New construction and proof techniques of projection algorithm for countable maximal monotone mappings and weakly relatively non-expansive mappings in a Banach space
Abstract
In a real uniformly convex and uniformly smooth Banach space, some new monotone projection iterative algorithms for countable maximal monotone mappings and countable weakly relatively non-expansive mappings are presented. Under mild assumptions, some strong convergence theorems are obtained. Compared to corresponding previous work, a new projection set involves projection instead of generalized projection, which needs calculating a Lyapunov functional. This may reduce the computational labor theoretically. Meanwhile, a new technique for finding the limit of the iterative sequence is employed by examining the relationship between the monotone projection sets and their projections. To check the effectiveness of the new iterative algorithms, a specific iterative formula for a special example is proved and its computational experiment is conducted by codes of Visual Basic Six. Finally, the application of the new algorithms to a minimization problem is exemplified.
Keywords
Maximal monotone mapping Weakly relatively non-expansive mapping Projection Limit of a sequence of sets Uniformly convex and uniformly smooth Banach spaceMSC
47H05 47H09 47H101 Introduction and preliminaries
Let E be a real Banach space with \(E^{*}\) its dual space. Suppose that C is a nonempty closed and convex subset of E. The symbol 〈⋅, ⋅〉 denotes the generalized duality pairing between E and \(E^{*}\). The symbols “→” and “⇀” denote strong and weak convergence either in E or in \(E^{*}\), respectively.
A Banach space E is said to be uniformly convex [1] if for any two sequences \(\{ x_{n} \} \) and \(\{ y_{n} \} \) in E such that \(\Vert x_{n} \Vert = \Vert y_{n} \Vert = 1\) and \(\lim_{n \to \infty} \Vert x_{n} + y_{n} \Vert = 2\), \(\lim _{n \to \infty } \Vert x_{n} - y_{n} \Vert = 0\) holds.
If E is uniformly convex, then it is strictly convex.
A Banach space E is said to be uniformly smooth [2] if \(\frac{\rho_{E}(t)}{t} \to 0\), as \(t \to 0\).
The Banach space E is uniformly smooth if and only if \(E^{*}\) is uniformly convex [2].
We say E has Property (H) if for every sequence \(\{ x_{n}\} \subset E\) which converges weakly to \(x \in E\) and satisfies \(\Vert x_{n} \Vert \to \Vert x \Vert \) as \(n \to \infty \) necessarily converges to x in the norm.
If E is uniformly convex and uniformly smooth, then E has Property (H).
Then the multi-valued mapping \(J:E \to 2^{E^{*}}\) is called the normalized duality mapping [1]. Now, we list some elementary properties of J.
Lemma 1.1
- (1)
IfEis a real reflexive and smooth Banach space, thenJis single valued;
- (2)
ifEis reflexive, thenJis surjective;
- (3)
ifEis uniformly smooth and uniformly convex, then\(J^{ - 1}\)is also the normalized duality mapping from\(E^{*}\)into E. Moreover, bothJand\(J^{ - 1}\)are uniformly continuous on each bounded subset ofEor\(E^{*}\), respectively;
- (4)
for\(x \in E\)and\(k \in ( - \infty, + \infty )\), \(J(kx) = kJ(x)\).
For a nonlinear mapping U, we use \(F(U)\) and \(N(U)\) to denote its fixed point set and null point set, respectively; that is, \(F(U) = \{ x \in D(U):Ux = x\}\) and \(N(U) = \{ x \in D(U):Ux = 0\}\).
Definition 1.2
([3])
A mapping \(T \subset E \times E^{*}\) is said to be monotone if, for \(\forall y_{i} \in Tx_{i}\), \(i = 1,2\), we have \(\langle x_{1} - x_{2},y_{1} - y_{2} \rangle \ge 0\). The monotone mapping T is called maximal monotone if \(R(J + \theta T) = E^{*}\) for \(\theta > 0\).
Definition 1.3
([4])
Definition 1.4
([5])
- (1)
an element \(p \in C\) is said to be an asymptotic fixed point of B if there exists a sequence \(\{ x_{n}\}\) in C which converges weakly to p such that \(x_{n} - Bx_{n} \to 0\), as \(n \to \infty \). The set of asymptotic fixed points of B is denoted by \(\hat{F}(B)\);
- (2)
\(B:C \to C\) is said to be strongly relatively non-expansive if \(\hat{F}(B) = F(B) \ne \emptyset \) and \(\varphi (p,Bx) \le \varphi (p,x)\) for \(x \in C\) and \(p \in F(B)\);
- (3)
an element \(p \in C\) is said to be a strong asymptotic fixed point of B if there exists a sequence \(\{ x_{n}\}\) in C which converges strongly to p such that \(x_{n} - Bx_{n} \to 0\), as \(n \to \infty \). The set of strong asymptotic fixed points of B is denoted by \(\tilde{F}(B)\);
- (4)
\(B:C \to C\) is said to be weakly relatively non-expansive if \(\tilde{F}(B) = F(B) \ne \emptyset \) and \(\varphi (p,Bx) \le \varphi (p,x)\) for \(x \in C\) and \(p \in F(B)\).
Remark 1.5
It is easy to see that strongly relatively non-expansive mappings are weakly relatively non-expansive mappings. However, an example in [6] shows that a weakly relatively non-expansive mapping is not a strongly relatively non-expansive mapping.
Lemma 1.6
([5])
LetEbe a uniformly convex and uniformly smooth Banach space andCbe a nonempty closed and convex subset of E. If\(B:C \to C\)is weakly relatively non-expansive, then\(F(B)\)is a closed and convex subset of E.
Lemma 1.7
([3])
- (1)
\(N(T)\)is a closed and convex subset ofE;
- (2)
if\(x_{n} \to x\)and\(y_{n} \in Tx_{n}\)with\(y_{n} \rightharpoonup y\), or\(x_{n} \rightharpoonup x\)and\(y_{n} \in Tx_{n}\)with\(y_{n} \to y\), then\(x \in D(T)\)and\(y \in Tx\).
Definition 1.8
([4])
- (1)
If E is a reflexive and strictly convex Banach space and C is a nonempty closed and convex subset of E, then for each \(x \in E\) there exists a unique element \(v \in C\) such that \(\Vert x - v \Vert = \inf \{ \Vert x - y \Vert :y \in C\}\). Such an element v is denoted by \(P_{C}x\) and \(P_{C}\) is called the metric projection of E onto C.
- (2)
Let E be a real reflexive, strictly convex, and smooth Banach space and C be a nonempty closed and convex subset of E, then for \(\forall x \in E\), there exists a unique element \(x_{0} \in C\) satisfying \(\varphi (x_{0},x) = \inf \{ \varphi (y, x) :y \in C\}\). In this case, \(\forall x \in E\), define \(\Pi_{C}:E \to C\) by \(\Pi_{C}x = x_{0}\), and then \(\Pi_{C}\) is called the generalized projection from E onto C.
It is easy to see that \(\Pi_{C}\) is coincident with \(P_{C}\) in a Hilbert space.
Maximal monotone mappings and weakly or strongly relatively non-expansive mappings are different types of important nonlinear mappings due to their practical background. Much work has been done in designing iterative algorithms either to approximate a null point of maximal monotone mappings or a fixed point of weakly or strongly relatively non-expansive mappings, see [5, 6, 7, 8, 9, 10] and the references therein. It is a natural idea to construct iterative algorithms to approximate common solutions of a null point of maximal monotone mappings and a fixed point of weakly or strongly relatively non-expansive mappings, which can be seen in [11, 12, 13, 14, 15] and the references therein. Now, we list some closely related work.
Under some mild assumptions, \(\{ x_{n}\}\) generated by (1.1), (1.2), or (1.3) is proved to be strongly convergent to \(\Pi_{N(T) \cap F(S)}(x _{1})\). Compared to projective iterative algorithms (1.1) and (1.2), iterative algorithm (1.3) is called monotone projection method since the projection sets \(H_{{n}}\), \(V_{n}\), and \(W_{n}\) are all monotone in the sense that \(H_{n + 1} \subset H_{n}\), \(V_{n + 1} \subset V_{n}\), and \(W_{{n} + 1} \subset W_{n}\) for \(n \in N\). Theoretically, the monotone projection method will reduce the computation task.
Under some assumptions, \(\{ x_{n}\}\) generated by (1.4) is proved to be strongly convergent to \(\Pi_{N(A) \cap F(S) \cap F(T)}(x_{1})\).
Inspired by the previous work, in Sect. 2.1, we shall construct some new iterative algorithms to approximate the common element of the sets of null points of countable maximal monotone mappings and the sets of fixed points of countable weakly relatively non-expansive mappings. New proof techniques can be found, restrictions are mild, and error is considered. In Sect. 2.2, an example is listed and a specific iterative formula is proved. Computational experiments which show the effectiveness of the new abstract iterative algorithms are conducted. In Sect. 2.3, an application to the minimization problem is demonstrated.
The following preliminaries are also needed in our paper.
Definition 1.9
([16])
- (1)
\(s\mbox{-}\lim \inf C_{n}\), which is called strong lower limit, is defined as the set of all \(x \in E\) such that there exists \(x_{n} \in C_{n}\) for almost all n and it tends to x as \(n \to \infty \) in the norm.
- (2)
\(w\mbox{-}\lim \sup C_{n}\), which is called weak upper limit, is defined as the set of all \(x \in E\) such that there exists a subsequence \(\{ C_{n_{k}}\}\) of \(\{ C_{n}\}\) and \(x_{n_{k}} \in C_{n_{k}}\) for every \(n_{k}\) and it tends to x as \(n_{k} \to \infty \) in the weak topology;
- (3)
if \(s\mbox{-}\lim \inf C_{n} = w\mbox{-}\lim \sup C_{n}\), then the common value is denoted by \(\lim C_{n}\).
Lemma 1.10
([16])
Let\(\{ C_{n}\}\)be a decreasing sequence of closed and convex subsets ofE, i.e., \(C_{n} \subset C_{m}\)if\(n \ge m\). Then\(\{ C_{n}\}\)converges in E and\(\lim C_{n} = \bigcap_{n = 1}^{\infty } C_{n}\).
Lemma 1.11
([17])
Suppose thatEis a real reflexive and strictly convex Banach space. If\(\lim C_{n}\)exists and is not empty, then\(\{ P_{c_{n}}x\}\)converges weakly to\(P_{\lim C_{n}}x\)for every\(x \in E\). Moreover, ifEhas Property (H), the convergence is in norm.
Lemma 1.12
([18])
LetEbe a real smooth and uniformly convex Banach space, and let\(\{ u_{n}\}\)and\(\{ v_{n}\}\)be two sequences of E. If either\(\{ u_{n}\}\)or\(\{ v_{n}\}\)is bounded and\(\varphi (u_{n},v_{n}) \to 0\), as\(n \to \infty \), then\(u_{n} - v_{n} \to 0\), as\(n \to \infty \).
Lemma 1.13
([19])
2 Strong convergence theorems and experiments
2.1 Strong convergence for infinite maximal monotone mappings and infinite weakly relatively non-expansive mappings
- (A1)
E is a real uniformly convex and uniformly smooth Banach space and \(J:E \to E^{*}\) is the normalized duality mapping;
- (A2)
\(T_{i} \subset E \times E^{*}\) is maximal monotone and \(S_{i}:E \to E\) is weakly relatively non-expansive for each \(i \in N\);
- (A3)
\(\{ s_{n,i}\}\) and \(\{ \tau_{n}\}\) are two real number sequences in (\(0, + \infty \)) for \(i,n \in N\). \(\{ \alpha_{n}\}\) is a real number sequence in (\(0,1\)) for \(n \in N\);
- (A4)
\(\{ \varepsilon_{n}\}\) is the error sequence in E.
Algorithm 2.1
Step 1. Choose \(u_{1},\varepsilon_{1} \in E\). Let \(s_{1,i} \in (0, + \infty )\) for \(i \in N\). \(\alpha_{1} \in (0,1)\) and \(\tau_{1} \in (0, + \infty )\). Set \(n = 1\), and go to Step 2.
Step 2. Compute \(v_{n,i} = (J + s_{n,i}T_{i})^{ - 1}J(u_{n} + \varepsilon_{n})\) and \(w_{n,i} = J^{ - 1}[\alpha_{n}Ju_{n} + (1 - \alpha_{n})JS_{i}v_{n,i}]\) for \(i \in N\). If \(v_{n,i} = u_{n} + \varepsilon_{n}\) and \(w_{n,i} = J^{ - 1}[\alpha_{n}Ju_{n} + (1 - \alpha_{n})J(u_{n} + \varepsilon_{n})]\) for all \(i \in N\), then stop; otherwise, go to Step 3.
Step 4. Choose any element \(u_{n + 1} \in U_{n + 1}\) for \(n \in N\).
Step 5. Set \(n = n + 1\), and return to Step 2.
Theorem 2.1
If, in Algorithm 2.1, \(v_{n,i} = u_{n} + \varepsilon_{n}\)and\(w_{n,i} = J^{ - 1}[\alpha_{n}Ju_{n} + (1 - \alpha _{n})J(u_{n} + \varepsilon_{n})]\)for all\(i \in N\), then\(u_{n} + \varepsilon_{n} \in (\bigcap_{i = 1}^{\infty } N(T_{i})) \cap ( \bigcap_{i = 1}^{\infty } F(S_{i}))\).
Proof
Since \(v_{n,i} = u_{n} + \varepsilon_{n}\), then from Step 2 in Algorithm 2.1, we know that \(Jv_{n,i} + s_{n,i}T_{i}v_{n,i} = Jv _{n,i}\) for all \(i \in N\), which implies that \(s_{n,i}T_{i}v_{n,i} = 0\) for \(i \in N\). Therefore, \(u_{n} + \varepsilon_{n} \in \bigcap_{i = 1} ^{\infty } N(T_{i})\).
Since \(w_{n,i} = J^{ - 1}[\alpha_{n}Ju_{n} + (1 - \alpha_{n})J(u_{n} + \varepsilon_{n})] = J^{ - 1}[\alpha_{n}Ju_{n} + (1 - \alpha_{n})JS _{i}v_{n,i}]\), then in view of Lemma 1.1\(v_{n,i} = S_{i}v_{n,i}\) for \(i,n \in N\). Thus \(v_{n,i} = u_{n} + \varepsilon_{n} \in \bigcap_{i = 1}^{\infty } F(S_{i})\), \(n \in N\).
This completes the proof. □
Theorem 2.2
Suppose\((\bigcap_{i = 1}^{\infty } N(T_{i})) \cap (\bigcap_{i = 1}^{\infty } F(S_{i})) \ne \emptyset, \inf_{n}s _{n,i} > 0\)for\(i \in N\), \(0 < \sup_{n}\alpha_{n} < 1\), \(\tau_{n} \to 0\), and\(\varepsilon_{n} \to 0\), as\(n \to \infty \). Then the iterative sequence\(u_{n} \to y_{0} = P_{\bigcap_{n = 1}^{\infty } W _{n}} (u_{1})\in (\bigcap_{i = 1}^{\infty } N(T_{i})) \cap (\bigcap_{i = 1}^{\infty } F(S_{i}))\), as\(n \to \infty \).
Proof
We split the proof into eight steps.
Step 1. \(V_{n}\) is a nonempty subset of E.
In fact, we shall prove that \((\bigcap_{i = 1}^{\infty } N(T_{i})) \cap (\bigcap_{i = 1}^{\infty } F(S_{i})) \subset V_{n}\), which ensures that \(V_{n} \ne \emptyset \).
For this, we shall use inductive method. Now, \(\forall p \in ( \bigcap_{i = 1}^{\infty } N(T_{i})) \cap (\bigcap_{i = 1}^{\infty } F(S _{i}))\).
Thus \(p \in V_{2,i}\), which ensures that \(p \in V_{2}\).
Then \(p \in V_{k + 2,i}\), which ensures that \(p \in V_{k + 2}\).
Therefore, by induction, \((\bigcap_{i = 1}^{\infty } N(T_{i})) \cap ( \bigcap_{i = 1}^{\infty } F(S_{i})) \subset V_{n}\) for \(n \in N\).
Step 2. \(W_{n}\) is a nonempty closed and convex subset of E for \(n \in N\).
Since \(\varphi (z,w_{n,i}) \le \alpha_{n} \varphi (z,u_{n}) + (1 - \alpha_{n})\varphi (z,v_{n,i})\) is equivalent to \(\langle z,2\alpha_{n}Ju_{n} + 2(1 - \alpha_{n})Jv_{n,i} - 2Jw_{n,i} \rangle \leq \alpha_{n}\Vert u _{n} \Vert ^{2} + (1 - \alpha_{n})\Vert v_{n,i} \Vert ^{2} - \Vert w_{n,i} \Vert ^{2}\), then it is easy to see that \(W_{n,i}\) is closed and convex for \(i,n \in N\). Thus \(W_{n}\) is closed and convex for \(n \in N\).
Next, we shall use inductive method to show that \((\bigcap_{i = 1} ^{\infty } N(T_{i})) \cap (\bigcap_{i = 1}^{\infty } F(S_{i})) \subset W_{n}\) for \(n \in N\), which ensures that \(W_{n} \ne \emptyset \) for \(n \in N\).
In fact, \(\forall p \in (\bigcap_{i = 1}^{\infty } N(T_{i})) \cap ( \bigcap_{i = 1}^{\infty } F(S_{i}))\).
Combining this with Step 1, we know that \(p \in W_{2,i}\) for \(i \in N\). Therefore, \(p \in W_{2}\).
Step 3. Set \(y_{n} = P_{W_{n + 1}}(u_{1})\). Then \(y_{n} \to y_{0} = P_{\bigcap_{n = 1}^{\infty } W_{n}} (u_{1})\), as \(n \to \infty \).
From the construction of \(W_{n}\) in Step 3 of Algorithm 2.1, \(W_{n + 1} \subset W_{n}\) for \(n \in N\). Lemma 1.10 implies that \(\lim W_{n}\) exists and \(\lim W_{n} = \bigcap_{n = 1}^{\infty } W_{n} \ne \emptyset \). Since E has Property (H), then Lemma 1.11 implies that \(y_{n} \to y_{0} = P_{\bigcap_{n = 1}^{\infty } W_{n}} (u_{1})\), as \(n \to \infty \).
Step 4. \(\{ u_{n}\}\) is well defined.
This ensures that \(U_{n + 1} \ne \emptyset \) for \(n \to \infty \).
Step 5. \(u_{n + 1} - y_{n} \to 0\) as \(n \to \infty \).
Step 6. \(u_{n} - v_{n,i} \to 0\) for \(i \in N\), as \(n \to \infty \).
Step 7. \(w_{n,i} - u_{n} \to 0\) for \(i \in N\), as \(n \to \infty \).
Step 8. \(y_{0} = P_{\bigcap_{n = 1}^{\infty } W_{n}} (u_{1}) \in (\bigcap_{i = 1}^{\infty } N(T_{i})) \cap (\bigcap_{i = 1}^{\infty } F(S _{i}))\).
Since \(v_{n,i} = (J + s_{n,i}T_{i})^{ - 1}J(u_{n} + \varepsilon_{n})\), then \(Jv_{n,i} + s_{n,i}T_{i}v_{n,i} = J(u_{n} + \varepsilon_{n})\). Since \(v_{n,i} \to y_{0}\), \(u_{n} \to y_{0}\), \(\varepsilon_{n} \to 0\) and \(\inf_{n}s_{n,i} > 0\), then \(T_{i}v_{n,i} \to 0\) for \(i \in N\), as \(n \to \infty \). Using Lemma 1.7, \(y_{0} \in \bigcap_{i = 1}^{\infty } N(T_{i})\).
Since \(w_{n,i} = J^{ - 1}[\alpha_{n}Ju_{n} + (1 - \alpha_{n})JS_{i}v_{n,i}]\), then in view of Lemma 1.1, \(S_{i}v_{n,i} \to y_{0}\), as \(n \to \infty \). Lemma 1.6 implies that \(y_{0} \in \bigcap_{i = 1}^{\infty } F(S_{i})\).
This completes the proof. □
Corollary 2.3
- (1)
Similar to Theorem 2.1, if\(v_{n} = u_{n} + \varepsilon_{n}\)and\(w_{n} = J^{ - 1}[\alpha_{n}Ju_{n} + (1 - \alpha_{n})J(u_{n} + \varepsilon_{n})]\)for all\(n \in N\), then\(u_{n} + \varepsilon_{n} \in N(T) \cap F(S)\).
- (2)
Suppose thatE, \(\{ \varepsilon_{n}\}\), \(\{ \tau_{n}\}\), and\(\{ \alpha_{n}\}\)satisfy the same conditions as those in Theorem 2.2. If\(N(T) \cap F(S) = \emptyset \)and\(\inf_{n}s_{n} > 0\), then the iterative sequence\(u_{n} \to y_{0} = P_{\bigcap_{n= 1}^{\infty } W_{n}} (u_{1}) \in N(T) \cap F(S)\), as\(n \to \infty \).
Algorithm 2.2
Theorem 2.4
If, in Algorithm 2.2, \(v_{n,i} = u_{n} + \varepsilon_{n}\)and\(w_{n,i} = J^{ - 1}[\alpha_{n}Ju_{1} + (1 - \alpha_{n})J(u_{n} + \varepsilon_{n})]\)for all\(i \in N\), then\(u_{n} + \varepsilon_{n} \in (\bigcap_{i = 1}^{\infty } N(T_{i} )) \cap (\bigcap_{i = 1}^{\infty } F(S_{i} ))\).
Proof
Similar to Theorem 2.1, the result follows. □
Theorem 2.5
We only change the condition that\(0 < \sup_{n} \alpha_{n} < 1\)in Theorem 2.2by\(\alpha_{n} \to 0\), as\(n \to \infty \). Then the iterative sequence\(u_{n} \to y_{0} = P_{\bigcap _{n = 1}^{\infty } W_{n}} (u_{1}) \in (\bigcap_{i = 1}^{\infty } N(T_{i} )) \cap (\bigcap_{i = 1}^{\infty } F(S_{i} ))\), as\(n \to \infty \).
Proof
Copy Steps 1, 3, 4, 5, and 6 in Theorem 2.2 and make slight changes in the following steps.
Step 2. \(W_{n}\) is a nonempty closed and convex subset of E for \(n \in N\).
Since \(\varphi (z,w_{n,i}) \le \alpha_{n}\varphi (z,u_{1}) + (1 - \alpha _{n})\varphi (z,v_{n,i})\) is equivalent to \(\langle z,2\alpha_{n}Ju _{1} + 2(1 - \alpha_{n})Jv_{n,i} - 2Jw_{n,i} \rangle \le \alpha _{n}\Vert u_{1} \Vert ^{2} + (1 - \alpha_{n})\Vert v_{n,i} \Vert ^{2} - \Vert w_{n,i} \Vert ^{2}\), then it is easy to see that \(W_{n,i}\) is closed and convex for \(i,n \in N\). Thus \(W_{n}\) is closed and convex for \(n \in N\).
Next, we shall use inductive method to show that \((\bigcap_{i = 1} ^{\infty } N(T_{i} )) \cap (\bigcap_{i = 1}^{\infty } F(S_{i} )) \subset W_{n}\) for \(n \in N\), which ensures that \(W_{n} \ne \emptyset \) for \(n \in N\).
In fact, \(\forall p \in (\bigcap_{i = 1}^{\infty } N(T_{i} )) \cap ( \bigcap_{i = 1}^{\infty } F(S_{i} ))\).
Combining this with Step 1, we know that \(p \in W_{2,i}\) for \(i \in N\). Therefore, \(p \in W_{2}\).
Step 7. \(w_{n,i} - u_{n} \to 0\) for \(i \in N\), as \(n \to \infty \).
Step 8. \(y_{0} = P_{\bigcap_{n = 1}^{\infty } W_{n}} (u_{1}) \in ( \bigcap_{i = 1}^{\infty } N(T_{i} )) \cap (\bigcap_{i = 1}^{\infty } F(S _{i} ))\).
In the same way as Step 8 in Theorem 2.2, we have \(y_{0} \in \bigcap_{i = 1}^{\infty } N(T_{i})\). Since \(w_{n,i} = J^{ - 1}[\alpha _{n}Ju_{1} + (1 - \alpha_{n})JS_{i}v_{n,i}]\), then \(S_{i}v_{n,i} \to y _{0}\), as \(n \to \infty \). Thus in view of Lemma 1.6, \(y_{0} \in \bigcap_{i = 1}^{\infty } F(S_{i})\).
This completes the proof. □
Corollary 2.6
- (1)
Similar to Theorem 2.4, if\(v_{n} = u_{n} + \varepsilon_{n}\)and\(w_{n} = J^{ - 1}[\alpha_{n}Ju_{1} + (1 - \alpha_{n})J(u_{n} + \varepsilon_{n})]\), then\(u_{n} + \varepsilon_{n} \in N(T) \cap F(S)\)for all\(n \in N\).
- (2)
Suppose thatE, \(\{ \varepsilon_{n}\}\), \(\{ \tau_{n}\}\), and\(\{ \alpha_{n}\}\)satisfy the same conditions as those in Theorem 2.5. If\(N(T) \cap F(S) = \emptyset \)and\(\inf_{n}s_{n} > 0\), then the iterative sequence\(u_{n} \to y_{0} = P_{\bigcap_{n = 1}^{\infty } W_{n}} (u_{1})\in N(T) \cap F(S)\)as\(n \to \infty \).
Remark 2.7
Compared to the existing related work, e.g., [12, 13, 14], strongly relatively non-expansive mappings are extended to weakly relatively non-expansive mappings. Moreover, in our paper, the discussion on this topic is extended to the case of infinite maximal monotone mappings and infinite weakly relatively non-expansive mappings.
Remark 2.8
Calculating the generalized projection \(\Pi_{H_{n} \cap V_{n} \cap W_{n}}(x_{1})\) in [12] or \(\Pi_{H_{n} \cap V_{n}}(x_{1})\) in [13] is replaced by calculating the projection \(P_{W_{n + 1}}(u_{1})\) in Step 3 in our Algorithms 2.1 and 2.2, which makes the computation easier.
Remark 2.9
A new proof technique for finding the limit \(y_{0} = P_{\bigcap_{n = 1}^{\infty } W_{n}} (u_{1})\) is employed in our paper by examining the properties of the projective sets \(W_{n}\) sufficiently, which is quite different from that for finding the limit \(\Pi_{N(T) \cap F(S)}(x_{1})\) in [12] or \(\Pi_{N(A) \cap F(S) \cap F(T)}(x _{1})\) in [13].
Remark 2.10
Theoretically, the projection is easier for calculating than the generalized projection in a general Banach space since the generalized projection involves a Lyapunov functional. In this sense, iterative algorithms constructed in our paper are new and more efficient.
2.2 Special cases in Hilbert spaces and computational experiments
Corollary 2.11
The results of Theorems 2.1 and 2.2 are true for this special case.
Corollary 2.12
The results of Theorems 2.4 and 2.5 are true for this special case.
Corollary 2.13
The results of Corollaries 2.3 and 2.6 are true for the special cases, respectively.
Remark 2.14
Take \(H = ( - \infty, + \infty )\), \(Tx = 2x\), and \(Sx = x\) for \(x \in ( - \infty, + \infty )\). Let \(\varepsilon_{n} = \alpha_{n} = \tau_{n} = \frac{1}{n}\) and \(s_{n} = 2^{n - 1}\) for \(n \in N\). Then T is maximal monotone and S is weakly relatively non-expansive. Moreover, \(N(T) \cap F(S) = \{ 0\}\).
Remark 2.15
Proof
From (2.6), \(v_{2} = \frac{u_{2} + \varepsilon_{2}}{1 + 2s_{2}} = \frac{4}{15} - \frac{\sqrt{22}}{60}\). Then \(0 < v_{2} < v_{1} < 1\). And it is easy to see \(v_{1} > \frac{1}{2^{1 + 1}(1 + 1)}\). Thus (2.12) is true for \(n + 1\).
Since \(0 < v_{k + 1} < 1 = u_{1}\), then \(P_{W_{k + 2}}(u_{1}) = v_{k + 1}\). Thus \(\vert u_{1} - z \vert \le \sqrt{ \vert P_{W_{k + 2}}(u _{1}) - u_{1} \vert ^{2} + \tau_{k + 2}}\) is equivalent to \(u_{1} - \sqrt{(u_{1} - v_{k + 1})^{2} + \tau_{k + 2}} \le z \le u _{1} + \sqrt{(u_{1} - v_{k + 1})^{2} + \tau_{k + 2}}\).
It is easy to check that \(u_{1} + \sqrt{(u_{1} - v_{k + 1})^{2} + \tau_{k + 2}} > 1 > v_{k + 1}\), and \(u_{1} - \sqrt{(u_{1} - v_{k + 1})^{2} + \tau_{k + 2}} < u_{1} - (u_{1} - v_{k + 1}) = v_{k + 1}\).
Now, we show that \(v_{k + 2} > 0\).
Next, we show that \(v_{k + 1} > \frac{1}{2^{k + 2}(k + 2)}\).
Finally, we show that \(v_{k + 2} < v_{k + 1}\).
From the definition of \(u_{k + 2}\), we have \(u_{k + 2} < \frac{1 + v _{k + 1} - (1 - v_{k + 1})}{2} = v_{k + 1}\). Then \(v_{k + 2} < \frac{v _{k + 1} + \frac{1}{k + 2}}{1 + 2^{k + 2}}\). Since \(v_{k + 1} > \frac{1}{2^{k + 2}(k + 2)}\), then \(\frac{v_{k + 1} + \frac{1}{k + 2}}{1 + 2^{k + 2}} - v_{k + 1} = \frac{\frac{1}{k + 2} - 2^{k + 2}v_{k + 1}}{1 + 2^{k + 2}} < 0\), which implies that \(v_{k + 2} < v_{k + 1}\).
Therefore, by induction, (2.12) is true for \(n \in N\). Since \(0 < v_{n + 1} < v_{n} < 1\), then \(\lim_{n \to \infty } v_{n}\) exists. Set \(a = \lim_{n \to \infty } v_{n}\). From (2.12), \(\lim_{n \to \infty } u_{n} = a\) and from (2.6), \(a = 0\). Then in view of (2.7), \(\lim_{n \to \infty } w_{n} = 0\). That is, \(\lim_{n \to \infty } w_{n} = \lim_{n\to\infty } v_{n} = \lim_{n \to \infty } u_{n} = 0\).
This completes the proof. □
Remark 2.16
Convergence of \(\{ u_{n}\}\), \(\{ v_{n}\}\), and \(\{ w_{n} \}\)
Numerical results of \(\{ u_{n}\}\), \(\{ v_{n}\}\), and \(\{ w_{n}\}\) with initial \(u_{1} = 1.0\)
n | \(v_{n}\) | \(w_{n}\) | \(u_{n}\) |
---|---|---|---|
1 | 0.666666666666667 | 1.00000000000000 | 1.00000000000000 |
2 | 0.188493070669609 | 0.315479212008828 | 0.442465353348047 |
3 | 0.047734978022387 | 0.063917141637640 | 0.096281468868147 |
4 | 0.013887781581545 | 0.006938907907725 | −0.01390771311373 |
5 | 0.005016751133393 | −0.00287604161289 | −0.03444721259803 |
6 | 0.002022073632571 | −0.00418691873111 | −0.03523188054954 |
7 | 0.000854971429905 | −0.00391942854572 | −0.03256582839944 |
8 | 0.000371596957448 | −0.00362300404227 | −0.02949958193595 |
9 | 0.000164574841194 | −0.00281862431655 | −0.02668421757849 |
10 | 0.000073908605586 | −0.002357850182411 | −0.02424367927438 |
Remark 2.17
Remark 2.18
Convergence of \(\{ u_{n}\}\), \(\{ v_{n}\}\), and \(\{ w_{n} \}\)
Numerical esults of \(\{ u_{n}\}\), \(\{ v_{n}\}\), and \(\{ w_{n}\}\) with initial \(u_{1} = 1.0\)
n | \(v_{n}\) | \(w_{n}\) | \(u_{n}\) |
---|---|---|---|
1 | 0.666666666666667 | 1.00000000000000 | 1.00000000000000 |
2 | 0.188493070669609 | 0.594246535334805 | 0.442465353348047 |
3 | 0.047734978022387 | 0.365156652014924 | 0.096281468868147 |
4 | 0.013887781581545 | 0.260415836186159 | −0.01390771311373 |
5 | 0.005016751133393 | 0.204013400906715 | −0.03444721259803 |
6 | 0.002022073632571 | 0.168351728027143 | −0.03523188054954 |
7 | 0.000854971429905 | 0.143589975511347 | −0.03256582839944 |
8 | 0.000371596957448 | 0.125325147337767 | −0.02949958193595 |
9 | 0.000164574841194 | 0.111257399858839 | −0.02668421757849 |
10 | 0.000073908605586 | 0.100066517745027 | −0.02424367927438 |
11 | 0.000033552200238 | 0.090939592909307 | −0.02216063262202 |
12 | 0.000015364834636 | 0.083347417765083 | −0.02038360583157 |
13 | 0.000007086981657 | 0.076929618752290 | −0.01885943628695 |
14 | 0.000003288762206 | 0.071431625279192 | −0.01754220267938 |
15 | 0.000001534136645 | 0.066668098527535 | −0.01639454294823 |
16 | 0.000000718881060 | 0.062500673950994 | −0.01538669196834 |
17 | 0.000000338196904 | 0.058823847714733 | −0.01449504667360 |
18 | 0.000000159662486 | 0.055555706347903 | −0.01370083322728 |
19 | 0.000000075612039 | 0.052631650579827 | −0.01298901840146 |
20 | 0.000000035908223 | 0.050000034112812 | −0.01234746359706 |
2.3 Applications to minimization problems
Theorem 2.19
- (1)
if\(v_{n} = u_{n} + \varepsilon_{n}\)and\(w_{n} = J^{ - 1}[\alpha_{n}Ju_{n} + (1 - \alpha_{n})J(u_{n} + \varepsilon_{n})]\)for all\(n \in N\), then\(u_{n} + \varepsilon_{n} \in N(\partial h) \cap F(S)\).
- (2)
If\(N(\partial h) \cap F(S) \ne \emptyset \)and\(\inf_{n}s _{n} > 0\), then the iterative sequence\(u_{n} \to y_{o} = P_{\bigcap_{n = 1}^{\infty }W_{n}} (u_{1}) \in N(\partial h) \cap F(S)\), as\(n \to \infty \).
Proof
Similar to [11], \(v_{n} = \arg \min_{z \in E}\{ h(z) + \frac{1}{2s_{n}}\Vert z \Vert ^{2} - \frac{1}{s_{n}} \langle z,J(u _{n} + \varepsilon_{n}) \rangle \}\) is equivalent to \(0 \in \partial h(v_{n}) + \frac{1}{s_{n}}Ju_{n} - \frac{1}{s_{n}}J(u_{n} + \varepsilon_{n})\). Then \(v_{n} = (J + s_{n}\partial h)^{ - 1}J(u_{n} + \varepsilon_{n})\). So, Corollary 2.3 ensures the desired results.
This completes the proof. □
Theorem 2.20
We only do the following changes in Theorem 2.19: \(w_{n} = J^{ - 1}[\alpha_{n}Ju_{1} + (1 - \alpha_{n})JSv_{n}]\)and\(W_{n + 1} = \{ z \in V_{n + 1}:\varphi (z,w_{n}) \le \alpha_{n}\varphi (z,u_{1}) + (1 - \alpha_{n})\varphi (z,v_{n})\} \cap W_{n}\). Then, under the assumptions of Corollary 2.6, we still have the result of Theorem 2.19.
Notes
Acknowledgements
Supported by the National Natural Science Foundation of China (11071053), Natural Science Foundation of Hebei Province (A2014207010), Key Project of Science and Research of Hebei Educational Department (ZD2016024), Key Project of Science and Research of Hebei University of Economics and Business (2016KYZ07), Youth Project of Science and Research of Hebei University of Economics and Business (2017KYQ09) and Youth Project of Science and Research of Hebei Educational Department (QN2017328).
Authors’ contributions
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
References
- 1.Takahashi, W.: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000) MATHGoogle Scholar
- 2.Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitz-Type Mappings with Applications. Springer, Berlin (2008) MATHGoogle Scholar
- 3.Pascali, D., Sburlan, S.: Nonlinear Mappings and Monotone Type. Sijthoff and Noordhoff, The Netherlands (1978) CrossRefMATHGoogle Scholar
- 4.Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Applied Mathematics, vol. 178, pp. 15–50. Dekker, New York (1996) Google Scholar
- 5.Zhang, J.L., Su, Y.F., Cheng, Q.Q.: Simple projection algorithm for a countable family of weak relatively nonexpansive mappings and applications. Fixed Point Theory Appl. 2012, Article ID 205 (2012) MathSciNetCrossRefMATHGoogle Scholar
- 6.Zhang, J.L., Su, Y.F., Cheng, Q.Q.: Hybrid algorithm of fixed point for weak relatively nonexpansive multivalued mappings and applications. Abstr. Appl. Anal. 2012, Article ID 479438 (2012) MathSciNetMATHGoogle Scholar
- 7.Matsushita, S., Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 134, 257–266 (2005) MathSciNetCrossRefMATHGoogle Scholar
- 8.Liu, Y.: Weak convergence of a hybrid type method with errors for a maximal monotone mapping in Banach spaces. J. Inequal. Appl. 2015, Article ID 260 (2015) MathSciNetCrossRefMATHGoogle Scholar
- 9.Su, Y.F., Li, M.Q., Zhang, H.: New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators. Appl. Math. Comput. 217, 5458–5465 (2011) MathSciNetMATHGoogle Scholar
- 10.Wei, L., Tan, R.: Iterative schemes for finite families of maximal monotone operators based on resolvents. Abstr. Appl. Anal. 2014, Article ID 451279 (2014). https://doi.org/10.1155/2014/451279 MathSciNetGoogle Scholar
- 11.Wei, L., Cho, Y.J.: Iterative schemes for zero points of maximal monotone operators and fixed points of nonexpansive mappings and their applications. Fixed Point Theory Appl. 2008, Article ID 168468 (2008) MathSciNetCrossRefMATHGoogle Scholar
- 12.Wei, L., Su, Y.F., Zhou, H.Y.: Iterative convergence theorems for maximal monotone operators and relatively nonexpansive mappings. Appl. Math. J. Chin. Univ. Ser. B 23(3), 319–325 (2008) MathSciNetCrossRefMATHGoogle Scholar
- 13.Klin-eam, C., Suantai, S., Takahashi, W.: Strong convergence of generalized projection algorithms for nonlinear operators. Abstr. Appl. Anal. 2009, Article ID 649831 (2009) MathSciNetCrossRefMATHGoogle Scholar
- 14.Wei, L., Su, Y.F., Zhou, H.Y.: Iterative schemes for strongly relatively nonexpansive mappings and maximal monotone operators. Appl. Math. J. Chin. Univ. Ser. B 25(2), 199–208 (2010) MathSciNetCrossRefMATHGoogle Scholar
- 15.Inoue, G., Takahashi, W., Zembayashi, K.: Strong convergence theorems by hybrid methods for maximal monotone operator and relatively nonexpansive mappings in Banach spaces. J. Convex Anal. 16(16), 791–806 (2009) MathSciNetMATHGoogle Scholar
- 16.Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3(4), 510–585 (1969) MathSciNetCrossRefMATHGoogle Scholar
- 17.Tsukada, M.: Convergence of best approximations in a smooth Banach space. J. Approx. Theory 40, 301–309 (1984) MathSciNetCrossRefMATHGoogle Scholar
- 18.Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13(3), 938–945 (2012) MathSciNetCrossRefMATHGoogle Scholar
- 19.Xu, H.K.: Inequalities in Banach spaces with applications. In: Nonlinear Analysis, vol. 16, pp. 1127–1138 (1991) Google Scholar
Copyright information
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.