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Strong convergence of a proximaltype algorithm for an occasionally pseudomonotone operator in Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 190 (2012)
Abstract
It is known that the proximal point algorithm converges weakly to a zero of a maximal monotone operator, but it fails to converge strongly. Then, in (Math. Program. 87:189202, 2000), Solodov and Svaiter introduced the new proximaltype algorithm to generate a strongly convergent sequence and established a convergence property for the algorithm in Hilbert spaces. Further, Kamimura and Takahashi (SIAM J. Optim. 13:938945, 2003) extended Solodov and Svaiter’s result to more general Banach spaces and obtained strong convergence of a proximaltype algorithm in Banach spaces. In this paper, by introducing the concept of an occasionally pseudomonotone operator, we investigate strong convergence of the proximal point algorithm in Hilbert spaces, and so our results extend the results of Kamimura and Takahashi.
MSC:47H05, 47J25.
1 Introduction
Let H be a real Hilbert space with inner product $\u3008\cdot ,\cdot \u3009$, and let $T:H\to {2}^{H}$ be a maximal monotone operator (or a multifunction) on H. We consider the classical problem:
Find $x\in H$ such that
A wide variety of the problems, such as optimization problems and related fields, minmax problems, complementarity problems, variational inequalities, equilibrium problems and fixed point problems, fall within this general framework. For example, if T is the subdifferential ∂f of a proper lower semicontinuous convex function $f:H\to (\mathrm{\infty},\mathrm{\infty})$, then T is a maximal monotone operator and the equation $0\in \partial f(x)$ is reduced to $f(x)=min\{f(z):z\in H\}$. One method of solving $0\in Tx$ is the proximal point algorithm. Let I denote the identity operator on H. Rockafellar’s proximal point algorithm generates, for any starting point ${x}_{0}=x\in H$, a sequence $\{{x}_{n}\}$ in H by the rule
where $\{{r}_{n}\}$ is a sequence of positive real numbers. Note that (1.2) is equivalent to
This algorithm was first introduced by Martinet [1] and generally studied by Rockafellar [2] in the framework of Hilbert spaces. Later, many authors studied the convergence of (1.2) in Hilbert spaces (see Agarwal et al. [3], Brezis and Lions [4], Cho et al. [5], Cholamjiak et al. [6], Güler [7], Lions [8], Passty [9], Qin et al. [10], Song et al. [11], Solodov and Svaiter [12], Wei and Cho [13] and the references therein). Rockafellar [2] proved that, if ${T}^{1}0\ne \mathrm{\varnothing}$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$, then the sequence $\{{x}_{n}\}$ generated by (1.2) converges weakly to an element of ${T}^{1}0$. Further, Rockafellar [2] posed an open question of whether the sequence $\{{x}_{n}\}$ generated by (1.2) converges strongly or not. This question was solved by Güler [7], who introduced an example for which the sequence $\{{x}_{n}\}$ generated by (1.2) converges weakly, but not strongly.
On the other hand, Kamimura and Takahashi [14, 15], Solodov and Svaiter [16] one decade ago modified the proximal point algorithm to generate a strongly convergent sequence. In 1999, Solodov and Svaiter [16] introduced the following algorithm $\{{x}_{n}\}$:
To explain how the sequence $\{{y}_{n}\}$ is generated, we formally state the above algorithm as follows.
Choose any ${x}_{0}\in H$ and $\sigma \in [0,1)$. At iteration n, having ${x}_{n}$, choose ${r}_{n}>0$ and find $({y}_{n},{v}_{n})$, an inexact solution of $0={v}_{n}+\frac{1}{{r}_{n}}({y}_{n}{x}_{n})$, ${v}_{n}\in T{y}_{n}$ with tolerance σ. Define ${H}_{n}$ and ${W}_{n}$ as in (1.3). Take ${x}_{n+1}={P}_{{H}_{n}\cap {W}_{n}}{x}_{0}$. Note that at each iteration, there are two subproblems to be solved: find an inexact solution of the proximal point subproblem and find the projection of ${x}_{0}$ onto ${H}_{n}\cap {W}_{n}$, the intersection of two halfspaces. By a classical result of Minty [17], the proximal subproblem always has an exact solution, which is unique. Notice that computing an approximate solution makes things easier. Hence, this part of the method is well defined. Regarding the projection step, it is easy to prove that ${H}_{n}\cap {W}_{n}$ is never empty, even when the solution set is empty. Therefore, the whole algorithm is well defined in the sense that it generates an infinite sequence $\{{x}_{n}\}$ and an associated sequence of pairs $\{({y}_{n},{v}_{n})\}$.
In 2003, Kamimura and Takahashi [18] extended Solodov and Svaiter’s result to more general Banach spaces like the spaces ${L}^{p}$ ($1<p<\mathrm{\infty}$) by further modifying the proximalpoint algorithm (1.2) in the following form in a smooth Banach space E:
to generate a strongly convergent sequence. They proved that if ${T}^{1}0\ne \mathrm{\varnothing}$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$, then the sequence $\{{x}_{n}\}$ generated by (1.3) converges strongly to a point ${P}_{{T}^{1}0}{x}_{0}$.
In this paper, by introducing the concept of an occasionally pseudomonotone operator, we investigate strong convergence of the proximal point algorithm in Hilbert spaces, and so our results extend the results of Kamimura and Takahashi.
2 Preliminaries and definitions
Let E be a real Banach space with norm $\parallel \cdot \parallel $, and let ${E}^{\ast}$ denote the dual space of E. Let $\u3008x,f\u3009$ denote the value of $f\in {E}^{\ast}$ at $x\in E$. Let $\{{x}_{n}\}$ be a sequence in E. We denote the strong convergence of $\{{x}_{n}\}$ to $x\in E$ by ${x}_{n}\to x$ and the weak convergence by ${x}_{n}\rightharpoonup x$, respectively.
Definition 2.1 A multivalued operator $T:E\to {2}^{{E}^{\ast}}$ with domain $D(T)=\{z\in E:Tz\ne \mathrm{\varnothing}\}$ and range $R(T)=\bigcup \{Tz:z\in D(T)\}$ is said to be monotone if $\u3008{x}_{1}{x}_{2},{y}_{1}{y}_{2}\u3009\ge 0$ for any ${x}_{i}\in D(T)$ and ${y}_{i}\in T{x}_{i}$, $i=1,2$. A monotone operator T is said to be maximal if its graph $G(T)=\{(x,y):y\in Tx\}$ is not properly contained in the graph of any other monotone operator.
Definition 2.2 A multivalued operator $T:E\to {2}^{{E}^{\ast}}$ with domain $D(T)$ and range $R(T)$ is said to be pseudomonotone (see also Karamardian [19]) if $\u3008{x}_{1}{x}_{2},{y}_{2}\u3009\ge 0$ implies $\u3008{x}_{1}{x}_{2},{y}_{1}\u3009\ge 0$ for any ${x}_{i}\in D(T)$ and ${y}_{i}\in T{x}_{i}$, $i=1,2$.
It is obvious that each monotone operator is pseudomonotone, but the converse is not true.
We now introduce the concept of occasionally pseudomonotone as follows.
Definition 2.3 A multivalued operator $T:E\to {2}^{{E}^{\ast}}$ is said to be occasionally pseudomonotone if, for any ${x}_{i}\in D(T)$, there exist ${y}_{i}\in T{x}_{i}$, $i=1,2$, such that $\u3008{x}_{1}{x}_{2},{y}_{2}\u3009\ge 0$ implies $\u3008{x}_{1}{x}_{2},{y}_{1}\u3009\ge 0$.
It is clear that every monotone operator is pseudomonotone and every pseudomonotone operator is occasionally pseudomonotone, but the converse implications need not be true. To this end, we observe the following examples.
Example 2.1 Let $E={\mathbb{R}}^{3}$ and $T:E\to {2}^{{E}^{\ast}}$ be a multivalued operator defined by
where
Then for any ${x}_{1}={({x}_{1}^{(1)},{x}_{2}^{(1)},{x}_{3}^{(1)})}^{T}$, ${x}_{2}={({x}_{1}^{(2)},{x}_{2}^{(2)},{x}_{3}^{(2)})}^{T}$ in ${\mathbf{R}}^{3}$, if ${y}_{1}={A}_{r}{x}_{1}$ and ${y}_{2}={A}_{r}{x}_{2}$, then we have
Thus, if $r\le 0$, then T is monotone. However, if $r>0$, then T is neither monotone nor pseudomonotone. Indeed, for ${x}_{1}=(0,1,0)$, then we have ${y}_{1}={A}_{r}{x}_{1}=(0,r,0)$, ${x}_{2}=(0,0,0)$ and $\u3008{x}_{1}{x}_{2},{y}_{2}\u3009=0\ge 0$, but $\u3008{x}_{1}{x}_{2},{y}_{1}\u3009=r<0$.
Further, we see that T is occasionally pseudomonotone. To effect this, for any ${x}_{1}={({x}_{1}^{(1)},{x}_{2}^{(1)},{x}_{3}^{(1)})}^{T}$ and ${x}_{2}={({x}_{1}^{(2)},{x}_{2}^{(2)},{x}_{3}^{(2)})}^{T}$ in ${\mathbf{R}}^{3}$, if ${y}_{i}={A}_{0}{x}_{i}$, $i=1,2$, then we have
Example 2.2 The rotation operator on ${\mathbf{R}}^{2}$ given by
is monotone and hence it is pseudomonotone. Thus, it follows that A is also occasionally pseudomonotone.
Maximality of pseudomonotone and occasionally pseudomonotone operators are defined as similar to maximality of a monotone operator. We denote by $L[{x}_{1},{x}_{2}]$ the ray passing through ${x}_{1}$, ${x}_{2}$.
A Banach space E is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. It is also said to be uniformly convex if ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0$ for any two sequences $\{{x}_{n}\}$, $\{{y}_{n}\}$ in E such that $\parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1$ and ${lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1$.
It is known that a uniformly convex Banach space is reflexive and strictly convex. The spaces ${\ell}^{1}$ and ${L}^{1}$ are neither reflexive nor strictly convex. Note also that a reflexive Banach space is not necessarily uniformly convex. For example, consider a finite dimensional Banach space in which the surface of the unit ball has a ‘flat’ part. We note that such a Banach space is reflexive because of finite dimension. But the ‘flat’ portion in the surface of the ball makes it nonuniformly convex. It is also well known that a Banach space E is reflexive if and only if every bounded sequence of elements of E contains a weakly convergent sequence.
Let $U=\{x\in E:\parallel x\parallel =1\}$. A Banach space E is said to be smooth if the limit
exists for all $x,y\in U$. It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for $x,y\in U$.
It is well known that the spaces ${\ell}^{p}$, ${L}^{p}$ and ${W}^{m}$ (Sobolev space) ($1<p<\mathrm{\infty}$, m is a positive integer) are uniformly convex and uniformly smooth Banach spaces.
For any $p\in (1,\mathrm{\infty})$, the mapping ${J}_{p}:E\to {2}^{{E}^{\ast}}$ defined by
is called the duality mapping with the gauge function $\phi (t)={t}^{p1}$. In particular, for $p=2$, the duality mapping ${J}_{2}$ with gauge function $\phi (t)=t$ is called the normalized duality mapping.
The following proposition gives some basic properties of the duality mapping.
Proposition 2.1 Let E be a real Banach space. For $1<p<\mathrm{\infty}$, the duality mapping ${J}_{p}:E\to {2}^{{E}^{\ast}}$ has the following properties:

(1)
${J}_{p}(x)\ne \varphi $ for all $x\in E$ and $D({J}_{p})=E$, where $D({J}_{p})$ denotes the domain of ${J}_{p}$;

(2)
${J}_{p}(x)={\parallel x\parallel}^{p2}\cdot {J}_{2}x$ for all $x\in E$ with $x\ne 0$;

(3)
${J}_{p}(\alpha x)={\alpha}^{p1}\cdot {J}_{2}x$ for all $\alpha \in [0,\mathrm{\infty})$;

(4)
${J}_{p}(x)={J}_{p}(x)$;

(5)
${\parallel x\parallel}^{p}{\parallel y\parallel}^{p}\ge p\u3008xy,j\u3009$ for all $x,y\in E$ and $j\in {J}_{p}y$;

(6)
If E is smooth, then ${J}_{p}$ is normtoweak∗ continuous;

(7)
If E is uniformly smooth, then ${J}_{p}$ is uniformly normtonorm continuous on each bounded subset of E;

(8)
${J}_{p}$ is bounded, i.e., for any bounded subset $A\subset E$, ${J}_{p}(A)$ is a bounded subset in ${E}^{\ast}$;

(9)
${J}_{p}$ can be equivalently defined as the subdifferential of the functional $\psi (x)={p}^{1}{\parallel x\parallel}^{p}$ (Asplund [20]), i.e.,
$${J}_{p}(x)=\partial \psi (x)=\{f\in {E}^{\ast}:\psi (y)\psi (x)\ge \u3008yx,f\u3009,\mathrm{\forall}y\in E\};$$ 
(10)
E is a uniformly smooth Banach space (equivalently, ${E}^{\ast}$ is a uniformly convex Banach space) if and only if ${J}_{p}$ is singlevalued and uniformly continuous on any bounded subset of E (see, for instance, Xu and Roach [21], Browder [22]).
Proposition 2.2 Let E be a real Banach space, and let ${J}_{p}:E\to {2}^{{E}^{\ast}}$, $1<p<\mathrm{\infty}$, be the duality mapping. Then for any $x,y\in E$,
Proof It is a straightforward consequence of the assertion (5) of Proposition 2.1 applied for x and $x+y$. Alternatively, from Proposition 2.1(9), it follows that ${J}_{p}(x)=\partial \psi (x)$ (subdifferential of the functional $\psi (x)$), where $\psi (x)={p}^{1}{\parallel x\parallel}^{p}$. Also, it follows from the definition of the subdifferential of ψ that
Now, substituting $\psi (x)$ by ${p}^{1}{\parallel x\parallel}^{p}$, we have
This completes the proof. □
Remark 2.1 If E is a uniformly smooth Banach space, it follows from Proposition 2.1(10) that ${J}_{p}$ ($1<p<\mathrm{\infty}$) is a singlevalued mapping. We now define the functions $\mathrm{\Psi},\varphi :E\times E\to \mathbb{R}$ by
and ϕ is the support function satisfying the following condition:
It is obvious from the definition of Ψ and Proposition 2.1(5) that
Also, we see that
In particular, for $p=2$, we have $\mathrm{\Psi}(x,y)\ge {(\parallel x\parallel \parallel y\parallel )}^{2}$.
Further, we can show the following two propositions.
Proposition 2.3 Let E be a smooth Banach space, and let $\{{y}_{n}\}$, $\{{z}_{n}\}$ be two sequences in E. If $\mathrm{\Psi}({y}_{n},{z}_{n})\to 0$, then ${y}_{n}{z}_{n}\to 0$.
Proof It follows from $\mathrm{\Psi}({y}_{n},{z}_{n})\to 0$ that
because of (2.2) and (2.3). Therefore, if $\{{z}_{n}\}$ is bounded, then $\{{y}_{n}\}$ (and also if $\{{y}_{n}\}$ is bounded, then $\{{z}_{n}\}$) is also bounded and ${y}_{n}{z}_{n}\to 0$. This completes the proof. □
Proposition 2.4 Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and let $x\in E$. Then there exists a unique element ${x}_{0}\in C$ such that
Proof Since E is reflexive and $\parallel {z}_{n}\parallel \to \mathrm{\infty}$ implies $\mathrm{\Psi}({z}_{n},x)\to \mathrm{\infty}$, there exists ${x}_{0}\in C$ such that $\mathrm{\Psi}({x}_{o},x)=inf\{\mathrm{\Psi}(z,x):z\in C\}$. Since E is strictly convex, ${\parallel \cdot \parallel}^{p}$ is a strictly convex function, that is,
for all ${x}_{1},{x}_{2}\in E$ with ${x}_{1}\ne {x}_{2}$, $1\le p<\mathrm{\infty}$ and $\lambda \in (0,1)$. Then the function $\mathrm{\Psi}(\cdot ,y)$ is also strictly convex. Therefore, ${x}_{0}\in C$ is unique. This completes the proof. □
For each nonempty closed convex subset C of a reflexive, convex and smooth Banach space E, we define the mapping ${R}_{C}$ of E onto C by ${R}_{C}x={x}_{0}$, where ${x}_{0}$ is defined by (2.5). For the case $p=2$, it is easy to see that the mapping is coincident with the metric projection in the setting of Hilbert spaces. In our discussion, instead of the metric projection, we make use of the mapping ${R}_{C}$. Finally, we prove two results concerning Proposition 2.4 and the mapping ${R}_{C}$. The first one is the usual analogue of a characterization of the metric projection in a Hilbert space.
Proposition 2.5 Let E be a smooth Banach space, let C be a convex subset of E, let $x\in E$ and ${x}_{0}\in C$. Then
if and only if
Proof First, we show that (2.6) ⇒ (2.7). Let $z\in C$ and $\lambda \in (0,1)$. It follows from $\mathrm{\Psi}({x}_{0},x)\le \mathrm{\Psi}((1\lambda ){x}_{0}+\lambda z,x)$ that
which implies
Taking $\lambda \downarrow 0$, since ${J}_{p}$ is normtoweak∗ continuous, we obtain
which shows (2.7).
Next, we show that (2.7) ⇒ (2.6). For any $z\in C$, we have
which proves (2.6). This completes the proof. □
Proposition 2.6 Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E, let $x\in E$ and ${R}_{C}x\in C$ with
Then we have
Proof It follows from Proposition 2.5 that
This completes the proof. □
3 Main results
Throughout this section, unless otherwise stated, we assume that $T:E\to {2}^{{E}^{\ast}}$ is a occasionally pseudomonotone maximal monotone operator. In this section, we study the following algorithm $\{{x}_{n}\}$ in a smooth Banach space E, which is an extension of (1.2):
where $\{{r}_{n}\}$ is a sequence of positive real numbers.
First, we investigate the condition under which the algorithm (3.1) is well defined. Rockafellar [23] proved the following theorem.
Theorem 3.1 Let E be a reflexive, strictly convex and smooth Banach space, and let $T:E\to {2}^{{E}^{\ast}}$ be a monotone operator. Then T is maximal if and only if $R({J}_{p}+rT)={E}^{\ast}$ for all $r>0$.
By the appropriate modification of arguments in Theorem 3.1, we can prove the following.
Theorem 3.2 Let E be a reflexive, strictly convex and smooth Banach space, and let $T:E\to {2}^{{E}^{\ast}}$ be an occasionally pseudomonotone operator. Then T is maximal if and only if $R({J}_{p}+rT)={E}^{\ast}$ for all $r>0$.
Using Theorem 3.2, we can show the following result.
Proposition 3.3 Let E be a reflexive, strictly convex and smooth Banach space. If ${T}^{1}0\ne \mathrm{\varnothing}$, then the sequence generated $\{{x}_{n}\}$ by (3.1) is well defined.
Proof From the definition of the sequence $\{{x}_{n}\}$, it is obvious that both ${H}_{n}$ and ${W}_{n}$ are closed convex sets. Let $w\in {T}^{1}0$. From Theorem 3.2, there exists $({y}_{0},{v}_{0})\in E\times {E}^{\ast}$ such that
Since T is occasionally pseudomonotone and $\u3008{y}_{0}w,0\u3009=0\ge 0$, from $Tw\ni 0$, it follows that
for some ${v}_{0}\in T{y}_{0}$. It follows that $w\in {H}_{0}$. On the other hand, it is clear that $w\in {W}_{0}=E$. Then $w\in {H}_{0}\cap {W}_{0}$ and so ${x}_{1}={R}_{{H}_{0}\cap {W}_{0}}{x}_{0}$ is well defined. Suppose that $w\in {H}_{n1}\cap {W}_{n1}$ is well defined for some $n\ge 1$. Again, by Theorem 3.2, we obtain $({y}_{n},{v}_{n})\in E\times {E}^{\ast}$ such that
Then since T is occasionally pseudomonotone and $\u3008{y}_{n}w,0\u3009=0\ge 0$, from $Tw\ni 0$, it follows that
for some ${v}_{n}\in T{y}_{n}$, and so $w\in {H}_{n}$. It follows from Proposition 2.5 that
which implies $w\in {W}_{n}$. Therefore, $w\in {H}_{n}\cap {W}_{n}$ and hence ${x}_{n1}={R}_{{H}_{n}\cap {W}_{n}}{x}_{0}$ is well defined. Then by induction, the sequence $\{{x}_{n}\}$ generated by (3.1) is well defined for each $n\ge 0$. This completes the proof. □
Remark 3.1 From the above proof, we obtain
Now, we are ready to prove our main theorem.
Theorem 3.4 Let E be a reflexive, strictly convex and uniformly smooth Banach space. If ${T}^{1}0\ne \mathrm{\varnothing}$, ϕ satisfies the condition (2.2) and $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ satisfies ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$, then the sequence $\{{x}_{n}\}$ generated by (3.1) converges strongly to ${R}_{{T}^{1}0}{x}_{0}$.
Proof It follows from the definition of ${W}_{n+1}$ and Proposition 2.5 that ${x}_{n+1}={R}_{{W}_{n+1}}{x}_{0}$. Further, from ${x}_{0}\in L({x}_{n},{R}_{{W}_{n+1}}{x}_{0})\cap {W}_{n1}$ and Proposition 2.6, we have
and hence
Since the sequence $\{\mathrm{\Psi}({x}_{n},{x}_{0})\}$ is monotone decreasing and bounded below by 0, it follows that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}\mathrm{\Psi}({x}_{n},{x}_{0})$ exists and, in particular, $\{\mathrm{\Psi}({x}_{n},{x}_{0})\}$ is bounded. Then by (2.3), $\{{x}_{n}\}$ is also bounded. This implies that there exists a subsequence $\{{x}_{{n}_{i}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{i}}\rightharpoonup w$ for some $w\in E$.
Now, we show that $w\in {T}^{1}0$. It follows from (3.2) that $\mathrm{\Psi}({x}_{n},{x}_{n+1})\to 0$. On the other hand, we have
Since ${R}_{{H}_{n}}{x}_{n}\in {H}_{n}$ and $0={v}_{n}+\frac{1}{{r}_{n}}({J}_{p}({y}_{n}){J}_{p}({x}_{n}))$, it follows that
and so $\mathrm{\Psi}({R}_{{H}_{n}}{x}_{n},{x}_{n})\ge \mathrm{\Psi}({y}_{n},{x}_{n})$. Further, since ${x}_{n+1}\in {H}_{n}$, we have
which yields that
Then it follows from $\mathrm{\Psi}({x}_{n},{x}_{n+1})\to 0$ that $\mathrm{\Psi}({y}_{n},{x}_{n})\to 0$. Consequently, by Proposition 2.3, we have ${y}_{n}{x}_{n}\to 0$, which implies ${y}_{{n}_{i}}\rightharpoonup w$. Moreover, since J is uniformly normtonorm continuous on bounded subsets and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$, we obtain
It follows from ${v}_{n}\in T{y}_{n}$ with ${v}_{n}\to 0$ and ${y}_{{n}_{i}}\rightharpoonup w$ that
Then, since T is occasionally pseudomonotone, it follows that $\u3008zw,{z}^{\prime}\u3009=0$ for some ${z}^{\prime}\in Tz$. Therefore, from the maximality of T, we obtain $w\in {T}^{1}0$. Let ${w}^{\ast}\in {R}_{{T}^{1}0}{x}_{0}$. Now, from ${x}_{n+1}={R}_{{H}_{n}\cap {W}_{n}}{x}_{0}$ and ${w}^{\ast}\in {T}^{1}0\subset L({x}_{n},{R}_{{W}_{n+1}}{x}_{0})\cap {H}_{n}\cap {W}_{n}$, we have
Then we have
which yields
Thus, from Proposition 2.5, we have
Then we obtain ${lim\hspace{0.17em}sup}_{i\to \mathrm{\infty}}\mathrm{\Psi}({x}_{{n}_{i}},{w}^{\ast})\le 0$, and hence $\mathrm{\Psi}({x}_{{n}_{i}},{w}^{\ast})\to 0$. It follows from Proposition 2.3 that ${x}_{{n}_{i}}\to {w}^{\ast}$. This means that the whole sequence $\{{x}_{n}\}$ generated by (3.1) converges weakly to ${w}^{\ast}$ and each weakly convergent subsequence of $\{{x}_{n}\}$ converges strongly to ${w}^{\ast}$. Therefore, $\{{x}_{n}\}$ converges strongly to ${w}^{\ast}\in {R}_{{T}^{1}0}{x}_{0}$. This completes the proof. □
4 An application
Let $f:E\to (\mathrm{\infty},\mathrm{\infty}]$ be a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined by
Using Theorem 3.4, we consider the problem of finding a minimizer of the function f.
Theorem 4.1 Let E be reflexive, strictly convex and uniformly smooth Banach space, and let $f:E\to (\mathrm{\infty},\mathrm{\infty}]$ be a proper convex lower semicontinuous function. Assume that $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ satisfies ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$ and $\{{x}_{n}\}$ is the sequence generated by
If ${(\partial f)}^{1}\ne \mathrm{\varnothing}$, then the sequence $\{{x}_{n}\}$ generated by (4.1) converges strongly to the minimizer of f.
Proof Since $f:E\to (\mathrm{\infty},\mathrm{\infty}]$ is a proper convex lower semicontinuous function, by Rockafelllar [23], the subdifferential ∂f of f is a maximal monotone operator and so it is also an occasionally pseudomonotone maximal operator. We also know that
is equivalent to the following:
This implies that
Thus, we have ${v}_{n}\in \partial f({y}_{n})$ such that $0={v}_{n}+\frac{1}{{r}_{n}}({J}_{p}({y}_{n}){J}_{p}({x}_{n}))$. Therefore, using Theorem 3.4, we get the conclusion. This completes the proof. □
5 Concluding remarks
We presented a modified proximaltype algorithm with the varied degree of rate of the convergence depending upon the choice of p ($1<p<\mathrm{\infty}$) for an occasionally pseudomonotone operator, which is a generalization of a monotone operator, to extend Kamimura and Takahashi’s result to more general Banach spaces which are not necessarily uniformly convex like locally uniformly Banach spaces. As an application, we consider the problem of finding a minimizer of a convex function in a more general setting of Banach spaces than what Kamimura and Takahashi have considered.
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The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 20120008170).
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Pathak, H.K., Cho, Y.J. Strong convergence of a proximaltype algorithm for an occasionally pseudomonotone operator in Banach spaces. Fixed Point Theory Appl 2012, 190 (2012). https://doi.org/10.1186/168718122012190
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Keywords
 proximal point algorithm
 monotone operator
 maximal monotone operator
 pseudomonotone operator
 occasionally pseudomonotone operator
 maximal pseudomonotone operator
 maximal occasionally pseudomonotone operator
 Banach space
 strong convergence