Onjai-uea and Phuengrattana Journal of Inequalities and Applications (2017) 2017:137 DOI 10.1186/s13660-017-1416-x
RESEARCH
Open Access
On solving split mixed equilibrium problems and fixed point problems of hybrid-type multivalued mappings in Hilbert spaces Nawitcha Onjai-uea1 and Withun Phuengrattana1,2* *
Correspondence: [email protected] Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand 2 Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand 1
Abstract In this paper, we introduce and study iterative algorithms for solving split mixed equilibrium problems and fixed point problems of λ-hybrid multivalued mappings in real Hilbert spaces and prove that the proposed iterative algorithm converges weakly to a common solution of the considered problems. We also provide an example to illustrate the convergence behavior of the proposed iteration process. MSC: 47N10; 47J20; 47J22; 46N10 Keywords: split mixed equilibrium problems; hybrid multivalued mappings; weak convergence; Hilbert spaces
1 Introduction Let H be a real Hilbert space with inner product ·, · and induced norm · . Let C be a nonempty closed convex subset of H, ϕ : C → R be a function, and F : C × C → R be a bifunction. The mixed equilibrium problem is to find x ∈ C such that F(x, y) + ϕ(y) – ϕ(x) ≥ ,
∀y ∈ C.
(.)
The solution set of mixed equilibrium problem is denoted by MEP(F, ϕ). In particular, if ϕ = , this problem reduces to the equilibrium problem, which is to find x ∈ C such that F(x, y) ≥ , ∀y ∈ C. The solution set of equilibrium problem is denoted by EP(F). The mixed equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, minimization problems, fixed point problems, Nash equilibrium problems in noncooperative games, and others; see, e.g., [–]. In , Censor and Elfving [] firstly introduced the following split feasibility problem in finite-dimensional Hilbert spaces: Let H , H be two Hilbert spaces and C, Q be nonempty closed convex subsets of H and H , respectively, and let A : H → H be a bounded linear operator. The split feasibility problem is formulated as finding a point x∗ © The Author(s) 2017. 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.
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with the property x∗ ∈ C
and
Ax∗ ∈ Q.
The split feasibility problem can extensively be applied in fields such as intensitymodulated radiation therapy, signal processing and image reconstruction, then the split feasibility problem has received so much attention by so many scholars; see [–]. In , Kazmi and Rizvi [] introduced and studied the following split equilibrium problem: let C ⊆ H and Q ⊆ H . Let F : C × C → R and F : Q × Q → R be nonlinear bifunctions and let A : H → H be a bounded linear operator. The split equilibrium problem is to find x∗ ∈ C such that F x∗ , x ≥ ,
∀x ∈ C and such that y∗ = Ax∗ ∈ Q solves F y∗ , y ≥ , ∀y ∈ Q. (.)
The solution set of the split equilibrium problem is denoted by SEP(F , F ) := x∗ ∈ C : x∗ ∈ EP(F ) and Ax∗ ∈ EP(F ) . The authors gave an iterative algorithm to find the common element of sets of solution of the split equilibrium problem and hierarchical fixed point problem; for more details refer to [, ]. In , Suantai et al. [] proposed the iterative algorithm to solve the problems for finding a common elements the set of solution of the split equilibrium problem and the fixed point of a nonspreading multivalued mapping in Hilbert space, given sequence {xn } by ⎧ ⎪ ⎪ ⎨x ∈ C arbitrarily, un = TrFn (I – γ A∗ (I – TrFn )A)xn , ⎪ ⎪ ⎩ xn+ ∈ αn xn + ( – αn )Sun , ∀n ∈ N,
(.)
where {αn } ⊂ (, ), rn ⊂ (, ∞) and γ ∈ (, L ) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A, C ⊂ H , Q ⊂ H , S : C → K(C) is a -nonspreading multivalued mapping, F : C × C → R and F : Q × Q → R are two bifunctions. The authors showed that under certain conditions, the sequence {xn } converges weakly to an element of F(S) ∩ SEP(F , F ). Several iterative algorithms have been developed for solving split feasibility problems and related split equilibrium problems; see, e.g., [–]. Motivated and inspired by the above results and related literature, we propose an iterative algorithm for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in real Hilbert spaces. Then we prove some weak convergence theorems which extend and improve the corresponding results of Kazmi and Rizvi [] and Suantai et al. [] and many others. We finally provide numerical examples for supporting our main result.
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2 Preliminaries Let C be a nonempty closed convex subset of a real Hilbert space H. We denote the strong convergence and the weak convergence of the sequence {xn } to a point x ∈ H by xn −→ x and xn x, respectively. It is also well known [] that a Hilbert space H satisfies Opial’s condition, that is, for any sequence {xn } with xn x, the inequality lim sup xn – x < lim sup xn – y n−→∞
n−→∞
holds for every y ∈ H with y = x. The following two lemmas are useful for our main results. Lemma . In a real Hilbert space H, the following inequalities hold: () x – y ≤ x – y – x – y, y, ∀x, y ∈ H; () x + y ≤ x + y, x + y, ∀x, y ∈ H; () tx + ( – t)y = tx + ( – t)y – t( – t)x – y , ∀t ∈ [, ], ∀x, y ∈ H; () If {xn } is a sequence in H which converges weakly to z ∈ H, then lim sup xn – y = lim sup xn – z + z – y , n→∞
n→∞
∀y ∈ H.
Lemma . ([]) Let H be a Hilbert space and {xn } be a sequence in H. Let u, v ∈ H be such that limn→∞ xn – u and limn→∞ xn – v exist. If {xnk } and {xmk } are subsequences of {xn } which converge weakly to u and v, respectively, then u = v. A single-valued mapping T : C −→ H is called δ-inverse strongly monotone [] if there exists a positive real number δ such that x – y, Tx – Ty ≥ δTx – Ty ,
∀x, y ∈ C.
For each γ ∈ (, δ], we see that I – γ T is a nonexpansive single-valued mapping, that is,
(I – γ T)x – (I – γ T)y ≤ x – y,
∀x, y ∈ C.
We denote by CB(C) and K(C) the collection of all nonempty closed bounded subsets and nonempty compact subsets of C, respectively. The Hausdorff metric H on CB(C) is defined by H(A, B) := max sup dist(x, B), sup dist(y, A) , x∈A
∀A, B ∈ CB(C),
y∈B
where dist(x, B) = inf{d(x, y) : y ∈ B} is the distance from a point x to a subset B. Let S : C → CB(C) be a multivalued mapping. An element x ∈ C is called a fixed point of S if x ∈ Sx. The set of all fixed points of S is denoted by F(S), that is, F(S) = {x ∈ C : x ∈ Sx}. Recall that a multivalued mapping S : C → CB(C) is called (i) nonexpansive if
H(Sx, Sy) ≤ x – y,
x, y ∈ C;
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(ii) quasi-nonexpansive if F(S) = ∅ and
H(Sx, Sp) ≤ x – p,
∀x ∈ C, ∀p ∈ F(S);
(iii) nonspreading [] if H(Sx, Sy) ≤ dist(y, Sx) + dist(x, Sy) ,
∀x, y ∈ C;
(iv) λ-hybrid [] if there exists λ ∈ R such that ( + λ)H(Sx, Sp) ≤ ( – λ)x – y + λ dist(y, Sx) + λ dist(x, Sy) ,
∀x, y ∈ C.
We note that -hybrid is nonexpansive, -hybrid is nonspreading, and if S is λ-hybrid with F(S) = ∅, then S is quasi-nonexpansive. It is well known [] that if S is λ-hybrid, then F(S) is closed. In addition, if S satisfies the condition: Sp = {p} for all p ∈ F(S), then F(S) is also convex. The following result is a demiclosedness principle for λ-hybrid multivalued mapping in a real Hilbert space. Lemma . ([]) Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → K(C) be a λ-hybrid multivalued mapping. If {xn } is a sequence in C such that xn x and yn ∈ Sxn with xn – yn → , then x ∈ Sx. For solving the mixed equilibrium problem, we assume that the bifunction F : C × C → R satisfies the following assumption: Assumption . Let C be a nonempty closed and convex subset of a Hilbert space H . Let F : C × C → R be the bifunction, ϕ : C → R ∪ {+∞} is convex and lower semicontinuous satisfies the following conditions: (A) F (x, x) = for all x ∈ C; (A) F is monotone, i.e., F (x, y) + F (y, x) ≤ , ∀x, y ∈ C; (A) for each x, y, z ∈ C, limt↓ F (tz + ( – t)x, y) ≤ F (x, y); (A) for each x ∈ C, y → F (x, y) is convex and lower semicontinuous; (B) for each x ∈ H and fixed r > , there exist a bounded subset Dx ⊆ C and yx ∈ C such that, for any z ∈ C \ Dx , F (z, yx ) + ϕ(yx ) – ϕ(z) + yx – z, z – x < ; r (B) C is a bounded set. Lemma . ([]) Let C be a nonempty closed and convex subset of a Hilbert space H . Let F : C × C → R be a bifunction satisfies Assumption . and let ϕ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function such that C ∩ dom ϕ = ∅. For r > and x ∈ H . Define a mapping TrF : H → C as follows:
TrF (x) = z ∈ C : F (z, y) + ϕ(y) – ϕ(z) + y – z, z – x ≥ , ∀y ∈ C , r
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for all x ∈ H . Assume that either (B) or (B) holds. Then the following conclusions hold: () for each x ∈ H , TrF = ∅; () TrF is single-valued; () TrF is firmly nonexpansive, i.e., for any x, y ∈ H ,
F
T x – T F y ≤ T F x – T F y, x – y ; r r r r () F(TrF ) = MEP(F , ϕ); () MEP(F , ϕ) is closed and convex. Further, assume that F : Q × Q → R satisfying Assumption . and φ : Q → R ∪ {+∞} is a proper lower semicontinuous and convex function such that Q ∩ dom φ = ∅, where Q is a nonempty closed and convex subset of a Hilbert space H . For each s > and w ∈ H , define a mapping TsF : H → Q as follows: TsF (v) =
w ∈ Q : F (w, d) + φ(d) – φ(w) + d – w, w – v ≥ , ∀d ∈ Q . r
Then we have the following: () for each v ∈ H , TsF = ∅; () TsF is single-valued; () TsF is firmly nonexpansive; () F(TsF ) = MEP(F , φ); () MEP(F , φ) is closed and convex.
3 Main results In this section, we prove the weak convergence theorems for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in real Hilbert spaces and give a numerical example to support our main result. We introduce the definition of split mixed equilibrium problems in real Hilbert spaces as follows. Definition . Let C be a nonempty closed convex subset of a real Hilbert space H and Q be a nonempty closed convex subset of a real Hilbert space H . Let F : C × C → R and F : Q × Q → R be nonlinear bifunctions, let ϕ : C → R ∪ {+∞} and φ : Q → R ∪ {+∞} be proper lower semicontinuous and convex functions such that C ∩ dom ϕ = ∅ and Q ∩ dom φ = ∅, and let A : H → H be a bounded linear operator. The split mixed equilibrium problem is to find x∗ ∈ C such that F x∗ , x + ϕ(x) – ϕ x∗ ≥ ,
∀x ∈ C,
(.)
and such that y∗ = Ax∗ ∈ Q solves F y∗ , y + φ(y) – φ y∗ ≥ ,
∀y ∈ Q.
The solution set of the split mixed equilibrium problem (.) and (.) is denoted by SMEP(F , ϕ, F , φ) := x∗ ∈ C : x∗ ∈ MEP(F , ϕ) and Ax∗ ∈ MEP(F , φ) .
(.)
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We now get our main result. Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H and Q be a nonempty closed convex subset of a real Hilbert space H . Let A : H → H be a bounded linear operator and S : C → K(C) a λ-hybrid multivalued mapping. Let F : C × C → R, F : Q × Q → R be bifunctions satisfying Assumption ., let ϕ : C → R ∪ {+∞} and φ : Q → R ∪ {+∞} be a proper lower semicontinuous and convex functions such that C ∩ dom ϕ = ∅ and Q ∩ dom φ = ∅, respectively, and F is upper semicontinuous in the first argument. Assume that = F(S) ∩ SMEP(F , ϕ, F , φ) = ∅ and Sp = {p} for all p ∈ F(S). Let {xn } be a sequence generated by x ∈ C and ⎧ F F ∗ ⎪ ⎪ ⎨un = Trn (I – γ A (I – Trn )A)xn , ⎪ ⎪ ⎩
yn = αn xn + ( – αn )wn , xn+ = βn wn + ( – βn )zn ,
wn ∈ Sun ,
(.)
zn ∈ Syn , ∀n ∈ N,
where {αn } ⊂ (, ), {βn } ⊂ (, ), {rn } ⊂ (, ∞), and γ ∈ (, L ) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A. Assume that the following conditions hold: (C) < lim infn→∞ βn ≤ lim supn→∞ βn < ; (C) < lim infn→∞ αn ≤ lim supn→∞ αn < ; (C) < lim infn→∞ rn . Then the sequence {xn } generated by (.) converges weakly to p ∈ . Proof First, we show that A∗ (I – TrFn )A is a L -inverse strongly monotone mapping. Since TrFn is firmly nonexpansive and I – TrFn is -inverse strongly monotone, we see that
∗
A I – T F Ax – A∗ I – T F Ay = A∗ I – T F (Ax – Ay), A∗ I – T F (Ax – Ay) rn rn rn rn = I – TrFn (Ax – Ay), AA∗ I – TrFn (Ax – Ay) ≤ L I – TrFn (Ax – Ay), I – TrFn (Ax – Ay)
= L I – T F (Ax – Ay)
rn
≤ L Ax – Ay, I – TrFn (Ax – Ay) = L x – y, A∗ I – TrFn Ax – A∗ I – TrFn Ay for all x, y ∈ H . This implies that A∗ (I – TrFn )A is a L -inverse strongly monotone mapping. Since γ ∈ (, L ), it follows that I – γ A∗ (I – TrFn )A is a nonexpansive mapping. Now, we divide the proof into five steps as follows: Step . Show that {xn } is bounded. Let q ∈ . Then we have q = TrFn q and q = (I – γ A∗ (I – TrFn )A)q. By nonexpansiveness of I – γ A∗ (I – TrFn )A, it implies that
un – q = TrFn I – γ A∗ I – TrFn A xn – TrFn I – γ A∗ I – TrFn A q
≤ I – γ A∗ I – TrFn A xn – I – γ A∗ I – TrFn A q
≤ xn – q.
(.)
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This implies that wn – q = dist(wn , Sq) ≤ H(Sun , Sq) ≤ un – q ≤ xn – q,
(.)
and so
yn – q = αn xn + ( – αn )wn – q
≤ αn xn – q + ( – αn )wn – q = xn – q.
(.)
It follows that zn – q = dist(zn , Sq) ≤ H(Syn , Sq) ≤ yn – q ≤ xn – q.
(.)
By (.) and (.), we have
xn+ – q = βn wn + ( – βn )zn – q
≤ βn wn – q + ( – βn )zn – q = xn – q.
(.)
This implies that {xn – q} is decreasing and bounded below, thus limn−→∞ xn – q exists for all q ∈ . Step . Show that limn→∞ wn – zn = . From Lemma .(), (.), (.), and Sq = {q}, we have
xn+ – q = βn wn + ( – βn )zn – q
≤ βn wn – q + ( – βn )zn – q – βn ( – βn )wn – zn ≤ xn – q – βn ( – βn )wn – zn .
(.)
This implies that βn ( – βn )wn – zn ≤ xn – q – xn+ – q . From Condition (C) and limn→∞ xn – q exists, we have lim wn – zn = .
n→∞
Step . Show that limn→∞ un – xn = and limn→∞ wn – un = . For q ∈ , we see that
un – q = TrFn I – γ A∗ I – TrFn A xn – TrFn q
≤ I – γ A∗ I – TrFn A xn – q
≤ xn – q + γ A∗ I – TrFn Axn + γ q – xn , A∗ I – TrFn Axn
(.)
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≤ xn – q + γ Axn – TrFn Axn , AA∗ I – TrFn Axn + γ A(q – xn ), Axn – TrFn Axn ≤ xn – q + Lγ Axn – TrFn Axn , Axn – TrFn Axn + γ A(q – xn ) + Axn – TrFn Axn – Axn – TrFn Axn , Axn – TrFn Axn
≤ xn – q + Lγ Axn – T F Axn
rn
+ γ Ap – TrFn Axn , Axn – TrFn Axn – Axn – TrFn Axn
≤ xn – q + Lγ Axn – TrFn Axn
Axn – T F Axn – Axn – T F Axn + γ rn rn
= xn – q + γ (Lγ – ) Axn – TrFn Axn . Thus, by (.) and (.), we have xn+ – q ≤ βn wn – q + ( – βn )zn – q ≤ βn un – q + ( – βn )xn – q
≤ βn xn – q + γ (Lγ – ) Axn – TrFn Axn + ( – βn )xn – q
≤ xn – q + γ (Lγ – )βn Axn – TrFn Axn .
(.)
Therefore, we have
–γ (Lγ – )βn Axn – TrFn Axn ≤ xn – q – xn+ – q . Since γ (Lγ – ) < , it follows by Condition (C) and the existence of limn→∞ xn – q that
lim Axn – TrFn Axn = .
(.)
n→∞
Since TrFn is firmly nonexpansive and I – γ A∗ (I – TrFn )A is nonexpansive, we have
un – q = TrFn I – γ A∗ I – TrFn A xn – TrFn q
≤ TrFn I – γ A∗ I – TrFn A xn – TrFn q, I – γ A∗ I – TrFn A xn – q = un – q, I – γ A∗ I – TrFn A xn – q =
un – q + I – γ A∗ I – TrFn A xn – q
– un – xn – γ A∗ I – T F Axn
rn
≤
un – q + xn – q – un – xn + γ A∗ I – TrFn Axn
– γ un – xn , A∗ I – TrFn Axn ,
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which implies that un – q ≤ xn – q – un – xn + γ un – xn , A∗ I – TrFn Axn
≤ xn – q – un – xn + γ un – xn A∗ I – TrFn Axn .
(.)
This implies by (.) and (.) that xn+ – q ≤ βn wn – q + ( – βn )zn – q ≤ βn un – q + ( – βn )xn – q
≤ βn xn – q – un – xn + γ un – xn A∗ I – TrFn Axn
+ ( – βn )xn – q . Therefore, we have
βn un – xn ≤ xn – q – xn+ – q + γβn un – xn A∗ I – TrFn Axn
≤ xn – q – xn+ – q + γβn M A∗ I – TrFn Axn , where M = sup{un – xn : n ∈ N}. This implies by Condition (C), (.), and the existence of limn→∞ xn – q that lim un – xn = .
n→∞
(.)
From (.), (.), and the definition of {yn }, we obtain xn+ – q ≤ βn wn – q + ( – βn )zn – q ≤ βn xn – q + ( – βn )yn – q = βn xn – q + ( – βn ) αn xn – q + ( – αn )wn – q – αn ( – αn )xn – wn ≤ βn xn – q + ( – βn ) xn – q – αn ( – αn )xn – wn = xn – q – αn ( – αn )( – βn )xn – wn . This implies that αn ( – αn )( – βn )xn – wn ≤ xn – q – xn+ – q . From Conditions (C), (C), and the existence of limn→∞ xn – q, we have lim wn – xn = .
n→∞
(.)
By (.) and (.), we get wn – un ≤ wn – xn + xn – un → as n → ∞.
(.)
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Step . Show that ωw (xn ) ⊂ , where ωw (xn ) = {x ∈ H : xni x, {xni } ⊂ {xn }}. Since {xn } is bounded and H is reflexive, ωw (xn ) is nonempty. Let p ∈ ωw (xn ) be an arbitrary element. Then there exists a subsequence {xni } ⊂ {xn } converging weakly to p. From (.), it implies that uni p as i → ∞. By (.) and Lemma ., we have p ∈ F(S). Next, we show that p ∈ MEP(F , ϕ). Since un = TrFn (I – γ A∗ (I – TrFn )A)xn , we have F (un , y) + ϕ(y) – ϕ(un ) +
y – un , un – xn – γ A∗ I – TrFn Axn ≥ , rn
∀y ∈ C,
which implies that F (un , y) + ϕ(y) – ϕ(un ) +
y – un , un – xn – y – un , γ A∗ I – TrFn Axn ≥ , rn rn
∀y ∈ C.
From Assumption .(A), we have ϕ(y) – ϕ(un ) +
y – un , γ A∗ I – TrFn Axn y – un , un – xn – rn rn
≥ –F (un , y) ≥ F (y, un ),
∀y ∈ C,
and hence ϕ(y) – ϕ(uni ) +
y – uni , uni – xni – y – uni , γ A∗ I – TrFn Axni ≥ F (y, uni ), i rni rni
∀y ∈ C. This implies by uni p, Condition (C), (.), (.), Assumption .(A), and the proper lower semicontinuity of ϕ that F (y, p) + ϕ(p) – ϕ(y) ≤ ,
∀y ∈ C.
Put yt = ty + ( – t)p for all t ∈ (, ] and y ∈ C. Consequently, we get yt ∈ C and hence F (yt , p) + ϕ(p) – ϕ(yt ) ≤ . So, by Assumption .(A)-(A), we have = F (yt , yt ) – ϕ(yt ) + ϕ(yt ) ≤ tF (yt , y) + ( – t)F (yt , p) + tϕ(y) + ( – t)ϕ(p) – ϕ(yt ) ≤ t F (yt , y) + ϕ(y) – ϕ(yt ) . Hence, we have F (yt , y) + ϕ(y) – ϕ(yt ) ≥ ,
∀y ∈ C.
Letting t → , by Assumption .(A) and the proper lower semicontinuity of ϕ, we have F (p, y) + ϕ(y) – ϕ(p) ≥ ,
∀y ∈ C.
This implies that p ∈ MEP(F , ϕ).
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Since A is a bounded linear operator, we have Axni Ap. Then it follows from (.) that TrFn Axni Ap i
as i → ∞.
(.)
By the definition of TrFni Axni , we have y – TrFn Axni , TrFn Axni – Axni ≥ , F TrFn Axni , y + φ(y) – φ TrFn Axni + i i i i rni
∀y ∈ Q.
Since F is upper semicontinuous in the first argument, it implies by (.) that F (Ap, y) + φ(y) – φ(Ap) ≥ ,
∀y ∈ Q.
This shows that Ap ∈ MEP(F , φ). Therefore, p ∈ SMEP(F , ϕ, F , φ) and hence p ∈ . Step . Show that {xn } converges weakly to an element of . It is sufficient to show that ωw (xn ) is a singleton set. Let p, q ∈ ωw (xn ) and {xnk }, {xnm } be two subsequences of {xn } such that xnk p and xnm q. From (.), we also have unk p and unm q. By (.) and Lemma ., we see that p, q ∈ F(S). Applying Lemma ., we obtain p = q. This completes the proof. If ϕ = φ = in (.) and (.), then the split mixed equilibrium problem reduces to split equilibrium problem. So, the following result can be obtained from Theorem . immediately. Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H and Q be a nonempty closed convex subset of a real Hilbert space H . Let A : H → H be a bounded linear operator and S : C → K(C) a λ-hybrid multivalued mapping. Let F : C × C → R, F : Q × Q → R be bifunctions satisfying Assumption ., and F is upper semicontinuous in the first argument. Assume that = F(S) ∩ SEP(F , F ) = ∅ and Sp = {p} for all p ∈ F(S). Let {xn } be a sequence generated by x ∈ C and ⎧ F F ∗ ⎪ ⎪ ⎨un = Trn (I – γ A (I – Trn )A)xn , ⎪ ⎪ ⎩
yn = αn xn + ( – αn )wn , xn+ = βn wn + ( – βn )zn ,
wn ∈ Sun ,
(.)
zn ∈ Syn , ∀n ∈ N,
where {αn } ⊂ (, ), {βn } ⊂ (, ), {rn } ⊂ (, ∞), and γ ∈ (, L ) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A. Assume that the following conditions hold: (C) < lim infn→∞ βn ≤ lim supn→∞ βn < ; (C) < lim infn→∞ αn ≤ lim supn→∞ αn < ; (C) < lim infn→∞ rn . Then the sequence {xn } generated by (.) converges weakly to p ∈ . Remark . (i) Theorems . and . extend the corresponding one of Suantai et al. [] and Kazmi and Rizvi [] to λ-hybrid multivalued mapping and to a split mixed equilibrium
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problem. In fact, we present a new iterative algorithm for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in a real Hilbert space. (ii) It is well known that the class of λ-hybrid multivalued mappings contains the classes of nonexpansive multivalued mappings, nonspreading multivalued mappings. Thus, Theorems . and . can be applied to these classes of mappings. We give an example to illustrate Theorem . as follows. Example . Let H = R, H = R, C = [–, ], and Q = (–∞, ]. Let A : H −→ H defined by Ax = x for each x ∈ H . Then A∗ y = y for each y ∈ H . So, L = is the spectral radius of A∗ A. Define a multivalued mapping S : C −→ K(C) by
Sx =
⎧ ⎨[–
|x| , ], |x|+
⎩{},
x ∈ [–, –); x ∈ [–, ].
It easy to see that S is -hybrid multivalued mapping with F(S) = {} and S() = {}. For each x, y ∈ C, define the bifunction F : C × C −→ R by F (x, y) = xy + y – x – x and define ϕ(x) = for each x ∈ C. For each u, v ∈ Q, define the bifunction F : Q × Q −→ R by F (u, v) = uv + v – u – u and define φ(u) = for each u ∈ Q. n n n , βn = n+ , rn = n+ , and γ = . It is easy to check that F , F , {αn }, {βn }, Choose αn = n+ {rn } satisfy all conditions in Theorem .. For each x ∈ C, we compute TrF Ax. Find z such that ≤ F (z, y) + ϕ(y) – ϕ(z) + y – z, z – Ax r x = zy + y – z – z + y – z, z – r x = (z + )(y – z) + (y – z) z – r x z– = (y – z) (z + ) + r for all y ∈ Q. Thus, by Lemma .(), it follows that z = each x ∈ C. Furthermore, we get
I – γ A∗ I – TrF A x = x – A∗ Ax – TrF Ax x x – r = x – A∗ – ( + r) x x – r =x– – ( + r) γ (x – r) γ – . =x – ( + r)
x–r . (+r)
That is, TrF Ax =
x–r (+r)
for
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Table 1 Numerical results of Example 3.5 for the algorithm (3.19) n
xn
xn – xn–1
1 2 3 4
–3.0000000e+00 –6.8786127e–02 0.0000000e+00 0.0000000e+00
2.9312139e+00 6.8786127e–02 0.0000000e+00
Next, we find u ∈ C such that F (u, v) + ϕ(y) – ϕ(z) + r v – u, u – s ≥ for all v ∈ C, where s = (I – γ A∗ (I – TrF )A)x. Note that ≤ F (u, v) + ϕ(y) – ϕ(z) + v – u, u – s = uv + v – u – u + v – u, u – s r r = (u + )(v – u) + (v – u)(u – s) r = (v – u) (u + ) + (u – s) . r Thus, by Lemma .(), it follows that u=
x – r s – r x – r . = – + r ( + r) ( + r)
Then the algorithm (.) becomes ⎧ ⎪ ⎪ ⎨un =
xn –rn xn –rn – (+r rn , (+rn ) n) n n yn = n+ xn + ( – n+ )wn , ⎪ ⎪ ⎩ n n xn+ = n+ wn + ( – n+ )zn ,
=
n , n+
(.) ∀n ∈ N,
where
wn =
⎧ ⎨[–
|un | , ], |un |+
⎩{},
un ∈ [–, –); un ∈ [–, ],
zn =
⎧ ⎨[–
|yn | , ], |yn |+
⎩{},
yn ∈ [–, –); yn ∈ [–, ].
| | if un ∈ [–, –) and zn = – |y|ynn|+ if yn ∈ [–, –). By using SciLab, We choose wn = – |u|unn|+ we compute the iterates of (.) for the initial point x = –. The numerical experiment’s results of our iteration for approximating the point are given in Table .
4 Conclusions The results presented in this paper extend and generalize the work of Suantai et al. [] and Kazmi and Rizvi []. The main aim of this paper is to propose an iterative algorithm to find an element for solving a class of split mixed equilibrium problems and fixed point problems for λ-hybrid multivalued mappings under weaker conditions. Some sufficient conditions for the weak convergence of such proposed algorithm are given. Also, in order to show the significance of the considered problem, some important applications are discussed.
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Acknowledgements The authors are thankful to the referees for careful reading and the useful comments and suggestions. Funding This work was supported by the Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University, Nakhon Pathom, Thailand. The second author was also supported by the Thailand Research Fund under the project RTA5780007. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 25 March 2017 Accepted: 5 June 2017 References 1. Ceng, LC, Yao, JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186-201 (2008) 2. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994) 3. Combettes, PI, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117-136 (2005) 4. Flam, SD, Antipin, AS: Equilibrium programming using proximal-like algorithm. Math. Program. 78, 29-41 (1997) 5. Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221-239 (1994) 6. Censor, Y, Bortfeld, T, Martin, B, Trofimov, A: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353-2365 (2006) 7. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071-2084 (2005) 8. Censor, Y, Motova, A, Segal, A: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244-1256 (2007) 9. Chan, T, Shen, J: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005) 10. Kazmi, KR, Rizvi, SH: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egypt. Math. Soc. 21, 44-51 (2013) 11. Bnouhachem, A: Algorithms of common solutions for a variational inequality, a split equilibrium problem and a hierarchical fixed point problem. Fixed Point Theory Appl. 2013, 278 (2013) 12. Bnouhachem, A: Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems. Sci. World J. 2014, 390956 (2014) 13. Suantai, S, Cholamjiak, P, Cho, YJ, Cholamjiak, W: On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces. Fixed Point Theory Appl. 2016, 35 (2016) 14. Deepho, J, Kumam, W, Kumam, P: A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems. J. Math. Model. Algorithms Oper. Res. 13(4), 405-423 (2014) 15. Kumam, W, Deepho, J, Kumam, P: Hybrid extragradient method for finding a common solution of the split feasibility and system of equilibrium problems. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 21(6), 367-388 (2014) 16. Deepho, J, Martinez-Moreno, J, Kumam, P: A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems. J. Nonlinear Sci. Appl. 9, 1475-1496 (2016) 17. Opial, Z: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591-597 (1967) 18. Suantai, S: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 311, 506-517 (2005) 19. Iiduka, H, Takahashi, W: Weak convergence theorem by Cesaro means for nonexpansive mappings and inverse-strongly monotone mappings. J. Nonlinear Convex Anal. 7, 105-113 (2006) 20. Suantai, S, Phuengrattana, W: Existence and convergence theorems for λ-hybrid mappings in Hilbert spaces. Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal. 22, 177-188 (2015) 21. Ma, Z, Wang, L, Chang, SS, Duan, W: Convergence theorems for split equality mixed equilibrium problems with applications. Fixed Point Theory Appl. 2015, 31 (2015)
RESEARCH
Open Access
On solving split mixed equilibrium problems and fixed point problems of hybrid-type multivalued mappings in Hilbert spaces Nawitcha Onjai-uea1 and Withun Phuengrattana1,2* *
Correspondence: [email protected] Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand 2 Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand 1
Abstract In this paper, we introduce and study iterative algorithms for solving split mixed equilibrium problems and fixed point problems of λ-hybrid multivalued mappings in real Hilbert spaces and prove that the proposed iterative algorithm converges weakly to a common solution of the considered problems. We also provide an example to illustrate the convergence behavior of the proposed iteration process. MSC: 47N10; 47J20; 47J22; 46N10 Keywords: split mixed equilibrium problems; hybrid multivalued mappings; weak convergence; Hilbert spaces
1 Introduction Let H be a real Hilbert space with inner product ·, · and induced norm · . Let C be a nonempty closed convex subset of H, ϕ : C → R be a function, and F : C × C → R be a bifunction. The mixed equilibrium problem is to find x ∈ C such that F(x, y) + ϕ(y) – ϕ(x) ≥ ,
∀y ∈ C.
(.)
The solution set of mixed equilibrium problem is denoted by MEP(F, ϕ). In particular, if ϕ = , this problem reduces to the equilibrium problem, which is to find x ∈ C such that F(x, y) ≥ , ∀y ∈ C. The solution set of equilibrium problem is denoted by EP(F). The mixed equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, minimization problems, fixed point problems, Nash equilibrium problems in noncooperative games, and others; see, e.g., [–]. In , Censor and Elfving [] firstly introduced the following split feasibility problem in finite-dimensional Hilbert spaces: Let H , H be two Hilbert spaces and C, Q be nonempty closed convex subsets of H and H , respectively, and let A : H → H be a bounded linear operator. The split feasibility problem is formulated as finding a point x∗ © The Author(s) 2017. 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.
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with the property x∗ ∈ C
and
Ax∗ ∈ Q.
The split feasibility problem can extensively be applied in fields such as intensitymodulated radiation therapy, signal processing and image reconstruction, then the split feasibility problem has received so much attention by so many scholars; see [–]. In , Kazmi and Rizvi [] introduced and studied the following split equilibrium problem: let C ⊆ H and Q ⊆ H . Let F : C × C → R and F : Q × Q → R be nonlinear bifunctions and let A : H → H be a bounded linear operator. The split equilibrium problem is to find x∗ ∈ C such that F x∗ , x ≥ ,
∀x ∈ C and such that y∗ = Ax∗ ∈ Q solves F y∗ , y ≥ , ∀y ∈ Q. (.)
The solution set of the split equilibrium problem is denoted by SEP(F , F ) := x∗ ∈ C : x∗ ∈ EP(F ) and Ax∗ ∈ EP(F ) . The authors gave an iterative algorithm to find the common element of sets of solution of the split equilibrium problem and hierarchical fixed point problem; for more details refer to [, ]. In , Suantai et al. [] proposed the iterative algorithm to solve the problems for finding a common elements the set of solution of the split equilibrium problem and the fixed point of a nonspreading multivalued mapping in Hilbert space, given sequence {xn } by ⎧ ⎪ ⎪ ⎨x ∈ C arbitrarily, un = TrFn (I – γ A∗ (I – TrFn )A)xn , ⎪ ⎪ ⎩ xn+ ∈ αn xn + ( – αn )Sun , ∀n ∈ N,
(.)
where {αn } ⊂ (, ), rn ⊂ (, ∞) and γ ∈ (, L ) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A, C ⊂ H , Q ⊂ H , S : C → K(C) is a -nonspreading multivalued mapping, F : C × C → R and F : Q × Q → R are two bifunctions. The authors showed that under certain conditions, the sequence {xn } converges weakly to an element of F(S) ∩ SEP(F , F ). Several iterative algorithms have been developed for solving split feasibility problems and related split equilibrium problems; see, e.g., [–]. Motivated and inspired by the above results and related literature, we propose an iterative algorithm for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in real Hilbert spaces. Then we prove some weak convergence theorems which extend and improve the corresponding results of Kazmi and Rizvi [] and Suantai et al. [] and many others. We finally provide numerical examples for supporting our main result.
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2 Preliminaries Let C be a nonempty closed convex subset of a real Hilbert space H. We denote the strong convergence and the weak convergence of the sequence {xn } to a point x ∈ H by xn −→ x and xn x, respectively. It is also well known [] that a Hilbert space H satisfies Opial’s condition, that is, for any sequence {xn } with xn x, the inequality lim sup xn – x < lim sup xn – y n−→∞
n−→∞
holds for every y ∈ H with y = x. The following two lemmas are useful for our main results. Lemma . In a real Hilbert space H, the following inequalities hold: () x – y ≤ x – y – x – y, y, ∀x, y ∈ H; () x + y ≤ x + y, x + y, ∀x, y ∈ H; () tx + ( – t)y = tx + ( – t)y – t( – t)x – y , ∀t ∈ [, ], ∀x, y ∈ H; () If {xn } is a sequence in H which converges weakly to z ∈ H, then lim sup xn – y = lim sup xn – z + z – y , n→∞
n→∞
∀y ∈ H.
Lemma . ([]) Let H be a Hilbert space and {xn } be a sequence in H. Let u, v ∈ H be such that limn→∞ xn – u and limn→∞ xn – v exist. If {xnk } and {xmk } are subsequences of {xn } which converge weakly to u and v, respectively, then u = v. A single-valued mapping T : C −→ H is called δ-inverse strongly monotone [] if there exists a positive real number δ such that x – y, Tx – Ty ≥ δTx – Ty ,
∀x, y ∈ C.
For each γ ∈ (, δ], we see that I – γ T is a nonexpansive single-valued mapping, that is,
(I – γ T)x – (I – γ T)y ≤ x – y,
∀x, y ∈ C.
We denote by CB(C) and K(C) the collection of all nonempty closed bounded subsets and nonempty compact subsets of C, respectively. The Hausdorff metric H on CB(C) is defined by H(A, B) := max sup dist(x, B), sup dist(y, A) , x∈A
∀A, B ∈ CB(C),
y∈B
where dist(x, B) = inf{d(x, y) : y ∈ B} is the distance from a point x to a subset B. Let S : C → CB(C) be a multivalued mapping. An element x ∈ C is called a fixed point of S if x ∈ Sx. The set of all fixed points of S is denoted by F(S), that is, F(S) = {x ∈ C : x ∈ Sx}. Recall that a multivalued mapping S : C → CB(C) is called (i) nonexpansive if
H(Sx, Sy) ≤ x – y,
x, y ∈ C;
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(ii) quasi-nonexpansive if F(S) = ∅ and
H(Sx, Sp) ≤ x – p,
∀x ∈ C, ∀p ∈ F(S);
(iii) nonspreading [] if H(Sx, Sy) ≤ dist(y, Sx) + dist(x, Sy) ,
∀x, y ∈ C;
(iv) λ-hybrid [] if there exists λ ∈ R such that ( + λ)H(Sx, Sp) ≤ ( – λ)x – y + λ dist(y, Sx) + λ dist(x, Sy) ,
∀x, y ∈ C.
We note that -hybrid is nonexpansive, -hybrid is nonspreading, and if S is λ-hybrid with F(S) = ∅, then S is quasi-nonexpansive. It is well known [] that if S is λ-hybrid, then F(S) is closed. In addition, if S satisfies the condition: Sp = {p} for all p ∈ F(S), then F(S) is also convex. The following result is a demiclosedness principle for λ-hybrid multivalued mapping in a real Hilbert space. Lemma . ([]) Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → K(C) be a λ-hybrid multivalued mapping. If {xn } is a sequence in C such that xn x and yn ∈ Sxn with xn – yn → , then x ∈ Sx. For solving the mixed equilibrium problem, we assume that the bifunction F : C × C → R satisfies the following assumption: Assumption . Let C be a nonempty closed and convex subset of a Hilbert space H . Let F : C × C → R be the bifunction, ϕ : C → R ∪ {+∞} is convex and lower semicontinuous satisfies the following conditions: (A) F (x, x) = for all x ∈ C; (A) F is monotone, i.e., F (x, y) + F (y, x) ≤ , ∀x, y ∈ C; (A) for each x, y, z ∈ C, limt↓ F (tz + ( – t)x, y) ≤ F (x, y); (A) for each x ∈ C, y → F (x, y) is convex and lower semicontinuous; (B) for each x ∈ H and fixed r > , there exist a bounded subset Dx ⊆ C and yx ∈ C such that, for any z ∈ C \ Dx , F (z, yx ) + ϕ(yx ) – ϕ(z) + yx – z, z – x < ; r (B) C is a bounded set. Lemma . ([]) Let C be a nonempty closed and convex subset of a Hilbert space H . Let F : C × C → R be a bifunction satisfies Assumption . and let ϕ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function such that C ∩ dom ϕ = ∅. For r > and x ∈ H . Define a mapping TrF : H → C as follows:
TrF (x) = z ∈ C : F (z, y) + ϕ(y) – ϕ(z) + y – z, z – x ≥ , ∀y ∈ C , r
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for all x ∈ H . Assume that either (B) or (B) holds. Then the following conclusions hold: () for each x ∈ H , TrF = ∅; () TrF is single-valued; () TrF is firmly nonexpansive, i.e., for any x, y ∈ H ,
F
T x – T F y ≤ T F x – T F y, x – y ; r r r r () F(TrF ) = MEP(F , ϕ); () MEP(F , ϕ) is closed and convex. Further, assume that F : Q × Q → R satisfying Assumption . and φ : Q → R ∪ {+∞} is a proper lower semicontinuous and convex function such that Q ∩ dom φ = ∅, where Q is a nonempty closed and convex subset of a Hilbert space H . For each s > and w ∈ H , define a mapping TsF : H → Q as follows: TsF (v) =
w ∈ Q : F (w, d) + φ(d) – φ(w) + d – w, w – v ≥ , ∀d ∈ Q . r
Then we have the following: () for each v ∈ H , TsF = ∅; () TsF is single-valued; () TsF is firmly nonexpansive; () F(TsF ) = MEP(F , φ); () MEP(F , φ) is closed and convex.
3 Main results In this section, we prove the weak convergence theorems for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in real Hilbert spaces and give a numerical example to support our main result. We introduce the definition of split mixed equilibrium problems in real Hilbert spaces as follows. Definition . Let C be a nonempty closed convex subset of a real Hilbert space H and Q be a nonempty closed convex subset of a real Hilbert space H . Let F : C × C → R and F : Q × Q → R be nonlinear bifunctions, let ϕ : C → R ∪ {+∞} and φ : Q → R ∪ {+∞} be proper lower semicontinuous and convex functions such that C ∩ dom ϕ = ∅ and Q ∩ dom φ = ∅, and let A : H → H be a bounded linear operator. The split mixed equilibrium problem is to find x∗ ∈ C such that F x∗ , x + ϕ(x) – ϕ x∗ ≥ ,
∀x ∈ C,
(.)
and such that y∗ = Ax∗ ∈ Q solves F y∗ , y + φ(y) – φ y∗ ≥ ,
∀y ∈ Q.
The solution set of the split mixed equilibrium problem (.) and (.) is denoted by SMEP(F , ϕ, F , φ) := x∗ ∈ C : x∗ ∈ MEP(F , ϕ) and Ax∗ ∈ MEP(F , φ) .
(.)
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We now get our main result. Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H and Q be a nonempty closed convex subset of a real Hilbert space H . Let A : H → H be a bounded linear operator and S : C → K(C) a λ-hybrid multivalued mapping. Let F : C × C → R, F : Q × Q → R be bifunctions satisfying Assumption ., let ϕ : C → R ∪ {+∞} and φ : Q → R ∪ {+∞} be a proper lower semicontinuous and convex functions such that C ∩ dom ϕ = ∅ and Q ∩ dom φ = ∅, respectively, and F is upper semicontinuous in the first argument. Assume that = F(S) ∩ SMEP(F , ϕ, F , φ) = ∅ and Sp = {p} for all p ∈ F(S). Let {xn } be a sequence generated by x ∈ C and ⎧ F F ∗ ⎪ ⎪ ⎨un = Trn (I – γ A (I – Trn )A)xn , ⎪ ⎪ ⎩
yn = αn xn + ( – αn )wn , xn+ = βn wn + ( – βn )zn ,
wn ∈ Sun ,
(.)
zn ∈ Syn , ∀n ∈ N,
where {αn } ⊂ (, ), {βn } ⊂ (, ), {rn } ⊂ (, ∞), and γ ∈ (, L ) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A. Assume that the following conditions hold: (C) < lim infn→∞ βn ≤ lim supn→∞ βn < ; (C) < lim infn→∞ αn ≤ lim supn→∞ αn < ; (C) < lim infn→∞ rn . Then the sequence {xn } generated by (.) converges weakly to p ∈ . Proof First, we show that A∗ (I – TrFn )A is a L -inverse strongly monotone mapping. Since TrFn is firmly nonexpansive and I – TrFn is -inverse strongly monotone, we see that
∗
A I – T F Ax – A∗ I – T F Ay = A∗ I – T F (Ax – Ay), A∗ I – T F (Ax – Ay) rn rn rn rn = I – TrFn (Ax – Ay), AA∗ I – TrFn (Ax – Ay) ≤ L I – TrFn (Ax – Ay), I – TrFn (Ax – Ay)
= L I – T F (Ax – Ay)
rn
≤ L Ax – Ay, I – TrFn (Ax – Ay) = L x – y, A∗ I – TrFn Ax – A∗ I – TrFn Ay for all x, y ∈ H . This implies that A∗ (I – TrFn )A is a L -inverse strongly monotone mapping. Since γ ∈ (, L ), it follows that I – γ A∗ (I – TrFn )A is a nonexpansive mapping. Now, we divide the proof into five steps as follows: Step . Show that {xn } is bounded. Let q ∈ . Then we have q = TrFn q and q = (I – γ A∗ (I – TrFn )A)q. By nonexpansiveness of I – γ A∗ (I – TrFn )A, it implies that
un – q = TrFn I – γ A∗ I – TrFn A xn – TrFn I – γ A∗ I – TrFn A q
≤ I – γ A∗ I – TrFn A xn – I – γ A∗ I – TrFn A q
≤ xn – q.
(.)
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This implies that wn – q = dist(wn , Sq) ≤ H(Sun , Sq) ≤ un – q ≤ xn – q,
(.)
and so
yn – q = αn xn + ( – αn )wn – q
≤ αn xn – q + ( – αn )wn – q = xn – q.
(.)
It follows that zn – q = dist(zn , Sq) ≤ H(Syn , Sq) ≤ yn – q ≤ xn – q.
(.)
By (.) and (.), we have
xn+ – q = βn wn + ( – βn )zn – q
≤ βn wn – q + ( – βn )zn – q = xn – q.
(.)
This implies that {xn – q} is decreasing and bounded below, thus limn−→∞ xn – q exists for all q ∈ . Step . Show that limn→∞ wn – zn = . From Lemma .(), (.), (.), and Sq = {q}, we have
xn+ – q = βn wn + ( – βn )zn – q
≤ βn wn – q + ( – βn )zn – q – βn ( – βn )wn – zn ≤ xn – q – βn ( – βn )wn – zn .
(.)
This implies that βn ( – βn )wn – zn ≤ xn – q – xn+ – q . From Condition (C) and limn→∞ xn – q exists, we have lim wn – zn = .
n→∞
Step . Show that limn→∞ un – xn = and limn→∞ wn – un = . For q ∈ , we see that
un – q = TrFn I – γ A∗ I – TrFn A xn – TrFn q
≤ I – γ A∗ I – TrFn A xn – q
≤ xn – q + γ A∗ I – TrFn Axn + γ q – xn , A∗ I – TrFn Axn
(.)
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≤ xn – q + γ Axn – TrFn Axn , AA∗ I – TrFn Axn + γ A(q – xn ), Axn – TrFn Axn ≤ xn – q + Lγ Axn – TrFn Axn , Axn – TrFn Axn + γ A(q – xn ) + Axn – TrFn Axn – Axn – TrFn Axn , Axn – TrFn Axn
≤ xn – q + Lγ Axn – T F Axn
rn
+ γ Ap – TrFn Axn , Axn – TrFn Axn – Axn – TrFn Axn
≤ xn – q + Lγ Axn – TrFn Axn
Axn – T F Axn – Axn – T F Axn + γ rn rn
= xn – q + γ (Lγ – ) Axn – TrFn Axn . Thus, by (.) and (.), we have xn+ – q ≤ βn wn – q + ( – βn )zn – q ≤ βn un – q + ( – βn )xn – q
≤ βn xn – q + γ (Lγ – ) Axn – TrFn Axn + ( – βn )xn – q
≤ xn – q + γ (Lγ – )βn Axn – TrFn Axn .
(.)
Therefore, we have
–γ (Lγ – )βn Axn – TrFn Axn ≤ xn – q – xn+ – q . Since γ (Lγ – ) < , it follows by Condition (C) and the existence of limn→∞ xn – q that
lim Axn – TrFn Axn = .
(.)
n→∞
Since TrFn is firmly nonexpansive and I – γ A∗ (I – TrFn )A is nonexpansive, we have
un – q = TrFn I – γ A∗ I – TrFn A xn – TrFn q
≤ TrFn I – γ A∗ I – TrFn A xn – TrFn q, I – γ A∗ I – TrFn A xn – q = un – q, I – γ A∗ I – TrFn A xn – q =
un – q + I – γ A∗ I – TrFn A xn – q
– un – xn – γ A∗ I – T F Axn
rn
≤
un – q + xn – q – un – xn + γ A∗ I – TrFn Axn
– γ un – xn , A∗ I – TrFn Axn ,
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which implies that un – q ≤ xn – q – un – xn + γ un – xn , A∗ I – TrFn Axn
≤ xn – q – un – xn + γ un – xn A∗ I – TrFn Axn .
(.)
This implies by (.) and (.) that xn+ – q ≤ βn wn – q + ( – βn )zn – q ≤ βn un – q + ( – βn )xn – q
≤ βn xn – q – un – xn + γ un – xn A∗ I – TrFn Axn
+ ( – βn )xn – q . Therefore, we have
βn un – xn ≤ xn – q – xn+ – q + γβn un – xn A∗ I – TrFn Axn
≤ xn – q – xn+ – q + γβn M A∗ I – TrFn Axn , where M = sup{un – xn : n ∈ N}. This implies by Condition (C), (.), and the existence of limn→∞ xn – q that lim un – xn = .
n→∞
(.)
From (.), (.), and the definition of {yn }, we obtain xn+ – q ≤ βn wn – q + ( – βn )zn – q ≤ βn xn – q + ( – βn )yn – q = βn xn – q + ( – βn ) αn xn – q + ( – αn )wn – q – αn ( – αn )xn – wn ≤ βn xn – q + ( – βn ) xn – q – αn ( – αn )xn – wn = xn – q – αn ( – αn )( – βn )xn – wn . This implies that αn ( – αn )( – βn )xn – wn ≤ xn – q – xn+ – q . From Conditions (C), (C), and the existence of limn→∞ xn – q, we have lim wn – xn = .
n→∞
(.)
By (.) and (.), we get wn – un ≤ wn – xn + xn – un → as n → ∞.
(.)
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Step . Show that ωw (xn ) ⊂ , where ωw (xn ) = {x ∈ H : xni x, {xni } ⊂ {xn }}. Since {xn } is bounded and H is reflexive, ωw (xn ) is nonempty. Let p ∈ ωw (xn ) be an arbitrary element. Then there exists a subsequence {xni } ⊂ {xn } converging weakly to p. From (.), it implies that uni p as i → ∞. By (.) and Lemma ., we have p ∈ F(S). Next, we show that p ∈ MEP(F , ϕ). Since un = TrFn (I – γ A∗ (I – TrFn )A)xn , we have F (un , y) + ϕ(y) – ϕ(un ) +
y – un , un – xn – γ A∗ I – TrFn Axn ≥ , rn
∀y ∈ C,
which implies that F (un , y) + ϕ(y) – ϕ(un ) +
y – un , un – xn – y – un , γ A∗ I – TrFn Axn ≥ , rn rn
∀y ∈ C.
From Assumption .(A), we have ϕ(y) – ϕ(un ) +
y – un , γ A∗ I – TrFn Axn y – un , un – xn – rn rn
≥ –F (un , y) ≥ F (y, un ),
∀y ∈ C,
and hence ϕ(y) – ϕ(uni ) +
y – uni , uni – xni – y – uni , γ A∗ I – TrFn Axni ≥ F (y, uni ), i rni rni
∀y ∈ C. This implies by uni p, Condition (C), (.), (.), Assumption .(A), and the proper lower semicontinuity of ϕ that F (y, p) + ϕ(p) – ϕ(y) ≤ ,
∀y ∈ C.
Put yt = ty + ( – t)p for all t ∈ (, ] and y ∈ C. Consequently, we get yt ∈ C and hence F (yt , p) + ϕ(p) – ϕ(yt ) ≤ . So, by Assumption .(A)-(A), we have = F (yt , yt ) – ϕ(yt ) + ϕ(yt ) ≤ tF (yt , y) + ( – t)F (yt , p) + tϕ(y) + ( – t)ϕ(p) – ϕ(yt ) ≤ t F (yt , y) + ϕ(y) – ϕ(yt ) . Hence, we have F (yt , y) + ϕ(y) – ϕ(yt ) ≥ ,
∀y ∈ C.
Letting t → , by Assumption .(A) and the proper lower semicontinuity of ϕ, we have F (p, y) + ϕ(y) – ϕ(p) ≥ ,
∀y ∈ C.
This implies that p ∈ MEP(F , ϕ).
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Since A is a bounded linear operator, we have Axni Ap. Then it follows from (.) that TrFn Axni Ap i
as i → ∞.
(.)
By the definition of TrFni Axni , we have y – TrFn Axni , TrFn Axni – Axni ≥ , F TrFn Axni , y + φ(y) – φ TrFn Axni + i i i i rni
∀y ∈ Q.
Since F is upper semicontinuous in the first argument, it implies by (.) that F (Ap, y) + φ(y) – φ(Ap) ≥ ,
∀y ∈ Q.
This shows that Ap ∈ MEP(F , φ). Therefore, p ∈ SMEP(F , ϕ, F , φ) and hence p ∈ . Step . Show that {xn } converges weakly to an element of . It is sufficient to show that ωw (xn ) is a singleton set. Let p, q ∈ ωw (xn ) and {xnk }, {xnm } be two subsequences of {xn } such that xnk p and xnm q. From (.), we also have unk p and unm q. By (.) and Lemma ., we see that p, q ∈ F(S). Applying Lemma ., we obtain p = q. This completes the proof. If ϕ = φ = in (.) and (.), then the split mixed equilibrium problem reduces to split equilibrium problem. So, the following result can be obtained from Theorem . immediately. Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H and Q be a nonempty closed convex subset of a real Hilbert space H . Let A : H → H be a bounded linear operator and S : C → K(C) a λ-hybrid multivalued mapping. Let F : C × C → R, F : Q × Q → R be bifunctions satisfying Assumption ., and F is upper semicontinuous in the first argument. Assume that = F(S) ∩ SEP(F , F ) = ∅ and Sp = {p} for all p ∈ F(S). Let {xn } be a sequence generated by x ∈ C and ⎧ F F ∗ ⎪ ⎪ ⎨un = Trn (I – γ A (I – Trn )A)xn , ⎪ ⎪ ⎩
yn = αn xn + ( – αn )wn , xn+ = βn wn + ( – βn )zn ,
wn ∈ Sun ,
(.)
zn ∈ Syn , ∀n ∈ N,
where {αn } ⊂ (, ), {βn } ⊂ (, ), {rn } ⊂ (, ∞), and γ ∈ (, L ) such that L is the spectral radius of A∗ A and A∗ is the adjoint of A. Assume that the following conditions hold: (C) < lim infn→∞ βn ≤ lim supn→∞ βn < ; (C) < lim infn→∞ αn ≤ lim supn→∞ αn < ; (C) < lim infn→∞ rn . Then the sequence {xn } generated by (.) converges weakly to p ∈ . Remark . (i) Theorems . and . extend the corresponding one of Suantai et al. [] and Kazmi and Rizvi [] to λ-hybrid multivalued mapping and to a split mixed equilibrium
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problem. In fact, we present a new iterative algorithm for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in a real Hilbert space. (ii) It is well known that the class of λ-hybrid multivalued mappings contains the classes of nonexpansive multivalued mappings, nonspreading multivalued mappings. Thus, Theorems . and . can be applied to these classes of mappings. We give an example to illustrate Theorem . as follows. Example . Let H = R, H = R, C = [–, ], and Q = (–∞, ]. Let A : H −→ H defined by Ax = x for each x ∈ H . Then A∗ y = y for each y ∈ H . So, L = is the spectral radius of A∗ A. Define a multivalued mapping S : C −→ K(C) by
Sx =
⎧ ⎨[–
|x| , ], |x|+
⎩{},
x ∈ [–, –); x ∈ [–, ].
It easy to see that S is -hybrid multivalued mapping with F(S) = {} and S() = {}. For each x, y ∈ C, define the bifunction F : C × C −→ R by F (x, y) = xy + y – x – x and define ϕ(x) = for each x ∈ C. For each u, v ∈ Q, define the bifunction F : Q × Q −→ R by F (u, v) = uv + v – u – u and define φ(u) = for each u ∈ Q. n n n , βn = n+ , rn = n+ , and γ = . It is easy to check that F , F , {αn }, {βn }, Choose αn = n+ {rn } satisfy all conditions in Theorem .. For each x ∈ C, we compute TrF Ax. Find z such that ≤ F (z, y) + ϕ(y) – ϕ(z) + y – z, z – Ax r x = zy + y – z – z + y – z, z – r x = (z + )(y – z) + (y – z) z – r x z– = (y – z) (z + ) + r for all y ∈ Q. Thus, by Lemma .(), it follows that z = each x ∈ C. Furthermore, we get
I – γ A∗ I – TrF A x = x – A∗ Ax – TrF Ax x x – r = x – A∗ – ( + r) x x – r =x– – ( + r) γ (x – r) γ – . =x – ( + r)
x–r . (+r)
That is, TrF Ax =
x–r (+r)
for
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Table 1 Numerical results of Example 3.5 for the algorithm (3.19) n
xn
xn – xn–1
1 2 3 4
–3.0000000e+00 –6.8786127e–02 0.0000000e+00 0.0000000e+00
2.9312139e+00 6.8786127e–02 0.0000000e+00
Next, we find u ∈ C such that F (u, v) + ϕ(y) – ϕ(z) + r v – u, u – s ≥ for all v ∈ C, where s = (I – γ A∗ (I – TrF )A)x. Note that ≤ F (u, v) + ϕ(y) – ϕ(z) + v – u, u – s = uv + v – u – u + v – u, u – s r r = (u + )(v – u) + (v – u)(u – s) r = (v – u) (u + ) + (u – s) . r Thus, by Lemma .(), it follows that u=
x – r s – r x – r . = – + r ( + r) ( + r)
Then the algorithm (.) becomes ⎧ ⎪ ⎪ ⎨un =
xn –rn xn –rn – (+r rn , (+rn ) n) n n yn = n+ xn + ( – n+ )wn , ⎪ ⎪ ⎩ n n xn+ = n+ wn + ( – n+ )zn ,
=
n , n+
(.) ∀n ∈ N,
where
wn =
⎧ ⎨[–
|un | , ], |un |+
⎩{},
un ∈ [–, –); un ∈ [–, ],
zn =
⎧ ⎨[–
|yn | , ], |yn |+
⎩{},
yn ∈ [–, –); yn ∈ [–, ].
| | if un ∈ [–, –) and zn = – |y|ynn|+ if yn ∈ [–, –). By using SciLab, We choose wn = – |u|unn|+ we compute the iterates of (.) for the initial point x = –. The numerical experiment’s results of our iteration for approximating the point are given in Table .
4 Conclusions The results presented in this paper extend and generalize the work of Suantai et al. [] and Kazmi and Rizvi []. The main aim of this paper is to propose an iterative algorithm to find an element for solving a class of split mixed equilibrium problems and fixed point problems for λ-hybrid multivalued mappings under weaker conditions. Some sufficient conditions for the weak convergence of such proposed algorithm are given. Also, in order to show the significance of the considered problem, some important applications are discussed.
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Acknowledgements The authors are thankful to the referees for careful reading and the useful comments and suggestions. Funding This work was supported by the Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University, Nakhon Pathom, Thailand. The second author was also supported by the Thailand Research Fund under the project RTA5780007. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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