mappings was provided. As an application, a simple proof of the weak convergence of the Douglas–Rachford (DR) algorithm [7, 8] was also provided in [6]: The DR algorithm belongs to the class of splitting methods for the problem of finding a zero of the sum of two maximally monotone operators $A,B:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$; see (25). The DR algorithm generates a sequence by an iterative application of the DR operator (see (26), (27) and the comment thereafter), which can be expressed in terms of the resolvents (see Definition 2.3) of A and B. If the solution set is nonempty, then the DR sequence converges weakly to a fixed point such that the resolvent of A maps it to a zero of $A+B$. Thus, we see that, in fact, we are interested in the image of the DR sequence under the resolvent of A. This image is often referred to as the shadow sequence. The resolvent of a maximally monotone operator is continuous (in fact, firmly nonexpansive), but not weakly continuous, in general. Hence, the convergence of the shadow sequence cannot be derived directly from the convergence of the DR sequence, unless the latter converges in norm. However, in general, norm convergence does not hold: In [9], an example of a DR iteration which does not converge in norm was explicitly constructed. Regardless of this fact, the weak convergence of the shadow sequence was established by Svaiter in [10]. A simpler and more accessible proof of the weak convergence of the shadow sequence was later given in [6] by employing a multi-operator demiclosedness principle. A demiclosedness principle for circumcenter mappings, a class of operators that is generally not continuous, was recently developed in [11].

In this paper, we present an extended demiclosedness principle for more general families of operators, which are not necessarily firmly nonexpansive, provided that they satisfy a (firm) nonexpansiveness balance condition. We are motivated by the adaptive Douglas–Rachford (aDR) algorithm which was recently studied in [12] in order to find a zero of the sum of a weakly monotone operator and a strongly monotone operator. Furthermore, the framework of [12] has been recently extended in [13] in order to hold for monotonicity and comonotonicity settings as well. In both studies [12, 13], the convergence of the shadow sequence generated by the aDR is guaranteed only under the assumption that the sum of the operators is strongly monotone. Moreover, the corresponding resolvents in the aDR are not necessarily firmly nonexpansive, and consequently, the demiclosedness principles of [6] cannot be directly applied in this framework. Our current approach is compatible with the framework of the aDR. Consequently, we employ our generalized demiclosedness principles in order to obtain weak convergence of the shadow sequence of the aDR in most cases. To this end, we employ and extend techniques and results from [6].

The remainder of this paper is organized as follows: In Sect. 2, we review preliminaries and basic results. New demiclosedness principles are provided in Sect. 3. In Sect. 4, we employ the demiclosedness principles from Sect. 3 in order to obtain the weak convergence of the shadow sequence of the adaptive Douglas–Rachford algorithm. Finally, in Sect. 5, we conclude our discussion.

2 Preliminaries

Throughout this paper, ${\mathcal {H}}$ is a real Hilbert space equipped with inner product $\langle \cdot , \cdot \rangle$ and induced norm $\Vert \cdot \Vert$. The weak convergence and strong convergence are denoted by $\rightharpoonup$ and $\rightarrow$, respectively. We set ${\mathbb {R}}_+:=\{r\in {\mathbb {R}}: r\ge 0\}$ and ${\mathbb {R}}_{++}:=\{r\in {\mathbb {R}}: r>0\}$. Given a set-valued operator $A:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$, the graph, the domain, the set of fixed points and the set of zeros of A, are denoted, respectively, by ${\text {gra}}A$, ${\text {dom}}A$, ${\text {Fix}}A$ and ${\text {zer}}A$; i.e.,

$$\begin{aligned} \begin{aligned} {\text {gra}}A&:=\big \{(x,u)\in {\mathcal {H}}\times {\mathcal {H}}: u\in A(x)\big \},&{\text {dom}}A&:=\big \{x\in {\mathcal {H}}: A(x)\ne \varnothing \big \},\\ {\text {Fix}}A&:=\big \{x\in {\mathcal {H}}: x\in A(x)\big \}&\text {and} \quad {\text {zer}}A&:=\big \{x\in {\mathcal {H}}: 0\in A(x)\big \}. \end{aligned} \end{aligned}$$

The identity mapping is denoted by ${\text {Id}}$ and the inverse of A is denoted by $A^{-1}$, i.e., ${\text {gra}}A^{-1}:=\{(u,x)\in {\mathcal {H}}\times {\mathcal {H}}: u\in A(x)\}$.

Definition 2.1

Let $D\subseteq {\mathcal {H}}$ be nonempty, let $T:D\rightarrow {\mathcal {H}}$ be a mapping, set $\tau >0$ and $\theta >0$. The mapping T is said to be

• (i) nonexpansive, if

  $$\begin{aligned} \big \Vert T(x)-T(y)\big \Vert \le \Vert x-y\Vert , \quad \forall x,y\in D; \end{aligned}$$

• (ii) firmly nonexpansive, if

  $$\begin{aligned} \big \Vert T(x)-T(y)\big \Vert ^2+\big \Vert ({\text {Id}}-T)(x)-({\text {Id}}-T)(y)\big \Vert ^2\le \Vert x-y\Vert ^2, \quad \forall x,y\in D, \end{aligned}$$

  equivalently,

  $$\begin{aligned} \big \langle x-y,T(x)-T(y)\big \rangle \ge \big \Vert T(x)-T(y)\big \Vert ^2, \quad \forall x,y\in D; \end{aligned}$$

• (iii) $\tau$-cocoercive, if $\tau T$ is firmly nonexpansive, i.e.,

  $$\begin{aligned} \big \langle x-y,T(x)-T(y)\big \rangle \ge \tau \big \Vert T(x)-T(y)\big \Vert ^2, \quad \forall x,y\in D; \end{aligned}$$

• (iv) conically $\theta$-averaged, if there exists a nonexpansive operator $R:D\rightarrow {\mathcal {H}}$ such that

  $$\begin{aligned} T=(1-\theta ){\text {Id}}+\theta R. \end{aligned}$$

Conically $\theta$-averaged mappings, introduced in [14] and originally named conically nonexpansive mappings, are natural extensions of the classical $\theta$-averaged mappings; more precisely, a conically $\theta$-averaged mapping is $\theta$-averaged whenever $\theta \in {]0,1[}$. Additional properties and further discussions of conically averaged mappings, such as the following result, are available in [13, 15].

Fact 2.1

Let $D\subseteq {\mathcal {H}}$ be nonempty, let $T:D\rightarrow {\mathcal {H}}$ and let $\theta ,\sigma >0$. Then the following assertions are equivalent:

• (i) T is conically $\theta$-averaged;

• (ii) $(1-\sigma ){\text {Id}}+\sigma T$ is conically $\sigma \theta$-averaged;

• (iii) For all $x,y\in D$,

  $$\begin{aligned} \big \Vert T(x)-T(y)\big \Vert ^2\le \Vert x-y\Vert ^2-\frac{1-\theta }{\theta }\big \Vert ({\text {Id}}-T)(x)-({\text {Id}}-T)(y)\big \Vert ^2. \end{aligned}$$

Proof

See [13, Proposition 2.2].$\square$

Lemma 2.1

Let $D\subseteq {\mathcal {H}}$ be nonempty, let $T:D\rightarrow {\mathcal {H}}$ and let $\tau ,\theta >0$.

• (i) If T is $\tau$-cocoercive, then it is $\tau '$-cocoercive for any $\tau '\in {]0,\tau ]}.$

• (ii) If T is conically $\theta$-averaged, then it is conically $\theta '$-averaged for any $\theta '\in {[\theta ,\infty [}$.

Proof

(i): Follows immediately from the definition of cocoercivity. (ii): Follows from the equivalence (i) $\iff$ (iii) in Fact 2.1.$\square$

Remark 2.1

We note that cocoercivity and conical averagedness generalize the notion of firm nonexpansiveness as follows:

• (i) By Definition 2.1(ii) and (iii) , the mapping T is firmly nonexpansive if and only if T is 1-cocoercive. Consequently, by employing Lemma 2.1(i), we see that whenever $\tau \ge 1$, a $\tau$-cocoercive mapping is firmly nonexpansive.

• (ii) Similarly, the mapping T is firmly nonexpansive if and only if it is conically $\frac{1}{2}$-averaged. Consequently, by employing Lemma 2.1(ii), we see that whenever $\theta \le \frac{1}{2}$, a conically $\theta$-averaged mapping is firmly nonexpansive.

In our study, we will employ the following generalized notions of monotonicity.

Definition 2.2

Let $A:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$ and let $\alpha \in {\mathbb {R}}$. Then A is said to be

• (i) $\alpha$-monotone, if

  $$\begin{aligned} \langle x-y,u-v\rangle \ge \alpha \Vert x-y\Vert ^2,\quad \forall (x,u),(y,v)\in {\text {gra}}A; \end{aligned}$$

• (ii) $\alpha$-comonotone, if $A^{-1}$ is $\alpha$-monotone, i.e.,

  $$\begin{aligned} \langle x-y,u-v\rangle \ge \alpha \Vert u-v\Vert ^2,\quad \forall (x,u),(y,v)\in {\text {gra}}A. \end{aligned}$$

The $\alpha$-monotone operator A is said to be maximally $\alpha$-monotone (resp. maximally $\alpha$-comonotone), if there is no $\alpha$-monotone (resp. $\alpha$-comonotone) operator $B:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$ such that ${\text {gra}}A$ is properly contained in ${\text {gra}}B$.

Remark 2.2

Common notions of monotonicity are related to the notions in Definition 2.2 as follows:

• In the case where $\alpha =0$, 0-monotonicity and 0-comonotonicity simply mean monotonicity (see, for example, [16, Definition 20.1]).

• In the case where $\alpha >0$, $\alpha$-monotonicity is also known as $\alpha$-strong monotonicity (see, for example, [16, Definition 22.1(iv)]). Similarly, $\alpha$-comonotonicity is $\alpha$-cocoercivity in Definition 2.1(iii).

• In the case where $\alpha <0$, $\alpha$-monotonicity and $\alpha$-comonotonicity are also known as $\alpha$-hypomonotonicity and $\alpha$-cohypomonotonicity, respectively (see, for example, [17, Definition 2.2]). In addition, $\alpha$-monotonicity is referred to as $\alpha$-weak monotonicity in [12].

We continue our preliminary discussion by recalling the definition of the resolvent.

Definition 2.3

Let $A:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$. The resolvent of A is the operator defined by

$$\begin{aligned} J_A:=({\text {Id}}+A)^{-1}. \end{aligned}$$

The relaxed resolvent of A with parameter $\lambda >0$ is defined by

$$\begin{aligned} J^{\lambda }_A:=(1-\lambda ){\text {Id}}+ \lambda J_A. \end{aligned}$$

When $\lambda =2$, we set $R_A:=J^2_A=2J_A-{\text {Id}}$, also known as the reflected resolvent of A.

We conclude our preliminary discussion by relating monotonicity and comonotonicity properties with corresponding properties for resolvents by recalling the following facts.

Fact 2.2

(resolvents of monotone operators) Let $A:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$ be $\alpha$-monotone, where $\alpha \in {\mathbb {R}}$. If $\gamma >0$ is such that $1+\gamma \alpha >0$, then

• (i) $J_{\gamma A}$ is single-valued and $(1+\gamma \alpha )$-cocoercive;

• (ii) ${\text {dom}}J_{\gamma A}={\mathcal {H}}$ if and only if A is maximally $\alpha$-monotone.

Proof

See [12, Lemma 3.3(ii) and Proposition 3.4].$\square$

Fact 2.3

(resolvents of comonotone operators) Let $A:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$ be $\alpha$-comonotone, where $\alpha \in {\mathbb {R}}$. If $\gamma >0$ is such that $\gamma +\alpha >0$, then

• (i) $J_{\gamma A}$ is at most single-valued and conically $\frac{\gamma }{2(\gamma +\alpha )}$-averaged;

• (ii) ${\text {dom}}J_{\gamma A}={\mathcal {H}}$ if and only if A is maximally $\alpha$-comonotone.

Proof

See [13, Propositions 3.7 and 3.8(i)].$\square$

3 New Demiclosedness Principles for Generalized Nonexpansive Mappings

In this section, we extend the multi-operator demiclosedness principles for firmly nonexpansive mappings from [6] to more general families of operators. To this end, we employ and extend techniques and results from [6] in a weighted inner product space. We begin our discussion by recalling the following fact.

Fact 3.1

Let $F:{\mathcal {H}}\rightarrow {\mathcal {H}}$ be a firmly nonexpansive mapping, let $\left( x_k\right) _{k=0}^{\infty }$ be a sequence in ${\mathcal {H}}$, and let $C,D\subseteq {\mathcal {H}}$ be closed affine subspaces such that $C-C=(D-D)^\perp$. Suppose that

$$\begin{aligned}&x_k\rightharpoonup x, \end{aligned}$$

(1a)

$$\begin{aligned}&F(x_k)\rightharpoonup y, \end{aligned}$$

(1b)

$$\begin{aligned}&F(x_k)-P_C(F(x_k))\rightarrow 0, \end{aligned}$$

(1c)

$$\begin{aligned}&(x_k-F(x_k))-P_D(x_k-F(x_k))\rightarrow 0. \end{aligned}$$

(1d)

Then $y\in C$, $x\in y+D$, and $y=F(x)$.

Proof

See [6, Corollary 2.7].$\square$

3.1 Demiclosedness Principles for Cocoercive Operators

Let $n\ge 2$ be an integer and set $\varvec{\omega }:=(\omega _1,\omega _2,\ldots ,\omega _n)\in {\mathbb {R}}_{++}^n$. We equip the space

$$\begin{aligned} \varvec{{\mathcal {H}}}:={\mathcal {H}}^n=\underbrace{{\mathcal {H}}\times {\mathcal {H}}\times \cdots \times {\mathcal {H}}}_{n}, \end{aligned}$$

(2a)

with the weighted inner product $\langle \cdot , \cdot \rangle _{\varvec{\omega }}$ defined by

$$\begin{aligned} \langle \varvec{x}, \varvec{y} \rangle _{\varvec{\omega }}:= \sum _{i=1}^n \omega _i\langle x_i,y_i\rangle ,\quad \forall \varvec{x}=(x_1,x_2,\ldots ,x_n), \varvec{y}=(y_1,y_2,\ldots ,y_n)\in \varvec{{\mathcal {H}}}. \end{aligned}$$

(2b)

Thus, $\varvec{{\mathcal {H}}}$ is a Hilbert space with the induced norm $\Vert \varvec{x}\Vert _{{\varvec{\omega }}}=\sqrt{\langle \varvec{x},\varvec{x}\rangle _{{\varvec{\omega }}}}$.

Let $\varvec{\tau }:=(\tau _1,\tau _2,\ldots ,\tau _n)\in {\mathbb {R}}^n\smallsetminus \big \{(0,\ldots ,0)\big \}$. Let ${{\varvec{C}}}\subset \varvec{{\mathcal {H}}}$ be the subspace defined by

$$\begin{aligned} {{\varvec{C}}}:=\big \{ (\tau _1x,\tau _2x,\ldots ,\tau _nx) : x\in {\mathcal {H}}\big \}. \end{aligned}$$

In the following lemma, we provide a formula for the projector $P_{{{\varvec{C}}}}$, which will be useful later.

Lemma 3.1

Let ${{\varvec{x}}}:=(x_1,x_2,\ldots ,x_n)\in \varvec{{\mathcal {H}}}$. Then the projection of ${{\varvec{x}}}$ onto ${{\varvec{C}}}$ is

$$\begin{aligned} P_{{{\varvec{C}}}}({{\varvec{x}}})= \left( \tau _1 {\overline{u}},\tau _2 {\overline{u}},\ldots ,\tau _n {\overline{u}}\right) , \quad \text {where}~{\overline{u}}:=\frac{\sum _{i=1}^n \omega _i\tau _ix_i}{\sum _{i=1}^n \omega _i\tau _i^2}. \end{aligned}$$

(3)

Proof

Fix ${{\varvec{x}}}:=(x_1,x_2,\ldots ,x_n)\in \varvec{{\mathcal {H}}}$. Since $\sum _{i=1}^{n}\omega _i\tau _i^2>0$, ${\overline{u}}$ is well defined by (3). Set ${{\varvec{y}}}=(\tau _1 {\overline{u}},\tau _2 {\overline{u}},\ldots ,\tau _n {\overline{u}})$. We prove that $(\varvec{x}-\varvec{y})\perp \varvec{z}$ for each $\varvec{z}\in {{\varvec{C}}}$; consequently, $P_{{{\varvec{C}}}}({{\varvec{x}}}) = {{\varvec{y}}}$. Indeed, let $\varvec{z}=(\tau _1 v,\tau _2 v,\ldots ,\tau _n v)\in {{\varvec{C}}}$. Then

$$\begin{aligned} \langle \varvec{z}, \varvec{x}-\varvec{y} \rangle _{\varvec{\omega }}&=\sum _{i=1}^{n}\omega _i\langle \tau _iv, x_i-\tau _i {\overline{u}}\rangle \\&=\left\langle v, \sum _{i=1}^{n} \omega _i\tau _i(x_i-\tau _i {\overline{u}})\right\rangle \\&=\left\langle v, \sum _{i=1}^{n} \omega _i\tau _i x_i- \bigg (\sum _{i=1}^{n} \omega _i\tau _i^2 \bigg ){\overline{u}}\right\rangle =\langle v, 0\rangle =0. \end{aligned}$$

$\square$

Theorem 3.1

(demiclosedness principle for cocoercive operators) Let $(\rho _1,\rho _2,\ldots ,\rho _n)$, $(\tau _1,\tau _2,\ldots ,\tau _n)\in {\mathbb {R}}_{++}^n$. For each $i\in \{1,2,\ldots ,n\}$, let $F_i:{\mathcal {H}}\rightarrow {\mathcal {H}}$ be $\tau _i$-cocoercive and let $\left( x_{i,k}\right) _{k=0}^{\infty }$ be a sequence in ${\mathcal {H}}$. Suppose that

$$\begin{aligned}&\forall i\in \{1,2,\ldots ,n\}, \qquad x_{i,k}\rightharpoonup x_i, \end{aligned}$$

(4a)

$$\begin{aligned}&\forall i\in \{1,2,\ldots ,n\}, \qquad F_i(x_{i,k})\rightharpoonup y, \end{aligned}$$

(4b)

$$\begin{aligned}&\sum _{i=1}^n\rho _i(x_{i,k}-\tau _iF_i(x_{i,k})) \rightarrow -\left( \sum _{i=1}^n\rho _i\tau _i\right) y+\sum _{i=1}^n\rho _ix_i, \end{aligned}$$

(4c)

$$\begin{aligned}&\forall i, j\in \{1,2,\ldots ,n\},\qquad F_i(x_{i,k})-F_j(x_{j,k})\rightarrow 0. \end{aligned}$$

(4d)

Then $F_1(x_1)=F_2(x_2)=\cdots =F_n(x_n)=y$.

Proof

For each $i\in \{1,2,\ldots ,n\}$, set

$$\begin{aligned} \omega _i:=\frac{\rho _i}{\tau _i} \end{aligned}$$

and equip $\varvec{{\mathcal {H}}}$ with the inner product $\langle \cdot , \cdot \rangle _{\varvec{\omega }}$ as in (2). Let ${{\varvec{F}}}:\varvec{{\mathcal {H}}}\rightarrow \varvec{{\mathcal {H}}}$ be the mapping defined by

$$\begin{aligned} {{\varvec{F}}}(\varvec{z})=(\tau _1F_1(z_1),\tau _2F_2(z_2),\ldots ,\tau _nF_n(z_n)), \quad \varvec{z}=(z_1,z_2,\ldots ,z_n)\in \varvec{{\mathcal {H}}}. \end{aligned}$$

Then for every ${{\varvec{u}}}=(u_1,u_2,\ldots ,u_n),{{\varvec{v}}}=(v_1,v_2,\ldots ,v_n)\in \varvec{{\mathcal {H}}}$, since $F_i$ is $\tau _i$ cocoercive, it follows that

$$\begin{aligned} \big \langle {{\varvec{u}}}-{{\varvec{v}}},{{\varvec{F}}}({{\varvec{u}}})-{{\varvec{F}}}({{\varvec{v}}})\big \rangle _{\varvec{\omega }}&=\sum _{i=1}^{n}\omega _i\langle u_i-v_i,\tau _i F_i(u_i) - \tau _i F_i(v_i)\rangle \\&\ge \sum _{i=1}^{n}\omega _i\tau _i^2\Vert F_i(u_i) - F_i(v_i)\Vert ^2 =\Vert {{\varvec{F}}}({{\varvec{u}}})-{{\varvec{F}}}({{\varvec{v}}})\Vert _{\varvec{\omega }}^2, \end{aligned}$$

which implies that ${{\varvec{F}}}$ is firmly nonexpansive. Set $\varvec{x}:=(x_1,x_2,\ldots ,x_n),\ \varvec{y}:=(\tau _1y,\tau _2y,\ldots ,\tau _ny)$ and, for each $k=0,1,2,\ldots$, set $\varvec{x}_k:=(x_{1,k},x_{2,k},\ldots ,x_{n,k})$. Then

$$\begin{aligned} {{\varvec{x}}}_k \rightharpoonup {{\varvec{x}}}~~\text {and}~~{{\varvec{F}}}({{\varvec{x}}}_k)\rightharpoonup {{\varvec{y}}}. \end{aligned}$$

(5)

Let ${{\varvec{C}}}$ and ${{\varvec{D}}}$ be the affine subspaces of $\varvec{{\mathcal {H}}}$ defined by

$$\begin{aligned} {{\varvec{C}}}:=\left\{ (\tau _1x,\tau _2x,\ldots ,\tau _nx) : x\in {\mathcal {H}}\right\} ~~\text {and}~~ {{\varvec{D}}}:= {{\varvec{x}}}- {{\varvec{y}}}+{{\varvec{C}}}^\perp . \end{aligned}$$

Then ${{\varvec{C}}}-{{\varvec{C}}}=({{\varvec{D}}}-{{\varvec{D}}})^\perp$. Consequently, by employing Lemma 3.1, we arrive at

$$\begin{aligned} P_{\varvec{C}}\left( {{\varvec{F}}}(\varvec{x}_k) \right) =(\tau _1 {\overline{v}}_k,\ldots , \tau _n {\overline{v}}_k),\quad \text {where } {\overline{v}}_k:=\frac{\sum _{i=1}^{n}\omega _i\tau _i^2 F_i(x_{i,k})}{\sum _{i=1}^{n}\omega _i\tau _i^2} =\frac{\sum _{i=1}^{n}\rho _i\tau _i F_i(x_{i,k})}{\sum _{i=1}^{n}\rho _i\tau _i}. \end{aligned}$$

Since ${\overline{v}}_k$ is a weighted average of the $F_i(x_{i,k})$’s, (4d) implies that

$$\begin{aligned} \forall i\in \{1,2,\ldots ,n\},\quad F_i(x_{i,k})-{\overline{v}}_k\rightarrow 0\quad \text {as}\quad k\rightarrow \infty ; \end{aligned}$$

consequently, we conclude that

$$\begin{aligned} {{\varvec{F}}}({{\varvec{x}}}_k)-P_{\varvec{C}}\left( {{\varvec{F}}}(\varvec{x}_k) \right) \rightarrow 0. \end{aligned}$$

(6)

We now employ the projections

$$\begin{aligned} P_{\varvec{C}}\left( {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k)\right) =(\tau _1{\overline{u}}_k,\ldots , \tau _n{\overline{u}}_k),\quad \text {where}~ {\overline{u}}_k=\frac{\sum _{i=1}^n \rho _i\left( x_{i,k}- \tau _iF_i(x_{i,k})\right) }{\sum _{i=1}^{n}\rho _i\tau _i}, \end{aligned}$$

and

$$\begin{aligned} P_{\varvec{C}}\left( {{\varvec{x}}}- {{\varvec{y}}}\right) =(\tau _1{\overline{u}},\ldots ,\tau _n{\overline{u}})-{{\varvec{y}}},\quad \text {where}~ {\overline{u}}=\frac{\sum _{i=1}^n \rho _i x_{i}}{\sum _{i}^{n}\rho _i\tau _i}. \end{aligned}$$

By invoking (4c), we see that ${\overline{u}}_k\rightarrow -y+{\overline{u}}$, which, in turn, implies that

$$\begin{aligned} P_{\varvec{C}}\left( \varvec{x}_k - {{\varvec{F}}}(\varvec{x}_k)\right) \rightarrow P_{\varvec{C}}\left( \varvec{x} - \varvec{y}\right) . \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{aligned} {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k)-&P_{{{\varvec{D}}}}\left( {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k) \right) \\&= {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k)- P_{{{\varvec{x}}}- {{\varvec{y}}}+{{\varvec{C}}}^\perp }\left( {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k) \right) \\&= {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k)- \left( {{\varvec{x}}}- {{\varvec{y}}}+ P_{{{\varvec{C}}}^\perp }\left( {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k)-({{\varvec{x}}}- {{\varvec{y}}}) \right) \right) \\&= ({\text {Id}}- P_{{{\varvec{C}}}^\perp })\left( {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k)\right) -({\text {Id}}- P_{{{\varvec{C}}}^\perp })\left( {{\varvec{x}}}- {{\varvec{y}}}\right) \\&= P_{{{\varvec{C}}}}\left( {{\varvec{x}}}_k - {{\varvec{F}}}({{\varvec{x}}}_k)\right) -P_{{{\varvec{C}}}}\left( {{\varvec{x}}}- {{\varvec{y}}}\right) \rightarrow 0. \end{aligned} \end{aligned}$$

(7)

Finally, since (5), (6) and (7) satisfy (1), we may employ Fact 3.1 in order to obtain ${{\varvec{y}}}= \varvec{F}({{\varvec{x}}})$, that is,

$$\begin{aligned} F_i(x_i)=y,\quad \forall i\in \{1,2,\ldots ,n\}, \end{aligned}$$

which concludes the proof.$\square$

As a consequence of Theorem 3.1, we obtain the demiclosedness principle for firmly nonexpansive operators [6, Theorem 2.10].

Corollary 3.1

(demiclosedness principle for firmly nonexpansive operators) For each ${i\in \{1,2,\ldots ,n\}}$ let $F_i:{\mathcal {H}}\rightarrow {\mathcal {H}}$ be firmly nonexpansive and let $\left( x_{i,k}\right) _{k=0}^{\infty }$ be a sequence in ${\mathcal {H}}$. Suppose further that

$$\begin{aligned}&\forall i\in \{1,2,\ldots ,n\}, \qquad x_{i,k}\rightharpoonup x_i, \end{aligned}$$

(8a)

$$\begin{aligned}&\forall i\in \{1,2,\ldots ,n\}, \qquad F_i(x_{i,k})\rightharpoonup y, \end{aligned}$$

(8b)

$$\begin{aligned}&\sum _{i=1}^n(x_{i,k}-F_i(x_{i,k}))\rightarrow -ny+\sum _{i=1}^nx_i, \end{aligned}$$

(8c)

$$\begin{aligned}&\forall i, j\in \{1,2,\ldots ,n\}, \qquad F_i(x_{i,k})-F_j(x_{j,k})\rightarrow 0. \end{aligned}$$

(8d)

Then $F_1(x_1)=F_2(x_2)=\cdots =F_n(x_n)=y$.

Proof

The proof follows by observing that firmly nonexpansive operators are cocoercive with constant $\tau =1$ and by setting $\tau _1=\tau _2=\cdots =\tau _n=1$ and $\rho _1=\rho _2=\cdots =\rho _n=1$ in Theorem 3.1.$\square$

Remark 3.1

By letting $n=1$ in Corollary 3.1, we obtain a special case of Fact 3.1 where $C={\mathcal {H}}$ and $D=\{x-y\}$, which, in turn, is equivalent to Browder’s original demiclosedness principle [1] (alternatively, see [16, Theorem 4.27]).

Remark 3.2

(Theorem 3.1 vs. Corollary 3.1) A demiclosedness principle for cocoercive mappings can be derived directly from Corollary 3.1 when we apply the latter to the firmly nonexpansive mappings $\tau _1F_1,\tau _2F_2,\dots ,\tau _nF_n$. However, this does not yield Theorem 3.1: In this case, the cocoercivity constants will appear in (8b) and (8d); however, they are neither a part of (4b) nor (4d).

Following Remark 3.2, in Theorem 3.1, the cocoercivity constants $\tau _i$ are not a part of the conditions in (4) except for the condition (4c). In the following result, we do not incorporate cocoercivity constants in any of the convergence conditions; however, in exchange, we do impose a certain balance condition on these constants.

Theorem 3.2

(closedness principle for balanced cocoercive operators) For each ${i\in \{1,2,\ldots ,n\}}$, let $F_i:{\mathcal {H}}\rightarrow {\mathcal {H}}$ be a $\tau _i$-cocoercive mapping where $\tau _i>0$ and let $\left( x_{i,k}\right) _{k=0}^{\infty }$ be a sequence in ${\mathcal {H}}$. Suppose that there exists $(\rho _1,\rho _2,\ldots ,\rho _n)\in {\mathbb {R}}_{++}^n$ such that the weighted average

$$\begin{aligned} \frac{\sum _{i=1}^n\rho _i\tau _i}{\sum _{i=1}^n\rho _i}\ge 1, \end{aligned}$$

(9)

and suppose further that

$$\begin{aligned}&\forall i\in \{1,2,\ldots ,n\}, \qquad x_{i,k}\rightharpoonup x_i, \end{aligned}$$

(10a)

$$\begin{aligned}&\forall i\in \{1,2,\ldots ,n\}, \qquad F_i(x_{i,k})\rightharpoonup y, \end{aligned}$$

(10b)

$$\begin{aligned}&\sum _{i=1}^n\rho _i(x_{i,k}-F_i(x_{i,k}))\rightarrow -\bigg (\sum _{i=1}^n\rho _i\bigg )y+\sum _{i=1}^n\rho _ix_i, \end{aligned}$$

(10c)

$$\begin{aligned}&\forall i, j\in \{1,2,\ldots ,n\}, \qquad F_i(x_{i,k})-F_j(x_{j,k})\rightarrow 0. \end{aligned}$$

(10d)

Then $F_1(x_1)=F_2(x_2)=\cdots =F_n(x_n)=y$.

Proof

In view of (9), we may choose $(\tau '_1,\tau _2',\ldots ,\tau _n'),\ \tau _i'\in ]0,\tau _i]$ such that

$$\begin{aligned} \frac{\sum _{i=1}^n\rho _i\tau '_i}{\sum _{i=1}^n\rho _i}= 1. \end{aligned}$$

By invoking Lemma 2.1(i), we see that for each $i\in \{1,2,\ldots ,n\}$, $F_i$ is $\tau '_i$-cocoercive. Consequently, we may assume without loss of generality that there is equality in (9), which we rewrite in the form

$$\begin{aligned} \rho _1(\tau _1-1)+\rho _2(\tau _2-1)+\cdots +\rho _n(\tau _n-1)= 0. \end{aligned}$$

(11)

Conditions (10a), (10b) and (10d) are the same as (4a), (4b) and (4d), respectively. Thus, in order to employ Theorem 3.1 3.1 and complete the proof, it suffices to prove that (4c) holds. Indeed, by combining (10c), (10d) and (11), we arrive at

$$\begin{aligned} \begin{aligned} \sum _{i=1}^n \rho _i\big (x_{i,k}-\tau _iF_i(x_{i,k})\big )&= \sum _{i=1}^n \rho _i\big (x_{i,k}-F_i(x_{i,k})\big )+\sum _{i=1}^n \rho _i(1-\tau _i)F_i(x_{i,k})\\&= \sum _{i=1}^n \rho _i\big (x_{i,k}-F_i(x_{i,k})\big )+\sum _{i=2}^n \rho _i(1-\tau _i)\big (F_i(x_{i,k})-F_1(x_{1,k})\big )\\&\rightarrow -\bigg (\sum _{i=1}^n\rho _i\bigg )y+\sum _{i=1}^n\rho _ix_i=-\bigg (\sum _{i=1}^n\rho _i\tau _i\bigg )y+\sum _{i=1}^n\rho _ix_i, \end{aligned} \end{aligned}$$

which is (4c).$\square$

Remark 3.3

(on the balance condition (9)) We note that the conditions in (11) for cocoercive mappings are a weighted version of the conditions in (9) for firmly nonexpansive mappings. However, the cocoercivity constants are required to be balanced as in (9); that is, the weighted average of the cocoercivity constants has to be at least 1, which is always true for firmly nonexpansive mappings (see Remark 2.1(i)).

3.2 Demiclosedness Principles for Conically Averaged Operators

In this section, we provide a demiclosedness principle for finitely many conically averaged operators. This is yet another generalization of the demiclosedness principle for firmly nonexpansive operators (Corollary 3.1), which we employ in our proof.

Theorem 3.3

(demiclosedness principle for conically averaged operators) For each ${i\in \{1,2,\ldots ,n\}}$, let $T_i:{\mathcal {H}}\rightarrow {\mathcal {H}}$ be conically $\theta _i$-averaged where $\theta _i>0$, and let $\left( x_{i,k}\right) _{k=0}^{\infty }$ be a sequence in ${\mathcal {H}}$. Suppose that

$$\begin{aligned}&\forall i\in \{1,\ldots ,n\},\quad x_{i,k}\rightharpoonup x_i, \end{aligned}$$

(12a)

$$\begin{aligned}&\forall i\in \{1,\ldots ,n\},\quad T_i(x_{i,k})\rightharpoonup 2\theta _i y + (1-2\theta _i)x_i, \end{aligned}$$

(12b)

$$\begin{aligned}&\sum _{i=1}^{n}\frac{x_{i,k}-T_i(x_{i,k})}{2\theta _i} \rightarrow -ny+\sum _{i=1}^{n}x_i, \end{aligned}$$

(12c)

$$\begin{aligned}&\forall i, j\in \{1,2,\ldots ,n\},\ (x_{i,k}-x_{j,k})- \left( \frac{x_{i,k}-T_i(x_{i,k})}{2\theta _i}-\frac{x_{j,k}-T_j(x_{j,k})}{2\theta _j}\right) \rightarrow 0. \end{aligned}$$

(12d)

Then $T_i(x_i)=2\theta _i y+(1-2\theta _i)x_i$ for all $i\in \{1,\ldots ,n\}$.

Proof

For each $i\in \{1,\ldots ,n\}$, set

$$\begin{aligned} F_i:=\left( 1-\frac{1}{2\theta _i}\right) {\text {Id}}+\frac{1}{2\theta _i} T_i ={\text {Id}}-\left( \frac{{\text {Id}}-T_i}{2\theta _i}\right) . \end{aligned}$$

Then Fact 2.1(ii) and Remark 2.1(ii) imply that $F_i$ is firmly nonexpansive. By employing (12a) and (12b), we see that

$$\begin{aligned} F_i(x_{i,k})=\Big (1-\frac{1}{2\theta _i}\Big )x_{i,k}+\frac{1}{2\theta _i} T_i(x_{i,k}) \rightharpoonup \Big (1-\frac{1}{2\theta _i}\Big )x_i + y+ \frac{1-2\theta _i}{2\theta _i}x_i = y. \end{aligned}$$

(13)

Next, by invoking (12c), we obtain

$$\begin{aligned} \sum _{i=1}^{n}(x_{i,k}-F_i(x_{i,k}))= \sum _{i=1}^n\frac{x_{i,k}-T_i(x_{i,k})}{2\theta _i}\rightarrow -ny+\sum _{i=1}^{n}x_i. \end{aligned}$$

(14)

Finally, by employing (12d), we see that for all $i,j\in \{1,\ldots ,n\}$,

$$\begin{aligned} \begin{aligned} F_i(x_{i,k})-F_j(x_{j,k})&=\Big (1-\frac{1}{2\theta _i}\Big )x_{i,k}-\Big (1-\frac{1}{2\theta _j}\Big )x_{j,k} +\frac{T_i(x_{i,k})}{2\theta _i}-\frac{T_j(x_{j,k})}{2\theta _j}\\&=(x_{i,k}-x_{j,k})+\frac{T_i(x_{i,k})-x_{i,k}}{2\theta _i} -\frac{T_j(x_{j,k})-x_{j,k}}{2\theta _j}\ \rightarrow 0. \end{aligned} \end{aligned}$$

(15)

Consequently, in view of (12a), (13), (14) and (15), we apply Corollary 3.1 to obtain

$$\begin{aligned} y=F_i(x_i)=\Big (1-\frac{1}{2\theta _i}\Big )x_i +\frac{1}{2\theta _i}T_i(x_i),\quad \forall i\in \{1,\ldots ,n\}, \end{aligned}$$

which concludes the proof.$\square$

Remark 3.4

(Theorem 3.3 vs. Corollary 3.1) Since firmly nonexpansive operators are conically $\theta$-averaged with $\theta =\frac{1}{2}$, it is clear that the assertion of Theorem 3.3 is more general than the one of Corollary 3.1. However, in view of the proof of Theorem 3.3, we conclude that the two assertions are equivalent.

We proceed in a similar manner to our discussion of demiclosedness principles for cocoercive operators; namely, we would like to have convergence conditions as in (13) of Theorem 3.3 that do not incorporate the conical average constants ${{\theta }_{i}}$. Indeed, in the following result, we provide such conditions while, yet again, imposing a balance condition on the ${\theta _{i}}$’s. We focus our attention on a result concerning two mappings, which we will use in applications.

Theorem 3.4

(demiclosedness principle for two balanced averaged operators) Let ${T_1,T_2:{\mathcal {H}}\rightarrow {\mathcal {H}}}$ be $\theta _1$- and $\theta _2$-averaged mappings where $\theta _1,\theta _2\in ]0,1[$, respectively, and suppose that there exist scalars $\rho _1,\rho _2>0$ such that

$$\begin{aligned} \theta _1\le \frac{\rho _2}{\rho _1+\rho _2} \quad \text {and}\quad \theta _2\le \frac{\rho _1}{\rho _1+\rho _2}. \end{aligned}$$

(16)

Let $(x_{1,k})_{k=0}^{\infty }$ and $(x_{2,k})_{k=0}^{\infty }$ be sequences in ${\mathcal {H}}$ such that

$$\begin{aligned}&x_{1,k}\rightharpoonup x_1 \quad \text { and }\quad x_{2,k}\rightharpoonup x_2, \end{aligned}$$

(17a)

$$\begin{aligned}&T_1(x_{1,k})\rightharpoonup y \quad \text { and }\quad T_2(x_{2,k})\rightharpoonup y, \end{aligned}$$

(17b)

$$\begin{aligned}&\rho _1(x_{1,k}-T_1(x_{1,k}))+\rho _2(x_{2,k}-T_2(x_{2,k}))\rightarrow 0, \end{aligned}$$

(17c)

$$\begin{aligned}&T_1(x_{1,k})-T_2(x_{2,k})\rightarrow 0. \end{aligned}$$

(17d)

Then $T_1(x_1)=T_2(x_2)=y$.

Proof

As in the proof of Theorem 3.2, by employing Lemma 2.1(ii), we may assume without the loss of generality that there is equality in (16), which we rewrite in the form

$$\begin{aligned} \left( \tfrac{1}{2}-\theta _1 \right) + \left( \tfrac{1}{2}-\theta _2 \right) = 0, \end{aligned}$$

(18a)

$$\begin{aligned} \text {and}\quad \rho _1\theta _1=\rho _2\theta _2. \end{aligned}$$

(18b)

By combining (17a), (17b) and (17c), we see that $\rho _1(x_1-y)+\rho _2(x_2-y)=0$, equivalently,

$$\begin{aligned} y=\frac{\rho _1x_1+\rho _2x_2}{\rho _1+\rho _2}=\theta _2x_1+\theta _1x_2. \end{aligned}$$

(19)

We set ${\overline{y}}:=\frac{1}{2}(x_1+x_2)$. By invoking (18a), it follows that

$$\begin{aligned} \begin{aligned} 2\theta _1{\overline{y}}+(1-2\theta _1)x_1=(1-\theta _1)x_1+\theta _1x_2=\theta _2x_1+\theta _1x_2,\\ 2\theta _2{\overline{y}}+(1-2\theta _2)x_2=\theta _2x_1+(1-\theta _2)x_2=\theta _2x_1+\theta _1x_2. \end{aligned} \end{aligned}$$

(20)

Consequently, (17b) and (19) imply that

$$\begin{aligned} \begin{aligned} T_1(x_{1,k})&\rightharpoonup 2\theta _1{\overline{y}}+(1-2\theta _1)x_1,\\ T_2(x_{2,k})&\rightharpoonup 2\theta _2{\overline{y}}+(1-2\theta _2)x_2. \end{aligned} \end{aligned}$$

(21)

Now, by (18b), (17c) and the definition of ${\overline{y}}$, it follows that

$$\begin{aligned} \begin{aligned}&\frac{x_{1,k}-T_1(x_{1,k})}{2\theta _1}+\frac{x_{2,k}-T_2(x_{2,k})}{2\theta _2}\\ {}&\qquad =\frac{1}{2\rho _2\theta _2}\big (\rho _1\left( x_{1,k}-T_1(x_{1,k})\right) +\rho _2\left( x_{2,k}-T_2(x_{2,k})\right) \big )\rightarrow 0=-2{\overline{y}}+x_1+x_2.\\ \end{aligned} \end{aligned}$$

(22)

In addition, from (18a) it follows that

$$\begin{aligned} \begin{aligned} (x_{1,k}&-x_{2,k})-\left( \frac{x_{1,k}-T_1(x_{1,k})}{2\theta _1}-\frac{x_{2,k}-T_2(x_{2,k})}{2\theta _2}\right) =\\&=\frac{T_1(x_{1,k})-(1-2\theta _1)x_{1,k}}{2\theta _1}+\frac{(1-2\theta _2)x_{2,k}-T_2(x_{2,k})}{2\theta _2}\\&=\frac{(1-2\theta _1)}{2\theta _1}\big (T_1(x_{1,k})-x_{1,k}\big )+T_1(x_{1,k})-T_2(x_{2,k})+\frac{(1-2\theta _2)}{2\theta _2}\big (x_{2,k}-T_2(x_{2,k})\big )\\&=(1-2\theta _2)\left( \frac{x_{1,k}-T_1(x_{1,k})}{2\theta _1}+\frac{x_{2,k}-T_2(x_{2,k})}{2\theta _2}\right) +\big (T_1(x_{1,k})-T_2(x_{2,k})\big ). \end{aligned} \end{aligned}$$

(23)

By combining (23) with (22) and (17d), we obtain

$$\begin{aligned} (x_{1,k}-x_{2,k})-\left( \frac{x_{1,k}-T_1(x_{1,k})}{2\theta _1}-\frac{x_{2,k}-T_2(x_{2,k})}{2\theta _2}\right) \rightarrow 0. \end{aligned}$$

(24)

Finally, in view of (17a), (21), (22) and (24), we employ Theorem 3.3 in order to obtain

$$\begin{aligned} T_1(x_1)=2\theta _1{\overline{y}}+(1-2\theta _1)x_1\quad \text {and}\quad T_2(x_2)=2\theta _2{\overline{y}}+(1-2\theta _2)x_2. \end{aligned}$$

By recalling (19) and (20), we arrive at $T_1(x_1)=T_2(x_2)=y$.$\square$

Remark 3.5

(on the balance condition) In Theorem 3.4, we did not provide an explicit balance condition for the averagedness constants as we did in (9). However, we did impose a stronger condition (16), which indeed implies

$$\begin{aligned} \frac{\rho _1\theta _1+\rho _2\theta _2}{\rho _1+\rho _2}\le \frac{2\rho _1\rho _2}{(\rho _1+\rho _2)^2}\le \frac{1}{2}; \end{aligned}$$

that is, the weighted average of the ${\theta _{i}}$’s is at most $\frac{1}{2}$. This is always true for firmly nonexpansive mappings (see Remark 2.1(ii)).

4 Applications to the Adaptive Douglas–Rachford Algorithm

We recall that the problem of finding a zero of the sum of two operators $A,B:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$ is

$$\begin{aligned} \text {find } x\in {\mathcal {H}}\text { such that } 0 \in Ax+Bx. \end{aligned}$$

(25)

In this section, we apply our generalized demiclosedness principles and derive the weak convergence of the shadow sequence of the adaptive Douglas–Rachford (aDR) algorithm, originally introduced in [12], in order to solve (25) for a weakly and a strongly monotone operators. The analysis is then extended in [13], which includes weakly and strongly comonotone operators as well.

Given $(\gamma ,\delta ,\lambda ,\mu ,\kappa )\in {\mathbb {R}}_{++}^5$, the aDR operator is defined by

$$\begin{aligned} T_{{\text {aDR}}}:=(1-\kappa ){\text {Id}}+\kappa R_2R_1, \end{aligned}$$

(26)

where

$$\begin{aligned} \begin{aligned} J_1&:=J_{\gamma A}=({\text {Id}}+\gamma A)^{-1},&R_1&:=J^{\lambda }_{\gamma A}=(1-\lambda ){\text {Id}}+\lambda J_1,\\ J_2&:=J_{\delta B}=({\text {Id}}+\delta B)^{-1},&R_2&:=J^{\mu }_{\delta A}=(1-\mu ){\text {Id}}+\mu J_2. \end{aligned} \end{aligned}$$

Set an initial point $x_0\in {\mathcal {H}}$. The aDR algorithm generates a sequence $\left( x_k\right) _{k=0}^{\infty }$ by the recurrence

$$\begin{aligned} x_{k+1}\in T_{{\text {aDR}}}(x_k),\quad k=0,1,2,\ldots . \end{aligned}$$

(27)

We observe that (26) coincides with the classical Douglas–Rachford operator in the case where $\lambda =\mu :=2$, $\gamma =\delta$, and $\kappa =1/2$. Similarly to the classical DR algorithm, the fixed points of $T_{{\text {aDR}}}$ are not explicit solutions of (25). Nonetheless, under the assumptions (see [13, Section 5])

$$\begin{aligned} (\lambda -1)(\mu -1)=1 \quad \text {and}\quad \delta =(\lambda -1)\gamma , \end{aligned}$$

(28a)

equivalently,

$$\begin{aligned} \lambda =1+\frac{\delta }{\gamma } \quad \text {and}\quad \mu =1+\frac{\gamma }{\delta }, \end{aligned}$$

(28b)

the fixed points are useful in order to obtain a solution as we show next.

Fact 4.1

(aDR and solutions to the inclusion problem) Suppose that $(\gamma ,\delta )\in {\mathbb {R}}_{++}^2$, that $\lambda ,\mu$ are defined by (29), that $\kappa >0$, and that $J_1$ is single-valued. Then

• (i) ${\text {Id}}-T_{{\text {aDR}}}=\kappa \mu (J_1-J_2R_1)$;

• (ii) $J_1({\text {Fix}}T_{{\text {aDR}}})={\text {zer}}(A+B)$.

Proof

See [12, Lemma 4.1].$\square$

Suppose that $\left( x_k\right) _{k=0}^{\infty }$ is generated by the aDR algorithm and converges weakly to the limit point $x^\star \in {\text {Fix}}T_{{\text {aDR}}}$. Then Fact 4.1(ii) asserts that the shadow limit point $J_1(x^\star )$ is a solution of (25). Our aim is to prove that, under certain assumptions, the shadow sequence $\left( J_1(x_k)\right) _{k=0}^{\infty }$ converges weakly to the shadow limit $J_1(x^\star )$.

In our analysis, we will employ the convergence results [13, Theorems 5.4 and 5.7]: Under the assumptions therein, the aDR operator (27) is shown to be averaged and, hence, single-valued. Consequently, in these cases, we will employ equality in (27).

4.1 Adaptive DR Algorithm for Monotone Operators

We begin our discussion with the case where the operators are maximally $\alpha$-monotone and maximally $\beta$-monotone. We will prove the weak convergence of the aDR algorithm shadow sequence by means of a generalized demiclosedness principle. To this end, we recall the following fact regarding the convergence of the aDR algorithm.

Fact 4.2

(aDR for monotone operators) Let $\alpha ,\beta \in {\mathbb {R}}$ such that $\alpha +\beta \ge 0$ and let $A,B:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$ be maximally $\alpha$-monotone and maximally $\beta$-monotone, respectively, with ${\text {zer}}(A+B)\ne \varnothing$. Let $(\gamma ,\delta ,\lambda ,\mu )\in {\mathbb {R}}^4_{++}$ satisfy (29) and either

$$\begin{aligned}&\delta (1+2\gamma \alpha )=\gamma ,&\text { if }\,\,\alpha +\beta =0, \end{aligned}$$

(29a)

$$\begin{aligned}&\text {or }\, (\gamma +\delta )^2<4\gamma \delta (1+\gamma \alpha )(1+\delta \beta ),&\text { if }\,\, \alpha +\beta >0. \end{aligned}$$

(29b)

Set $\kappa \in {]0,{\overline{\kappa }}[}$ where

$$\begin{aligned} {\overline{\kappa }}:=\left\{ \begin{array}{l@{\quad }l} 1, &{} \text {if } \alpha +\beta =0;\\ \displaystyle \frac{4\gamma \delta (1+\gamma \alpha )(1+\delta \beta )-(\gamma +\delta )^2}{2\gamma \delta (\gamma +\delta )(\alpha +\beta )}, &{} \text {if } \alpha +\beta >0. \end{array}\right. \end{aligned}$$

(30)

Set a starting point $x_0\in {\mathcal {H}}$ and $(x_k)_{k=0}^{\infty }$ by $x_{k+1}=T_{{\text {aDR}}}(x_k),\ k=0,1,2,\ldots .$ Then

• (i) $x_k-x_{k+1}\rightarrow 0;$

• (ii) $x_k\rightharpoonup x^\star \in {\text {Fix}}T_{{\text {aDR}}}\text { with } J_1(x^\star )\in {\text {zer}}(A+B).$

Proof

Combine [13, Theorem 5.7] with [13, Corollary 2.10] and Fact 4.1(ii).$\square$

Theorem 4.1

(weak convergence of the shadow sequence under monotonicity) Suppose that A and B are maximally $\alpha$-monotone and maximally $\beta$-monotone, respectively, where $\alpha +\beta \ge 0$ and ${\text {zer}}(A+B)\ne \varnothing$. Let $(\gamma ,\delta ,\lambda ,\mu )\in {\mathbb {R}}_{++}^4$ satisfy (29) and (30). Let $\kappa \in {]0,{\overline{\kappa }}[}$ where ${\overline{\kappa }}$ is defined by (30). Set a starting point $x_0\in {\mathcal {H}}$ and $(x_k)_{k=0}^{\infty }$ by $x_{k+1}=T_{{\text {aDR}}}(x_k),\ k=0,1,2,\ldots .$ Then the shadow sequence $(J_1(x_k))_{k=0}^{\infty }$ converges weakly

$$\begin{aligned} J_1(x_k)\rightharpoonup J_1(x^\star )\in {\text {zer}}(A+B). \end{aligned}$$

Proof

Fact 4.2(ii) asserts that

$$\begin{aligned} x_k\rightharpoonup x^\star \in {\text {Fix}}T_{{\text {aDR}}}\text { with } J_1(x^\star )\in {\text {zer}}(A+B). \end{aligned}$$

(31)

Consequently, $(x_k)_{k=0}^{\infty }$ is bounded. By combining (29) and (30), it follows that $1+\gamma \alpha >0$ and $1+\delta \beta >0$ (see [13, Theorem 5.7]). Thus, we employ Fact 2.2 which asserts that $J_1$ and $J_2$ are $\tau _1$-cocoercive and $\tau _2$-cocoercive, respectively, with full domain, where

$$\begin{aligned} \tau _1:=1+\gamma \alpha \quad \text {and}\quad \tau _2:=1+\delta \beta . \end{aligned}$$

Due to the cocoerciveness of $J_1$, the shadow sequence $\left( J_1(x_k)\right) _{k=0}^{\infty }$ is bounded and has a weak converging subsequence, say,

$$\begin{aligned} J_1\left( x_{k_j}\right) \rightharpoonup y^\star . \end{aligned}$$

(32)

Set $z_k:= R_1(x_k)$, for each $k=0,1,2,\ldots$. Then

$$\begin{aligned} z_{k_j} \rightharpoonup (1-\lambda )x^\star + \lambda y^\star =:z^\star . \end{aligned}$$

(33)

Moreover, Fact 4.2(i) and Fact 4.1(i) imply that

$$\begin{aligned} J_1(x_k) - J_2(z_k)\rightarrow 0. \end{aligned}$$

(34)

which, when combined with (32), implies that

$$\begin{aligned} J_2(z_{k_j})\rightharpoonup y^\star . \end{aligned}$$

(35)

Thus, on the one hand, $J_1(x_{k_j}) - J_2(z_{k_j})\rightarrow 0$ while, on the other hand,

$$\begin{aligned} \begin{aligned} J_1(x_{k_j}) - J_2(z_{k_j})&= J_1(x_{k_j}) - R_1(x_{k_j})+ R_1(x_{k_j}) - J_2(z_{k_j})\\&= (\lambda -1)\left( x_{k_j}- J_{1}(x_{k_j}) \right) + \left( z_{k_j}- J_{2}(z_{k_j})\right) \\&\rightharpoonup (\lambda -1)(x^\star -y^\star )+(z^\star - y^\star )= -\lambda y^\star +(\lambda -1)x^\star +z^\star . \end{aligned} \end{aligned}$$

(36)

By combining (31)–(36), we see that the sequences $(x_{k_j})_{j=0}^{\infty }$ and $(z_{k_j})_{j=0}^{\infty }$ satisfy the conditions in (11) by setting

$$\begin{aligned} \rho _1:=\lambda -1>0 \quad \text {and}\quad \rho _2:=1>0. \end{aligned}$$

(37)

With this choice of the parameters $\rho _i$’s, we observe that the balance condition (9) is satisfied as well. Indeed, (29) implies that

$$\begin{aligned} \rho _1(\tau _1-1)+\rho _2(\tau _2-1)=(\lambda -1)\gamma \alpha +\delta \beta =\delta (\alpha +\beta )\ge 0. \end{aligned}$$

Consequently, we apply Theorem 3.2 in order to obtain $y^\star =J_1(x^\star ).$$\square$

Remark 4.1

We observe that Theorem 4.1 guarantees the weak convergence of the shadow sequence whenever the original sequence converges weakly. In particular, this is guaranteed under the conditions on the parameters in Fact 4.2. However, this is a meaningful contribution only in the case where $\alpha +\beta =0$: In the case where $\alpha +\beta >0$, it is known that the shadow sequence converges not only weakly but, in fact, strongly (see [12, Theorem 4.5] and [13, Remark 5.8]).

4.2 Adaptive DR Algorithm for Comonotone Operators

We now address the weak convergence of the shadow sequence in the case where the operators are comonotone. To this end, we recall the following result regarding the convergence of the aDR algorithm.

Fact 4.3

(aDR for comonotone operators) Let $\alpha ,\beta \in {\mathbb {R}}$ such that $\alpha +\beta \ge 0$ and let $A,B:{\mathcal {H}}\rightrightarrows {\mathcal {H}}$ be maximally $\alpha$-comonotone and maximally $\beta$-comonotone, respectively, such that ${\text {zer}}(A+B)\ne \varnothing$. Let $(\gamma ,\delta ,\lambda ,\mu )\in {\mathbb {R}}^4_{++}$ satisfy (29) and either

$$\begin{aligned}&\delta =\gamma +2\alpha ,&\text { if }\,\, \alpha +\beta =0, \end{aligned}$$

(38a)

$$\begin{aligned}&\text {or } \quad (\gamma +\delta )^2<4(\gamma +\alpha )(\delta +\beta ),&\text { if }\,\, \alpha +\beta >0. \end{aligned}$$

(38b)

Set $\kappa \in {]0,{\overline{\kappa }}[}$ where

$$\begin{aligned} {\overline{\kappa }}:=\left\{ \begin{array}{l@{\quad }l} 1, &{} \text {if } \alpha +\beta =0;\\ \displaystyle \frac{4(\gamma +\alpha )(\delta +\beta )-(\gamma +\delta )^2}{2(\gamma +\delta )(\alpha +\beta )}, &{} \text {if } \alpha +\beta >0. \end{array}\right. \end{aligned}$$

(39)

Set a starting point $x_0\in {\mathcal {H}}$ and $(x_k)_{k=0}^{\infty }$ by $x_{k+1}=T_{{\text {aDR}}}(x_k),\ k=0,1,2,\ldots .$ Then

• (i) $x_k-x_{k+1}\rightarrow 0;$

• (ii) $x_k\rightharpoonup x^\star \in {\text {Fix}}T_{{\text {aDR}}}\text { with } J_1(x^\star )\in {\text {zer}}(A+B).$

Proof

Combine [13, Theorem 5.4] with [13, Corollary 2.10] and Fact 4.1(ii).$\square$

Theorem 4.2

(weak convergence of the shadow sequence under comonotonicity) Suppose that A and B are maximally $\alpha$-comonotone and maximally $\beta$-comonotone such that $\alpha +\beta \ge 0$ and ${\text {zer}}(A+B)\ne \varnothing$. Let $(\gamma ,\delta ,\lambda ,\mu )\in {\mathbb {R}}_{++}^4$ satisfy (29) and suppose that

$$\begin{aligned} (\gamma +\delta ) \le \min \{2(\gamma +\alpha ),2(\delta +\beta )\}. \end{aligned}$$

(40)

Let $\kappa \in {]0,{\overline{\kappa }}[}$ where ${\overline{\kappa }}$ is defined by (39). Set a starting point $x_0\in {\mathcal {H}}$ and $(x_k)_{k=0}^{\infty }$ by $x_{k+1}=T_{{\text {aDR}}}(x_k),\ k=0,1,2,\ldots .$ Then the shadow sequence $(J_1(x_k))_{k=0}^{\infty }$ converges weakly

$$\begin{aligned} J_1(x_k)\rightharpoonup J_1(x^\star )\in {\text {zer}}(A+B). \end{aligned}$$

Proof

We claim that (40) implies (38). Indeed, if $\alpha +\beta =0$, we obtain

$$\begin{aligned} (\gamma +\delta ) \le \min \{2(\gamma +\alpha ),2(\delta -\alpha )\}&\ \iff \ (\gamma +\delta )\le 2(\gamma +\alpha ) \quad \text {and}\\&\qquad \quad \quad \,\, (\gamma +\delta )\le 2(\delta -\alpha )\\&\ \iff \ \delta = \gamma +2\alpha , \end{aligned}$$

which is (38a). On the other hand, observe that (40) trivially implies that

$$\begin{aligned} (\gamma +\delta )^2\le (\min \{2(\gamma +\alpha ),2(\delta +\beta )\})^2 \le 4(\gamma +\alpha )(\delta +\beta ). \end{aligned}$$

Suppose that $(\gamma +\delta )^2=(\min \{2(\gamma +\alpha ),2(\delta +\beta )\})^2 = 4(\gamma +\alpha )(\delta +\beta )$. Then

$$\begin{aligned} (\gamma +\delta )=2(\gamma +\alpha )=2(\beta +\delta ), \end{aligned}$$

which implies that $\alpha +\beta =0$. Thus, (38b) holds whenever $\alpha +\beta >0$ which concludes the proof of our claim. Consequently, we employ Fact 4.3 in order to obtain

$$\begin{aligned} x_k\rightharpoonup x^\star \in {\text {Fix}}T_{{\text {aDR}}}\text { with } J_1(x^\star )\in {\text {zer}}(A+B). \end{aligned}$$

We conclude that $(x_k)_{k=0}^{\infty }$ is bounded. Now, (29) and (38) imply that $\gamma +\alpha >0$ and $\delta +\beta >0$ (see [13, Theorem 5.4]). Thus, by Fact 2.3, $J_1$ and $J_2$ are conically $\theta _1$-averaged and $\theta _2$-averaged, respectively, with full domain, where

$$\begin{aligned} \theta _1:= \frac{\gamma }{2(\gamma +\alpha )} \quad \text {and}\quad \theta _2:= \frac{\delta }{2(\delta +\beta )}. \end{aligned}$$

(41)

Since $J_1$ is conically averaged, the shadow sequence $\left( J_1(x_k)\right) _{k=0}^{\infty }$ is bounded and has a weakly convergent subsequence, say, $J_1(x_{k_j})\rightharpoonup y^\star$. By the same arguments as in the proof of Theorem 4.1, while employing Theorem 3.4 instead of Theorem 3.2, we arrive at

$$\begin{aligned} y^\star =J_1(x^\star ), \end{aligned}$$

which concludes the proof. To this end, it remains to verify that (16) holds, and then, Theorem 3.4 is applicable. Indeed, by (41), (37) and (29),

$$\begin{aligned} \theta _1\le \frac{\rho _2}{\rho _1+\rho _2} \quad \text {and}\quad \theta _2\le \frac{\rho _1}{\rho _1+\rho _2}&\iff \frac{\gamma }{2(\gamma +\alpha )}\le \frac{1}{\lambda } \quad \text {and}\quad \frac{\delta }{2(\delta +\beta )}\le \frac{\lambda -1}{\lambda }\\&\iff \frac{\gamma }{2(\gamma +\alpha )}\le \frac{1}{\lambda } \quad \text {and}\quad \frac{\gamma }{2(\delta +\beta )}\le \frac{1}{\lambda } \\&\iff \gamma \lambda \le \min \{2(\gamma +\alpha ),2(\delta +\beta )\}\\&\iff (\gamma +\delta ) \le \min \{2(\gamma +\alpha ),2(\delta +\beta )\}, \end{aligned}$$

which is (40).$\square$

Remark 4.2

We note that, in contrast to Theorem 4.1, Theorem 4.2 guarantees the weak convergence of the shadow sequence under more restrictive conditions on the parameters than the conditions in Fact 4.3, namely in the case where $\alpha +\beta >0$. Nevertheless, Theorem 4.2 covers new ground since the convergence of the shadow sequence in the aDR for comonotone operators has not been previously addressed. Although outside the scope of this work, we believe that the convergence of the shadow sequence may be strong whenever $\alpha +\beta >0$, as in the case of monotone operators.

5 Conclusions

In this paper, we extend the multi-operator demiclosedness principle [6] to more general classes of operators such as cocoercive and conically averaged operators. The new findings are natural and consistent with existing theory and are later justified by applications in which we show the weak convergence of the shadow sequence of the adaptive Douglas–Rachford algorithm. It remains of interest to find new connections between the demiclosedness principle and other classes of algorithms and optimization problems.