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# An algorithm for a common minimum-norm zero of a finite family of monotone mappings in Banach spaces

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 566 (2013)

## Abstract

We introduce an iterative process which converges strongly to a common minimum-norm point of solutions of a finite family of monotone mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

**MSC:**47H05, 47H09, 47H10, 47J05, 47J25.

## 1 Introduction

In many problems, it is quite often to seek a particular solution of the minimum-norm solution of a given nonlinear problem. In an abstract way, we may formulate such problems as finding a point {x}^{\ast} with the property

where *C* is a nonempty closed convex subset of a real Hilbert space *H*. In other words, {x}^{\ast} is the (nearest point or metric) projection of the origin onto *C*,

where {P}_{C} is the metric (or nearest point) projection from *H* onto *C*. For instance, the split feasibility problem (SFP), introduced in [1, 2], is to find a point

where *C* and *Q* are closed convex subsets of Hilbert spaces {H}_{1} and {H}_{2}, respectively, and *A* is a linear bounded operator from {H}_{1} to {H}_{2}. We note that problem (1.3) can be extended to a problem of finding

where A:D(A)\to {E}^{\ast} and B:D(B)\to {E}^{\ast} are *monotone mappings* on a subset of a Banach space *E*. The problem has been addressed by many authors in view of the applications in image recovery and signal processing; see, for example, [3–5] and the references therein.

A mapping A:C\to {E}^{\ast} is said to be *monotone* if for each x,y\in C, the following inequality holds:

where *C* is a nonempty subset of a real Banach space *E* with {E}^{\ast} as its dual. *A* is said to be *maximal monotone* if its graph is not properly contained in the graph of any other monotone mapping. A mapping A:C\to {E}^{\ast} is said to be *γ*-*inverse strongly monotone* if there exists a positive real number *γ* such that

and it is called *strongly monotone* if there exists k>0 such that

An operator A:C\to E is called *accretive* if there exists j(x-y)\in J(x-y) such that

where *J* is the normalized duality mapping from *E* into {2}^{{E}^{\ast}} defined for each x\in E by

It is well known that *E* is smooth if and only if *J* is single-valued, and if *E* is uniformly smooth, then *J* is uniformly continuous on bounded subsets of *E* (see [6]). *A* is called *m*-*accretive* if it is accretive and R(I+rA), the range of (I+rA), is *E* for all r>0; and an accretive mapping *A* is said to satisfy *range condition* if

for some nonempty closed convex subset *C* of a real Banach space *H*.

Clearly, the class of monotone mappings includes the class of strongly monotone and the class of *γ*-inverse strongly monotone mappings. However, we observe that accretive mappings and monotone mappings have different natures in Banach spaces more general than Hilbert spaces.

When *A* and *B* are *maximal monotone* mappings in Hilbert spaces, Bauschke *et al.* [7] proved that sequences generated from the method of alternating resolvents given by

where {J}_{\mu}^{A}:={(I+\mu A)}^{-1} is the *resolvent of* *A*, converge weakly to a point of {A}^{-1}(0)\cap {B}^{-1}(0) provided that {A}^{-1}(0)\cap {B}^{-1}(0) is nonempty. Note that strong convergence of these methods fails in general (see a counter example by Hundal [8]).

With regard to a finite family of *m*-accretive mappings, Zegeye and Shahzad [9] proved that under appropriate conditions, an iterative process of Halpern type defined by

where {\alpha}_{n}\in (0,1) for all n\ge 0, u,{x}_{0}\in H, {S}_{r}:={a}_{0}I+{a}_{1}{J}_{r}^{1}+{a}_{2}{J}_{r}^{2}+\cdots +{a}_{N}{J}_{r}^{N} with {J}_{r}^{i}={(I+r{A}_{i})}^{-1} for {a}_{i}\in (0,1), i=0,1,\dots ,N, and {\sum}_{i=1}{a}_{i}^{N}=1, converges strongly to a point in {\bigcap}_{i=1}^{N}{A}^{-1}(0) nearest to *u*, where \{{A}_{i}:i=1,2,\dots ,N\} is the set of a finite family of *m*-accretive mappings in a strictly convex and reflexive (real) Banach space *E* which has a uniformly Gâteaux differentiable norm.

In 2009, Hu and Liu [10] also proved that under appropriate conditions, an iterative process of Halpern type defined by

where {\alpha}_{n},{\delta}_{n},{\gamma}_{n}\in (0,1) with {\alpha}_{n}+{\delta}_{n}+{\gamma}_{n}=1, for all n\ge 0, u={x}_{0}\in H, {S}_{{r}_{n}}:={a}_{0}I+{a}_{1}{J}_{{r}_{n}}^{1}+{a}_{2}{J}_{{r}_{n}}^{2}+\cdots +{a}_{N}{J}_{{r}_{n}}^{N} with {J}_{r}^{i}={(I+r{A}_{i})}^{-1}, for {a}_{i}\in (0,1), i=0,1,\dots ,N, and {\sum}_{i=1}{a}_{i}^{N}=1, and \{{r}_{n}\}\subset (0,\mathrm{\infty}), for {A}_{i}, i=1,2,\dots ,N, accretive mappings satisfying range condition (1.9), converges strongly to a point in {\bigcap}_{i=1}^{N}{A}^{-1}(0) nearest to *u* in a strictly convex and reflexive (real) Banach space *E* which has a uniformly Gâteaux differentiable norm.

*A natural question arises whether we can have the results of Zegeye and Shahzad* [9]*and Hu and Liu* [10]*for the class of monotone mappings or not, in Banach spaces more general than Hilbert spaces?*

Let *C* be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space *E*. Let {A}_{i}:C\to {E}^{\ast} for i=1,2,\dots ,N be continuous monotone mappings satisfying range condition (2.1) with F:={\bigcap}_{i=1}^{N}{A}_{i}^{-1}(0)\ne \mathrm{\varnothing}.

It is our purpose in this paper to introduce an iterative scheme (see (3.1)) which converges strongly to the common minimum-norm zero of the family \{{A}_{i},i=1,2,\dots ,N\}. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

## 2 Preliminaries

Let *E* be a normed linear space with dimE\ge 2. The *modulus of smoothness* of *E* is the function {\rho}_{E}:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) defined by

The space *E* is said to be *smooth* if {\rho}_{E}(\tau )>0, \mathrm{\forall}\tau >0, and *E* is called *uniformly smooth* if and only if {lim}_{t\to {0}^{+}}\frac{{\rho}_{E}(t)}{t}=0.

The *modulus of convexity* of *E* is the function {\delta}_{E}:(0,2]\to [0,1] defined by

*E* is called *uniformly convex* if and only if {\delta}_{E}(\u03f5)>0 for every \u03f5\in (0,2].

Let *C* be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space *E* with dual {E}^{\ast}. A monotone mapping *A* is said to satisfy *range condition* if we have that

for some nonempty closed convex subset *C* of a smooth, strictly convex, and reflexive Banach space *E*. In the sequel, the resolvent of a monotone mapping A:C\to {E}^{\ast} shall be denoted by {Q}_{r}^{A}:={(J+rA)}^{-1}J for r>0. We know the following lemma.

**Lemma 2.1** [11]

*Let* *E* *be a smooth and strictly convex Banach space*, *C* *be a nonempty*, *closed*, *and convex subset of* *E*, *and* A\subset E\times {E}^{\ast} *be a monotone mapping satisfying* (2.1). *Let* {Q}_{{r}_{n}}^{A} *be the resolvent of* *A* *for* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *such that* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *If* \{{x}_{n}\} *is a bounded sequence of* *C* *such that* {Q}_{{r}_{n}}^{A}{x}_{n}\rightharpoonup z, *then* z\in {A}^{-1}(0).

Let *E* be a smooth Banach space with dual {E}^{\ast}. Let the Lyapunov function \varphi :E\times E\to \mathbb{R}, introduced by Alber [12], be defined by

where *J* is the normalized duality mapping. If E=H, a Hilbert space, then the duality mapping becomes the identity map on *H*. We observe that in a Hilbert space *H*, (2.2) reduces to \varphi (x,y)={\parallel x-y\parallel}^{2} for x,y\in H.

In the sequel, we shall make use of the following lemmas.

**Lemma 2.2** [13]

*Let* *E* *be a smooth and strictly convex Banach space*, *and* *C* *be a nonempty*, *closed*, *and convex subset of* *E*. *Let* A\subset E\times {E}^{\ast} *be a monotone mapping satisfying* (2.1), {A}^{-1}(0) *be nonempty and* {Q}_{r}^{A} *be the resolvent of* *A* *for some* r>0. *Then*, *for each* r>0, *we have that*

*for all* p\in {A}^{-1}(0) *and* x\in C.

**Lemma 2.3** [14]

*Let* *E* *be a smooth and strictly convex Banach space*, *C* *be a nonempty*, *closed*, *and convex subset of* *E*, *and* *T* *be a mapping from* *C* *into itself such that* F(T) *is nonempty and* \varphi (p,Tx)\le \varphi (p,x) *for all* p\in F(T) *and* x\in C. *Then* F(T) *is closed and convex*.

**Lemma 2.4** [15]

*Let* *E* *be a real smooth and uniformly convex Banach space*, *and let* \{{x}_{n}\} *and* \{{y}_{n}\} *be two sequences of* *E*. *If either* \{{x}_{n}\} *or* \{{y}_{n}\} *is bounded and* \varphi ({x}_{n},{y}_{n})\to 0 *as* n\to \mathrm{\infty}, *then* {x}_{n}-{y}_{n}\to 0 *as* n\to \mathrm{\infty}.

We make use of the function V:E\times {E}^{\ast}\to \mathbb{R} defined by

studied by Alber [12]. That is, V(x,{x}^{\ast})=\varphi (x,{J}^{-1}{x}^{\ast}) for all x\in E and {x}^{\ast}\in {E}^{\ast}.

**Lemma 2.5** [12]

*Let* *E* *be a reflexive*, *strictly convex*, *and smooth Banach space with* {E}^{\ast} *as its dual*. *Then*

*for all* x\in E *and* {x}^{\ast},{y}^{\ast}\in {E}^{\ast}.

Let *E* be a reflexive, strictly convex, and smooth Banach space, and let *C* be a nonempty, closed, and convex subset of *E*. The *generalized projection mapping*, introduced by Alber [12], is a mapping {\mathrm{\Pi}}_{C}:E\to C that assigns an arbitrary point x\in E to the minimizer, \overline{x}, of \varphi (\cdot ,x) over *C*, that is, {\mathrm{\Pi}}_{C}x=\overline{x}, where \overline{x} is the solution to the minimization problem

**Lemma 2.6** [12]

*Let* *C* *be a nonempty*, *closed*, *and convex subset of a real reflexive*, *strictly convex*, *and smooth Banach space* *E*, *and let* x\in E. *Then*, \mathrm{\forall}y\in C,

**Lemma 2.7** [12]

*Let* *C* *be a convex subset of a real smooth Banach space* *E*. *Let* x\in E. *Then* {x}_{0}={\mathrm{\Pi}}_{C}x *if and only if*

**Lemma 2.8** [16]

*Let* *E* *be a uniformly convex Banach space and* {B}_{R}(0) *be a closed ball of* *E*. *Then there exists a continuous strictly increasing convex function* g:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) *with* g(0)=0 *such that*

*for* {\alpha}_{i}\in (0,1) *such that* {\sum}_{i=0}^{N}{\alpha}_{i}=1 *and* {x}_{i}\in {B}_{R}(0):=\{x\in E:\parallel x\parallel \le R\} *for some* R>0.

**Lemma 2.9** [17]

*Let* \{{a}_{n}\} *be a sequence of nonnegative real numbers satisfying the following relation*:

*where* \{{\beta}_{n}\}\subset (0,1) *and* \{{\delta}_{n}\}\subset R *satisfy the following conditions*: {lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty}, *and* {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0. *Then* {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.

**Lemma 2.10** [18]

*Let* \{{a}_{n}\} *be sequences of real numbers such that there exists a subsequence* \{{n}_{i}\} *of* \{n\} *such that* {a}_{{n}_{i}}<{a}_{{n}_{i}+1} *for all* i\in \mathbb{N}. *Then there exists a nondecreasing sequence* \{{m}_{k}\}\subset \mathbb{N} *such that* {m}_{k}\to \mathrm{\infty}, *and the following properties are satisfied by all* (*sufficiently large*) *numbers* k\in \mathbb{N}:

*In fact*, {m}_{k} *is the largest number* *n* *in the set* \{1,2,\dots ,k\} *such that the condition* {a}_{n}\le {a}_{n+1} *holds*.

## 3 Main result

We now prove the following theorem.

**Theorem 3.1** *Let* *C* *be a nonempty*, *closed*, *and convex subset of a smooth and uniformly convex real Banach space* *E*. *Let* {A}_{i}:C\to {E}^{\ast}, *for* i=1,2,\dots ,N, *be continuous monotone mappings satisfying* (2.1). *Assume that* \mathcal{F}:={\bigcap}_{i=1}^{N}{A}_{i}^{-1}(0) *is nonempty*. *Let* \{{x}_{n}\} *be a sequence generated by*

*where* {\alpha}_{n}\in (0,1), {\{{\beta}_{i}\}}_{i=1}^{N}\subset [c,d]\subset (0,1) *and* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *satisfy the following conditions*: {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, {\sum}_{i=0}^{N}{\beta}_{i}=1, *and* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *Then* \{{x}_{n}\} *converges strongly to the minimum*-*norm point of* ℱ.

*Proof* From Lemmas 2.2 and 2.3 we get that {A}_{i}^{-1}(0) is closed and convex. Thus, {\mathrm{\Pi}}_{\mathcal{F}}(0) is well defined. Let p={\mathrm{\Pi}}_{\mathcal{F}}(0). Then from (3.1), Lemma 2.6 and the property of *ϕ*, we get that

Moreover, from (3.1), Lemma 2.8, Lemma 2.2 and (3.2), we get that

for each i\in \{1,2,\dots ,N\}. Thus, by induction,

which implies that \{{x}_{n}\} and hence \{{y}_{n}\} are bounded. Now let {z}_{n}=(1-{\alpha}_{n}){x}_{n}. Then we note that {y}_{n}={\mathrm{\Pi}}_{C}{z}_{n}. Using Lemma 2.6, Lemma 2.5 and the property of *ϕ*, we obtain that

Furthermore, from (3.3) and (3.5) we have that

Now, following the method of proof of Lemma 3.2 of Maingé [18], we consider two cases as follows.

Case 1. Suppose that there exists {n}_{0}\in \mathbb{N} such that \{\varphi (p,{x}_{n})\} is nonincreasing for all n\ge {n}_{0}. In this situation, \{\varphi (p,{x}_{n})\} is convergent. Then from (3.6) we have that

which implies, by the property of *g*, that

and hence, since {J}^{-1} is uniformly continuous on bounded sets, we obtain that

for each i\in \{1,2,\dots ,N\}.

Furthermore, Lemma 2.6, the property of *ϕ* and the fact that {\alpha}_{n}\to 0, as n\to \mathrm{\infty}, imply that

and hence from Lemma 2.4 we get that

Since \{{z}_{n}\} is bounded and *E* is reflexive, we choose a subsequence \{{z}_{{n}_{i}}\} of \{{z}_{n}\} such that {z}_{{n}_{i}}\rightharpoonup z and {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\u3008{z}_{n}-p,Jp\u3009={lim}_{i\to \mathrm{\infty}}\u3008{z}_{{n}_{i}}-p,Jp\u3009. Then from (3.12) we get that

Thus, from (3.10) and Lemma 2.1, we obtain that z\in {A}_{i}^{-1}(0) for each i\in \{1,2,\dots ,N\} and hence z\in {\bigcap}_{i=1}^{N}{A}_{i}^{-1}(0).

Therefore, by Lemma 2.7, we immediately obtain that {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\u3008{z}_{n}-p,Jp\u3009={lim}_{i\to \mathrm{\infty}}\u3008{z}_{{n}_{i}}-p,Jp\u3009=\u3008z-p,Jp\u3009\ge 0. It follows from Lemma 2.9 and (3.7) that \varphi (p,{x}_{n})\to 0 as n\to \mathrm{\infty}. Consequently, from Lemma 2.4 we obtain that {x}_{n}\to p.

Case 2. Suppose that there exists a subsequence \{{n}_{i}\} of \{n\} such that

for all i\in \mathbb{N}. Then, by Lemma 2.10, there exists a nondecreasing sequence \{{m}_{k}\}\subset \mathbb{N} such that {m}_{k}\to \mathrm{\infty}, \varphi (p,{x}_{{m}_{k}})\le \varphi (p,{x}_{{m}_{k}+1}), and \varphi (p,{x}_{k})\le \varphi (p,{x}_{{m}_{k}+1}) for all k\in \mathbb{N}. Then, from (3.6) and the fact that {\alpha}_{n}\to 0, we obtain that

for each i\in \{1,2,\dots ,N\}. Thus, following the method of proof of Case 1, we obtain that {y}_{{m}_{k}}-{Q}_{{r}_{{m}_{k}}}^{{A}_{i}}{y}_{{m}_{k}}\to 0, {x}_{{m}_{k}}-{y}_{{m}_{k}}\to 0, {x}_{{m}_{k}}-{z}_{{m}_{k}}\to 0 as k\to \mathrm{\infty}, and hence we obtain that

Then from (3.7) we have that

Now, since \varphi (p,{x}_{{m}_{k}})\le \varphi (p,{x}_{{m}_{k}+1}), inequality (3.15) implies that

In particular, since {\alpha}_{{m}_{k}}>0, we get

Then from (3.14) we obtain \varphi (p,{x}_{{m}_{k}})\to 0 as k\to \mathrm{\infty}. This together with (3.15) gives \varphi (p,{x}_{{m}_{k}+1})\to 0 as k\to \mathrm{\infty}. But \varphi (p,{x}_{k})\le \varphi (p,{x}_{{m}_{k}+1}) for all k\in \mathbb{N}, thus we obtain that {x}_{k}\to p. Therefore, from the above two cases, we can conclude that \{{x}_{n}\} converges strongly to *p*, which is the common minimum-norm zero of the family \{{A}_{i},i=1,2,\dots ,N\}, and the proof is complete. □

We would like to mention that the method of proof of Theorem 3.1 provides the following theorem.

**Theorem 3.2** *Let* *C* *be a nonempty*, *closed*, *and convex subset of a smooth and uniformly convex real Banach space* *E*. *Let* {A}_{i}:C\to {E}^{\ast}, *for* i=1,2,\dots ,N, *be continuous monotone mappings satisfying* (2.1). *Assume that* \mathcal{F}:={\bigcap}_{i=1}^{N}{A}_{i}^{-1}(0) *is nonempty*. *Let* \{{x}_{n}\} *be a sequence generated by*

*where* {\alpha}_{n}\in (0,1), {\{{\beta}_{i}\}}_{i=1}^{N}\subset [c,d]\subset (0,1), *and* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *satisfy* {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, {\sum}_{i=0}^{N}{\beta}_{i}=1, *and* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *Then* \{{x}_{n}\} *converges strongly to* {\mathrm{\Pi}}_{\mathcal{F}}(u).

If in Theorem 3.1, N=1, then we get the following corollary.

**Corollary 3.3** *Let* *C* *be a nonempty*, *closed*, *and convex subset of a smooth and uniformly convex real Banach space* *E*. *Let* A:C\to {E}^{\ast} *be a continuous monotone mapping satisfying* (2.1). *Assume that* {A}^{-1}(0) *is nonempty*. *Let* \{{x}_{n}\} *be a sequence generated by*

*where* {\alpha}_{n}\in (0,1), \beta \in (0,1), *and* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *satisfy* {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, *and* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *Then* \{{x}_{n}\} *converges strongly to the minimum*-*norm element of* {A}^{-1}(0).

We remark that if *A* is a maximal monotone mapping, then {A}^{-1}(0) is closed and convex (see [6] for more details). The following lemma is well known.

**Lemma 3.4** [19]

*Let* *E* *be a smooth*, *strictly convex*, *and reflexive Banach space*, *let* *C* *be a nonempty closed convex subset of* *E*, *and let* A\subset E\times {E}^{\ast} *be a monotone mapping*. *Then* *A* *is maximal if and only if* R(J+rA)={E}^{\ast} *for all* r>0.

We note from the above lemma that if *A* is maximal, then it satisfies condition (2.1) and hence we have the following corollary.

**Corollary 3.5** *Let* *C* *be a nonempty*, *closed*, *and convex subset of a smooth and uniformly convex real Banach space* *E*. *Let* {A}_{i}:C\to {E}^{\ast}, i=1,2,\dots ,N, *be maximal monotone mappings*. *Assume that* \mathcal{F}:={\bigcap}_{i=1}^{N}{A}_{i}^{-1}(0) *is nonempty*. *Let* \{{x}_{n}\} *be a sequence generated by*

*where* {\alpha}_{n}\in (0,1), {\{{\beta}_{i}\}}_{i=1}^{N}\subset [c,d]\subset (0,1) *and* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *satisfy* {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, {\sum}_{i=0}^{N}{\beta}_{i}=1 *and* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *Then* \{{x}_{n}\} *converges strongly to the minimum*-*norm element of* ℱ.

If in Corollary 3.5, N=1, then we get the following corollary.

**Corollary 3.6** *Let* *C* *be a nonempty*, *closed and convex subset of a smooth and uniformly convex real Banach space* *E*. *Let* A:C\to {E}^{\ast} *be a maximal monotone mapping*. *Assume that* {A}^{-1}(0) *is nonempty*. *Let* \{{x}_{n}\} *be a sequence generated by*

*where* {\alpha}_{n}\in (0,1), \beta \in (0,1), *and* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *satisfy* {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, *and* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *Then* \{{x}_{n}\} *converges strongly to the minimum*-*norm element of* {A}^{-1}(0).

If E=H, a real Hilbert space, then *E* is uniformly convex and smooth real Banach space. In this case, J=I, identity map on *H*, and {\mathrm{\Pi}}_{C}={P}_{C}, projection mapping from *H* onto *C*. Furthermore, (2.1) reduces to (1.9). Thus, the following corollaries hold.

**Corollary 3.7** *Let* *C* *be a nonempty*, *closed*, *and convex subset of a real Hilbert space* *H*. *Let* {A}_{i}:C\to {E}^{\ast}, *for* i=1,2,\dots ,N, *be continuous monotone mappings satisfying* (1.9). *Assume that* \mathcal{F}:={\bigcap}_{i=1}^{N}{A}_{i}^{-1}(0) *is nonempty*. *Let* \{{x}_{n}\} *be a sequence generated by*

*where* {Q}_{r}^{A}:={(I+rA)}^{-1}, {\alpha}_{n}\in (0,1), {\{{\beta}_{i}\}}_{i=1}^{N}\subset [c,d]\subset (0,1), *and* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *satisfy* {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, {\sum}_{i=0}^{N}{\beta}_{i}=1, *and* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *Then* \{{x}_{n}\} *converges strongly to the minimum*-*norm element of* ℱ.

**Corollary 3.8** *Let* *C* *be a nonempty*, *closed*, *and convex subset of a real Hilbert space* *H*. *Let* {A}_{i}:C\to H, i=1,2,\dots ,N, *be maximal monotone mappings*. *Assume that* \mathcal{F}:={\bigcap}_{i=1}^{N}{A}_{i}^{-1}(0) *is nonempty*. *Let* \{{x}_{n}\} *be a sequence generated by*

*where* {Q}_{r}^{A}:={(I+rA)}^{-1}, {\alpha}_{n}\in (0,1), {\{{\beta}_{i}\}}_{i=1}^{N}\subset [c,d]\subset (0,1), *and* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *satisfy* {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, {\sum}_{i=0}^{N}{\beta}_{i}=1, *and* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *Then* \{{x}_{n}\} *converges strongly to the minimum*-*norm element of* ℱ.

## 4 Application

In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional which has minimum-norm in Banach spaces. The following is deduced from Corollary 3.6.

**Theorem 4.1** *Let* *E* *be a uniformly convex and uniformly smooth real Banach space*. *Let* {f}_{i} *be a continuously Fréchet differentiable convex functional on* *E*, *and let* \u25bd{f}_{i} *be maximal monotone with* \mathcal{F}:={\bigcap}_{i=1}^{N}{(\u25bd{f}_{i})}^{-1}(0)\ne \mathrm{\varnothing}, *where* {(\u25bd{f}_{i})}^{-1}(0)=\{z\in E:{f}_{i}(z)={min}_{y\in E}{f}_{i}(y)\}, *for* i=1,2,\dots ,N. *Let* \{{x}_{n}\} *be a sequence generated by*

*where* {\alpha}_{n}\in (0,1), {\{{\beta}_{i}\}}_{i=1}^{N}\subset [c,d]\subset (0,1), *and* \{{r}_{n}\}\subset (0,\mathrm{\infty}) *satisfy* {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, {\sum}_{i=0}^{N}{\beta}_{i}=1, *and* {lim}_{n\to \mathrm{\infty}}{r}_{n}=\mathrm{\infty}. *Then* \{{x}_{n}\} *converges strongly to the minimum*-*norm element of* ℱ.

**Remark 4.2** Theorem 3.1 provides convergence scheme to the common minimum-norm zero of a finite family of monotone mappings which improves the results of Bauschke *et al.* [7] to Banach spaces more general than Hilbert spaces. We also note that our results complement the results of Zegeye and Shahzad [9] and Hu and Liu [10] which are convergence results for accretive mappings.

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## Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support.

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Zegeye, H., Shahzad, N. An algorithm for a common minimum-norm zero of a finite family of monotone mappings in Banach spaces.
*J Inequal Appl* **2013**, 566 (2013). https://doi.org/10.1186/1029-242X-2013-566

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DOI: https://doi.org/10.1186/1029-242X-2013-566