Abstract
We introduce a Mann-type hybrid steepest-descent iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of solutions of finitely many variational inequalities, and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality.
Similar content being viewed by others
1 Introduction
Let H be a real Hilbert space with inner product \(\langle\cdot,\cdot \rangle\) and norm \(\|\cdot\|\), C be a nonempty closed convex subset of H and \(P_{C}\) be the metric projection of H onto C. Let \(S:C\to C\) be a self-mapping on C. We denote by \(\operatorname{Fix}(S)\) the set of fixed points of S and by R the set of all real numbers. A mapping \(A:C\to H\) is called L-Lipschitz continuous if there exists a constant \(L\geq 0\) such that
A mapping \(T:C\to C\) is called ξ-strictly pseudocontractive if there exists a constant \(\xi\in[0,1)\) such that
Let \(A:C\to H\) be a nonlinear mapping on C. We consider the following variational inequality problem (VIP) [1]: find a point \(\bar{x}\in C\) such that
The solution set of VIP (1.1) is denoted by \(\operatorname{VI}(C,A)\).
The general mixed equilibrium problem (GMEP) (see, e.g., [2]) is to find \(x\in C\) such that
where \({\varTheta },h:C\times C\to{\mathbf{R}}\) are two bi-functions. We denote the set of solutions of GMEP (1.2) by \(\operatorname{GMEP}({\varTheta },h)\). We assume as in [3] that \({\varTheta }:C\times C\to{\mathbf{R}}\) is a bi-function satisfying conditions (θ1)-(θ3) and \(h:C\times C\to{\mathbf{R}}\) is a bi-function with restrictions (h1)-(h3), where
- (θ1):
-
\({\varTheta }(x,x)=0\) for all \(x\in C\);
- (θ2):
-
Θ is monotone (i.e., \({\varTheta }(x,y)+{\varTheta }(y,x)\leq0\), \(\forall x,y\in C\)) and upper hemicontinuous in the first variable, i.e., for each \(x,y,z\in C\),
$$\limsup_{t\to0^{+}}{\varTheta } \bigl(tz+(1-t)x,y \bigr)\leq{ \varTheta }(x,y); $$ - (θ3):
-
Θ is lower semicontinuous and convex in the second variable;
- (h1):
-
\(h(x,x)=0\) for all \(x\in C\);
- (h2):
-
h is monotone and weakly upper semicontinuous in the first variable;
- (h3):
-
h is convex in the second variable.
For \(r>0\) and \(x\in H\), let \(T_{r}:H\to2^{C}\) be a mapping defined by
called the resolvent of Θ and h.
Let \(F_{1},F_{2}:C\to H\) be two mappings. Consider the following general system of variational inequalities (GSVI) of finding \((x^{*},y^{*})\in C\times C\) such that
where \(\nu_{1}>0\) and \(\nu_{2}>0\) are two constants. The solution set of GSVI (1.3) is denoted by \(\operatorname{GSVI}(C,F_{1},F_{2})\).
If C is the fixed point set \(\operatorname{Fix}(T)\) of a nonexpansive mapping T and S is another nonexpansive mapping (not necessarily with fixed points), then VIP (1.1) becomes the variational inequality problem of finding \(x^{*}\in\operatorname{Fix}(T)\) such that
This problem, introduced by Mainge and Moudafi [4, 5], is called the hierarchical fixed point problem. It is clear that if S has fixed points, then they are solutions of VIP (1.4).
During the 1980s and 1990s, the system of variational inequalities used as tools to solve Nash equilibrium problems. See, for example, [6–8] and the references therein. On the similar lines, the results of this paper can be applicable to solve Nash equilibrium problem for two person game. In the recent past, several iterative methods have been proposed and analyzed to three nonlinear problems, namely, system of variational inequalities, generalized mixed equilibrium problems and variational inequalities; see, for example, [9–11] and the references therein.
In this paper, we will introduce a Mann-type hybrid steepest-descent iterative algorithm for finding a common element of the solution set \(\operatorname{GMEP}({\varTheta },h)\) of GMEP (1.2), the solution set \(\operatorname{GSVI}(C,F_{1},F_{2})\) (i.e., Ξ) of GSVI (1.3), the solution set \(\bigcap^{M}_{k=1}\operatorname{VI}(C,A_{k})\) of finitely many variational inequalities for inverse-strongly monotone mappings \(A_{k}:C\to H\), \(k=1,\ldots,M\), and the common fixed point set \(\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\operatorname{Fix}(T)\) of finitely many nonexpansive mappings \(S_{i}:C\to C\), \(i=1,\ldots,N\) and a strictly pseudocontractive mapping \(T:C\to C\), in the setting of the infinite-dimensional Hilbert space. The iterative algorithm is based on Korpelevich’s extragradient method, the viscosity approximation method [12], Mann’s iteration method, and the hybrid steepest-descent method. Our aim is to prove that the iterative algorithm converges strongly to a common element of these sets, which also solves some hierarchical variational inequality. We observe that related results have been derived in [4, 5, 13, 14].
2 Preliminaries
Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by \(\langle\cdot, \cdot\rangle\) and \(\|\cdot\|\), respectively. Let C be a nonempty, closed, and convex subset of H. We write \(x_{n}\rightharpoonup x\) to indicate that the sequence \(\{x_{n}\}\) converges weakly to x and \(x_{n}\to x\) to indicate that the sequence \(\{x_{n}\}\) converges strongly to x. Moreover, we use \(\omega_{w}(x_{n})\) to denote the weak ω-limit set of the sequence \(\{x_{n}\}\) and \(\omega_{s}(x_{n})\) to denote the strong ω-limit set of the sequence \(\{x_{n}\}\), i.e.,
and
The metric (or nearest point) projection from H onto C is the mapping \(P_{C}:H\to C\) which assigns to each point \(x\in H\) the unique point \(P_{C}x\in C\) satisfying the property
The following properties of projections are useful for our purpose.
Proposition 2.1
Given any \(x\in H\) and \(z\in C\). One has
-
(i)
\(z=P_{C}x \Leftrightarrow\langle x-z,y-z\rangle\leq0\), \(\forall y\in C\);
-
(ii)
\(z=P_{C}x \Leftrightarrow\|x-z\|^{2}\leq\|x-y\|^{2}-\|y-z\|^{2}\), \(\forall y\in C\);
-
(iii)
\(\langle P_{C}x-P_{C}y,x-y\rangle\geq\|P_{C}x-P_{C}y\|^{2}\), \(\forall y\in H\), which hence implies that \(P_{C}\) is nonexpansive and monotone.
Definition 2.1
A mapping \(T:H\to H\) is said to be
-
(a)
nonexpansive if
$$\|Tx-Ty\|\leq\|x-y\|,\quad \forall x,y\in H; $$ -
(b)
firmly nonexpansive if \(2T-I\) is nonexpansive, or equivalently, if T is 1-inverse-strongly monotone (1-ism),
$$\langle x-y,Tx-Ty\rangle\geq\|Tx-Ty\|^{2}, \quad \forall x,y\in H; $$alternatively, T is firmly nonexpansive if and only if T can be expressed as
$$T=\frac{1}{2}(I+S), $$where \(S:H\to H\) is nonexpansive; projections are firmly nonexpansive.
Definition 2.2
A mapping \(A:C\to H\) is said to be
-
(i)
monotone if
$$\langle Ax-Ay,x-y\rangle\geq0,\quad \forall x,y\in C; $$ -
(ii)
η-strongly monotone if there exists a constant \(\eta>0\) such that
$$\langle Ax-Ay,x-y\rangle\geq\eta\|x-y\|^{2}, \quad \forall x,y\in C; $$ -
(iii)
α-inverse-strongly monotone if there exists a constant \(\alpha>0\) such that
$$\langle Ax-Ay,x-y\rangle\geq\alpha\|Ax-Ay\|^{2},\quad \forall x,y\in C. $$
It is obvious that if \(A:C\to H\) is α-inverse-strongly monotone, then A is monotone and \(\frac{1}{\alpha}\)-Lipschitz continuous. Moreover, we also have, for all \(u,v\in C\) and \(\lambda>0\),
Proposition 2.2
(see [15])
For given \(\bar{x},\bar{y}\in C\), \((\bar{x},\bar{y})\) is a solution of the GSVI (1.3) if and only if x̄ is a fixed point of the mapping \(G:C\to C\) defined by
where \(\bar{y}=P_{C}(I-\nu_{2}F_{2})\bar{x}\).
In particular, if the mapping \(F_{j}:C\to H\) is \(\zeta _{j}\)-inverse-strongly monotone for \(j=1,2\), then the mapping G is nonexpansive provided \(\nu_{j}\in(0,2\zeta_{j}]\) for \(j=1,2\). We denote by Ξ the fixed point set of the mapping G.
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 2.1
Let X be a real inner product space. Then there holds the following inequality
Lemma 2.2
Let H be a real Hilbert space. Then the following hold:
-
(a)
\(\|x-y\|^{2}=\|x\|^{2}-\|y\|^{2}-2\langle x-y,y\rangle\) for all \(x,y\in H\);
-
(b)
\(\|\lambda x+\mu y\|^{2}=\lambda\|x\|^{2}+\mu\|y\|^{2}-\lambda\mu\|x-y\| ^{2}\) for all \(x,y\in H\) and \(\lambda,\mu\in[0,1]\) with \(\lambda+\mu=1\);
-
(c)
if \(\{x_{n}\}\) is a sequence in H such that \(x_{n}\rightharpoonup x\), it follows that
$$\limsup_{n\to\infty}\|x_{n}-y\|^{2}=\limsup _{n\to\infty}\|x_{n}-x\|^{2}+\|x-y\| ^{2},\quad \forall y\in H. $$
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in \((0,1]\) and let \(\mu>0\). Associating with a nonexpansive mapping \(T:C\to C\), we define the mapping \(T^{\lambda}:C\to H\) by
where \(F:C\to H\) is an operator such that, for some positive constants \(\kappa,\eta>0\), F is κ-Lipschitzian and η-strongly monotone on C; that is, F satisfies the conditions:
for all \(x,y\in C\).
In the sequel, we let \(\operatorname{GMEP}({\varTheta },h)\) denote the solution set of GMEP (1.2).
Lemma 2.3
(see [3])
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \({\varTheta }:C\times C\to{\mathbf{R}}\) be a bi-function satisfying conditions (θ1)-(θ3) and \(h:C\times C\to{\mathbf{R}}\) is a bi-function with restrictions (h1)-(h3). Moreover, let us suppose that
-
(H)
for fixed \(r>0\) and \(x\in C\), there exist a bounded \(K\subset C\) and \(\hat{x}\in K\) such that for all \(z\in C\setminus K\), \(-{\varTheta } (\hat{x},z)+h(z,\hat{x})+\frac{1}{r}\langle\hat{x}-z,z-x\rangle<0\).
For \(r>0\) and \(x\in H\), the mapping \(T_{r}:H\to2^{C}\) (i.e., the resolvent of Θ and h) has the following properties:
-
(i)
\(T_{r}x\neq\emptyset\);
-
(ii)
\(T_{r}x\) is a singleton;
-
(iii)
\(T_{r}\) is firmly nonexpansive;
-
(iv)
\(\operatorname{GMEP}({\varTheta },h)=\operatorname{Fix}(T_{r})\) and it is closed and convex.
Recall that a set-valued mapping \(T:D(T)\subset H\to2^{H}\) is called monotone if for all \(x,y\in D(T)\), \(f\in Tx\) and \(g\in Ty\) imply
A set-valued mapping T is called maximal monotone if T is monotone and \((I+\lambda T)D(T)=H\) for each \(\lambda>0\), where I is the identity mapping of H. We denote by \(G(T)\) the graph of T. It is known that a monotone mapping T is maximal if and only if, for \((x,f)\in H\times H\), \(\langle f-g,x-y\rangle\geq0\) for every \((y,g)\in G(T)\) implies \(f\in Tx\).
3 Main results
We now propose the following Mann-type hybrid steepest-descent iterative scheme:
for all \(n\geq0\), where
-
\(F:C\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\) and \(f:C\to C\) is an l-Lipschitzian mapping with constant \(l\geq0\);
-
\(A_{k}:C\to H\) is \(\eta_{k}\)-inverse-strongly monotone, \(\{\lambda_{k,n}\} \subset[a_{k},b_{k}]\subset(0,2\eta_{k})\), \(\forall k\in \{1,\ldots,M\}\), and \({\varLambda }^{M}_{n}:=P_{C}(I-\lambda_{M,n}A_{M})\cdots P_{C}(I-\lambda_{1,n}A_{1})\);
-
\(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse-strongly monotone and \(G:=P_{C}(I-\nu _{1}F_{1})P_{C}(I-\nu_{2}F_{2})\) with \(\nu_{j}\in(0,2\zeta_{j})\) for \(j=1,2\);
-
\(T:C\to C\) is a ξ-strict pseudocontraction and \(S_{i}:C\to C\) is a nonexpansive mapping for each \(i=1,\ldots,N\);
-
\({\varTheta },h:C\times C\to{\mathbf{R}}\) are two bi-functions satisfying the hypotheses of Lemma 2.3;
-
\(0<\mu<2\eta/\kappa^{2}\) and \(0\leq\gamma l<\tau\) with \(\tau:=1-\sqrt {1-\mu(2\eta-\mu\kappa^{2})}\);
-
\(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) are sequences in \((0,1)\) with \(0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty}\beta_{n}<1\);
-
\(\{\gamma_{n}\}\), \(\{\delta_{n}\}\) are sequences in \([0,1]\) with \(\beta _{n}+\gamma_{n}+\delta_{n}=1\), \(\forall n\geq0\);
-
\(\{\beta_{n,i}\}^{N}_{i=1}\) are sequences in \((0,1)\) and \((\gamma _{n}+\delta_{n})\xi\leq\gamma_{n}\), \(\forall n\geq0\);
-
\(\{r_{n}\}\) is a sequence in \((0,\infty)\) with \(\liminf_{n\to\infty }r_{n}>0\) and \(\liminf_{n\to\infty}\delta_{n}>0\).
We start our main result from the following series of propositions.
Proposition 3.1
Let us suppose that \({\varOmega }=\operatorname{Fix}(T)\cap\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\bigcap^{M}_{k=1}\operatorname{VI} (C,A_{k})\cap \operatorname{GMEP}({\varTheta },h)\cap{ \varXi }\neq\emptyset\). Then the sequences \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{y_{n,i}\}\) for all i, \(\{u_{n}\}\) are bounded.
Proof
Since \(0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to \infty}\beta_{n}<1\), we may assume, without loss of generality, that \(\{\beta_{n}\}\subset[c,d]\subset(0,1)\). For simplicity, we write
for all \(n\geq0\). Then \(y_{n}=P_{C}v_{n}\). Also, we set \(\tilde{y}_{n}={ \varLambda }^{M}_{n}y_{n}\),
for all \(k\in\{1,2,\ldots,M\}\) and \(n\geq0\), and \({\varLambda }^{0}_{n}=I\), where I is the identity mapping on H.
First of all, take a fixed \(p\in{ \varOmega }\) arbitrarily. We observe that
For all i from \(i=2\) to \(i=N\), by induction, one proves that
Thus we obtain that for every \(i=1,\ldots,N\),
Since for each \(k\in\{1,\ldots,M\}\), \(I-\lambda_{k,n}A_{k}\) is nonexpansive and \(p=P_{C}(I-\lambda_{k,n}A_{k})p\), we have
For simplicity, we write \(\tilde{p}=P_{C}(p-\nu_{2}F_{2}p)\), \(\tilde{y}_{n,N}=P_{C}(y_{n,N}-\nu_{2}F_{2}y_{n,N})\) and \(z_{n}=P_{C}(\tilde{y}_{n,N}-\nu_{1}F_{1}\tilde{y}_{n,N})\) for each \(n\geq0\). Then \(z_{n}=Gy_{n,N}\) and
Since \(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse-strongly monotone and \(0<\nu _{j}<2\zeta_{j}\) for each \(j=1,2\), we know that, for all \(n\geq0\),
Also, since \(Gp=p\) and G is nonexpansive, utilizing Lemma 3.1 of [16] we have from (3.1) and (3.4)
Taking into account \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) and utilizing [17], we obtain from (3.1), (3.3), and (3.5) that
By induction, we get
This implies that \(\{x_{n}\}\) is bounded and so are \(\{F_{2}y_{n,N}\}\), \(\{ F_{1}\tilde{y}_{n,N}\}\), \(\{\tilde{y}_{n,N}\}\), \(\{z_{n}\}\), \(\{u_{n}\}\), \(\{v_{n}\}\), \(\{y_{n}\}\), \(\{y_{n,i}\}\) for each \(i=1,\ldots,N\). Since \(\| T\tilde{y}_{n}-p\|\leq\frac{1+\xi}{1-\xi}\|\tilde{y}_{n}-p\|\), \(\{T\tilde{y}_{n}\}\) is also bounded. □
Proposition 3.2
Let us suppose that \({\varOmega }\neq\emptyset\). Moreover, let us suppose that the following hold:
-
(H0)
\(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum^{\infty}_{n=0}\alpha _{n}=\infty\);
-
(H1)
\(\sum^{\infty}_{n=1}|\lambda_{k,n}-\lambda_{k,n-1}|<\infty\) or \(\lim_{n\to\infty}\frac{|\lambda_{k,n}-\lambda_{k,n-1}|}{ \alpha_{n}}=0\) for each \(k=1,\ldots,M\);
-
(H2)
\(\sum^{\infty}_{n=1}|\alpha_{n}-\alpha_{n-1}|<\infty\) or \(\lim_{n\to \infty}\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}=0\);
-
(H3)
\(\sum^{\infty}_{n=1}|\beta_{n,i}-\beta_{n-1,i}|<\infty\) or \(\lim_{n\to\infty}\frac{|\beta_{n,i}-\beta_{n-1,i}|}{\alpha_{n}}=0\) for each \(i=1,\ldots,N\);
-
(H4)
\(\sum^{\infty}_{n=1}|r_{n}-r_{n-1}|<\infty\) or \(\lim_{n\to\infty}\frac {|r_{n}-r_{n-1}|}{\alpha_{n}}=0\);
-
(H5)
\(\sum^{\infty}_{n=1}|\beta_{n}-\beta_{n-1}|<\infty\) or \(\lim_{n\to \infty}\frac{|\beta_{n}-\beta_{n-1}|}{\alpha_{n}}=0\);
-
(H6)
\(\sum^{\infty}_{n=1}|\frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma _{n-1}}{1-\beta_{n-1}}|<\infty\) or \(\lim_{n\to\infty}\frac{1}{\alpha_{n}}|\frac{\gamma_{n}}{1-\beta_{n}}-\frac {\gamma_{n-1}}{1-\beta_{n-1}}|=0\).
Then \(\lim_{n\to\infty}\|x_{n+1}-x_{n}\|=0\), i.e., \(\{x_{n}\}\) is asymptotically regular.
Proof
First, it is known that \(\{\beta_{n}\}\subset[c,d]\subset (0,1)\) as in the proof of Proposition 3.1. Taking into account \(\liminf_{n\to\infty}r_{n}>0\), we may assume, without loss of generality, that \(\{r_{n}\}\subset[\epsilon,\infty)\) for some \(\epsilon>0\). First, we write \(x_{n}=\beta_{n-1}x_{n-1}+(1-\beta _{n-1})w_{n-1}\), \(\forall n\geq1\), where \(w_{n-1}= \frac{x_{n}-\beta_{n-1}x_{n-1}}{1-\beta_{n-1}}\). It follows that for all \(n\geq1\),
Since \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) for all \(n\geq0\), we have
Next, we estimate \(\|y_{n}-y_{n-1}\|\). Observe that
Also, we observe that
Simple calculations show that
Then, passing to the norm we get from (3.9)
where \(\sup_{n\geq0}\|\gamma f(y_{n,N})-\mu Fz_{n}\|\leq\widetilde{M}\) for some \(\widetilde{M}>0\). In the meantime, by the definition of \(y_{n,i}\) one obtains, for all \(i=N,\ldots,2\),
In the case \(i=1\), we have
Substituting (3.12) in all (3.11)-type expressions one obtains for \(i=2,\ldots,N\),
This together with (3.10) implies that
Furthermore, utilizing (2.1), we obtain
where \(\sup_{n\geq1}\{\sum^{M}_{k=1}\|A_{k}{\varLambda }^{k-1}_{n}y_{n}\|\}\leq \widetilde{M}_{0}\) for some \(\widetilde{M}_{0}>0\).
By [3], we know that
where \(L=\sup_{n\geq0}\|u_{n}-x_{n}\|\). So, substituting (3.15) in (3.13), we obtain
where \(\sup_{n\geq0}\{L+\widetilde{M}+\sum^{N}_{k=2}\|S_{k}u_{n}-y_{n,k-1}\|+\| S_{1}u_{n}-u_{n}\|\}\leq\widetilde{M}_{1}\) for some \(\widetilde{M}_{1}>0\). This together with (3.7), (3.8), and (3.14), implies that
where \(\sup_{n\geq0}\{\widetilde{M}_{0}+\widetilde{M}_{1}+\|\tilde{y}_{n}\|+\| T\tilde{y}_{n}\|\}\leq\widetilde{M}_{2}\) for some \(\widetilde{M}_{2}>0\).
Further, we observe that
Simple calculations show that
Then, passing to the norm we get from (3.16)
where \(\sup_{n\geq0}\{\widetilde{M}_{2}+\|x_{n}-w_{n}\|\}\leq\widetilde{M}_{3}\) for some \(\widetilde{M}_{3}>0\). By hypotheses (H0)-(H6) and Lemma 2.1 of [16], we obtain the claim. □
Proposition 3.3
Let us suppose that \({\varOmega }\neq\emptyset\). Let us suppose that \(\{x_{n}\}\) is asymptotically regular. Then \(\|x_{n}-u_{n}\|=\|x_{n}-T_{r_{n}}x_{n}\|\to0\) and \(\|y_{n}-\tilde{y}_{n}\|\to0\) as \(n\to\infty\).
Proof
Take a fixed \(p\in{ \varOmega }\) arbitrarily. We recall that, by the firm nonexpansivity of \(T_{r_{n}}\), a standard calculation shows that for \(p\in\operatorname{GMEP}({\varTheta },h)\),
Observe that
for each \(k\in\{1,2,\ldots,M\}\).
Utilizing Lemma 2.1 and Lemma 3.1 of [16], we obtain from \(0\leq\gamma l<\tau\), (3.1), (3.4), and (3.18)
Since \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) for all \(n\geq0\), utilizing [17] we have from (3.19) and (3.20)
So, we deduce from \(\{\beta_{n}\}\subset[c,d]\subset(0,1)\) and \(\{\lambda _{k,n}\}\subset[a_{k},b_{k}]\subset(0,2\eta_{k})\), \(k=1,\ldots,M\), that
By Propositions 3.1 and 3.2 we know that the sequences \(\{x_{n}\}\), \(\{y_{n}\} \), and \(\{y_{n,N}\}\) are bounded, and that \(\{x_{n}\}\) is asymptotically regular. Therefore, from \(\alpha_{n}\to0\) we obtain
for each \(k\in\{1,\ldots,M\}\).
By Proposition 2.1(iii), we deduce that for each \(k\in\{1,2,\ldots,M\}\),
which immediately leads to
From (3.4), (3.20), (3.21), and (3.23) we conclude that
which together with \(\{\beta_{n}\}\subset[c,d]\subset(0,1)\) and \(\{\lambda _{k,n}\}\subset[a_{k},b_{k}]\subset(0,2\eta_{k})\), \(k=1,\ldots,M\), yields
Since \(\alpha_{n}\to0\), and \(\{x_{n}\}\), \(\{y_{n}\}\), and \(\{y_{n,N}\}\) are bounded, we obtain from (3.22) and the asymptotical regularity of \(\{x_{n}\}\) (due to Proposition 3.2),
Note that
Thus,
□
Remark 3.1
By the last proposition we have \(\omega_{w}(x_{n})=\omega _{w}(u_{n})\) and \(\omega_{s}(x_{n})=\omega_{s}(u_{n})\), i.e., the sets of strong/weak cluster points of \(\{x_{n}\}\) and \(\{u_{n}\}\) coincide.
Of course, if \(\beta_{n,i}\to\beta_{i}\neq0\) as \(n\to\infty\), for all indices i, the assumptions of Proposition 3.2 are enough to assure that
In the next proposition, we estimate the case in which at least one sequence \(\{\beta_{n,k_{0}}\}\) is a null sequence.
Proposition 3.4
Let us suppose that \({\varOmega }\neq\emptyset\). Let us suppose that (H0) holds. Moreover, for an index \(k_{0}\in\{1,\ldots,N\}\), \(\lim_{n\to\infty}\beta_{n,k_{0}}=0\) and the following hold:
-
(H7)
for each \(i\in\{1,\ldots,N\}\) and \(k\in\{1,\ldots,M\}\),
$$\begin{aligned} { \lim_{n\to\infty}}\frac{|\beta_{n,i}-\beta _{n-1,i}|}{\alpha_{n}\beta_{n,k_{0}}} &={ \lim_{n\to\infty}} \frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}\beta_{n,k_{0}}} ={ \lim_{n\to\infty}}\frac{|\beta_{n}-\beta_{n-1}|}{\alpha_{n}\beta_{n,k_{0}}} ={ \lim _{n\to\infty}}\frac{|r_{n}-r_{n-1}|}{\alpha_{n}\beta_{n,k_{0}}} \\ &={ \lim_{n\to\infty}}\frac{1}{\alpha_{n}\beta_{n,k_{0}}}\biggl|\frac{\gamma _{n}}{1-\beta_{n}}- \frac{\gamma_{n-1}}{1-\beta_{n-1}}\biggr| \\ &={ \lim_{n\to\infty}}\frac{|\lambda_{k,n}-\lambda_{k,n-1}|}{\alpha _{n}\beta_{n,k_{0}}}=0; \end{aligned}$$ -
(H8)
there exists a constant \(b>0\) such that \(\frac{1}{\alpha_{n}}|\frac {1}{\beta_{n,k_{0}}}-\frac{1}{\beta_{n-1,k_{0}}}|< b\) for all \(n\geq1\).
Then
Proof
We start by (3.17). Dividing both terms by \(\beta _{n,k_{0}}\) we have
So, by (H8) we have
Therefore, utilizing Lemma 2.1 of [16], from (H0), (H7), and the asymptotical regularity of \(\{x_{n}\}\) (due to Proposition 3.2), we deduce that
□
Proposition 3.5
Let us suppose that \({\varOmega }\neq\emptyset\). Let us suppose that (H0)-(H6) hold. Then \(\|z_{n}-y_{n,N}\|\to0\) as \(n\to\infty\).
Proof
Let \(p\in{ \varOmega }\). In terms of the firm nonexpansivity of \(P_{C}\) and the \(\zeta_{j}\)-inverse-strong monotonicity of \(F_{j}\) for \(j=1,2\), we obtain from \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\), and (3.4)
and
Thus, we have
and
Consequently, from (3.4), (3.24) and (3.28), it follows that
which yields
Since \(\lim_{n\to\infty}\alpha_{n}=0\), \(\lim_{n\to\infty}\|x_{n+1}-x_{n}\| =0\), and \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{y_{n,N}\}\), and \(\{\tilde{y}_{n,N}\}\) are bounded, we deduce from (3.22) that
Furthermore, from (3.4), (3.24), and (3.29), it follows that
which leads to
Since \(\lim_{n\to\infty}\alpha_{n}=0\), \(\lim_{n\to\infty}\|x_{n+1}-x_{n}\| =0\), and \(\{x_{n}\}\), \(\{z_{n}\}\), \(\{y_{n}\}\), \(\{y_{n,N}\}\), and \(\{\tilde{y}_{n,N}\}\) are bounded, we deduce from (3.22) that
Note that
Hence from (3.30) and (3.31) we get
□
Proposition 3.6
Let us suppose that \({\varOmega }\neq\emptyset\). Let us suppose that \(0<\liminf_{n\to\infty}\beta_{n,i} \leq\limsup_{n\to\infty}\beta_{n,i}<1\) for each \(i=1,\ldots,N\). Moreover, suppose that (H0)-(H6) are satisfied. Then, \(\lim_{n\to \infty}\|S_{i}u_{n}-u_{n}\|=0\) for each \(i=1,\ldots,N\) provided \(\|Ty_{n}-y_{n}\|\to \) as \(n\to\infty\).
Proof
First of all, observe that
By Proposition 3.2 we know that \(\{x_{n}\}\) is asymptotically regular. Utilizing [17] we have from \((\gamma_{n}+\delta_{n}) \xi\leq\gamma_{n}\),
which together with (3.26) and \(\|Ty_{n}-y_{n}\|\to0\), implies that
Let us show that for each \(i\in\{1,\ldots,N\}\), one has \(\| S_{i}u_{n}-y_{n,i-1}\|\to0\) as \(n\to\infty\). Let \(p\in{ \varOmega }\). When \(i=N\), by Lemma 2.2(b) we have from (3.2), (3.4), and (3.20)
So, we have
Since \(\alpha_{n}\to0\), \(0<\liminf_{n\to\infty}\beta_{n,N}\leq\limsup_{n\to \infty}\beta_{n,N}<1\) and \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\) (due to (3.33)), it is known that \(\{\|S_{N}u_{n}-y_{n,N-1}\|\}\) is a null sequence.
Let \(i\in\{1,\ldots,N-1\}\). Then one has
and so, after \((N-i+1)\) iterations,
Again we obtain
Since \(\alpha_{n}\to0\), \(0<\liminf_{n\to\infty}\beta_{n,i}\leq\limsup_{n\to \infty}\beta_{n,i}<1\) for each \(i=1,\ldots,N-1\), and \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\) (due to (3.33)), it is known that
Obviously for \(i=1\), we have \(\|S_{1}u_{n}-u_{n}\|\to0\).
To conclude, we have
from which \(\|S_{2}u_{n}-u_{n}\|\to0\). Thus by induction \(\|S_{i}u_{n}-u_{n}\|\to0\) for all \(i=2,\ldots,N\) since it is enough to observe that
□
Remark 3.2
As an example, we consider \(M=1\), \(N=2\) and the sequences:
-
(a)
\(\lambda_{1,n}=\eta_{1}-\frac{1}{n}\), \(\forall n>\frac{1}{\eta_{1}}\);
-
(b)
\(\alpha_{n}=\frac{1}{\sqrt{n}}\), \(r_{n}=2-\frac{1}{n}\), \(\forall n>1\);
-
(c)
\(\beta_{n}=\beta_{n,1}=\frac{1}{2}-\frac{1}{n}\), \(\beta_{n,2}=\frac {1}{2}-\frac{1}{n^{2}}\), \(\forall n>2\).
Then they satisfy the hypotheses on the parameter sequences in Proposition 3.6.
Proposition 3.7
Let us suppose that \({\varOmega }\neq\emptyset\) and \(\beta_{n,i}\to\beta_{i}\) for all i as \(n\to\infty\). Suppose there exists \(k\in\{1,\ldots,N\}\) such that \(\beta_{n,k}\to0\) as \(n\to\infty\). Let \(k_{0} \in\{1,\ldots,N\}\) be the largest index such that \(\beta_{n,k_{0}}\to0\) as \(n\to\infty\). Suppose that
-
(i)
\(\frac{\alpha_{n}}{\beta_{n,k_{0}}}\to0\) as \(n\to\infty\);
-
(ii)
if \(i\leq k_{0}\) and \(\beta_{n,i}\to0\) then \(\frac{\beta _{n,k_{0}}}{\beta_{n,i}}\to0\) as \(n\to\infty\);
-
(iii)
if \(\beta_{n,i}\to\beta_{i}\neq0\) then \(\beta_{i}\) lies in \((0,1)\).
Moreover, suppose that (H0), (H7), and (H8) hold. Then, \(\lim_{n\to \infty}\|S_{i}u_{n}-u_{n}\|=0\) for each \(i=1,\ldots,N\) provided \(\|Ty_{n}-y_{n}\|\to0\) as \(n\to\infty\).
Proof
First of all we note that if (H7) holds then also (H1)-(H6) are satisfied. So \(\{x_{n}\}\) is asymptotically regular.
Let \(k_{0}\) be as in the hypotheses. As in Proposition 3.6, for every index \(i\in\{1,\ldots,N\}\) such that \(\beta_{n,i}\to\beta_{i} \neq0\) (which leads to \(0<\liminf_{n\to\infty}\beta_{n,i}\leq\limsup_{n\to\infty}\beta_{n,i}<1\)), one has \(\|S_{i}u_{n}-y_{n,i-1}\| \to0\) as \(n\to\infty\).
For all the other indices \(i\leq k_{0}\), we can prove that \(\| S_{i}u_{n}-y_{n,i-1}\|\to0\) as \(n\to\infty\) in a similar manner. By the relation (due to (3.21) and (3.34))
we immediately obtain that
By Proposition 3.4 or by hypothesis (ii) on the sequences, we have
So, the conclusion follows. □
Remark 3.3
Let us consider \(M=1\), \(N=3\) and the following sequences:
-
(a)
\(\alpha_{n}=\frac{1}{n^{1/2}}\), \(r_{n}=2-\frac{1}{n^{2}}\), \(\forall n>1\);
-
(b)
\(\lambda_{1,n}=\eta_{1}-\frac{1}{n^{2}}\), \(\forall n>\frac{1}{\eta^{1/2}_{1}}\);
-
(c)
\(\beta_{n,1}=\frac{1}{n^{1/4}}\), \(\beta_{n}=\beta_{n,2}=\frac {1}{2}-\frac{1}{n^{2}}\), \(\beta_{n,3}=\frac{1}{n^{1/3}}\), \(\forall n>1\).
It is easy to see that all hypotheses (i)-(iii), (H0), (H7), and (H8) of Proposition 3.7 are satisfied.
Remark 3.4
Under the hypotheses of Proposition 3.7, analogously to Proposition 3.6, one can see that
Corollary 3.1
Let us suppose that the hypotheses of either Proposition 3.6 or Proposition 3.7 are satisfied. Then \(\omega_{w}(x_{n})=\omega_{w}(u_{n})=\omega_{w}(y_{n,1})\), \(\omega_{s}(x_{n})=\omega _{s}(u_{n})=\omega_{s}(y_{n,1})\), and \(\omega_{w}(x_{n})\subset{ \varOmega }\).
Proof
By Remark 3.1, we have \(\omega_{w}(x_{n})=\omega_{w}(u_{n})\) and \(\omega_{s}(x_{n})=\omega_{s}(u_{n})\). Note that by Remark 3.4,
In the meantime, it is known that
Hence we have
Furthermore, it follows from (3.1) that
which, together with \(\lim_{n\to\infty}\|S_{N}u_{n}-y_{n,N-1}\|=0\), yields
Combining (3.35) and (3.36), we conclude that
which, together with \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\), leads to
Now we observe that
By Propositions 3.3 and 3.6, \(\|x_{n}-u_{n}\|\to0\) and \(\|S_{1}u_{n}-u_{n}\|\to0\) as \(n\to\infty\), and hence
So we get \(\omega_{w}(x_{n})=\omega_{w}(y_{n,1})\) and \(\omega_{s}(x_{n})=\omega _{s}(y_{n,1})\).
Let \(p\in\omega_{w}(x_{n})\). Then there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup p\). Since \(p\in\omega_{w}(u_{n})\), by Proposition 3.6 and [18] (demiclosedness principle), we have \(p\in\operatorname{Fix}(S_{i})\) for each \(i=1,\ldots,N\), i.e., \(p\in\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\). Taking into account \(p\in\in\omega_{w}(y_{n,N})\) (due to (3.38)) and \(\|y_{n,N}-Gy_{n,N}\|\to0\) (due to (3.32)), by [18] we know that \(p\in\operatorname{Fix}(G)=: {\varXi }\). Also, since \(p\in\omega_{w}(y_{n})\) (due to (3.33)), in terms of \(\|Ty_{n}-y_{n}\|\to0\) and Proposition 2.1 of [19], we get \(p\in\operatorname{Fix}(T)\). Moreover, by [20] and Proposition 3.3 we know that \(p\in\operatorname{GMEP}({\varTheta },h)\). Next we prove that \(p\in\bigcap^{M}_{m=1}\operatorname{VI}(C,A_{m})\). As a matter of fact, from (3.25) and (3.33) we know that \(y_{n_{i}}\rightharpoonup p\) and \({\varLambda }^{m}_{n_{i}}y_{n_{i}}\rightharpoonup p\) for each \(m=1,\ldots,M\). Let
where \(m\in\{1,2,\ldots,M\}\). Let \((v,u)\in G(\widetilde{T}_{m})\). Since \(u-A_{m}v\in N_{C}v\) and \({\varLambda }^{m}_{n}y_{n}\in C\), we have
On the other hand, from \({\varLambda }^{m}_{n}y_{n}=P_{C}(I-\lambda_{m,n}A_{m}){ \varLambda }^{m-1}_{n}y_{n}\) and \(v\in C\), we have
and hence
Therefore, we have
From (3.25) and since \(A_{m}\) is Lipschitz continuous, we obtain
From \({\varLambda }^{m}_{n_{i}}y_{n_{i}}\rightharpoonup p\), \(\{\lambda_{m,n}\} \subset[a_{m},b_{m}]\subset(0,2\eta_{m})\), \(\forall m\in\{1,2,\ldots,M\}\) and (3.25), we have
Since \(\widetilde{T}_{m}\) is maximal monotone, we have \(p\in\widetilde{T}^{-1}_{m}0\) and hence \(p\in\operatorname{VI}(C,A_{m})\), \(m=1,2,\ldots,M\), which implies \(p\in\bigcap^{M}_{m=1}\operatorname{VI}(C,A_{m})\). Consequently, it is known that \(p\in \operatorname{Fix}(T)\cap\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i}) \cap\bigcap^{M}_{m=1}\operatorname{VI}(C,A_{m})\cap\operatorname{GMEP}({\varTheta },h)\cap{ \varXi }=:{\varOmega }\). □
Theorem 3.1
Let us suppose that \({\varOmega }\neq\emptyset\). Let \(\{\alpha_{n}\}\), \(\{\beta_{n,i}\}\), \(i=1,\ldots,N\), be sequences in \((0,1)\) such that \(0<\liminf_{n\to\infty}\beta_{n,i}\leq\limsup_{n\to\infty}\beta_{n,i}<1\) for each index i. Moreover, let us suppose that (H0)-(H6) hold. Then the sequences \(\{x_{n}\}\), \(\{y_{n}\} \), and \(\{u_{n}\}\) defined by scheme (3.1), all converge strongly to \(x^{*}=P_{\varOmega }(I-(\mu F-\gamma f))x^{*}\) if and only if \(\|y_{n}-Ty_{n}\|\to0\) as \(n\to\infty\), where \(x^{*}=P_{ \varOmega }(I-(\mu F-\gamma f))x^{*}\) is the unique solution of the hierarchical VIP
Proof
First of all, we note that \(F:C\to H\) is η-strongly monotone and κ-Lipschitzian on C and \(f:C \to C\) is an l-Lipschitz continuous mapping with \(0\leq\gamma l<\tau\). Observe that
It is clear that
Hence we deduce that \(\mu F-\gamma f\) is \((\mu\eta-\gamma l)\)-strongly monotone. In the meantime, it is easy to see that \(\mu F- \gamma f\) is \((\mu\kappa+\gamma l)\)-Lipschitz continuous with constant \(\mu\kappa+\gamma l>0\). Thus, there exists a unique solution \(x^{*}\) in Ω to the VIP (3.39).
Now, observe that there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\} \) such that
Since \(\{x_{n_{i}}\}\) is bounded, there exists a subsequence \(\{ x_{n_{i_{j}}}\}\) of \(\{x_{n_{i}}\}\) which converges weakly to some \(p\in H\). Without loss of generality, we may assume that \(x_{n_{i}}\rightharpoonup p\). Then by Corollary 3.1, we get \(p\in\omega_{w} (x_{n})\subset{ \varOmega }\). Hence, from (3.39) and (3.40), we have
Since (H1)-(H6) hold, the sequence \(\{x_{n}\}\) is asymptotically regular (according to Proposition 3.2). In terms of (3.33) and Proposition 3.3, \(\|x_{n}-y_{n}\|\to0\) and \(\|x_{n}-u_{n}\|\to0\) as \(n\to\infty\).
Let us show that \(\|x_{n}-x^{*}\|\to0\) as \(n\to\infty\). Indeed, putting \(p=x^{*}\), we deduce from (3.3), (3.4), (3.20), and (3.21) that
Since \(\sum^{\infty}_{n=0}\alpha_{n}=\infty\), \(\{\beta_{n}\}\subset[c,d]\subset (0,1)\) and \(\|x_{n}-y_{n}\|\to0\), we obtain \(\sum^{\infty}_{n=0}\alpha_{n}(1-\beta_{n})\frac {\tau^{2}-(\gamma l)^{2}}{ \tau}\geq\sum^{\infty}_{n=0}\alpha_{n}(1-d)\frac{\tau^{2}-(\gamma l)^{2}}{\tau }=\infty\) and
(due to (3.41)). Applying Lemma 2.1 of [16] to (3.42), we infer that the sequence \(\{x_{n}\}\) converges strongly to \(x^{*}\). This completes the proof. □
In a similar way, we can conclude another theorem as follows.
Theorem 3.2
Let us suppose that \({\varOmega }\neq\emptyset\). Let \(\{\alpha_{n}\}\), \(\{\beta_{n,i}\}\), \(i=1,\ldots,N\), be sequences in \((0,1)\) such that \(\beta_{n,i}\to\beta_{i}\) for each index i as \(n\to\infty\). Suppose that there exists \(k\in\{1,\ldots,N\}\) for which \(\beta_{n,k}\to0\) as \(n\to\infty\). Let \(k_{0}\in\{1,\ldots,N\}\) the largest index for which \(\beta_{n,k_{0}}\to0\). Moreover, let us suppose that (H0), (H7), and (H8) hold and
-
(i)
\(\frac{\alpha_{n}}{\beta_{n,k_{0}}}\to0\) as \(n\to\infty\);
-
(ii)
if \(i\leq k_{0}\) and \(\beta_{n,i}\to\beta_{i}\) then \(\frac{\beta _{n,k_{0}}}{\beta_{n,i}}\to0\) as \(n\to\infty\);
-
(iii)
if \(\beta_{n,i}\to\beta_{i}\neq0\) then \(\beta_{i}\) lies in \((0,1)\).
Then the sequences \(\{x_{n}\}\), \(\{y_{n}\}\), and \(\{u_{n}\}\) defined by scheme (3.1) all converge strongly to \(x^{*}=P_{\varOmega }(I-(\mu F-\gamma f))x^{*}\) if and only if \(\|y_{n}-Ty_{n}\| \to0\) as \(n\to\infty\), where \(x^{*}=P_{\varOmega } (I-(\mu F-\gamma f))x^{*}\) is the unique solution of the hierarchical VIP
Remark 3.5
According to the above argument for Theorems 3.1 and 3.2, we can readily see that if, in scheme (3.1), the iterative step \(y_{n}=P_{C}[\alpha_{n}\gamma f(y_{n,N})+(I-\alpha _{n}\mu F)Gy_{n,N}]\) is replaced by the iterative one, \(y_{n}=P_{C}[\alpha_{n}\gamma f(x_{n})+(I-\alpha_{n}\mu F)Gy_{n,N}]\), then Theorems 3.1 and 3.2 remain valid.
References
Lions, JL: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)
Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(1-4), 123-145 (1994)
Cianciaruso, F, Marino, G, Muglia, L, Yao, Y: A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl. 2010, Article ID 383740 (2010)
Moudafi, A, Mainge, P-E: Towards viscosity approximations of hierarchical fixed point problems. Fixed Point Theory Appl. 2006, Article ID 95453 (2006)
Moudafi, A, Mainge, P-E: Strong convergence of an iterative method for hierarchical fixed point problems. Pac. J. Optim. 3(3), 529-538 (2007)
Ansari, QH, Yao, JC: A fixed point theorem and its applications to the system of variational inequalities. Bull. Aust. Math. Soc. 59, 433-442 (1999)
Ansari, QH, Schaible, S, Yao, JC: System of vector equilibrium problems and its applications. J. Optim. Theory Appl. 107(3), 547-557 (2000)
Ansari, QH, Yao, JC: Systems of generalized variational inequalities and their applications. Appl. Anal. 76(3-4), 203-217 (2000)
Zeng, L-C, Ansari, QH, Schaible, S, Yao, J-C: Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theory 12(2), 293-308 (2011)
Ceng, L-C, Ansari, QH, Schaible, S: Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems. J. Glob. Optim. 53(1), 69-96 (2012)
Latif, A, Ceng, L-C, Ansari, QH: Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of equilibrium problem and fixed point problems. Fixed Point Theory Appl. 2012, 186 (2012)
Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241(1), 46-55 (2000)
Ceng, LC, Guu, SM, Yao, JC: Finding common solutions of a variational inequality, a general system of variational inequalities, and a fixed-point problem via a hybrid extragradient method. Fixed Point Theory Appl. 2011, Article ID 626159 (2011)
Ceng, LC, Kong, ZR, Wen, CF: On general systems of variational inequalities. Comput. Math. Appl. 66, 1514-1532 (2013)
Ceng, LC, Wang, CY, Yao, JC: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67(3), 375-390 (2008)
Xu, HK, Kim, TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119(1), 185-201 (2003)
Yao, Y, Liou, YC, Kang, SM: Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput. Math. Appl. 59, 3472-3480 (2010)
Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336-346 (2007)
Marino, G, Muglia, L, Yao, Y: Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points. Nonlinear Anal. 75, 1787-1798 (2012)
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under Grant No. (34-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Finally, the authors thank the honorable reviewers and respectable editor for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article.
Rights and permissions
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.
About this article
Cite this article
Latif, A., Alofi, A.S., Al-Mazrooei, A.E. et al. Mann-type hybrid steepest-descent method for three nonlinear problems. J Inequal Appl 2015, 282 (2015). https://doi.org/10.1186/s13660-015-0807-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0807-0