Abstract
In this paper, a regularization projection algorithm is investigated for solving common elements of an equilibrium problem, a variational inequality problem and a fixed point problem of a strictly pseudocontractive mapping. Strong convergence theorems are established in the framework of real Hilbert spaces.
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1 Introduction and preliminaries
Let H be a real Hilbert space with the inner product \(\langle\cdot,\cdot\rangle\) and the norm \(\|\cdot\|\). Let C be a nonempty closed convex subset of H. \(P_{C}\) stands for the metric projection from H onto C. Let F be a bifunction of \(C\times C\) into ℝ, where ℝ denotes the set of real numbers. Consider the following equilibrium problem in the terminology of Blum and Oettli [1]:
In this paper, the solution set of problem (1.1) is denoted by \(\operatorname{FP}(F)\).
To study problem (1.1), we may assume that F satisfies the following conditions:
-
(A1)
\(F(x,x)=0\) for all \(x\in C\);
-
(A2)
F is monotone, i.e., \(F(x,y)+F(y,x)\leq0\) for all \(x,y\in C\);
-
(A3)
for each \(x,y,z\in C\),
$$\limsup_{t\downarrow0}F\bigl(tz+(1-t)x,y\bigr)\leq F(x,y); $$ -
(A4)
for each \(x\in C\), \(y\mapsto F(x,y)\) is convex and lower semi-continuous.
Let S be a self-mapping of C. \(F(S)\) stands for the fixed point set of S. Recall that S is said to be β-contractive if there exists a constant \(\beta\in[0,1)\) such that
S is said to be nonexpansive if
S is said to be κ-strictly pseudocontractive if there exists a constant \(\kappa\in[0,1)\) such that
The class of κ-strictly pseudocontractive mappings was introduced by Browder and Petryshyn [2]. It is clear that every nonexpansive mapping is a 0-strictly pseudocontractive mapping.
Let \(A:C\rightarrow H\) be a mapping. Recall that A is said to be monotone if
A is said to be strongly monotone if there exists a constant \(\alpha>0\) such that
For such a case, we say that A is α-strongly monotone. A is said to be inverse-strongly monotone if there exists a constant \(\alpha>0\) such that
For such a case, we say that A is α-inverse-strongly monotone. It is clear that A is α-inverse-strongly monotone if and only if \(A^{-1}\) is α-strongly monotone.
Recall that the classical variational inequality is to find \(x\in C\) such that
It is known that \(x\in C\) is a solution of (1.2) if and only if x is a fixed point of \(P_{C}(I-r A)\), where \(r>0\) is a constant and I is an identity mapping. From now on, the solution set of (1.2) is denoted by \(\operatorname{VI}(C,A)\).
A set-valued mapping \(T:H\rightarrow2^{H}\) is said to be monotone if for all \(x,y\in H\), \(f\in Tx\) and \(g\in Ty\) imply \(\langle x-y,f-g\rangle\geq0\). A monotone mapping \(T:H\rightarrow2^{H}\) is maximal if the graph \(G(T)\) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if, for any \((x,f)\in H\times H\), \(\langle x-y,f-g\rangle\geq0\) for all \((y,g)\in G(T)\) implies \(f\in Tx\).
Recently, Iiduka and Takahshi [3] investigated variational inequality (1.2) and fixed points of a nonexpansive mapping based on a Halpern-like algorithm. To be more clear, they proved the following result.
Theorem IT
Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inverse-strongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that \(F(S)\cap \operatorname{VI}(C,A)\neq\emptyset\). Suppose \(x_{1}=x\in C\) and \(\{x_{n}\}\) is given by
for every \(n=1,2,\ldots\) , where \(\{\alpha_{n}\}\) is a sequence in \([0,1)\) and \(\{\lambda_{n}\}\) is a sequence in \([0,2\alpha]\). If \(\{\alpha_{n}\}\) and \(\{\lambda_{n}\}\) are chosen so that \(\lambda_{n}\in [a,b]\) for some a, b with \(0< a <b<2\alpha\), \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty \), \(\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty\) and \(\sum_{n=1}^{\infty}|\lambda_{n+1}-\lambda_{n}|<\infty\), then \(\{x_{n}\}\) converges strongly to \(P_{F(S)\cap \operatorname{VI}(C,A)}x\).
Since equilibrium problem (1.1) provides a unified model of several problems such as variational inequalities, fixed point problems and inclusion problems. In [4], Takahashi and Takahashi further studied fixed points of a nonexpansive mapping and equilibrium problem (1.1) based on the viscosity approximation method, which was introduced by Moudafi [5]. To be more clear, they proved the following result.
Theorem TT
Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from \(C\times C\) to R satisfying (A1)-(A4), and let S be a nonexpansive mapping of C into H such that \(F(S)\cap \operatorname{EP}(F)\neq\emptyset\). Let f be a contraction of H into itself, and let \(\{x_{n}\}\) and \(\{u_{n}\}\) be sequences generated by \(x_{1}\in H\) and
for every \(n=1,2,\ldots\) , where \(\{\alpha_{n}\}\subset[0,1]\) and \(\{r_{n}\} \subset(0,\infty)\) satisfy \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty \), \(\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty\), \(\liminf_{n\rightarrow\infty}r_{n}>0\) and \(\sum_{n=1}^{\infty}|r_{n+1}-r_{n}|<\infty \). Then \(\{x_{n}\}\) and \(\{u_{n}\}\) converge strongly to \(z\in F(S)\cap \operatorname{EP}(F)\), where \(z=P_{F(S)\cap \operatorname{EP}(F)}f(z)\).
Subsequently, many authors investigated common solution problems based on hybrid projection methods due to real world applications, for example, image restoration and digital signal processing; see [6–20] and the references therein. In view of the complexity of convex, hybrid projection methods, they are not easy to implement. In this paper, we study a regularization projection algorithm for solving equilibrium problem (1.1), variational inequality (1.2) and a fixed point problem of a κ-strictly pseudocontractive mapping. Possible computation errors are taken into account. Strong convergence theorems are established in the framework of real Hilbert spaces.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1
[21]
Let C be a nonempty closed convex subset of H. Let A be a monotone mapping of C into H, and let \(N_{C}v\) be a normal cone to C at \(v\in C\), i.e.,
and define a mapping T on C by
Then T is maximal monotone and \(0\in Tv\) if and only if \(\langle Av,u-v\rangle\geq0\) for all \(u\in C\).
Lemma 1.2
[2]
Let C be a nonempty closed convex subset of H, and let \(S:C\rightarrow H\) be a κ-strictly pseudocontractive mapping. Then \(I-S\) is demi-closed at zero.
Lemma 1.3
[1]
Let C be a nonempty closed convex subset of H, and let \(F:C\times C\rightarrow\mathbb{R}\) be a bifunction satisfying (A1)-(A4). Then, for any \(r>0\) and \(x\in H\), there exists \(z\in C\) such that
Further, define
for all \(r>0\) and \(x\in H\). Then the following hold:
-
(a)
\(T_{r}\) is single-valued;
-
(b)
\(T_{r}\) is firmly nonexpansive, i.e., for any \(x,y\in H\),
$$\|T_{r}x-T_{r}y\|^{2}\leq\langle T_{r}x-T_{r}y, x-y\rangle; $$ -
(c)
\(F(T_{r})=\operatorname{EP}(F)\);
-
(d)
\(\operatorname{EP}(F)\) is closed and convex.
Lemma 1.4
[2]
Let C be a nonempty closed convex subset of H, and let \(S:C\rightarrow H\) be a κ-strictly pseudocontractive mapping. Define a mapping T by \(T=\delta I+(1-\delta )S\), where δ is a constant in \((0,1)\). If \(\delta\in[\kappa ,1)\) then T is nonexpansive and \(F(T)=F(S)\).
Lemma 1.5
[22]
Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be bounded sequences in H, and let \(\{\beta_{n}\}\) be a sequence in \([0,1]\) with \(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty }\beta_{n}<1\). Suppose \(x_{n+1}=(1-\beta_{n})y_{n}+\beta_{n}x_{n}\) for all integers \(n\geq 0\) and
Then \(\lim_{n\rightarrow\infty}\|y_{n}-x_{n}\|=0\).
Lemma 1.6
[23]
Assume that \(\{\alpha_{n}\}\) is a sequence of nonnegative real numbers such that
where \(\{\gamma_{n}\}\) is a sequence in \((0,1)\) and \(\{\mu_{n}\}\), \(\{\delta _{n}\}\) are real sequences such that
-
(i)
\(\sum_{n=1}^{\infty}\gamma_{n}=\infty\) and \(\sum_{n=1}^{\infty}\mu_{n}<\infty\);
-
(ii)
\(\limsup_{n\rightarrow\infty}\delta_{n}/\gamma_{n}\leq0\) or \(\sum_{n=1}^{\infty}|\delta_{n}|<\infty\).
Then \(\lim_{n\rightarrow\infty}\alpha_{n}=0\).
2 Main results
Theorem 2.1
Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping, and let F be a bifunction from \(C\times C\) to ℝ which satisfies (A1)-(A4). Let \(S:C\rightarrow H\) be a κ-strictly pseudocontractive mapping, and let f be a β-contraction on H. Assume that \(\Omega=F(S)\cap \operatorname{VI}(C,A)\cap \operatorname{EP} (F)\neq\emptyset\). Let \(\{ r_{n}\}\) and \(\{s_{n}\}\) be positive real number sequences. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process:
where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions:
-
(a)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
-
(b)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
-
(c)
\(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\), \(\lim_{n\rightarrow\infty }|\delta_{n+1}-\delta_{n}|=0 \) and \(\kappa\leq\delta_{n}\leq\delta<1\);
-
(d)
\(\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0 \) and \(\liminf_{n\rightarrow\infty}r_{n}> 0\);
-
(e)
\(\lim_{n\rightarrow\infty}|s_{n+1}-s_{n}|=0\), \(0< s\leq s_{n}\leq s'<2\alpha\),
where δ, s, \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality
Proof
For any \(x,y\in C\), we see that
By using condition (e), we see that \(\|(I-s_{n}A)x-(I-s_{n}A)y\|\leq\|x-y\|\). This proves that \(I-s_{n}A\) is nonexpansive. Put \(S_{n}=\delta_{n}I+(1-\delta_{n})S\). It follows from Lemma 1.4 that \(S_{n}\) is nonexpansive and \(F(S_{n})=F(S)\). Let \(p\in\Omega\) be fixed arbitrarily. Hence, we have
It follows that \(\|x_{n}-p\|\leq\max\{\|x_{1}-p\|,\frac{\|f(p)-p\|}{1-\beta}\}+\sum_{n=1}^{\infty}\|e_{n}\|\). This shows that \(\{x_{n}\}\) is bounded, so are \(\{y_{n}\}\) and \(\{z_{n}\}\). Let \(\lambda_{n}=\frac{x_{n+1}-\beta_{n}x_{n}}{1-\beta_{n}}\). It follows that
Hence, we have
Since \(z_{n}=T_{r_{n}}x_{n}\), we find that \(F(z_{n},z)+\frac{1}{r_{n}}\langle z-z_{n},z_{n}-x_{n}\rangle\geq0\), \(\forall z\in C \) and \(F(z_{n+1},z)+\frac{1}{r_{n+1}}\langle z-z_{n+1},z_{n+1}-x_{n+1}\rangle\geq0\), \(\forall z\in C\). It follows that \(F(z_{n},z_{n+1})+\frac{1}{r_{n}}\langle z_{n+1}-z_{n},z_{n}-x_{n}\rangle\geq0 \) and \(F(z_{n+1},z_{n})+\frac{1}{r_{n+1}}\langle z_{n}-z_{n+1},z_{n+1}-x_{n+1}\rangle\geq0\), \(\forall z\in C\). By using condition (A2), we find that \(\langle z_{n+1}-z_{n},\frac{z_{n}-x_{n}}{r_{n}}-\frac {z_{n+1}-x_{n+1}}{r_{n+1}}\rangle\geq0\). Hence, we have
This implies that \(\langle z_{n+1}-z_{n}, x_{n+1}-x_{n} +(1-\frac {r_{n}}{r_{n+1}})(z_{n+1}-x_{n+1})\rangle\geq\|z_{n+1}-z_{n}\|^{2}\). Hence, we have
It follows that
This implies that
Substituting (2.2) into (2.1), we find that
It follows from conditions (a)-(e) that
By using Lemma 1.5, we see that \(\lim_{n\rightarrow\infty}\|\lambda_{n}-x_{n}\|=0\), which in turn implies that
Since \(T_{r_{n}}\) is firmly nonexpansive, we find that
That is,
It follows that
where \(f_{n}=\|e_{n}\|(\|e_{n}\|+2\|z_{n}-p\|)\). This further implies that
By using conditions (a) and (b), we find from (2.3) that
Since A is α-inverse-strongly monotone, we find that
It follows that
This yields that
By using (2.3), we find from conditions (a), (c) and (d) that
Since \(P_{C}\) is firmly nonexpansive, we find that
which yields that
It follows that
This in turn leads to
By use of (2.3) and (2.5), we find from restrictions (a), (b), (c) and (e) that
On the other hand, we have
By using conditions (a) and (b), we find from (2.3) that
Since \(S_{n}\) is nonexpansive, we find that
It follows from (2.4), (2.6) and (2.7) that
Notice that
By using condition (c), we find that
Now, we are in a position to show \(\limsup_{n\rightarrow\infty}\langle f(q)-q,x_{n}-q\rangle\leq0\), where \(q=P_{\Omega}f(q)\). To show it, we can choose a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that
Since \(\{x_{n_{i}}\}\) is bounded, we can choose a subsequence \(\{x_{n_{i_{j}}}\}\) of \(\{x_{n_{i}}\}\) which converges weakly to some point \(\bar{x}\). We may assume, without loss of generality, that \(\{x_{n_{i}}\}\) converges weakly to \(\bar{x}\).
Next, we prove \(\bar{x}\in\Omega\). First, we show \(x\in \operatorname{EP}(F)\). Notice that
By using condition (A2), we see that \(\frac{1}{r_{n}}\langle y-y_{n},y_{n}-x_{n}\rangle\geq F(y,y_{n})\), \(\forall y\in C\). Replacing n by \(n_{i}\), we arrive at
By using (2.4) and (2.6), we find that \(\{y_{n_{i}}\}\) converges weakly to \(\bar{x}\). It follows that \(0\geq F(y,\bar{x})\). For each t with \(0< t\leq1\), let \(z_{t}=tz+(1-t)\bar{x}\), where \(z\in C\). It follows that \(z_{t}\in C\). Hence, we have \(F(z_{t}, \bar{x})\leq0\). It follows that
which yields that \(F(z_{t},z)\geq0\), \(\forall z\in C\). Letting \(t\downarrow0\), we obtain from condition (A3) that \(F(\bar{x},z)\geq0\), \(\forall z\in C\). This implies that \(\bar{x}\in \operatorname{EP}(F)\).
Next, we show that \(\bar{x}\in \operatorname{VI}(C,A)\). Let T be a maximal monotone mapping defined by
For any given \((x,y)\in \operatorname{Graph}(T)\), we have \(y-Ax\in N_{C}x\). Since \(y_{n}\in C\), we have \(\langle x-y_{n}, y-Ax\rangle\geq0\). Since \(y_{n}=P_{C}(z_{n}-s_{n}Az_{n}+e_{n})\), we see that \(\langle x-y_{n}, y_{n}-(I-s_{n}A)z_{n}-e_{n}\rangle\geq0 \) and hence
It follows that
Since A is Lipschitz continuous, we see that \(\langle x-\bar{x},y\rangle\geq0\). Notice that T is maximal monotone and hence \(0\in T\bar{x}\). This shows that \(\bar{x}\in \operatorname{VI}(C,A)\). By using Lemma 1.2, we find that \(\bar {x}\in F(S)\). It follows that \(\limsup_{n\rightarrow\infty}\langle f(q)-q,x_{n}-q\rangle\leq0\),
It follows that
where \(K=\sup_{n\geq1}\{\|x_{n}-q\|\}\). By using Lemma 1.6, we find that \(\lim_{n\rightarrow\infty}\|x_{n}-q\| =0\). This completes the proof. □
For nonexpansive mappings, we have the following result.
Corollary 2.2
Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping, and let F be a bifunction from \(C\times C\) to ℝ which satisfies (A1)-(A4). Let \(S:C\rightarrow H\) be nonexpansive, and let f be a β-contraction on H. Assume that \(\Omega=F(S)\cap \operatorname{VI}(C,A)\cap \operatorname{EP} (F)\neq\emptyset\). Let \(\{ r_{n}\}\) and \(\{s_{n}\}\) be positive real number sequences. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process:
where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions:
-
(a)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
-
(b)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
-
(c)
\(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\);
-
(d)
\(\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0 \) and \(\liminf_{n\rightarrow\infty}r_{n}> 0\);
-
(e)
\(\lim_{n\rightarrow\infty}|s_{n+1}-s_{n}|=0\), \(0< s\leq s_{n}\leq s'<2\alpha\),
where s, \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality
Further, if S is an identity, we have the following result.
Corollary 2.3
Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping, and let F be a bifunction from \(C\times C\) to ℝ which satisfies (A1)-(A4). Let f be a β-contraction on H. Assume that \(\Omega= \operatorname{VI}(C,A)\cap \operatorname{EP} (F)\neq\emptyset\). Let \(\{r_{n}\}\) and \(\{s_{n}\}\) be positive real number sequences. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process:
where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions:
-
(a)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
-
(b)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
-
(c)
\(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\);
-
(d)
\(\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0 \) and \(\liminf_{n\rightarrow\infty}r_{n}> 0\);
-
(e)
\(\lim_{n\rightarrow\infty}|s_{n+1}-s_{n}|=0\), \(0< s\leq s_{n}\leq s'<2\alpha\),
where s, \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality
Putting \(F(x,y)=0\) and \(r_{n}=1\), we find the following result.
Corollary 2.4
Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping. Let \(S:C\rightarrow H\) be a κ-strictly pseudocontractive mapping, and let f be a β-contraction on H. Assume that \(\Omega=F(S)\cap \operatorname{VI}(C,A)\neq\emptyset\). Let \(\{s_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process:
where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions:
-
(a)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
-
(b)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
-
(c)
\(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\), \(\lim_{n\rightarrow\infty }|\delta_{n+1}-\delta_{n}|=0 \) and \(\kappa\leq\delta_{n}\leq\delta<1\);
-
(d)
\(\lim_{n\rightarrow\infty}|s_{n+1}-s_{n}|=0\), \(0< s\leq s_{n}\leq s'<2\alpha\),
where δ, s, \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality
Corollary 2.5
Let C be a closed convex subset of a real Hilbert space H. Let F be a bifunction from \(C\times C\) to ℝ which satisfies (A1)-(A4). Let \(S:C\rightarrow H\) be a κ-strict pseudocontraction, and let f be a β-contraction on H. Assume that \(\Omega=F(S)\cap \operatorname{EP} (F)\neq\emptyset\). Let \(\{r_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process:
where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions:
-
(a)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
-
(b)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
-
(c)
\(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\), \(\lim_{n\rightarrow\infty }|\delta_{n+1}-\delta_{n}|=0 \) and \(\kappa\leq\delta_{n}\leq\delta<1\);
-
(d)
\(\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0 \) and \(\liminf_{n\rightarrow\infty}r_{n}> 0\),
where δ is a real constant. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality
Remark 2.6
Comparing Theorem 2.1, Corollaries 2.2-2.5 with Theorems IT, TT and the corresponding results in [6–20], we have the following.
-
(a)
We extend the nonlinear mapping from the class of nonexpansive mappings to the class of κ-strictly pseudocontractive mappings.
-
(b)
Possible computation errors are taken into account.
-
(c)
The conditions imposed on control sequences \(\{\alpha_{n}\}\) and \(\{s_{n}\}\) are relaxed.
-
(d)
The common element is also a unique solution of variational inequality (1.2).
Theorem 2.7
Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping, and let f be a β-contraction on H. Assume that \(\operatorname{VI}(C,A)\neq\emptyset\). Let \(\{s_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in \(x_{1} \in C\), \(x_{n+1} = \alpha_{n} f(x_{n}) + \beta_{n} x_{n} + \gamma_{n} y_{n}\), where \(y_{n} = P_{C}(x_{n} -s_{n} Ax_{n} + e_{n})\), where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions:
-
(a)
\(\lim_{n\rightarrow\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
-
(b)
\(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\);
-
(c)
\(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\), \(\lim_{n\rightarrow\infty }|s_{n+1}-s_{n}|=0\), \(0< s\leq s_{n}\leq s'<2\alpha\),
where s and \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to \(q=P_{\operatorname{VI}(C,A)}f(q)\).
Remark 2.8
To construct a mathematical model which is as close as possible to a real complex problem, we often have to use more than one constraint. Solving such problems, we have to obtain some solution which is simultaneously the solution of two or more subproblems. This is a common problem in diverse areas of mathematics and physical sciences. It consists of trying to find a solution satisfying certain constraints. In this paper, we investigate the problem of solving common solutions of an equilibrium problem, a variational inequality problem and a fixed point problem of a strictly pseudocontractive mapping based on a regularization projection algorithm. Possible computation errors are also taken into account. Strong convergence theorems are established without compact assumptions and additional metric projections in the framework of real Hilbert spaces.
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant No. (58-130-35-HiCi). The authors, therefore, acknowledge technical and financial support of KAU. The authors are grateful to the editor and the anonymous reviewers for useful suggestions which improved the contents of the article.
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Bin Dehaish, B.A., Latif, A., Bakodah, H.O. et al. A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces. J Inequal Appl 2015, 51 (2015). https://doi.org/10.1186/s13660-014-0541-z
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DOI: https://doi.org/10.1186/s13660-014-0541-z