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An approximate solution to the fixed point problems for an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, cocoercive quasivariational inclusions problems and mixed equilibrium problems in Hilbert spaces
Fixed Point Theory and Applications volume 2012, Article number: 214 (2012)
Abstract
We introduce a new iterative scheme by modifying Mann’s iteration method to find a common element for the set of common fixed points of an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem of the iterative scheme to a common element of the three aforementioned sets is obtained based on the shrinking projection method which extends and improves that of Ezeora and Shehu (Thai J. Math. 9(2):399-409, 2011) and many others.
MSC:46C05, 47H09, 47H10, 49J30, 49J40.
1 Introduction
Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by and , respectively. For a sequence in H, we denote the strong convergence and the weak convergence of to by and , respectively.
Recall that is the metric projection of H onto C; that is, for each , there exists the unique point such that . A mapping is called nonexpansive if for all , and uniformly L-Lipschitzian if there exists a constant such that for each , for all , and a mapping is called a contraction if there exists a constant such that for all . A point is a fixed point of T provided that . We denote by the set of fixed points of T; that is, . If C is a nonempty bounded closed convex subset of H and T is a nonexpansive mapping of C into itself, then is nonempty (see [1]). Recall that a mapping is said to be
-
(i)
monotone if
(1.1) -
(ii)
k-Lipschitz continuous if there exists a constant such that
(1.2)
if , then A is nonexpansive,
-
(iii)
α-strongly monotone if there exists a constant such that
(1.3) -
(iv)
α-inverse-strongly monotone (or α-cocoercive) if there exists a constant such that
(1.4)
if , then T is called firmly nonexpansive; it is obvious that any α-inverse-strongly monotone mapping T is monotone and -Lipschitz continuous,
-
(v)
κ-strictly pseudocontractive [2] if there exists a constant such that
(1.5)
In brief, we use κ-SPC to denote κ-strictly pseudocontractive. It is obvious that T is nonexpansive if and only if T is 0-SPC,
-
(vi)
asymptotically κ-SPC [3] if there exists a constant and a sequence of nonnegative real numbers with such that
(1.6)
for all . If , then T is asymptotically nonexpansive with for all ; that is, T is asymptotically nonexpansive [4] if there exists a sequence with such that
for all . It is known that the class of κ-SPC mappings and the class of asymptotically κ-SPC mappings are independent (see [5]). And the class of asymptotically nonexpansive mappings is reduced to the class of asymptotically nonexpansive mappings in the intermediate sense with for all and for some ; that is, T is an asymptotically nonexpansive mapping in the intermediate sense if there exists a sequence of nonnegative real numbers with such that
for all ,
-
(vii)
asymptotically κ-SPC mapping in the intermediate sense [6] if there exists a constant and a sequence of nonnegative real numbers with such that
If we define
then , and the last inequality is reduced to
for all . It is obvious that if for all , then the class of asymptotically κ-SPC mappings in the intermediate sense is reduced to the class of asymptotically κ-SPC mappings; and if for all , then the class of asymptotically κ-SPC mappings in the intermediate sense is reduced to the class of asymptotically nonexpansive mappings; and if for all , then the class of asymptotically κ-SPC mappings in the intermediate sense is reduced to the class of nonexpansive mappings; and the class of asymptotically nonexpansive mappings in the intermediate sense with of nonnegative real numbers such that is reduced to the class of asymptotically κ-SPC mappings in the intermediate sense with for all and for some . Some methods have been proposed to solve the fixed point problem of an asymptotically κ-SPC mapping in the intermediate sense (1.9); related work can also be found in [6–13] and the references therein.
Example 1.1 (Sahu et al. [6])
Let and . For each , we define by
where . Then
-
(1)
T is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, T is an asymptotically κ-SPC mapping in the intermediate sense.
-
(2)
T is not continuous. Therefore, T is not an asymptotically κ-SPC and asymptotically nonexpansive mapping.
Example 1.2 (Hu and Cai [7])
Let , , and . For each , we define by
Then
-
(1)
T is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, T is an asymptotically κ-SPC mapping in the intermediate sense.
-
(2)
T is continuous but not uniformly L-Lipschitzian. Therefore, T is not an asymptotically κ-SPC mapping.
Example 1.3 Let and . For each , we define by
where . Then
-
(1)
T is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, T is an asymptotically κ-SPC mapping in the intermediate sense.
-
(2)
T is not continuous. Therefore, T is not an asymptotically κ-SPC and asymptotically nonexpansive mapping.
Iterative methods are often used to solve the fixed point equation . The most well-known method is perhaps the Picard successive iteration method when T is a contraction. Picard’s method generates a sequence successively as for all with chosen arbitrarily, and this sequence converges in norm to the unique fixed point of T. However, if T is not a contraction (for instance, if T is nonexpansive), then Picard’s successive iteration fails, in general, to converge. Instead, Mann’s iteration method for a nonexpansive mapping T (see [14]) prevails, generates a sequence recursively by
where chosen arbitrarily and the sequence lies in the interval .
Mann’s algorithm for nonexpansive mappings has been extensively investigated (see [2, 15, 16] and the references therein). One of the well-known results is proven by Reich [16] for a nonexpansive mapping T on C, which asserts the weak convergence of the sequence generated by (1.10) in a uniformly convex Banach space with a Frechet differentiable norm under the control condition . Recently, Marino and Xu [17] developed and extended Reich’s result to a SPC mapping in a Hilbert space setting. More precisely, they proved the weak convergence of Mann’s iteration process (1.10) for a κ-SPC mapping T on C, and subsequently, this result was improved and carried over the class of asymptotically κ-SPC mappings by Kim and Xu [18].
It is known that Mann’s iteration (1.10) is in general not strongly convergent (see [19]). The way to guarantee strong convergence has been proposed by Nakajo and Takahashi [20]. They modified Mann’s iteration method (1.10), which is to find a fixed point of a nonexpansive mapping by the hybrid method, called the shrinking projection method (or the CQ method), as the following theorem.
Theorem NT Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that . Suppose that chosen arbitrarily and is the sequence defined by
where . Then converges strongly to .
Subsequently, Marino and Xu [21] introduced an iterative scheme for finding a fixed point of a κ-SPC mapping as the following theorem.
Theorem MX Let C be a nonempty closed convex subset of a real Hilbert space H and let be a κ-SPC mapping for some . Assume that . Suppose that chosen arbitrarily and is the sequence defined by
where . Then the sequence converges strongly to .
Quite recently, Kim and Xu [18] have improved and carried Theorem MX over a wider class of asymptotically κ-SPC mappings as the following theorem.
Theorem KX Let C be a nonempty closed convex subset of a real Hilbert space H and let be an asymptotically κ-SPC mapping for some with a bounded sequence such that . Assume that is a nonempty bounded subset of C. Suppose that chosen arbitrarily and is the sequence defined by
where as , and such that . Then the sequence converges strongly to .
The domain of the function is the set
Let be a proper extended real-valued function and let Φ be a bifunction from into ℝ, where ℝ is the set of real numbers. The so-called mixed equilibrium problem is to find such that
The set of solutions of problem (1.11) is denoted by , that is,
It is obvious that if x is a solution of problem (1.11), then . If , then problem (1.11) is reduced to finding such that
We denote by the set of solutions of the equilibrium problem. The theory of equilibrium problems has played an important role in the study of a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity, and optimization and has numerous applications, including but not limited to problems in economics, game theory, finance, traffic analysis, circuit network analysis, and mechanics. The ideas and techniques of this theory are being used in a variety of diverse areas and have proved to be productive and innovative. Problem (1.12) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instance, [22, 23] and the references therein. Some methods have been proposed to solve equilibrium problem (1.12); related work can also be found in [7, 12, 24–34].
For solving the mixed equilibrium problem, let us assume that the bifunction , the function , and the set C satisfy the following conditions:
(A1) for all ;
(A2) Φ is monotone; that is, for all ;
(A3) for each ,
(A4) for each , is convex and lower semicontinuous;
(A5) for each , is weakly upper semicontinuous;
(B1) for each and , there exists a bounded subset and such that for any ,
(B2) C is a bounded set.
Variational inequality theory provides us with a simple, natural, general, and unified framework for studying a wide class of unrelated problems arising in elasticity, structural analysis, economics, optimization, oceanography, and regional, physical, and engineering sciences, etc. (see [35–41] and the references therein). In recent years, variational inequalities have been extended and generalized in different directions, using novel and innovative techniques, both for their own sake and for their applications. A useful and important generalization of variational inequalities is a variational inclusion.
Let be a single-valued nonlinear mapping and let be a set-valued mapping. We consider the following quasivariational inclusion problem, which is to find a point such that
where θ is the zero vector in H. The set of solutions of problem (1.13) is denoted by .
A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if the graph of of T is not properly contained in the graph of any other monotone mappings. It is known that a monotone mapping T is maximal if and only if for , for all implies .
Definition 1.4 (see [42])
Let be a multi-valued maximal monotone mapping. Then the single-valued mapping defined by , for all , is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping.
Recently, Qin et al. [43] introduced the following algorithm for a finite family of asymptotically -SPC mappings on C. Let and be a sequence in . Let be an integer. The sequence generated in the following way:
is called the explicit iterative scheme of a finite family of asymptotically -SPC mappings on C, where . Since, for each , it can be written as , where , is a positive integer and as .
For each , let be a finite family of asymptotically -SPC mappings with the sequence such that . One has
for all , and we can rewrite (1.14) in the following compact form:
To be more precise, they introduced an iterative scheme for finding a common fixed point of a finite family of asymptotically -SPC mappings as the following theorem.
Theorem QCKS Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an integer. For each , let be a finite family of asymptotically -SPC mappings defined as in (1.15), when with the sequence such that . Let and . Assume that is a nonempty bounded subset of C. For chosen arbitrarily, suppose that is generated iteratively by
where as such that . If the control sequence is chosen such that . Then the sequence converges strongly to .
Recall that a discrete family of self-mappings of C is said to be an asymptotically κ-SPC semigroup [44] if the following conditions are satisfied:
-
(1)
, where I denotes the identity operator on C;
-
(2)
, , ;
-
(3)
there exists a constant and a sequence of nonnegative real numbers with such that
(1.16)
for all . Note that, for a single asymptotically κ-SPC mapping , (1.16) immediately reduces to (1.6) by taking for all such that .
On the other hand, Tianchai [24] introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of common fixed points for a discrete asymptotically κ-SPC semigroup which is a subclass of the class of infinite families for the asymptotically κ-SPC mapping as the following theorem.
Theorem T Let C be a nonempty closed convex subset of a real Hilbert space H, Φ be a bifunction from into ℝ satisfying the conditions (A1)-(A5), and let be a proper lower semicontinuous and convex function with the assumption that either (B1) or (B2) holds. Let be an asymptotically κ-SPC semigroup on C for some with the sequence such that . Assume that is a nonempty bounded subset of C. For chosen arbitrarily, suppose that , and are generated iteratively by
where , such that , for some and . Then the sequences , , and converge strongly to .
Recently, Saha et al. [6] modified Mann’s iteration method (1.10) for finding a fixed point of the asymptotically κ-SPC mapping in the intermediate sense which is not necessarily uniformly Lipschitzian (see, e.g., [6, 7]) as the following theorem.
Theorem SXY Let C be a nonempty closed convex subset of a real Hilbert space H and let be a uniformly continuous and asymptotically κ-SPC mapping in the intermediate sense defined as in (1.9), when , with the sequences such that . Assume that is a nonempty bounded subset of C. Suppose that chosen arbitrarily and is the sequence defined by
where as , and such that . Then the sequence converges strongly to .
Let be an integer. For each , let be a finite family of asymptotically -SPC mappings in the intermediate sense with the sequences such that . One has
for all such that , where , is a positive integer and as .
Subsequently, Hu and Cai [7] modified Ishikawa’s iteration method (see [45]) for finding a common element of the set of common fixed points for a finite family of asymptotically -SPC mappings in the intermediate sense and the set of solutions of the equilibrium problems (see also Duan and Zhao [8]) as the following theorem.
Theorem HC Let C be a nonempty closed convex subset of a real Hilbert space H, be a bifunction satisfying the conditions (A1)-(A4), and let be a ξ-cocoercive mapping. Let be an integer. For each , let be a finite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.17) when with the sequences such that . Let , and . Assume that is a nonempty bounded subset of C. For chosen arbitrarily, suppose that and are generated iteratively by
where as such that . If the control sequences , and are chosen such that , , and , then the sequences and converge strongly to .
In this paper, we study the sequences generated by modifying Mann’s iteration method (1.10) for an infinite family of asymptotically -SPC mappings in the intermediate sense. For each , let be an infinite family of asymptotically -SPC mappings in the intermediate sense with the sequences such that . One has
for all . For each , let and be a sequence in , and let the sequences be generated in the following way:
Quite recently, Ezeora and Shehu [9] introduced an iterative scheme for finding a common fixed point of an infinite family of asymptotically -SPC mappings in the intermediate sense as the following theorem.
Theorem ES Let C be a nonempty closed convex subset of a real Hilbert space H. For each , let be an infinite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.18) when with the sequences such that . Assume that is a nonempty bounded subset of C. For chosen arbitrarily, suppose that is generated iteratively by
where (), and () such that . Then the sequence converges strongly to .
Inspired and motivated by the works mentioned above, in this paper, we introduce a new iterative scheme (3.1) below by modifying Mann’s iteration method (1.10) to find a common element for the set of common fixed points of an infinite family of asymptotically -SPC mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem of the iterative scheme to a common element of the three aforementioned sets is obtained based on the shrinking projection method which extends and improves that of Ezeora and Shehu [9] and many others.
2 Preliminaries
We collect the following lemmas which be used in the proof of the main results in the next section.
Lemma 2.1 (see [1])
Let C be a nonempty closed convex subset of a Hilbert space H. Then the following inequality holds:
Lemma 2.2 (see [46])
Let H be a Hilbert space. For all and such that , one has
Lemma 2.3 (see [25])
Let C be a nonempty closed convex subset of a real Hilbert space H, satisfying the conditions (A1)-(A5), and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For , define a mapping as follows:
for all . Then the following statements hold:
-
(1)
for each , ;
-
(2)
is single-valued;
-
(3)
is firmly nonexpansive; that is, for any ,
-
(4)
;
-
(5)
is closed and convex.
Lemma 2.4 (see [42])
Let be a maximal monotone mapping and let be an α-inverse-strongly monotone mapping. Then, the following statements hold:
-
(1)
B is a -Lipschitz continuous and monotone mapping.
-
(2)
is a solution of quasivariational inclusion (1.13) if and only if , for all , that is,
-
(3)
If , then is a closed convex subset in H, and the mapping is nonexpansive, where I is the identity mapping on H.
-
(4)
The resolvent operator associated with M is single-valued and nonexpansive for all .
-
(5)
The resolvent operator is 1-inverse-strongly monotone; that is,
Lemma 2.5 (see [47])
Let be a maximal monotone mapping and let be a Lipschitz continuous mapping. Then the mapping is a maximal monotone mapping.
Lemma 2.6 (see [6])
Let C be a nonempty closed convex subset of a real Hilbert space H and let be a uniformly continuous and asymptotically κ-strictly pseudocontractive mapping in the intermediate sense defined as in (1.9) when with the sequences such that . Then the following statements hold:
-
(1)
, for all and .
-
(2)
If is a sequence in C such that and . Then .
-
(3)
is demiclosed at zero in the sense that if is a sequence in C such that as , and , then .
-
(4)
is closed and convex.
3 Main results
Theorem 3.1 Let H be a real Hilbert space, Φ be a bifunction from into ℝ satisfying the conditions (A1)-(A5), and let be a proper lower semicontinuous and convex function with the assumption that either (B1) or (B2) holds. Let be a maximal monotone mapping and let be a ξ-cocoercive mapping. For each , let be an infinite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.18) when with the sequences such that . Assume that is a nonempty bounded subset of H. For chosen arbitrarily, suppose that is generated iteratively by
where () and satisfying the following conditions:
(C1) and () such that for each , and ();
(C2) and for some .
Then the sequence converges strongly to .
Proof Pick and fix . From (3.1), by the definition of in Lemma 2.3, we have
and by , Lemmas 2.3(4) and 2.4(2), we have
From Lemma 2.3(3), we know that is nonexpansive. Therefore, by (3.2) and (3.3), we have
Let . From Lemmas 2.4(3) and 2.4(4), we know that and are nonexpansive. Therefore, by (3.3), we have
By (3.3), (3.4), (3.5), Lemma 2.2, and the asymptotically -SPC in the intermediate sense of , we have
where and .
Firstly, we show that is closed and convex for all . It is obvious that is closed and, by mathematical induction, that is closed for all , that is, is closed for all . Since, for any , is equivalent to
for all . Therefore, for any and , we have
for all . Since is convex, by putting in (3.7), (3.8), and (3.9), we have that is convex. Suppose that is given and is convex for some . It follows by putting in (3.7), (3.8), and (3.9) that is convex. Therefore, by mathematical induction, we have that is convex for all , that is, is convex for all . Hence, we obtain that is closed and convex for all .
Next, we show that for all . It is obvious that . Therefore, by (3.1) and (3.6), we have for all i, and so , and note that , and so . Hence, we have . Since is a nonempty closed convex subset of H, there exists a unique element such that . Suppose that is given such that , and for some . Therefore, by (3.1) and (3.6), we have for all i, and so . Since , therefore, by Lemma 2.1, we have
for all . Thus, by (3.1), we have , and so . Hence, we have . Since is a nonempty closed convex subset of H, there exists a unique element such that . Therefore, by mathematical induction, we obtain that for all , and so for all , and we can define for all . Hence, we obtain that the iteration (3.1) is well defined, and by Lemmas 2.3(5), 2.4(3), and 2.6(4), we also obtain that is well defined.
Next, we show that is bounded. Since for all , we have
for all . It follows by that for all . This implies that is bounded, and so are , , and for each .
Next, we show that as . Since , therefore, by (3.10), we have for all . This implies that is a bounded nondecreasing sequence, there exists the limit of , that is,
for some . Since , therefore, by (3.1), we have
It follows that
Therefore, by (3.11), we obtain
Next, we show that is a Cauchy sequence. Observe that
It follows that
for each . Therefore, by (3.1), we have
Thus, we have
Hence, by (3.11), we obtain that as , which implies that is a Cauchy sequence in H, and then there exists a point such that as .
Next, we show that as . From (3.1), we have . Therefore, we have
It follows by (3.13) and that
Since,
Therefore, by (3.13) and (3.17), we obtain
Next, we show that and as . By (3.2), (3.3), and the firmly nonexpansiveness of in Lemma 2.3(3), we have
which implies that
By (3.4), (3.6), and (3.20), we have
which implies that
Therefore, by (3.19) and , we obtain
Since
therefore, by (3.13) and (3.22), we obtain
Next, we show that . From (3.4) and (3.6), we have
which implies that
Therefore, by (3.19), (3.22), and , we obtain
From (3.24) and (3.26), by Lemma 2.6(2), we have
Therefore, by (3.27) and the uniform continuity of , it is easy to see, by mathematical induction on , that
for each . Since, for any , we have
such that , where I is the identity mapping on H. Therefore, by (3.27) and (3.28), we obtain
From (3.22) and , we have as . Therefore, from (3.29), by Lemma 2.6(3), we obtain that for all ; that is, .
Next, we show that . From (3.1), we have
It follows by the condition (A2) that
Hence,
From (3.22) and , we have as . Therefore, we obtain
For a constant t with and , let . Since , thus, . So, from (3.31), we have
By (3.32), the conditions (A1) and (A4), and the convexity of φ, we have
which implies that
Therefore, by the condition (A3) and the weakly lower semicontinuity of φ, we have as for all , and hence we obtain that .
Next, we show that . From (3.1), we have
which implies that
Therefore, by (3.19), (3.22), and (3.26), we obtain
From Lemma 2.4(1), we have that B is -Lipschitz continuous. Therefore, by Lemma 2.5, we have that is maximal monotone. Let , that is,
Since , we have . Therefore,
By (3.34), (3.35), and M is maximal monotone, we have
It follows by the monotonicity of B that
From (3.33), we have , and since , by (3.22) and (3.33), we have as . Therefore, by (3.33) and (3.36), we obtain that as . It follows from the maximal monotonicity of that ; that is, , and so .
Finally, we show that . Since . Therefore, by (3.1), we have
It follows by as that
Therefore, by Lemma 2.1, we obtain that . This completes the proof. □
Remark 3.2 The iteration (3.1) is different from the iterative scheme of Ezeora and Shehu [9] as follows:
-
1.
The sequence is a projection sequence of onto for all such that
-
2.
The proof to the strong convergence of the sequence is simple by a Cauchy sequence.
-
3.
An approximate solution to a common element for the set of common fixed points of an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems by iteration is obtained.
We define the condition (B3) as the condition (B1) such that . If , then Theorem 3.1 is reduced immediately to the following result.
Corollary 3.3 Let H be a real Hilbert space and let Φ be a bifunction from into ℝ satisfying the conditions (A1)-(A5) with the assumption that either (B2) or (B3) holds. Let be a maximal monotone mapping and let be a ξ-cocoercive mapping. For each , let be an infinite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.18) when with the sequences such that . Assume that is a nonempty bounded subset of H. For chosen arbitrarily, suppose that is generated iteratively by
where () and satisfying the following conditions:
(C1) and () such that for each , and ();
(C2) and for some .
Then the sequence converges strongly to .
If , then Corollary 3.3 is reduced immediately to the following result.
Corollary 3.4 Let H be a real Hilbert space, be a maximal monotone mapping, and let be a ξ-cocoercive mapping. For each , let be an infinite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.18) when with the sequences such that . Assume that is a nonempty bounded subset of H. For chosen arbitrarily, suppose that is generated iteratively by
where () and satisfying the following conditions:
(C1) and () such that for each , and ();
(C2) .
Then the sequence converges strongly to .
If and , then Theorem 3.1 is reduced immediately to the following result.
Corollary 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H, Φ be a bifunction from into ℝ satisfying the conditions (A1)-(A5), and let be a proper lower semicontinuous and convex function with the assumption that either (B1) or (B2) holds. For each , let be an infinite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.18) when with the sequences such that . Assume that is a nonempty bounded subset of C. For chosen arbitrarily, suppose that is generated iteratively by
where () and satisfying the following conditions:
(C1) () such that ;
(C2) for some .
Then the sequence converges strongly to .
If , then Corollary 3.5 is reduced immediately to the following result.
Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H and let Φ be a bifunction from into ℝ satisfying the conditions (A1)-(A5) with the assumption that either (B2) or (B3) holds. For each , let be an infinite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.18) when with the sequences such that . Assume that is a nonempty bounded subset of C. For chosen arbitrarily, suppose that is generated iteratively by
where () and satisfying the following conditions:
(C1) () such that ;
(C2) for some .
Then the sequence converges strongly to .
If , then Corollary 3.6 is reduced immediately to the following result.
Corollary 3.7 Let C be a nonempty closed convex subset of a real Hilbert space H. For each , let be an infinite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.18) when with the sequences such that . Assume that is a nonempty bounded subset of C. For chosen arbitrarily, suppose that is generated iteratively by
where (), and () such that . Then the sequence converges strongly to .
Recall that for each , a mapping is said to be asymptotically nonexpansive if there exists a sequence with such that
for all . If and for all and , then Corollary 3.7 is reduced immediately to the following result.
Corollary 3.8 Let C be a nonempty closed convex subset of a real Hilbert space H. For each , let be an infinite family of asymptotically nonexpansive mappings defined as in (3.37) with the sequence such that . Assume that is a nonempty bounded subset of C. For chosen arbitrarily, suppose that is generated iteratively by
where (), , and () such that . Then the sequence converges strongly to .
4 Applications
We introduce the equilibrium problem to the optimization problem:
where C is a nonempty closed convex subset of a real Hilbert space H and is proper convex and lower semicontinuous. We denote by the set of solutions of problem (4.1). We define the condition (B4) as the condition (B3) such that is a bifunction defined by for all . Observe that . We obtain that Corollary 3.3 is reduced immediately to the following result.
Theorem 4.1 Let H be a real Hilbert space and let be a proper lower semicontinuous and convex function with the assumption that either (B2) or (B4) holds. Let be a maximal monotone mapping and let be a ξ-cocoercive mapping. For each , let be an infinite family of uniformly continuous and asymptotically -SPC mappings in the intermediate sense defined as in (1.18) when with the sequences such that . Assume that is a nonempty bounded subset of H. For chosen arbitrarily, suppose that is generated iteratively by
where () and satisfying the following conditions:
(C1) and () such that for each , and ();
(C2) and for some .
Then the sequence converges strongly to .
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Tianchai, P. An approximate solution to the fixed point problems for an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, cocoercive quasivariational inclusions problems and mixed equilibrium problems in Hilbert spaces. Fixed Point Theory Appl 2012, 214 (2012). https://doi.org/10.1186/1687-1812-2012-214
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DOI: https://doi.org/10.1186/1687-1812-2012-214