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Convergence theorems of convex combination methods for treating daccretive mappings in a Banach space and nonlinear equation
Journal of Inequalities and Applications volume 2014, Article number: 482 (2014)
Abstract
mdAccretive mappings, which are totally different from maccretive mappings in nonHilbertian Banach spaces, belong to another type of nonlinear mappings with practical backgrounds. The purpose of this paper is to present some new iterative schemes by means of convex combination methods to approximate the common zeros of finitely many mdaccretive mappings. Some strong and weak convergence theorems are obtained in a real uniformly smooth and uniformly convex Banach space by using the techniques of the Lyapunov functional and retraction. The restrictions are weaker than in the previous corresponding works. Moreover, an example of mdaccretive mapping is exemplified, from which we can see the connections between mdaccretive mappings and the nonlinear elliptic equations.
MSC:47H05, 47H09, 47H10.
1 Introduction and preliminaries
Let E be a real Banach space with norm \parallel \cdot \parallel and let {E}^{\ast} denote the dual space of E. We use ‘→’ and ‘⇀’ to denote strong and weak convergence, respectively. We denote the value of f\in {E}^{\ast} at x\in E by \u3008x,f\u3009.
The normalized duality mapping J from E to {2}^{{E}^{\ast}} is defined by
We call J weakly sequentially continuous if \{{x}_{n}\} is sequence in E which converges weakly to x it follows that \{J{x}_{n}\} converges in weak^{∗} to Jx.
A mapping T:D(T)=E\to {E}^{\ast} is said to be demicontinuous [1] on E if T{x}_{n}\rightharpoonup Tx, as n\to \mathrm{\infty}, for any sequence \{{x}_{n}\} strongly convergent to x in E. A mapping T:D(T)=E\to {E}^{\ast} is said to be hemicontinuous [1] on E if w\text{}{lim}_{t\to 0}T(x+ty)=Tx, for any x,y\in E. A mapping T:E\to E is said to be nonexpansive if \parallel TxTy\parallel \le \parallel xy\parallel, for \mathrm{\forall}x,y\in E.
The Lyapunov functional \phi :E\times E\to {R}^{+} is defined as follows [2]:
for \mathrm{\forall}x,y\in E.
It is obvious from the definition of φ that
for all x,y\in E. We also know that
for each x,y,z\in E; see [3, 4].
We use Fix(S) to denote the set of fixed points of a mapping S:E\to E. That is, Fix(S):=\{x\in E:Sx=x\}. A mapping S:E\to E is said to be generalized nonexpansive [4] if Fix(S)\ne \mathrm{\varnothing} and \phi (Sx,p)\le \phi (x,p), for \mathrm{\forall}x\in E and p\in Fix(S).
Let C be a nonempty closed subset of E and let Q be a mapping of E onto C. Then Q is said to be sunny [4] if Q(Q(x)+t(xQ(x)))=Q(x), for all x\in E and t\ge 0. A mapping Q:E\to C is said to be a retraction [4] if Q(z)=z for every z\in C. If E is a smooth and strictly convex Banach space, then a sunny generalized nonexpansive retraction of E onto C is uniquely decided, which is denoted by {R}_{C}.
Let I denote the identity operator on E. A mapping A:D(A)\subset E\to E is said to be accretive if \u3008AxAy,J(xy)\u3009\ge 0, for \mathrm{\forall}x,y\in D(A) and it is called maccretive if R(I+\lambda A)=E, for \mathrm{\forall}\lambda >0.
If A is accretive, we can define, for each r>0, a singlevalued mapping {J}_{r}^{A}:R(I+rA)\to D(A) by {J}_{r}^{A}={(I+rA)}^{1}, which is called the resolvent of A. And, {J}_{r}^{A} is a nonexpansive mapping [1]. In the process of constructing iterative schemes to approximate zeros of an accretive mapping A, the nonexpansive property of {J}_{r}^{A} plays an important role.
A mapping A:D(A)\subset E\to E is said to be daccretive [5] if \u3008AxAy,JxJy\u3009\ge 0, for \mathrm{\forall}x,y\in D(A). And it is called mdaccretive if R(I+\lambda A)=E, for \mathrm{\forall}\lambda >0. However, the resolvent of an mdaccretive mapping is not a nonexpansive mapping.
An operator B\subset E\times {E}^{\ast} is said to be monotone if \u3008{x}_{1}{x}_{2},{y}_{1}{y}_{2}\u3009\ge 0, for \mathrm{\forall}{y}_{i}\in B{x}_{i}, i=1,2. A monotone operator B is said to be maximal monotone if R(J+\lambda B)={E}^{\ast}, for \mathrm{\forall}\lambda >0. An operator B\subset E\times {E}^{\ast} is said to be strictly monotone if \u3008{x}_{1}{x}_{2},{y}_{1}{y}_{2}\u3009>0, for \mathrm{\forall}{x}_{1}\ne {x}_{2}, \mathrm{\forall}{y}_{i}\in B{x}_{i}, i=1,2.
It is clear that in the frame of Hilbert spaces, (m)accretive mappings, (m)daccretive mappings and (maximal) monotone operators are the same. But in the frame of nonHilbertian Banach spaces, they belong to different classes of important nonlinear operators, which have practical backgrounds. During the past 50 years or so, a large number of researches have been done on the topics of constructing iterative schemes to approximate the zeros of maccretive mappings and maximal monotone operators. However, rarely related work on daccretive mappings can be found.
As we know, in 2000, Alber and Reich [5] presented the following iterative schemes for the daccretive mapping T in a real uniformly smooth and uniformly convex Banach space:
and
They proved that the iterative sequences \{{x}_{n}\} generated by (1.3) and (1.4) converge weakly to the zero point of T under the assumptions that T is uniformly bounded and demicontinuous.
In 2007, Guan [6] presented the following projective method for the mdaccretive mapping A in a real uniformly smooth and uniformly convex Banach space:
where {J}_{{r}_{n}}^{A}={(I+{r}_{n}A)}^{1}, and {\mathrm{\Pi}}_{{C}_{n}\cap {Q}_{n}} is the generalized projection from D(A) onto {C}_{n}\cap {Q}_{n}. It was shown that the iterative sequence \{{x}_{n}\} generated by (1.5) converges strongly to the zero point of A under the assumptions that A is demicontinuous, the normalized duality mapping J is weakly sequentially continuous, and {J}_{{r}_{n}}^{A} satisfies
for \mathrm{\forall}x\in E and p\in {A}^{1}0. The restrictions are extremely strong and it is hardly for us to find such an mdaccretive mapping which both is demicontinuous and satisfies (1.6).
The socalled block iterative scheme for solving the problem of image recovery proposed by Aharoni and Censor [7] inspired us. In a finitedimensional space H, the block iterative sequence \{{x}_{n}\} is generated in the following way: {x}_{1}=x\in H and
where {P}_{i} is a nonexpansive retraction from H onto {C}_{i}, and {\{{C}_{i}\}}_{i=1}^{m} is a family of nonempty closed and convex subsets of H. \{{\omega}_{n,i}\}\subset [0,1], {\sum}_{i=1}^{m}{\omega}_{n,i}=1, and \{{\alpha}_{n,i}\}\subset (1,1), for i=1,2,\dots ,m and n\ge 1.
In [8], Kikkawa and Takahashi applied the block iterative method to approximate the common fixed point of finite nonexpansive mappings {\{{T}_{i}\}}_{i=1}^{m} in Banach spaces in the following way and obtained the weak convergence theorems: {x}_{1}=x\in C, and
where \{{\omega}_{n,i}\}\subset [0,1], {\sum}_{i=1}^{m}{\omega}_{n,i}=1, and \{{\alpha}_{n,i}\}\subset [0,1], for i=1,2,\dots ,m and n\ge 1.
In this paper, we shall borrow the idea of block iterative method which highlights the convex combination techniques. Our main work can be divided into three parts. In Section 2, we shall construct iterative schemes by convex combination techniques for approximating common zeros of mdaccretive mappings. Some weak convergence theorems are obtained in a Banach space. In Section 3, we shall construct iterative schemes by convex combination and retraction techniques for approximating common zeros of mdaccretive mappings. Some strong convergence theorems are obtained in a Banach space. In Section 4, we shall present a nonlinear elliptic equation from which we can define an mdaccretive mapping. Our main contributions lie in the following aspects:

(i)
The restrictions are weaker. The semicontinuity of the daccretive mapping A and the inequality of (1.6) are no longer needed.

(ii)
The Lyapunov functional is employed in the process of estimating the convergence of the iterative sequence. This is mainly because the resolvent of a daccretive mapping is not nonexpansive.

(iii)
The connection between a nonlinear elliptic equation and an mdaccretive mapping is set up, from which we cannot only find a good example of mdaccretive mapping but also see the iterative construction of the solution of the nonlinear elliptic equation.
In order to prove our convergence theorems, we also need the following lemmas.
The duality mapping J:E\to {2}^{{E}^{\ast}} has the following properties:

(i)
If E is a real reflexive and smooth Banach space, then J:E\to {E}^{\ast} is singlevalued.

(ii)
If E is reflexive, then J is a surjection.

(iii)
If E is a real uniformly smooth and uniformly smooth Banach space, then {J}^{1}:{E}^{\ast}\to E is also a duality mapping. Moreover, J and {J}^{1} are uniformly continuous on each bounded subset of E and {E}^{\ast}, respectively.

(iv)
E is strictly convex if and only if J is strictly monotone.
Lemma 1.2 [10]
Let E be a real smooth and uniformly convex Banach space, B\subset E\times {E}^{\ast} be a maximal monotone operator, then {B}^{1}0 is a closed and convex subset of E and the graph of B, G(B) is demiclosed in the following sense: \mathrm{\forall}\{{x}_{n}\}\subset D(B) with {x}_{n}\rightharpoonup x in E, and \mathrm{\forall}{y}_{n}\in B{x}_{n} with {y}_{n}\to y in {E}^{\ast} it follows that x\in D(B) and y\in Bx.
Lemma 1.3 [2]
Let E be a real reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed subset of E, and let {R}_{C}:E\to C be a sunny generalized nonexpansive retraction. Then, for \mathrm{\forall}u\in C and x\in E, \phi (x,{R}_{C}x)+\phi ({R}_{C}x,u)\le \phi (x,u).
Lemma 1.4 [3]
Let E be a real smooth and uniformly convex Banach space, and let \{{x}_{n}\} and \{{y}_{n}\} be two sequences in E. If either \{{x}_{n}\} or \{{y}_{n}\} is bounded and \phi ({x}_{n},{y}_{n})\to 0, as n\to \mathrm{\infty}, then {x}_{n}{y}_{n}\to 0, as n\to \mathrm{\infty}.
Lemma 1.5 [11]
Let \{{a}_{n}\} and \{{b}_{n}\} be two sequences of nonnegative real numbers and {a}_{n+1}\le {a}_{n}+{b}_{n}, for \mathrm{\forall}n\ge 1. If {\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}<+\mathrm{\infty}, then {lim}_{n\to \mathrm{\infty}}{a}_{n} exists.
2 Weak convergence theorems
Theorem 2.1 Let E be a real uniformly smooth and uniformly convex Banach space. Let {A}_{i}:E\to E be a finite family of mdaccretive mappings, \{{\omega}_{n,i}\},\{{\eta}_{n,i}\}\subset (0,1], \{{\alpha}_{n,i}\},\{{\beta}_{n,i}\}\subset [0,1), \{{r}_{n,i}\},\{{s}_{n,i}\}\subset (0,+\mathrm{\infty}), for i=1,2,\dots ,m. {\sum}_{i=1}^{m}{\omega}_{n,i}=1 and {\sum}_{i=1}^{m}{\eta}_{n,i}=1. Let D:={\bigcap}_{i=1}^{m}{A}_{i}^{1}0\ne \mathrm{\varnothing}. Suppose that the normalized duality mapping J:E\to {E}^{\ast} is weakly sequentially continuous. Let \{{x}_{n}\} be generated by the following iterative algorithm:
Suppose the following conditions are satisfied:

(i)
{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n,i}<1, {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n,i}<1, for i=1,2,\dots ,m;

(ii)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\eta}_{n,i}>0, {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\omega}_{n,i}>0, for i=1,2,\dots ,m;

(iii)
{inf}_{n\ge 1}{r}_{n,i}>0, {inf}_{n\ge 1}{s}_{n,i}>0, for i=1,2,\dots ,m.
Then \{{x}_{n}\} converges weakly to a point {v}_{0}\in D.
Proof For i=1,2,\dots ,m, let {J}_{{r}_{n,i}}^{{A}_{i}}={(I+{r}_{n,i}{A}_{i})}^{1} and {J}_{{s}_{n,i}}^{{A}_{i}}={(I+{s}_{n,i}{A}_{i})}^{1}.
We split the proof into the following six steps.
Step 1. For p\in D, {J}_{{r}_{n,i}}^{{A}_{i}} and {J}_{{s}_{n,i}}^{{A}_{i}} satisfy the following two inequalities, respectively:
In fact, using (1.2), we know that for \mathrm{\forall}p\in D,
Since {A}_{i} is daccretive and \frac{{x}_{n}{J}_{{r}_{n,i}}^{{A}_{i}}{x}_{n}}{{r}_{n,i}}={A}_{i}{J}_{{r}_{n,i}}^{{A}_{i}}{x}_{n},
From (2.4) we know that (2.2) is true. So is (2.3).
Step 2. \{{x}_{n}\} is bounded.
\mathrm{\forall}p\in D, using (2.2) and (2.3), we have
Lemma 1.5 implies that {lim}_{n\to \mathrm{\infty}}\phi ({x}_{n},p) exists. Then (1.1) ensures that \{{x}_{n}\} is bounded.
Step 3. {A}_{i}{J}^{1}\subset {E}^{\ast}\times E is maximal monotone, for each i, 1\le i\le m.
Since {A}_{i} is daccretive, then \mathrm{\forall}x,y\in {E}^{\ast},
Therefore, {A}_{i}{J}^{1} is monotone, for each i, 1\le i\le m.
Since R(I+\lambda {A}_{i})=E, \lambda >0, then \mathrm{\forall}y\in E, there exists x\in E satisfying x+\lambda {A}_{i}x=y, \lambda >0. Using Lemma 1.1(ii), there exists {x}^{\ast}\in {E}^{\ast} such that {J}^{1}{x}^{\ast}=x. Thus {J}^{1}{x}^{\ast}+\lambda {A}_{i}{J}^{1}{x}^{\ast}=y, which implies that R({J}^{1}+\lambda {A}_{i}{J}^{1})=E, \lambda >0. Thus {A}_{i}{J}^{1} is maximal monotone, for each i, 1\le i\le m.
Step 4. {({A}_{i}{J}^{1})}^{1}0\ne \mathrm{\varnothing}, for each i, 1\le i\le m.
Since D\ne \mathrm{\varnothing}, then there exists x\in E such that {A}_{i}x=0, where i=1,2,\dots ,m. Using Lemma 1.1(ii) again, there exists {x}^{\ast}\in {E}^{\ast} such that {J}^{1}{x}^{\ast}=x. Thus {A}_{i}{J}^{1}{x}^{\ast}=0, for each i, 1\le i\le m. That is, {x}^{\ast}\in {({A}_{i}{J}^{1})}^{1}0, for each i, 1\le i\le m.
Step 5. \omega ({x}_{n})\subset D, where \omega ({x}_{n}) denotes the set of all of the weak limit points of the weakly convergent subsequences of \{{x}_{n}\}.
Since \{{x}_{n}\} is bounded, there exists a subsequence of \{{x}_{n}\}, for simplicity, we still denote it by \{{x}_{n}\} such that {x}_{n}\rightharpoonup x, as n\to \mathrm{\infty}.
For \mathrm{\forall}p\in D, using (2.2) and (2.3), we have
which implies that
Using the assumptions and the result of Step 2, we know that \phi ({x}_{n},{J}_{{r}_{n,i}}^{{A}_{i}}{x}_{n})\to 0, as n\to \mathrm{\infty}, for i=1,2,\dots ,m. Then Lemma 1.4 ensures that {x}_{n}{J}_{{r}_{n,i}}^{{A}_{i}}{x}_{n}\to 0, as n\to \mathrm{\infty}, for i=1,2,\dots ,m.
Let {u}_{i}={A}_{i}v, since {A}_{i} is daccretive, then
Since both \{{x}_{n}\} and \{{J}_{{r}_{n,i}}^{{A}_{i}}{x}_{n}\} are bounded, then letting n\to \mathrm{\infty} and using Lemma 1.1(iii), we have
i=1,2,\dots ,m. That is, \u3008{A}_{i}{J}^{1}(Jv),JvJx\u3009\ge 0, for i=1,2,\dots ,m. From Step 3 and Lemma 1.2, we know that Jx\in {({A}_{i}{J}^{1})}^{1}0, which implies that x\in {A}_{i}^{1}0. And then x\in D.
Step 6. {x}_{n}\rightharpoonup {v}_{0}, as n\to \mathrm{\infty}, where {v}_{0} is the unique element in D.
From Steps 2 and 5, we know that there exists a subsequence \{{x}_{{n}_{i}}\} of \{{x}_{n}\} such that {x}_{{n}_{i}}\rightharpoonup {v}_{0}\in D, as i\to \mathrm{\infty}. If there exists another subsequence \{{x}_{{n}_{j}}\} of \{{x}_{n}\} such that {x}_{{n}_{j}}\rightharpoonup {v}_{1}\in D, as j\to \mathrm{\infty}, then from Step 2, we know that
Similarly,
From (2.5) and (2.6), we have \u3008{v}_{1}{v}_{0},J{v}_{1}J{v}_{0}\u3009=0, which implies that {v}_{0}={v}_{1} since J is strictly monotone.
This completes the proof. □
Remark 2.1 If E reduces to the Hilbert space H, then (2.1) becomes the iterative scheme for approximating common zeros of maccretive mappings.
Remark 2.2 The iterative scheme (2.1) can be regarded as twostep block iterative scheme.
Remark 2.3 If m=1, then (2.1) becomes to the following one approximating the zero point of an mdaccretive mapping A:
If, moreover, {\alpha}_{n}\equiv 0, {\beta}_{n}\equiv 0, then (2.7) becomes the socalled doublebackward iterative scheme for the mdaccretive mapping A:
3 Strong convergence theorems
Theorem 3.1 Let E be a real uniformly smooth and uniformly convex Banach space. Let {A}_{i}:E\to E be a finite family of mdaccretive mappings, \{{\omega}_{n,i}\},\{{\eta}_{n,i}\}\subset (0,1], \{{\alpha}_{n,i}\},\{{\beta}_{n,i}\}\subset [0,1), \{{r}_{n,i}\},\{{s}_{n,i}\}\subset (0,+\mathrm{\infty}), for i=1,2,\dots ,m. {\sum}_{i=1}^{m}{\omega}_{n,i}=1 and {\sum}_{i=1}^{m}{\eta}_{n,i}=1. Let D:={\bigcap}_{i=1}^{m}{A}_{i}^{1}0\ne \mathrm{\varnothing}. Suppose the normalized duality mapping J:E\to {E}^{\ast} is weakly sequentially continuous. Let \{{x}_{n}\} be generated by the iterative scheme:
Suppose the following conditions are satisfied:

(i)
{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n,i}<1, {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n,i}<1, for i=1,2,\dots ,m;

(ii)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\eta}_{n,i}>0, {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\omega}_{n,i}>0, for i=1,2,\dots ,m;

(iii)
{inf}_{n\ge 0}{r}_{n,i}>0, {inf}_{n\ge 0}{s}_{n,i}>0, for i=1,2,\dots ,m.
Then \{{x}_{n}\} converges strongly to {p}_{0}={R}_{D}{x}_{0}, where {R}_{D} is the sunny generalized nonexpansive retraction from E onto D, as n\to \mathrm{\infty}.
Proof We split the proof into six steps.
Step 1. \{{x}_{n}\} is well defined.
Noticing that
then from Lemma 1.1(iii), we can easily know that {H}_{n} (n\ge 0) is a closed subset of E.
For \mathrm{\forall}p\in D, using (2.2), we know that
which implies that p\in {H}_{n}. Thus \mathrm{\varnothing}\ne D\subset {H}_{n}, for n\ge 0.
Since {H}_{n} is a nonempty closed subset of E, there exists a unique sunny generalized nonexpansive retraction from E onto {H}_{n}, which is denoted by {R}_{{H}_{n}}. Therefore, \{{x}_{n}\} is well defined.
Step 2. \{{x}_{n}\} is bounded.
Using Lemma 1.3, \phi ({x}_{n+1},p)\le \phi ({x}_{0},p), \mathrm{\forall}p\in D\subset {H}_{n+1}. Thus \{\phi ({x}_{n},p)\} is bounded and then (1.1) ensures that \{{x}_{n}\} is bounded.
Step 3. \omega ({x}_{n})\subset D, where \omega ({x}_{n}) denotes the set of all of the weak limit points of the weakly convergent subsequences of \{{x}_{n}\}.
Since \{{x}_{n}\} is bounded, there exists a subsequence of \{{x}_{n}\}, for simplicity, we still denote it by \{{x}_{n}\} such that {x}_{n}\rightharpoonup x, as n\to \mathrm{\infty}.
Since {x}_{n+1}\in {H}_{n+1}\subset {H}_{n}, using Lemma 1.3, we have
which implies that {lim}_{n\to \mathrm{\infty}}\phi ({x}_{0},{x}_{n}) exists. Thus \phi ({x}_{n},{x}_{n+1})\to 0. Lemma 1.4 implies that {x}_{n}{x}_{n+1}\to 0, as n\to \mathrm{\infty}.
Since {x}_{n+1}\in {H}_{n+1}\subset {H}_{n}, then \phi ({u}_{n},{x}_{n+1})\le \phi ({x}_{n},{x}_{n+1})\to 0, which implies that {x}_{n}{u}_{n}\to 0, as n\to \mathrm{\infty}.
\mathrm{\forall}p\in D, using (2.2) again, we have
Then
as n\to \mathrm{\infty}. Lemma 1.4 implies that {x}_{n}{J}_{{r}_{n,i}}^{{A}_{i}}{x}_{n}\to 0, as n\to \mathrm{\infty}, where i=1,2,\dots ,m.
The remaining part is similar to that of Step 5 in Theorem 2.1, then we have \omega ({x}_{n})\subset D.
Step 4. \{{x}_{n}\} is a Cauchy sequence.
If, on the contrary, there exist two subsequences \{{n}_{k}\} and \{{m}_{k}\} of \{n\} such that \parallel {x}_{{n}_{k}+{m}_{k}}{x}_{{n}_{k}}\parallel \ge {\epsilon}_{0}, \mathrm{\forall}k\ge 1.
Since {lim}_{n\to \mathrm{\infty}}\phi ({x}_{0},{x}_{n}) exists, using Lemma 1.3 again,
as k\to \mathrm{\infty}. Lemma 1.4 implies that {lim}_{k\to \mathrm{\infty}}\parallel {x}_{{n}_{k}+{m}_{k}}{x}_{{n}_{k}}\parallel =0, which makes a contradiction. Thus \{{x}_{n}\} is a Cauchy sequence.
Step 5. D is a closed subset of E.
Let {z}_{n}\in D with {z}_{n}\to z, as n\to \mathrm{\infty}. Then {A}_{i}{z}_{n}=0, for i=1,2,\dots ,m. In view of Lemma 1.1(ii), there exists {z}_{n}^{\ast}\in {E}^{\ast} such that {z}_{n}={J}^{1}{z}_{n}^{\ast}. Using Lemma 1.1(iii), {z}_{n}^{\ast}\to Jz, as n\to \mathrm{\infty}. Since {A}_{i}{J}^{1}{z}_{n}^{\ast}=0, {z}_{n}^{\ast}\to Jz and {A}_{i}{J}^{1} is maximal monotone, Lemma 1.2 ensures that Jz\in {({A}_{i}{J}^{1})}^{1}0. Thus, z\in {A}_{i}^{1}0, for i=1,2,\dots ,m. And then z\in D. Therefore, D is closed subset of E, which ensures there exists a unique sunny generalized nonexpansive retraction {R}_{D} from E onto D.
Step 6. {x}_{n}\to {q}_{0}={R}_{D}{x}_{0}, as n\to \mathrm{\infty}.
Since \{{x}_{n}\} is a Cauchy sequence, there exists {q}_{0}\in E such that {x}_{n}\to q, as n\to \mathrm{\infty}. From Step 5, {q}_{0}\in D.
Now, we prove that {q}_{0}={R}_{D}{x}_{0}.
Using Lemma 1.3, we have the following two inequalities:
and
Letting n\to +\mathrm{\infty}, from (3.3), we know that
From (3.2) and (3.4), 0\le \phi ({R}_{D}{x}_{0},{q}_{0})\le \phi ({q}_{0},{R}_{D}{x}_{0})\le 0. Thus \phi ({R}_{D}{x}_{0},{q}_{0})=0. So in view of Lemma 1.4, q={R}_{D}{x}_{0}.
This completes the proof. □
Remark 3.1 Combining the techniques of convex combination and the retraction, the strong convergence of iterative scheme (3.1) is obtained.
Remark 3.2 It is obvious that the restrictions in Theorems 2.1 and 3.1 are weaker.
4 Connection between nonlinear mappings and nonlinear elliptic equations
We have mentioned that in a Hilbert space, mdaccretive mappings and maccretive mappings are the same, while in a nonHilbertian Banach space, they are different. So, there are many examples of mdaccretive mappings in Hilbert spaces. Can we find one mapping that is (m)daccretive but not (m)accretive?
In Section 4.1, we shall review the work done in [12], where an maccretive mapping is constructed for discussing the existence of solution of one kind nonlinear elliptic equations.
In Section 4.2, we shall construct an mdaccretive mapping based on the same nonlinear elliptic equation presented in Section 4.1, from which we can see that it is quite different from the maccretive mapping defined in Section 4.1.
4.1 mAccretive mappings and nonlinear elliptic equations
The following nonlinear elliptic boundary value problem is extensively studied in [12, 13]:
In (4.1), Ω is a bounded conical domain of a Euclidean space {R}^{N} with its boundary \mathrm{\Gamma}\in {C}^{1} (see [14]). f\in {L}^{s}(\mathrm{\Omega}) is a given function, ϑ is the exterior normal derivative of Γ, g:\mathrm{\Omega}\times R\to R is a given function satisfying Carathéodory’s conditions such that the mapping u\in {L}^{s}(\mathrm{\Omega})\to g(x,u(x))\in {L}^{s}(\mathrm{\Omega}) is defined and there exists a function T(x)\in {L}^{s}(\mathrm{\Omega}) such that g(x,t)t\ge 0, for t\ge T(x) and x\in \mathrm{\Omega}. {\beta}_{x} is the subdifferential of a proper, convex, and semilowercontinuous function. \alpha :{R}^{N}\to {R}^{N} is a monotone and continuous function, and there exist positive constants {k}_{1}, {k}_{2}, and {k}_{3} such that, for \mathrm{\forall}\xi ,{\xi}^{\prime}\in {R}^{N}, the following conditions are satisfied:

(i)
\alpha (\xi )\le {k}_{1}{\xi }^{p1};

(ii)
\alpha (\xi )\alpha ({\xi}^{\prime})\le {k}_{2}{\xi }^{p2}\xi {{\xi}^{\prime}}^{p2}{\xi}^{\prime};

(iii)
\u3008\u3008\alpha (\xi ),\xi \u3009\u3009\ge {k}_{3}{\xi }^{p},
where \u3008\u3008\cdot ,\cdot \u3009\u3009 denotes the inner product in {R}^{N}.
In [12], they first present the following definitions.
Definition 4.1 [12]
Define the mapping {B}_{p}:{W}^{1,p}(\mathrm{\Omega})\to {({W}^{1,p}(\mathrm{\Omega}))}^{\ast} by
for any u,v\in {W}^{1,p}(\mathrm{\Omega}).
Definition 4.2 [12]
Define the mapping {\mathrm{\Phi}}_{p}:{W}^{1,p}(\mathrm{\Omega})\to R by {\mathrm{\Phi}}_{p}(u)={\int}_{\mathrm{\Gamma}}{\phi}_{x}(u{}_{\mathrm{\Gamma}}(x))\phantom{\rule{0.2em}{0ex}}d\mathrm{\Gamma}(x), for any u\in {W}^{1,p}(\mathrm{\Omega}).
Definition 4.3 [12]
Define a mapping A:{L}^{2}(\mathrm{\Omega})\to {2}^{{L}^{2}(\mathrm{\Omega})} as follows:
For u\in D(A), Au=\{f\in {L}^{2}(\mathrm{\Omega})\mid f\in {B}_{p}u+\partial {\mathrm{\Phi}}_{p}(u)\}.
Definition 4.4 [12]
Define a mapping {A}_{s}:{L}^{s}(\mathrm{\Omega})\to {2}^{{L}^{s}(\mathrm{\Omega})} as follows:

(i)
If s\ge 2, then
D({A}_{s})=\{u\in {L}^{s}(\mathrm{\Omega})\mid \text{there exists an}f\in {L}^{s}(\mathrm{\Omega})\text{such that}f\in {B}_{p}u+\partial {\mathrm{\Phi}}_{p}(u)\}.
For u\in D({A}_{s}), we set {A}_{s}u=\{f\in {L}^{s}(\mathrm{\Omega})\mid f\in {B}_{p}u+\partial {\mathrm{\Phi}}_{p}(u)\}.

(ii)
If 1<s<2, then define {A}_{s}:{L}^{s}(\mathrm{\Omega})\to {2}^{{L}^{s}(\mathrm{\Omega})} as the {L}^{s}closure of A:{L}^{2}(\mathrm{\Omega})\to {2}^{{L}^{2}(\mathrm{\Omega})} defined in Definition 4.3.
Then they get the following important result in [12].
Proposition 4.1 [12]
Both A and {A}_{s} are maccretive mapping.
Later, by using the perturbations on ranges of maccretive mappings, the sufficient condition on the existence of solution of (4.1) is discussed.
Theorem 4.1 [12]
If f\in {L}^{s}(\mathrm{\Omega}) (\frac{2N}{N+1}<p\le s<+\mathrm{\infty}) satisfies the following condition:
then (4.1) has a solution in {L}^{s}(\mathrm{\Omega}).
The meaning of {\beta}_{}(x), {\beta}_{+}(x), {g}_{}(x), and {g}_{+}(x) can be found in the following two definitions.
For t\in R and x\in \mathrm{\Gamma}, let {\beta}_{x}^{0}(t)\in {\beta}_{x}(t) be the element with least absolute value if {\beta}_{x}(t)\ne \mathrm{\varnothing} and {\beta}_{x}^{0}(t)=\pm \mathrm{\infty}, where t>0 or <0, respectively, in the case {\beta}_{x}(t)=\mathrm{\varnothing}. Finally, let {\beta}_{\pm}(x)={lim}_{t\to \pm \mathrm{\infty}}{\beta}_{x}^{0}(t) (in the extended sense) for x\in \mathrm{\Gamma}. Then {\beta}_{\pm}(x) define measurable functions on Γ.
Define {g}_{+}(x)={lim\hspace{0.17em}inf}_{t\to +\mathrm{\infty}}g(x,t) and {g}_{}(x)={lim\hspace{0.17em}sup}_{t\to \mathrm{\infty}}g(x,t).
4.2 Examples of mdaccretive mappings
Now, based on nonlinear elliptic problem (4.1), we are ready to give the example of mdaccretive mapping in the sequel.
Lemma 4.1 [10]
Let E be a Banach space, if B:E\to {2}^{{E}^{\ast}} is an everywhere defined, monotone, and hemicontinuous mapping, then B is maximal monotone.
Definition 4.7 Let 1<p\le 2 and \frac{1}{p}+\frac{1}{{p}^{\prime}}=1.
Define the mapping B:{W}^{1,{p}^{\prime}}(\mathrm{\Omega})\to {({W}^{1,{p}^{\prime}}(\mathrm{\Omega}))}^{\ast} by
for any u,v\in {W}^{1,{p}^{\prime}}(\mathrm{\Omega}).
Proposition 4.2 B:{W}^{1,{p}^{\prime}}(\mathrm{\Omega})\to {({W}^{1,{p}^{\prime}}(\mathrm{\Omega}))}^{\ast} (1<p\le 2) is maximal monotone.
Proof We split the proof into four steps.
Step 1. B is everywhere defined.
In fact, for u,v\in {W}^{1,{p}^{\prime}}(\mathrm{\Omega}), from the property (i) of α, we have
Thus B is everywhere defined.
Step 2. B is monotone.
Since α is monotone, then, for u,v\in D(B),
which implies that B is monotone.
Step 3. B is hemicontinuous.
To show that B is hemicontinuous. It suffices to prove that for u,v,w\in {W}^{1,{p}^{\prime}}(\mathrm{\Omega}) and t\in [0,1], \u3008w,B(u+tv)Bu\u3009 as t\to 0.
In fact, since α is continuous,
as t\to 0.
Step 4. B is maximal monotone.
Lemma 4.1 implies that B is maximal monotone.
This completes the proof. □
Remark 4.1 [10]
There exists a maximal monotone extension of B from {L}^{{p}^{\prime}}(\mathrm{\Omega}) to {L}^{p}(\mathrm{\Omega}), which is denoted by \tilde{B}.
Definition 4.8 For 1<p\le 2, the normalized duality mapping J:{L}^{{p}^{\prime}}(\mathrm{\Omega})\to {L}^{p}(\mathrm{\Omega}) is defined by
for u\in {L}^{{p}^{\prime}}(\mathrm{\Omega}).
Define the mapping A:{L}^{p}(\mathrm{\Omega})\to {L}^{p}(\mathrm{\Omega}) (1<p\le 2) as follows:
Proposition 4.3 The mapping A:{L}^{p}(\mathrm{\Omega})\to {L}^{p}(\mathrm{\Omega}) (1<p\le 2) is mdaccretive.
Proof Since \tilde{B} is monotone, for \mathrm{\forall}u,v\in D(A),
Thus A is daccretive.
In view of Remark 4.1, \tilde{B} is maximal monotone, then R(J+\lambda \tilde{B})={L}^{p}(\mathrm{\Omega}), for \mathrm{\forall}\lambda >0.
For \mathrm{\forall}f\in {L}^{p}(\mathrm{\Omega}), there exists u\in {L}^{{p}^{\prime}}(\mathrm{\Omega}) such that Ju+\lambda \tilde{B}u=f. Using Lemma 1.1 again, there exists {u}^{\ast}\in {L}^{p}(\mathrm{\Omega}) such that u={J}^{1}{u}^{\ast}. Then {u}^{\ast}+\lambda \tilde{B}{J}^{1}{u}^{\ast}=f. Thus f\in R(I+\lambda A) and then R(I+\lambda A)={L}^{p}(\mathrm{\Omega}), for \lambda >0.
Thus A is mdaccretive.
This completes the proof. □
Proposition 4.4 {A}^{1}0=\{u\in {L}^{p}(\mathrm{\Omega}):u(x)\equiv \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\}.
Proof It is obvious that \{u\in {L}^{p}(\mathrm{\Omega}):u(x)\equiv \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\}\subset {A}^{1}0.
On the other hand, if u(x)\in {A}^{1}0, then Au(x)\equiv 0. Let {u}^{\ast}\in {L}^{{p}^{\prime}}(\mathrm{\Omega}) be such that u=J{u}^{\ast}. From the property (iii) of α, we have
which implies that u=J{u}^{\ast}\equiv \mathrm{const}.
Thus {A}^{1}0\subset \{u\in {L}^{p}(\mathrm{\Omega}):u(x)\equiv \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\}.
This completes the proof. □
Remark 4.2 From Propositions 4.2 and 4.3, we know that the restriction on the mdaccretive mapping in Theorem 2.1 or 3.1 that {A}^{1}0\ne \mathrm{\varnothing} is valid.
Remark 4.3 If (4.1) is reduced to the following:
then it is not difficult to see that u\in {A}^{1}0 is exactly the solution of (4.2), from which we cannot only see the connections between the zeros of an mdaccretive mapping and the nonlinear equation but also see that the work on designing the iterative schemes to approximate zeros of nonlinear mappings is meaningful.
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Acknowledgements
Li Wei is supported by the National Natural Science Foundation of China (No. 11071053), Natural Science Foundation of Hebei Province (No. A2014207010), Key Project of Science and Research of Hebei Educational Department (ZH2012080) and Key Project of Science and Research of Hebei University of Economics and Business (2013KYZ01).
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The main idea was proposed by LW, and YL and RPA participated in the research. All authors read and approved the final manuscript.
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Wei, L., Liu, Y. & Agarwal, R.P. Convergence theorems of convex combination methods for treating daccretive mappings in a Banach space and nonlinear equation. J Inequal Appl 2014, 482 (2014). https://doi.org/10.1186/1029242X2014482
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DOI: https://doi.org/10.1186/1029242X2014482
Keywords
 Lyapunov functional
 daccretive mapping
 common zeros
 retraction
 nonlinear elliptic equation