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

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

$$x^{\ast }\in C\text{such that}\parallel x^{\ast }\parallel =\underset{x\in C}{min}\{\parallel x\parallel \},$$

(1.1)

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,

$$x^{\ast }=P_C(0),$$

(1.2)

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

$$x^{\ast }\in C\text{such that}A{x}^{\ast }\in Q,$$

(1.3)

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

$$x\in D(A)\cap D(B)\text{such that}x\in A^{-1}(0)\cap B^{-1}(0),$$

(1.4)

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:

$$〈x-y,Ax-Ay〉\ge 0,$$

(1.5)

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

$$〈x-y,Ax-Ay〉\ge \gamma {\parallel Ax-Ay\parallel }^2\text{for all }x,y\in C,$$

(1.6)

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

$$〈x-y,Ax-Ay〉\ge k{\parallel x-y\parallel }^2\text{for all }x,y\in C.$$

(1.7)

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

$$〈Ax-Ay,j(x-y)〉\ge 0\text{for all }x,y\in C,$$

(1.8)

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

$$Jx:=\{f^{\ast }\in E^{\ast }:〈x,f^{\ast }〉={\parallel x\parallel }^2={\parallel f^{\ast }\parallel }^2\}.$$

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

$$D(A)\subseteq C\subseteq \bigcap _{r>0}R(I+rA)$$

(1.9)

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

$$\{\begin{array}{cc}{x}_{2n+1}=J_{\beta }^A(x_{2n}),\hfill & n\ge 0,\hfill \\ x_{2n}=J_{\mu }^B(x_{2n-1}),\hfill & n\ge 0,\hfill \end{array}$$

(1.10)

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

$$x_{n+1}={\alpha }_{n}u+(1-{\alpha }_n)S_{r_n}{x}_n,n\ge 0,$$

(1.11)

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

$$x_{n+1}={\alpha }_{n}u+{\delta }_n{x}_n+{\gamma }_n{S}_{r_n}{x}_n,n\ge 0,$$

(1.12)

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

$${\rho }_E(\tau ):=sup\{\frac{\parallel x+y\parallel +\parallel x-y\parallel }{2}-1:\parallel x\parallel =1;\parallel y\parallel =\tau \}.$$

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

$${\delta }_E(ϵ):=inf\{1-\parallel \frac{x+y}{2}\parallel :\parallel x\parallel =\parallel y\parallel =1;ϵ=\parallel x-y\parallel \}.$$

E is called uniformly convex if and only if ${\delta }_E(ϵ)>0$ for every $ϵ\in (0,2]$.

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

$$D(A)\subseteq C\subseteq \bigcap _{r>0}{J}^{-1}R(J+rA)$$

(2.1)

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

$$\varphi (y,x)={\parallel y\parallel }^2-2〈y,Jx〉+{\parallel x\parallel }^2\text{for }x,y\in E,$$

(2.2)

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

$$\varphi (p,Q_r^{A}x)+\varphi (Q_r^{A}x,x)\le \varphi (p,x)$$

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

$$V(x,x^{\ast })={\parallel x\parallel }^2-2〈x,x^{\ast }〉+{\parallel x^{\ast }\parallel }^2\text{for all }x\in E\text{ and }{x}^{\ast }\in E,$$

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

$$V(x,x^{\ast })+2〈J^{-1}{x}^{\ast }-x,y^{\ast }〉\le V(x,x^{\ast }+y^{\ast })$$

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

$$\varphi (\overline{x},x)=min\{\varphi (y,x),y\in C\}.$$

(2.3)

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, $\mathrm{\forall }y\in C$,

$$\varphi (y,{\mathrm{\Pi }}_{C}x)+\varphi ({\mathrm{\Pi }}_{C}x,x)\le \varphi (y,x).$$

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

$$〈z-x_0,Jx-J{x}_0〉\le 0,\mathrm{\forall }z\in C.$$

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

$${\parallel {\alpha }_0{x}_0+{\alpha }_1{x}_1+\cdots +{\alpha }_N{x}_N\parallel }^2\le \sum _{i=0}^N{\alpha }_i{\parallel x_i\parallel }^2-{\alpha }_i{\alpha }_{j}g(\parallel x_i-x_j\parallel )$$

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:

$$a_{n+1}\le (1-{\beta }_n)a_n+{\beta }_n{\delta }_n,\mathrm{\forall }n\ge n_0,$$

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 $k\in \mathbb{N}$:

$$a_{m_k}\le a_{m_k+1}\mathit{\text{and}}{a}_k\le a_{m_k+1}.$$

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

$$\{\begin{array}{c}{x}_0\in C,\mathit{\text{chosen arbitrarily}},\hfill \\ y_n={\mathrm{\Pi }}_C[(1-{\alpha }_n)x_n],\hfill \\ x_{n+1}=J^{-1}({\beta }_{0}J{y}_n+{\sum }_{i=1}^N{\beta }_{i}J{Q}_{r_n}^{A_i}{y}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.1)

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

$$\begin{array}{rcl}\varphi (p,y_n)& =& \varphi (p,{\mathrm{\Pi }}_C(1-{\alpha }_n)x_n)\le \varphi (p,(1-{\alpha }_n)x_n)\\ =& \varphi (p,J^{-1}({\alpha }_{n}J0+(1-{\alpha }_n)J{x}_n))\\ =& {\parallel p\parallel }^2-2〈p,{\alpha }_{n}J0+(1-{\alpha }_n)J{x}_n〉+{\parallel {\alpha }_{n}J0+(1-{\alpha }_n)J{x}_n\parallel }^2\\ \le & {\parallel p\parallel }^2-2{\alpha }_n〈p,J0〉-2(1-{\alpha }_n)〈p,J{x}_n〉\\ +{\alpha }_n{\parallel J0\parallel }^2+(1-{\alpha }_n){\parallel J{x}_n\parallel }^2\\ =& {\alpha }_n\varphi (p,0)+(1-{\alpha }_n)\varphi (p,x_n).\end{array}$$

(3.2)

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

$$\begin{array}{rcl}\varphi (p,x_{n+1})& =& \varphi (p,J^{-1}({\beta }_{0}J{y}_n+\sum _{i=1}^N{\beta }_{i}J{Q}_{r_n}^{A_i}{y}_n))\\ =& {\parallel p\parallel }^2-2〈p,{\beta }_{0}J{y}_n+\sum _{i=1}^N{\beta }_{i}J{Q}_{r_n}^{A_i}{y}_n〉+{\parallel {\beta }_{0}J{y}_n+\sum _{i=1}^N{\beta }_{i}J{Q}_{r_n}^{A_i}{y}_n\parallel }^2\\ \le & {\parallel p\parallel }^2-2{\beta }_0〈p,J{y}_n〉-2\sum _{i=1}^N{\beta }_i〈p,J{Q}_{r_n}^{A_i}{y}_n〉\\ +{\beta }_0{\parallel y_n\parallel }^2+\sum _{i=1}^N{\beta }_i{\parallel Q_{r_n}^{A_i}{y}_n\parallel }^2-{\beta }_0{\beta }_{i}g\left(\parallel J{y}_n-J{Q}_{r_n}^{A_i}{y}_n\parallel \right)\\ =& {\beta }_0\varphi (p,y_n)+\sum _{i=1}^N{\beta }_i\varphi (p,Q_{r_n}^{A_i}{y}_n)-{\beta }_0{\beta }_{i}g\left(\parallel J{y}_n-J{Q}_{r_n}^{A_i}{y}_n\parallel \right)\\ \le & {\beta }_0\varphi (p,y_n)+(1-{\beta }_0)\varphi (p,y_n)-{\beta }_0{\beta }_{i}g\left(\parallel J{y}_n-J{Q}_{r_n}^{A_i}{y}_n\parallel \right)\\ \le & \varphi (p,y_n)-{\beta }_0{\beta }_{i}g\left(\parallel J{y}_n-J{Q}_{r_n}^{A_i}{y}_n\parallel \right)\le \varphi (p,y_n)\end{array}$$

(3.3)

$$\le {\alpha }_n\varphi (p,0)+(1-{\alpha }_n)\varphi (p,x_n)$$

(3.4)

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

$$\varphi (p,x_{n+1})\le max\{\varphi (p,0),\varphi (p,x_0)\},\mathrm{\forall }n\ge 0,$$

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

$$\begin{array}{rcl}\varphi (p,y_n)& \le & \varphi (p,z_n)=V(p,J{z}_n)\\ \le & V(p,J{z}_n-{\alpha }_n(J0-Jp))-2〈z_n-p,-{\alpha }_n(J0-Jp)〉\\ =& \varphi (p,J^{-1}({\alpha }_{n}Jp+(1-{\alpha }_n)J{x}_n))-2{\alpha }_n〈z_n-p,Jp〉\\ \le & {\alpha }_n\varphi (p,p)+(1-{\alpha }_n)\varphi (p,x_n)-2{\alpha }_n〈z_n-p,Jp〉\\ =& (1-{\alpha }_n)\varphi (p,x_n)-2{\alpha }_n〈z_n-p,Jp〉\\ \le & (1-{\alpha }_n)\varphi (p,x_n)-2{\alpha }_n〈z_n-p,Jp〉.\end{array}$$

(3.5)

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

$$\begin{array}{rcl}\varphi (p,x_{n+1})& \le & (1-{\alpha }_n)\varphi (p,x_n)-2{\alpha }_n〈z_n-p,Jp〉\\ -{\beta }_0{\beta }_{i}g\left(\parallel J{y}_n-J{Q}_{r_n}^{A_i}{y}_n\parallel \right)\end{array}$$

(3.6)

$$\le (1-{\alpha }_n)\varphi (p,x_n)-2{\alpha }_n〈z_n-p,Jp〉.$$

(3.7)

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

$${\beta }_0{\beta }_{i}g\left(\parallel J{y}_n-J{Q}_{r_n}^{A_i}{y}_n\parallel \right)\to 0,$$

(3.8)

which implies, by the property of g, that

$$J{y}_n-J{Q}_{r_n}^{A_i}{y}_n\to 0\text{as }n\to \mathrm{\infty },$$

(3.9)

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

$$y_n-Q_{r_n}^{A_i}{y}_n\to 0\text{as }n\to \mathrm{\infty },$$

(3.10)

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

$$\begin{array}{rl}\varphi (x_n,y_n)& =\varphi (x_n,{\mathrm{\Pi }}_C{z}_n)\\ \le \varphi (x_n,z_n)\\ =\varphi (x_n,J^{-1}({\alpha }_{n}J0+(1-{\alpha }_n)J{x}_n))\\ \le {\alpha }_n\varphi (x_n,0)+(1-{\alpha }_n)\varphi (x_n,x_n)\\ \le {\alpha }_n\varphi (x_n,0)+(1-{\alpha }_n)\varphi (x_n,x_n)\to 0\text{as }n\to \mathrm{\infty },\end{array}$$

(3.11)

and hence from Lemma 2.4 we get that

$$x_n-y_n\to 0,x_n-z_n\to 0\text{as }n\to \mathrm{\infty }.$$

(3.12)

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 ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}〈z_n-p,Jp〉={lim}_{i\to \mathrm{\infty }}〈z_{n_i}-p,Jp〉$. Then from (3.12) we get that

$$y_{n_i}\rightharpoonup z\text{as }i\to \mathrm{\infty }.$$

(3.13)

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 ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}〈z_n-p,Jp〉={lim}_{i\to \mathrm{\infty }}〈z_{n_i}-p,Jp〉=〈z-p,Jp〉\ge 0$. 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

$$\varphi (p,x_{n_i})<\varphi (p,x_{n_i+1})$$

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

$$g\left(\parallel J{y}_{m_k}-J{Q}_{r_{m_k}}^{A_i}{y}_{m_k}\parallel \right)\to 0\text{as }k\to \mathrm{\infty },$$

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

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈z_{m_k}-p,Jp〉\ge 0.$$

(3.14)

Then from (3.7) we have that

$$\varphi (p,x_{m_k+1})\le (1-{\alpha }_{m_k})\varphi (p,x_{m_k})-2{\alpha }_{m_k}〈z_{m_k}-p,Jp〉.$$

(3.15)

Now, since $\varphi (p,x_{m_k})\le \varphi (p,x_{m_k+1})$, inequality (3.15) implies that

$$\begin{array}{rl}{\alpha }_{m_k}\varphi (p,x_{m_k})& \le \varphi (p,x_{m_k})-\varphi (p,x_{m_k+1})-2{\alpha }_{m_k}〈z_{m_k}-p,Jp〉\\ \le -2{\alpha }_{m_k}〈z_{m_k}-p,Jp〉.\end{array}$$

In particular, since ${\alpha }_{m_k}>0$, we get

$$\varphi (p,x_{m_k})\le -2〈z_{m_k}-p,Jp〉.$$

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

$$\{\begin{array}{c}u=x_0\in C,\mathit{\text{chosen arbitrarily}},\hfill \\ y_n={\mathrm{\Pi }}_C{J}^{-1}({\alpha }_{n}Ju+(1-{\alpha }_n)J{x}_n),\hfill \\ x_{n+1}=J^{-1}({\beta }_{0}J{y}_n+{\sum }_{i=1}^N{\beta }_{i}J{Q}_{r_n}^{A_i}{y}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.16)

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

$$\{\begin{array}{c}{x}_0\in C,\mathit{\text{chosen arbitrarily}},\hfill \\ y_n={\mathrm{\Pi }}_C[(1-{\alpha }_n)x_n],\hfill \\ x_{n+1}=J^{-1}(\beta J{y}_n+(1-\beta )J{Q}_{r_n}^A{y}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.17)

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

$$\{\begin{array}{c}{x}_0\in C,\mathit{\text{chosen arbitrarily}},\hfill \\ y_n={\mathrm{\Pi }}_C[(1-{\alpha }_n)x_n],\hfill \\ x_{n+1}=J^{-1}({\beta }_{0}J{y}_n+{\sum }_{i=1}^N{\beta }_{i}J{Q}_{r_n}^{A_i}{y}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.18)

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

$$\{\begin{array}{c}{x}_0\in C,\mathit{\text{chosen arbitrarily}},\hfill \\ y_n={\mathrm{\Pi }}_C[(1-{\alpha }_n)x_n],\hfill \\ x_{n+1}=J^{-1}(\beta J{y}_n+(1-\beta )J{Q}_{r_n}^A{y}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.19)

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

$$\{\begin{array}{c}{x}_0\in C,\mathit{\text{chosen arbitrarily}},\hfill \\ y_n=P_C[(1-{\alpha }_n)x_n],\hfill \\ x_{n+1}={\beta }_0{y}_n+{\sum }_{i=1}^N{\beta }_i{Q}_{r_n}^{A_i}{y}_n,\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.20)

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

$$\{\begin{array}{c}{x}_0\in C,\mathit{\text{chosen arbitrarily}},\hfill \\ y_n=P_C[(1-{\alpha }_n)x_n],\hfill \\ x_{n+1}={\beta }_0{y}_n+{\sum }_{i=1}^N{\beta }_i{Q}_{r_n}^{A_i}{y}_n,\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(3.21)

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 $▽f_i$ be maximal monotone with $\mathcal{F}:={\bigcap }_{i=1}^N{(▽f_i)}^{-1}(0)\ne \mathrm{\varnothing }$, where ${(▽f_i)}^{-1}(0)=\{z\in E:f_i(z)={min}_{y\in E}{f}_i(y)\}$, for $i=1,2,\dots ,N$. Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_0\in C,\mathit{\text{chosen arbitrarily}},\hfill \\ y_n={\mathrm{\Pi }}_C[(1-{\alpha }_n)x_n],\hfill \\ x_{n+1}=J^{-1}({\beta }_{0}J{y}_n+{\sum }_{i=1}^N{\beta }_{i}J{(J+r_n▽f_i)}^{-1}J{y}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

(4.1)

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.