Perturbed Ishikawa-hybrid quasi-proximal point algorithm with accretive mappings for a fuzzy system
© Li et al.; licensee Springer. 2013
Received: 9 May 2013
Accepted: 26 September 2013
Published: 8 November 2013
A new class of fuzzy general nonlinear set-valued mixed quasi-variational inclusions frameworks for a perturbed Ishikawa-hybrid quasi-proximal point algorithm using the notion of -accretive is developed. Convergence analysis for the algorithm of solving a fuzzy nonlinear set-valued inclusions problem and existence analysis of a solution for the problem is explored along with some results on the resolvent operator corresponding to an -accretive mapping due to Lan et al. The result that the sequence generated by the perturbed Ishikawa-hybrid quasi-proximal point algorithm converges linearly to a solution of the fuzzy general nonlinear set-valued mixed quasi-variational inclusions with the convergence rate ε is proved.
The set-valued inclusions problem, which was introduced and discussed by Bella , Huang et al.  and Jeong , is a useful extension of the mathematics analysis. And the variational inclusion (inequality) is an important context in the set-valued inclusions problem. It provides us with a unified, natural, novel, innovative and general technique to study a wide class of problems arising in different branches of mathematical and engineering sciences. Various variational inclusions have been intensively studied in recent years. Ding , Verma , Huang , Fang and Huang , Lan et al. , Fang et al. , Zhang et al.  introduced the concepts of η-subdifferential operators, maximal η-monotone operators, H-monotone operators, A-monotone operators, -monotone operators, -accretive mappings, -monotone operators and defined resolvent operators associated with them, respectively. Moreover, by using the resolvent operator technique, many authors constructed some approximation algorithms for some nonlinear variational inclusions in Hilbert spaces or Banach spaces. In 2008, Li  studied the existence of solutions and the stability of a perturbed Ishikawa iterative algorithm for nonlinear mixed quasi-variational inclusions involving -accretive mappings in Banach spaces by using the resolvent operator technique.
On the other hand, in many scientific and engineering applications, the fuzzy set concept and the variational inequalities with fuzzy mappings play an important role. The fuzziness appears when we need to perform, on manifold, calculations with imprecision variables. The concept of fuzzy sets was introduced initially by Zadeh  in 1965. Since then, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and application [13, 14]. Various concepts of fuzzy metrics on an ordinary set were considered in [15, 16], and many authors studied fixed point theory for ordinary mappings in such fuzzy metric spaces [17–21]. A number of metrics are used on the subspaces of fuzzy sets. The sendograph metric [22, 23] and the d1-metric induced by the Hausdorff metric [24, 25] were studied most frequently. Some attention was also given to Lp-type metrics [26, 27]. These results have a very important application in quantum particle physics, particularly in connection with both string and e1-theory, which were given and studied by El-Naschie [28, 29].
From 1989, Chang and Zhu  introduced and investigated a class of variational inequalities for fuzzy mappings. Afterwards, Chang and Huang , Ding and Jong , Jin , Li  and others studied several kinds of variational inequalities (inclusions) for fuzzy mappings.
Recently, Verma has developed a hybrid version of the Eckstein-Bertsekas  proximal point algorithm, introduced the algorithm based on the -maximal monotonicity framework  and studied convergence of the algorithm. The author showed the general nonlinear set-valued inclusions problem based on an -accretive framework and suggested and discussed an Ishikawa-hybrid proximal point algorithm for solving the inclusions problem. On the other hand, from 1989, the generalized nonlinear quasi-variational inequalities (inclusions) with fuzzy mappings have been studied widely by a number of authors, who have more and more achievements [17–37], and we refer to [1–56] and references contained therein.
Inspired and motivated by recent research work in this field, in this paper, a fuzzy general nonlinear set-valued mixed quasi-variational inclusions framework for a perturbed Ishikawa-hybrid quasi-proximal point algorithm using the notion of -accretive is developed. Convergence analysis for the algorithm of solving a fuzzy nonlinear set-valued inclusions problem and existence analysis of a solution for the problem are explored along with some results on the resolvent operator corresponding to an -accretive mapping due to Lan et al. . The result that the sequence generated by the perturbed Ishikawa-hybrid quasi-proximal point algorithm converges linearly to a solution of the fuzzy general nonlinear set-valued mixed quasi-variational inclusions with the convergence rate ε is proved.
where is a constant.
Remark 2.1 In particular, is the usual normalized duality mapping, and (for all ). If is strictly convex , or X is a uniformly smooth Banach space, then is single-valued. In what follows, we always denote the single-valued generalized duality mapping by in a real uniformly smooth Banach space X unless otherwise stated.
Let us recall the following results and concepts.
- (i)accretive if
strictly accretive if A is accretive and if and only if ;
- (iii)r-strongly η-accretive if there exists a constant such that
- (iv)α-Lipschitz continuous if there exists a constant such that
- (v)A single-valued mapping is said to be -Lipschitz continuous if there exist constants such that
- (vi)Let be single-valued mappings, be a set-valued mapping. g is said to be -S-relaxed cocoercive with respect to A mapping, if there exist two constants such that for any , the following holds:
- (i)accretive if
- (ii)η-accretive if
- (iii)r-strongly accretive if there exists a constant such that
- (iv)m-relaxed η-accretive if there exists a constant such that
A-accretive if M is accretive and for all ;
-accretive if M is m-relaxed η-accretive and for all .
- (vii)h-Lipschitz continuous with constants ξ if
where is the Hausdorff metric in .
Based on the literature , we can define the resolvent operator as follows.
Definition 2.5 ()
where is a constant.
Remark 2.6 The -accretive mappings are more general than the -monotone mappings and m-accretive mappings in a Banach space or a Hilbert space, and the resolvent operators associated with -accretive mappings include as special cases the corresponding resolvent operators associated with -monotone operators, m-accretive mappings, A-monotone operators, η-subdifferential operators [3–11, 30–34].
Lemma 2.7 ()
In the study of characteristic inequalities in q-uniformly smooth Banach spaces, Xu  proved the following result.
Lemma 2.8 ()
3 Fuzzy general nonlinear set-valued mixed quasi variational inclusions with -accretive mappings
Let X be a real q-uniformly smooth Banach space with the dual space , be the dual pair between X and , denote the family of all the nonempty subsets of X, and denote the family of all nonempty closed bounded subsets of X.
Let be a collection of all fuzzy sets over X. A mapping is called a fuzzy mapping. For each , (denote it by in the sequel) is a fuzzy set on X and is the membership function of y in . For , , the set is called a q-cut set of .
Let for any , be a fuzzy mapping satisfying the condition :
there exists a function such that for all , , where denotes the family of all nonempty bounded closed subsets of X.
By using the fuzzy mapping , we can define a set-valued mapping by for each , and is called the set-valued mapping induced by the fuzzy mapping for any in the sequel, respectively.
Let ; be single-valued mappings. Let be a set-valued -accretive mapping and for each . We consider the fuzzy general nonlinear set-valued mixed quasi-variational inclusions with -accretive mappings (FGNSVMQVI):
where problem (3.1) is called fuzzy general nonlinear set-valued mixed quasi-variational inclusions with -accretive mappings.
If for any , is a set-valued mapping, we can define the fuzzy mapping by , where is the characteristic function of . Taking () for all , problem (3.1) is equivalent to the following problem:
which is called a new class of general nonlinear set-valued mixed quasi-variational inclusions with -accretive mappings (GNSVMQVI).
- (1)A special case of problem (3.1) is the following:
For a suitable choice of A, f, g, η, F, G, () and the space X, a number of known classes of fuzzy variational inclusions and fuzzy variational inequalities in [30–32, 37] can be obtained as special cases of the fuzzy general nonlinear set-valued mixed quasi-variational inclusions (3.1).
- (2)A special case of problem (3.2) is the following:
- (i)For any , find and such that(3.3)
If is a Hilbert space, is the zero operator in X, are the identity operators in X, , and , then problem (3.2) becomes the parametric usual variational inclusion with an -maximal monotone mapping M, which was studied by Verma .
If X is a real Banach space, is the identity operator in X, and , then problem (3.3) becomes the parametric usual variational inclusion with an -accretive mapping, which was studied by Li .
Furthermore, these types of fuzzy variational inclusions and variational inclusions can enable us to study many important nonlinear problems arising in mechanics, physics, optimization and control, nonlinear programming, economics, finance, regional, structural, transportation, elasticity and applied sciences in a general and unified framework.
4 The existence of solutions
Now, we study the existence of solutions for problem (3.1).
is a solution of problem (3.1), where , that is, ().
- (ii)For and any , the following relation holds:(4.1)
where , that is, ().
- (iii)For and any , the following relation holds:(4.2)
where is a constant, and , that is, ().
Proof This directly follows from the definition of . □
where is the same as in Lemma 2.8, and , then problem (3.1) has a solution , which ().
where (). This completes the proof. □
5 Perturbed Ishikawa-hybrid quasi-proximal point algorithm and solvability of problem (3.1)
Based on Lemma 4.1, we develop a perturbed Ishikawa-hybrid quasi-proximal point algorithm for finding iterative sequence solving problem (3.1) as follows.
where (), and are errors to take into account a possible inexact computation of the proximal point.
where is the Hausdorff metric in , .
Remark 5.2 For a suitable choice of the mappings A, η, f, g, F, G, , , space X, and nonnegative sequences , , , , , and , Algorithm 5.1 can be degenerated to a number of algorithms involving many known algorithms which are due to classes of variational inequalities and variational inclusions [10, 11, 30–34, 45, 47, 50, 56].
where is the same as in Lemma 2.8, , and the convergence rate is , and problem (3.1) has a solution , where ().
and the convergence rate is ε.
Therefore, the sequence generated by Algorithm 5.1 converges linearly to a solution with convergence rate ε in a Banach space.
Hence and therefore .
for any . By Lemma 4.1, we know that problem (3.1) has a solution , where (). This completes the proof. □
This work was supported by the Mathematical Tianyuan Foundation of China (Grant No. 11126087), the National Natural Science Foundation of China (Grant No. 11201512) and the Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001).
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