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Variational inequality problems over split fixed point sets of strict pseudo-nonspreading mappings and quasi-nonexpansive mappings with applications
Fixed Point Theory and Applications volume 2014, Article number: 198 (2014)
Abstract
In this paper, we first establish a strong convergence theorem for a variational inequality problem over split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of quasi-nonexpansive mappings. As applications, we establish a strong convergence theorem of split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of strict pseudo-nonspreading mappings without semicompact assumption on the strict pseudo-nonspreading mappings. We also study the variational inequality problems over split common solutions of fixed points for a finite family of strict pseudo-nonspreading mappings, fixed points of a countable family of pseudo-contractive mappings (or strict pseudo-nonspreading mappings) and solutions of a countable family of nonlinear operators. We study fixed points of a countable family of pseudo-contractive mappings with hemicontinuity assumption, neither Lipschitz continuity nor closedness assumption is needed.
1 Introduction
The split feasibility problem (SFP) in finite dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. Since then, the split feasibility problem (SFP) has received much attention due to its applications in signal processing, image reconstruction, with particular progress in intensity-modulated radiation therapy, approximation theory, control theory, biomedical engineering, communications, and geophysics. For example, one can see [2–5].
Let C be a nonempty closed convex subset of a real Hilbert space with the inner product and the norm .
Let f be a contraction on C, let be a nonexpansive mapping and be a sequence in . In 2004, Xu [6] proved that under some condition on , the sequence generated by
strongly converges to in which is the unique solution of the variational inequality
for all .
Xu [7] studied the following minimization problem over the set of fixed point set of a nonexpansive operator T on a real Hilbert space :
where a is a given point in and B is a strongly positive bounded linear operator on . In [7], Xu proved that the sequence defined by the following iterative method:
converges strongly to the unique solution of the minimization problem of a quadratic function.
Let , be two real Hilbert spaces, be a bounded linear operator, be an infinite family of -strictly pseudo-nonspreading mappings and be a finite family of -strict pseudo-nonspreading mappings, and , Chang et al. [8] studied the following iterative sequence.
Let be defined by
where , is a constant, I is an identity function on .
Chang et al. [8] proved a weak convergence theorem to the solution of the following problem: Find , .
In addition, if there exists some positive integer m such that is semicompact, then , converge strongly to a point . In 2013, Tang [9] studied common solutions of fixed points of continuous pseudo-contractive mappings and zero points of the sum of monotone mappings. In 2014, Zegeye and Shahzad [10] and Yao et al. [11] studied common solutions of fixed points of Lipschitz continuous pseudo-contractive mappings and zero points of the sum of monotone mappings. Wangkeree and Nammanee [12] studied common solutions of fixed points of continuous pseudo-contractive mappings and solutions of a continuous monotone variational inequality.
In 2013, Cheng et al. [13] constructed a three-step iteration and obtained a convergence theorem for a countable family of uniformly Lipschitz and uniformly closed pseudo-contractive mappings. Deng [14] constructed another iteration and obtained a convergence theorem for a countable family of closed and Lipschitz pseudo-contractive mappings. Both the results of Cheng et al. [13] and Deng [14] used too strong conditions to study common fixed points of a countable family of pseudo-contractive mappings. The fixed point theorems of pseudo-contractive mappings in the literature assumed the Lipschitz continuous condition; see, for example, [15, 16].
Motivated and inspired by the above results, in this paper, we first establish a strong convergence theorem for a variational inequality problem over split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of quasi-nonexpansive mappings. As applications, we establish a strong convergence theorem of split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of strict pseudo-nonspreading mappings without semicompact assumption on the strict pseudo-nonspreading mappings. We also study the variational inequality problems over split common solutions of a fixed point for a finite family of strict pseudo-nonspreading mappings, fixed points of a countable family of pseudo-contractive mappings (or strict pseudo-nonspreading mappings) and solutions of a countable family of nonlinear operators. We study a fixed point of a countable family of pseudo-contractive mappings with hemicontinuity assumption, neither Lipschitz continuity nor closedness assumptions is needed.
2 Preliminaries
Throughout this paper, let ℕ be the set of positive integers, and let ℝ be the set of real numbers, be a (real) Hilbert space. Let and denote the inner product and the norm of , respectively. Let C be a nonempty closed convex subset of . We denote the strong convergence and the weak convergence of to by and , respectively.
Let be a mapping, and let denote the set of fixed points of T. A mapping is called
-
(i)
pseudo-contractive if for each , we have .
Note that the above inequality can be equivalently written as
-
(ii)
a k-strictly pseudo-contractive mapping if there exists a constant such that for all ;
-
(iii)
nonexpansive if for all ;
-
(iv)
firmly nonexpansive if for every ;
-
(v)
quasi-nonexpansive if and for all , ;
-
(vi)
nonspreading [17] if for all , that is, for all ;
-
(vii)
α-strictly pseudo-nonspreading [18] if there exists such that
for all ;
-
(viii)
strongly monotone if there exists such that for all ;
-
(ix)
Lipschitz continuous if there exists such that for all ;
-
(x)
hemicontinuous ([19], p.204) if, for all , the mapping defined by is continuous, where has a weak topology;
-
(xi)
α-inverse-strongly monotone if for all and ;
-
(xi)
monotone if for all .
We also know that (i) if V is an α-inverse-strongly monotone mapping and , then is a nonexpansive mapping; (ii) if V is a monotone mapping, then is a pseudo-contractive mapping.
Let be a mapping. Then is called an asymptotic fixed point of T [20] if there exists such that , and . We denote by the set of asymptotic fixed points of T. A mapping is said to be demiclosed if it satisfies .
Let be a multivalued mapping. The effective domain of B is denoted by , that is, .
Then is called
-
(i)
a monotone operator on if for all , , and ;
-
(ii)
a maximal monotone operator on if B is a monotone operator on and its graph is not properly contained in the graph of any other monotone operator on .
For a maximal monotone operator B on and , we may define a single-valued operator , which is called the resolvent of B for r. Let .
The following lemmas are needed in this paper.
A mapping is said to be averaged if , where and is nonexpansive. In this case, we also say that T is α-averaged. A firmly nonexpansive mapping is -averaged.
Let be a mapping. Then the following are satisfied:
-
(i)
T is nonexpansive if and only if the complement is -ism.
-
(ii)
If S is υ-ism, then, for , γS is -ism.
-
(iii)
S is averaged if and only if the complement is υ-ism for some .
-
(iv)
If S and T are both averaged, then the product (composite) ST is averaged.
-
(v)
If the mappings are averaged and have a common fixed point, then .
Lemma 2.2 [22]
Let be a -strongly monotone and L-Lipschitz continuous operator with and . Let and such that . Then is a -strongly monotone and L-Lipschitz continuous mapping. Furthermore, there exists a unique fixed point in C satisfying . This point is also a unique solution of the hierarchical variational inequality
Lemma 2.3 [19]
Let be maximal monotone.
-
(i)
For each , is single-valued and firmly nonexpansive.
-
(ii)
and .
Lemma 2.4 [23]
Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :
In fact, .
Lemma 2.5 [24]
Let be a sequence of nonnegative real numbers, be a sequence of real numbers in with , be a sequence of nonnegative real numbers with , be a sequence of real numbers with . Suppose that for each . Then .
The equilibrium problem is to find such that
where is a bifunction.
This problem includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, minimax inequalities, and saddle point problems as special cases. (For example, one can see [25] and related literature.) The solution set of equilibrium problem (EP) is denoted by .
For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:
(A1) for each .
(A2) g is monotone, i.e., for any .
(A3) For each , .
(A4) For each , the scalar function is convex and lower semicontinuous.
We have the following result from Blum and Oettli [25].
Theorem 2.1 [25]
Let be a bifunction which satisfies conditions (A1)-(A4). Then, for each and each , there exists such that
for all .
In 2005, Combettes and Hirstoaga [26] established the following important properties of a resolvent operator.
Theorem 2.2 [26]
Let be a function satisfying conditions (A1)-(A4). For , define by
for all . Then the following hold:
-
(i)
is single-valued.
-
(ii)
is firmly nonexpansive, that is, for all .
-
(iii)
.
-
(iv)
is a closed and convex subset of C.
We call such the resolvent of g for .
Lemma 2.6 [27]
Let E be a uniformly convex Banach space, be a constant. Then there exists a continuous, strictly increasing and convex function with such that
for all , , with .
Takahashi et al. [28] showed the following result.
Lemma 2.7 [28]
Let be a bifunction satisfying the conditions (A1)-(A4). Define as follows:
Then and is a maximal monotone operator with the domain of . Furthermore, for any and , the resolvent of g coincides with the resolvent of , i.e., .
Lemma 2.8 [18]
Let H be a real Hilbert space, C be a nonempty and closed convex subset of H, and be a k-strictly pseudo-nonspreading mapping. Then the following hold:
-
(i)
If , then is closed and convex.
-
(ii)
T is demiclosed.
Lemma 2.9 [8]
Let H be a real Hilbert space, C be a nonempty and closed convex subset of H, and be a k-strictly pseudo-nonspreading mapping and . Let , . Then is a quasi-nonexpansive mapping.
3 Convergence theorems of hierarchical problems
Let C and Q be nonempty closed convex subsets of real Hilbert spaces and , respectively. For each , let be a maximal monotone mapping on such that the domains of are included in C, and let for each . For , , let be a -inverse-strongly monotone mapping of C into . Let be a bounded linear operator and be the adjoint of A, and let R be the spectral radius of the operator . Let be a sequence, and let V be a -strongly monotone and L-Lipschitz continuous operator with and . Let and be countable families of quasi-nonexpansive mappings from C into itself with demiclosed property, and let be a finite family of -strictly pseudo-nonspreading mappings with . Let be a countable family of -strictly pseudo-nonspreading mappings from C into itself with , and let be a countable family of pseudo-contractive mappings. Throughout this paper, we use these notations and assumptions unless specified otherwise.
In this paper, we first study the variational inequality problem over split common solutions for a fixed point of a finite family of strict pseudo-nonspreading mappings, and a solution of a countable family of quasi-nonexpansive mappings.
Theorem 3.1 Suppose that . Take as follows:
Let be defined by
for each , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof Take any and let be fixed. Then and . For each , by Lemma 2.6, we have
In Theorem 3.1 in [8], Chang et al. showed that
Let and , following the same argument as in the proof of Theorem 3.1 in [22], we have
where . By mathematical induction, we know
This implies that the sequence is bounded. Furthermore, , and are bounded for all .
Following the same argument as in the proof of Theorem 3.1 in [22] again, we have
We will divide the proof into two cases as follows.
Case 1: There exists a natural number N such that for each . So, exists. Hence, it follows from (3), (i) and (ii) that
Following the same argument as in the proof of Theorem 3.1 in [22] again, we have
and
By (1) and (2), we have
for all .
Therefore,
for all .
Thus, by (4), (8), and condition (iii), we have
for all .
From the properties of g, we conclude that
for all .
By Lemma 2.8 and the assumption, we have that is a nonempty closed convex subset of H. Hence, by Lemma 2.2, we can take such that
This point is also a unique solution of the hierarchical variational inequality
We show that
Without loss of generality, there exists a subsequence of such that for some and
By (7), we have that
Since A is a bounded linear operator, this implies that . Also, by equation (10), we have
Hence there exist a positive integer and a subsequence of with such that
Since , it follows from Lemma 2.8 that is demiclosed at 0. This implies that .
By equations (13) and (14), we have
For each , by is demiclosed, equations (10) and (16), we have . Hence, . Thus, from (11) and (12) we have
Following the same argument as in the proof of Theorem 3.1 in [22] again, we have that
By (17), (18), assumptions, and Lemma 2.5, we know that , where
Case 2: Suppose that there exists of such that for all . By Lemma 2.4, there exists a nondecreasing sequence in ℕ such that and
Following the same argument as in the proof of Theorem 3.1 in [22] again, we have
Thus, the proof is completed. □
Remark 3.1 Theorem 3.1 improves Theorem 3.1 in [29], Theorem 3.3 in [30] and Theorem 3.1 in [31] in the sense that our convergence is for the common fixed point of a countable family of quasi-nonexpansive mappings and for the split feasibility problem.
Applying Theorem 3.1, we study a variational inequality problem over split common solutions for a fixed point of a finite family of strict pseudo-nonspreading mappings and a common fixed point of countable families of mappings.
Theorem 3.2 Suppose that . Take as follows:
Let be defined by
where , , for each , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof By , there exists such that . For each , is a -strictly pseudo-nonspreading mapping and , by Lemma 2.9, we have is quasi-nonexpansive. For each , by Lemma 2.8 and , we see that is demiclosed.
For each , let and in Theorem 3.1, then is quasi-nonexpansive with demiclosed property and . It follows from Theorem 3.1 that , where . This point is also a unique solution of the following hierarchical variational inequality:
□
Applying Theorem 3.1, we study split common solutions for common fixed points of a finite family of strict pseudo-nonspreading mappings, and common fixed points of a countable family of strict pseudo-nonspreading mappings. Our result improves Theorem 3.1 in [8].
Theorem 3.3 Suppose that . Take as follows:
Let be defined by
where , , for each , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , and ;
-
(iv)
for some .
Then , where .
Proof By , there exists such that . For each , is a -strictly pseudo-nonspreading mapping and , by Lemma 2.9, we have is quasi-nonexpansive and . For each , by Lemma 2.8 and , we see that is demiclosed.
For each , let and for all in Theorem 3.1, then is quasi-nonexpansive with demiclosed property and . It follows from Theorem 3.1 that , where . □
Remark 3.2 Theorem 3.3 improves Theorem 3.1 in [8]. Indeed, Theorem 3.1 in [8] establishes a weak convergence. In Theorem 3.1 in [8], if there exists some positive integer m such that is semicompact, then converges strongly to . However, Theorem 3.3 is a strong convergence theorem without semicompact assumption.
Let for all and for all in Theorem 3.2, we have the following theorem.
Theorem 3.4 Suppose that for each is a nonspreading mapping, and let be a finite family of nonspreading mappings. Let . Take as follows:
Let be defined by
for each , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof For each , is a nonspreading mapping, hence is a quasi-nonexpansive mapping and a -strictly pseudo-nonspreading mapping with . For each , is a nonspreading mapping, hence is a -strictly pseudo-nonspreading mapping with .
Let for all and for all in Theorem 3.2. It follows from Theorem 3.2 and that , where . This point is also a unique solution of the following hierarchical variational inequality:
□
-
(I)
We study split common fixed points for a finite family of -strictly pseudo-nonspreading mappings, zeros for a countable family of mappings which are the sum of two monotone mappings and common fixed points for a countable family of quasi-nonexpansive mappings:
and
Theorem 3.5 Suppose that . Take as follows:
Let be defined by
for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for all and for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof By Theorem 4.1 in [32] and Lemma 2.1(iv), (v), we have that
for each .
For each , let and let . Then is a quasi-nonexpansive mapping and
Thus, . By Theorem 3.1, we have that , where . This point is also a unique solution of the following hierarchical variational inequality:
and the proof is completed. □
Remark 3.3 Theorem 3.5 improves Theorem 10 in [33] in the sense that our convergence is for the fixed point of a countable family of quasi-nonexpansive mappings and for the split feasibility problem.
-
(II)
We study split common fixed points for a finite family of -strictly pseudo-nonspreading mappings, zeros for a countable family of mappings which are the sum of two monotone mappings and common fixed points of a countable family of -strictly pseudo-nonspreading mappings:
and
Theorem 3.6 Suppose that . Take as follows:
Let be defined by
for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for all and for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof For each , let . Then is a quasi-nonexpansive mapping and for each . Then Theorem 3.6 follows immediately from Theorem 3.5. □
-
(III)
We study split common fixed points for a finite family of -strictly pseudo-nonspreading mappings, zeros for a countable family of monotone mappings and common fixed points for a countable family of -strictly pseudo-nonspreading mappings:
and
Theorem 3.7 Suppose that . Take as follows:
Let be defined by
where , , for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for all and for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof For each , is a -strictly pseudo-nonspreading mapping and , by Lemma 2.9, we have is quasi-nonexpansive. By Lemma 2.1, we see that is averaged for each .
For each , let and let in Theorem 3.1. Then is a quasi-nonexpansive mapping and
Thus, . Then Theorem 3.7 follows immediately from Theorem 3.1. □
-
(IV)
Let be a bifunction satisfying conditions (A1)-(A4). We study split common solutions for a countable family of equilibrium problems, common fixed points for a countable family of quasi-nonexpansive mappings and common fixed points of a finite family of -strictly pseudo-nonspreading mappings.
That is,
for all , and
Theorem 3.8 For each , let be as in Lemma 2.7. Suppose that . Take as follows:
Let be defined by
for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof For each , by Lemma 2.7, we know that and is a maximal monotone operator with the domain of . For each , by Lemma 2.3, we know that is firmly nonexpansive, so is quasi-nonexpansive. For each , let , and let in Theorem 3.1. Then is a quasi-nonexpansive mapping and
. By Theorem 3.1, we have that , where . This point is also a unique solution of the following hierarchical variational inequality:
The proof is completed. □
Remark 3.4 Theorem 3.8 improves Theorem 3.1 in [34] and Theorem 15 in [33] in the sense that our convergence is for the split feasibility problem and for the common solutions of a countable family of equilibrium problems and fixed points for a countable family of quasi-nonexpansive mappings.
-
(V)
Let be a bifunction satisfying conditions (A1)-(A4). We study split common solutions for a countable family of equilibrium problems, common fixed points for a countable family of -strictly pseudo-nonspreading mappings and common fixed points of a finite family of -strictly pseudo-nonspreading mappings:
for all , and
Theorem 3.9 For each , let be as in Lemma 2.7, and suppose that . Take as follows:
Let be defined by
where , , for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof For each , let and . Then is a quasi-nonexpansive mapping and
. By Theorem 3.1, we have that , where . This point is also a unique solution of the following hierarchical variational inequality:
and the proof is completed. □
-
(VI)
For each , let be a hemicontinuous monotone mapping. We study split common solutions for a countable family of variational inequality problems, common fixed points of a countable family of quasi-nonexpansive mappings and common fixed points of a finite family of -strictly pseudo-nonspreading mappings:
where
and
For each , let for all in Theorem 3.9, we have the following theorem.
Theorem 3.10 For each , let be as in Lemma 2.7 and suppose that is bounded on any line segment of C; that is, for each , there exists such that for all . Suppose that . Take as follows:
Let be defined by
where for all , for each , , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof Since is a hemicontinuous monotone mapping, it is sufficient to show that for each , satisfies conditions (A1)-(A4). Since is monotone, we have
and
Therefore (A2) is satisfied. Since is a hemicontinuous mapping, for any , the mapping defined by is continuous, where has a weak topology. Since is bounded on any line segment of C and each ,
it is easy to see that . This shows that condition (A3) is satisfied. It is easy to see that (A1) and (A4) are satisfied. Then Theorem 3.10 follows immediately from Theorem 3.8. □
Remark 3.5 (i) It is easy to see that if is continuous on C, then is hemicontinuous and bounded on any line segment of C.
(ii) It is easy to see that Theorem 3.10 is true if for each , is a -inverse strongly monotone mapping or is a continuous strongly monotone mapping. Theorem 3.10 improves Theorem 3.1 of Qing and Shang [35] in the sense that our convergence is for the split feasibility problem and for the common solutions of a countable family of variational inequality problems of hemicontinuous monotone mappings and fixed points for a countable family of quasi-nonexpansive mappings.
(VII) For each , let be a hemicontinuous monotone mapping. We study split common solutions for a countable family of variational inequality problems, common fixed points for a countable family of -strictly pseudo-nonspreading mappings and common fixed points of a finite family of -strictly pseudo-nonspreading mappings:
and
Applying Theorem 3.9 and following the same argument as in Theorem 3.10, we have the following result.
Theorem 3.11 For each , let be as in Lemma 2.7 and suppose that is bounded on any line segment of C. Suppose that . Take as follows:
Let be defined by
where , , for all , , for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
-
(VIII)
For each , let be a hemicontinuous pseudo-contractive mapping. We study split common fixed points for a countable family of hemicontinuous pseudo-contractive mappings, a countable family of quasi-nonexpansive mappings and a finite family of -strictly pseudo-nonspreading mappings:
and
For , let for all in Theorem 3.11, we have the following theorem.
Theorem 3.12 For each , let be as in Lemma 2.7 and suppose that is bounded on any line segment of C. Suppose that . Take as follows:
Let be defined by
where for all , for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Proof Let . Since, for each , is a pseudo-contraction, it is easy to see that is monotone. Then Theorem 3.12 follows from Theorem 3.10. □
Remark 3.6 Theorem 3.12 improves Theorem 3.1 of Cheng et al. in [13], Theorem 3.1 of Deng [14], Theorem 3.1 of Zegeye and Shahzad in [16] in the sense that our convergence is for the split feasibility problem and for the common fixed points of a countable family of quasi-nonexpansive mappings and a countable family hemicontinuous pseudo-contractive mappings without any closedness or Lipschitz continuity assumption on these pseudo-contractive mappings.
-
(IX)
For each , let be a hemicontinuous pseudo-contractive mapping. We study split common fixed points for a countable family of hemicontinuous pseudo-contractive mappings, a countable family of -strictly pseudo-nonspreading mappings and a finite family of -strictly pseudo-nonspreading mappings:
and
For each , let in Theorem 3.12, we have the following theorem.
Theorem 3.13 Suppose that for each , is bounded on any line segment of C and . Take as follows:
Let be defined by
where , , for all , for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
-
(X)
For each , let be a hemicontinuous pseudo-contractive mapping. We study a split feasibility problem for common fixed points of a countable family of hemicontinuous pseudo-contractive mappings, zeros for a countable family of mappings which are the sum of two monotone mappings and common fixed points of a finite family of -strictly pseudo-nonspreading mappings.
That is,
and
For each , let in Theorem 3.12, we have the following theorem.
Theorem 3.14 Suppose that for each , is bounded on any line segment of C and . Take as follows:
Let be defined by
where for all , for each , , , and . Assume the following:
-
(i)
;
-
(ii)
and ;
-
(iii)
, , , and ;
-
(iv)
for all and for some .
Then , where . This point is also a unique solution of the following hierarchical variational inequality:
Remark 3.7 Theorem 3.14 improves Theorem 3.1 of Zegeye and Shahzad [10], Theorem 3.1 of Yao et al. [11], Theorem 4.1 of Takahashi et al. [28], Theorem 2.1 of Cho et al. [36] and Theorem 3.1 of Tang [9] in the sense that our convergence is for the split feasibility problem and for the common solutions of a countable family of variational inclusion problems and fixed points of a countable family hemicontinuous pseudo-contractive mappings without any closedness or Lipschitz continuity assumption on these pseudo-contractive mappings.
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Yu, ZT., Lin, LJ. Variational inequality problems over split fixed point sets of strict pseudo-nonspreading mappings and quasi-nonexpansive mappings with applications. Fixed Point Theory Appl 2014, 198 (2014). https://doi.org/10.1186/1687-1812-2014-198
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DOI: https://doi.org/10.1186/1687-1812-2014-198