Abstract
In ordered Banach spaces, characterizations of ordered -weak-ANODD set-valued mappings are introduced and studied, which is applied to giving an approximate solution for a new class of general nonlinear mixed-order quasi-variational inclusions involving ⊕ operator. By using the resolvent operator associated with an -weak-ANODD set-valued mapping and fixed point theory, an existence theorem of solutions and an approximation algorithm for this kind of inclusions are established and discussed in ordered Banach spaces, and the relation between the first-valued point and the solution of the problems is shown. The results obtained seem to be general in nature.
MSC:49J40, 47H06.
Similar content being viewed by others
1 Introduction
Generalized nonlinear ordered variational inequalities (ordered equations) have wide applications in many fields including, for example, mathematics, physics, optimization and control, nonlinear programming, economics, and engineering sciences.
The variational inclusion, which was introduced and studied by Hassouni and Moudafi [1], is a useful and important extension of the variational inequality. In 1989, Chang and Zhu [2] introduced and investigated a class of variational inequalities for fuzzy mappings. Afterwards, Chang and Huang [3], Ding and Jong [4], Jin [5], Li [6] and others studied several kinds of variational inequalities (inclusions) for fuzzy mappings. A generalized random multivalued quasi-complementarity problem was introduced and studied by Chang and Huang [7], and the random variational inclusion (inequalities, equalities, quasi-variational inclusions, quasi-complementarity) problems were studied by Ahmad and Bazán [8], Chang [9], Cho et al. [10]. In recent years, Huang and Fang [11] introduced the concept of generalized m-accretive mapping, studied the properties of the resolvent operator with the generalized m-accretive mapping; and furthermore, Huang and Fang [12] studied a class of generalized monotone mappings, maximal η-monotone mappings, and defined an associated resolvent operator in 2003. Using resolvent operator methods, they developed some iterative algorithms to approximate the solution of a class of general variational inclusions involving maximal η-monotone operators. Huang and Fang’s method extended the resolvent operator method associated with an η-subdifferential operator due to Ding and Luo [13]. In [14], Fang and Huang introduced another class of generalized monotone operators, H-monotone operators, defined an associated resolvent operator, established the Lipschitz continuity of the resolvent operator, and studied a class of variational inclusions in Hilbert spaces using the resolvent operator associated with H-monotone operators. In a recent paper [15], Fang, Huang and Thompson further introduced a new class of generalized monotone operators, -monotone operators, which provided a unifying framework for classes of maximal monotone operators, maximal η-monotone operators, and H-monotone operators. Very recently, Lan et al. [16] introduced a new concept of -accretive mappings, which generalized the existing monotone or accretive operators, and studied some properties of mappings. They also studied a class of variational inclusions using the resolvent operator associated with -accretive mappings.
On the other hand, in 1972, a number of solutions of nonlinear equations were introduced and studied by Amann [17], and in recent years, the nonlinear mapping fixed point theory and application have been intensively studied in ordered Banach spaces [18–20]. Therefore, it is very important and natural that generalized nonlinear ordered variational inequalities (ordered equation) are studied and discussed.
In 2008 the author introduced and studied the approximation algorithm and the approximation solution for a class of generalized nonlinear ordered variational inequalities and ordered equations to find such that ( and are single-valued mappings) in ordered Banach spaces [21]. By using the B-restricted-accretive method of the mapping A with constants , , the author introduced and studied a new class of general nonlinear ordered variational inequalities and equations to find such that (, and are single-valued mappings), and established an existence theorem and an approximation algorithm of solutions for this kind of generalized nonlinear ordered variational inequalities (equations) in ordered Banach spaces [22]. By using the resolvent operator associated with an RME set-valued mapping, the author introduced and studied a class of nonlinear inclusion problems for ordered MR set-valued mappings to find such that ( is a set-valued mapping), and the existence theorem of solutions and an approximation algorithm for this kind of nonlinear inclusion problems for ordered extended set-valued mappings in ordered Hilbert spaces [23]. In 2012, the author introduced and studied a class of nonlinear inclusion problems to find such that ( is a set-valued mapping) for ordered -NODM set-valued mappings, and then, applying the resolvent operator associated with -NODM set-valued mappings, established the existence theorem on the solvability and a general algorithm applied to the approximation solvability of the nonlinear inclusion problem of this class of nonlinear inclusion problems, based on the existence theorem and the new -NODM model in an ordered Hilbert space [24]. In Banach spaces, the author proved sensitivity analysis of the solution for a new class of general nonlinear ordered parametric variational inequalities to find such that (, and are single-valued mappings) in 2012 [25]. In this field, the obtained results seem to be general in nature. Now, it is of excellent interest that we are studying the characterizations of ordered -weak-ANODD set-valued mappings, which is applied to solving an approximate solution for a new class of general nonlinear mixed-order quasi-variational inclusions involving ⊕ operator in ordered Banach spaces. For details, we refer the reader to [1–48] and the references therein.
Let X be a real ordered Banach space with a norm , a zero θ, a normal cone P, a normal constant N and a partial ordered relation ≤ defined by the cone P [21]. Let be a single-valued ordered compression mapping, and let and
be two set-valued mappings. We consider the following problem:
For , find such that
Problem (1.1) is called a general nonlinear mixed-order quasi-variational inclusions (GNMOQVI) involving ⊕ operator in an ordered Banach space.
In recent years, the nonlinear mapping fixed point theory and its applications have been intensively studied in ordered Banach spaces [21–26]. And very recently, the author introduced and studied the approximation theory, the approximation algorithm and the approximation solution for generalized nonlinear ordered variational inequalities (ordered equations, inclusion problems) in ordered Hilbert spaces or ordered Banach spaces [21–25].
Inspired by recent research work in this field, we introduce a new class of general nonlinear mixed-order quasi-variational inclusion problems involving ⊕ operator in an ordered Banach space. By using the resolvent operator techniques [27–37, 44, 45, 48] associated with an ordered -weak-ANODD set-valued mapping with a strongly comparison mapping A, an existence theorem of solutions and an approximation algorithm for this kind of problems are studied, and the relation between the first-valued point and the solution of the problems is discussed. The results obtained seem to be general in nature.
2 Preliminaries
Let X be a real ordered Banach space with a norm , a zero θ, a normal cone P, a normal constant N and a partial ordered relation ≤ defined by the cone P. For arbitrary , and express the least upper bound of the set and the greatest lower bound of the set on the partial ordered relation ≤, respectively. Suppose that and exist. Let us recall some concepts and results.
Let X be a real Banach space with a norm , θ be a zero element in X.
-
(i)
A nonempty closed convex subset P of X is said to be a cone if
-
(1)
for any and any , holds,
-
(2)
if and , then ;
-
(ii)
P is said to be a normal cone if and only if there exists a constant , a normal constant of P, such that for , holds;
-
(iii)
For arbitrary , if and only if ;
-
(iv)
For , x and y are said to be comparative to each other if and only if (or ) holds (denoted by for and ).
Lemma 2.2 [18]
If , then and exist, , and .
Lemma 2.3 [18]
If for any natural number n, and (), then .
Let X be an ordered Banach space, let P be a cone of X, let ≤ be a relation defined by the cone P in Definition 2.1(iii). For , the following relations hold:
-
(1)
the relation ≤ in X is a partial ordered relation in X;
-
(2)
;
-
(3)
;
-
(4)
;
-
(5)
let λ be real, then ;
-
(6)
if x, y, and w can be comparative to each other, then ;
-
(7)
let exist, and if and , then ;
-
(8)
if x, y, z, w can be compared with each other, then ;
-
(9)
if and , then ;
-
(10)
if , then ;
-
(11)
if , then ;
-
(12)
;
-
(13)
if and , and , then and .
Proof (1)-(8) come from Lemma 2.5 in [21] and Lemma 2.3 in [22], and (8)-(13) directly follow from (1)-(8). □
Definition 2.5 [24]
Let X be a real ordered Banach space, let be a single-valued mapping, and let be a set-valued mapping.
-
(1)
A single-valued mapping A is said to be a γ-order non-extended mapping if there exists a constant such that
-
(2)
A single-valued mapping A is said to be a strongly comparison mapping if A is a comparison mapping, and if and only if for any .
3 Characterizations of ordered -weak-ANODD set-valued mappings in ordered Banach spaces
Definition 3.1 Let X be a real ordered Banach space, let be a single-valued mapping, and let be a set-valued mapping.
-
(1)
A set-valued mapping M is said to be a weak-comparison mapping if for any , , and if , then there exist and such that ().
-
(2)
A weak-comparison mapping M is said to be an α-week-non-ordinary difference mapping with respect to A if for each , there exist a constant and and such that
-
(3)
A weak-comparison mapping M is said to be a λ-order different weak-comparison mapping with respect to B if there exists a constant , , and there exist , such that
-
(4)
A weak-comparison mapping M is said to be an ordered -weak-ANODD mapping with respect to B if M is an α-weak-non-ordinary difference with respect to A and a λ-order different weak-comparison mapping with respect to B, and for .
Remark 3.2 Let X be a real ordered Banach space, let be a single-valued mapping, and let be a set-valued mapping, then the following properties hold obviously:
-
(i)
A λ-order different comparison mapping must be a λ-order monotone mapping;
-
(ii)
An ordered -ANODD mapping must be an ordered -ANODM mapping [24];
-
(iii)
A comparison, an α-non-ordinary difference mapping, a λ-order different comparison mapping, or an ordered -ANODD mapping, a set-valued mapping M must be a weak-comparison mapping, an α-weak-non-ordinary difference mapping, a λ-order different weak-comparison mapping, or an ordered -weak-ANODD mapping, respectively.
Definition 3.3 [24]
Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, let A be a γ-order non-extended mapping, and let M be an α-non-ordinary difference mapping with respect to A. The resolvent operator of M is defined by
where are three constants.
Lemma 3.4 Let X be a real ordered Banach space. If A is a γ-order non-extended mapping, M is an α-weak-non-ordinary difference mapping with respect to A and , then is an α-weak-non-ordinary difference mapping with respect to A and an inverse mapping of is a single-valued mapping (), that is, the resolvent operator of exists.
Proof Let X be a real ordered Banach space, let A be a γ-order non-extended mapping, and let M be an α-weak-non-ordinary difference mapping with respect to A, then for each , there exist a constant and and such that
By using Lemma 2.4(11), we consider
Therefore, is surely an α-weak-non-ordinary difference mapping with respect to A.
Let , and let x and y be two elements in . It follows that and from . And
Since is an α-weak-non-ordinary difference mapping with respect to A, and A is a γ-order non-extended mapping, it follows
and from Lemma 2.4(5)(11)(13). It follows that from Definition 2.5(1). Thus is a single-valued mapping and the resolvent operator of exists. The proof is completed. □
Lemma 3.5 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, let ≤ be an ordered relation defined by the cone P, let the operator ⊕ be an XOR operator. If the resolvent operator of M exists, and M is a λ-order different weak-comparison mapping with respect to and A is a strongly comparison mapping, then the resolvent operator is a comparison mapping.
Proof Let X be a real ordered Banach space, and let the resolvent operator of M exist. If is a λ-order different weak-comparison mapping with respect to , and for any , , it follows from Definition 3.1(1) that there exist and such that . Then we have
and
Therefore, by using and Lemma 2.4(1). It follows that from strong comparability of A. The proof is completed. □
Lemma 3.6 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, and let ≤ be an ordered relation defined by the cone P. Let A be a γ-order non-extended mapping, and M be an ordered -weak-ANODD mapping with respect to . If , then for the resolvent operator , the following relation holds:
Proof Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, and let ≤ be an ordered relation defined by the cone P. Let A be a γ-order non-extended mapping and M be an α-weak-non-ordinary difference mapping with respect to A, it follows that exists from Lemma 3.5.
For any , let , then there exist and such that for . Since M is an ordered -weak-ANODM mapping with respect to , the following relation holds by Lemma 2.4(7) and the condition :
Since A is a γ-order non-extended mapping and , we have
and
The proof is completed. □
Theorem 3.7 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, and let ≤ be an ordered relation defined by the cone P. Let A be a γ-order non-extended mapping, and let M be an ordered -weak-ANODD mapping with respect to . If , then the resolvent operator is continuous.
Proof Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, let ≤ be an ordered relation defined by the cone P. Let A be a γ-order non-extended mapping, and let be an ordered -weak-ANODD mapping with respect to . If , then
holds.
Let the sequence and , then we have
and
Therefore, by [21]. If , then obviously. The proof is completed. □
4 Approximate solution for GNMOQVI problem (1.1)
In this section, we show the algorithm of approximation sequences for finding a solution of general nonlinear mixed-order quasi-variational inclusions problem (1.1) involving ⊕ operator in an ordered Banach space, and we discuss the convergence and the relation between the first-valued and the solution of problem (1.1) in X, real Banach spaces.
Theorem 4.1 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, let ≤ be an ordered relation defined by the cone P, and let the operator ⊕ be an XOR operator. Let be two single-valued ordered compression mappings and , . If A is a γ-order non-extended strongly comparison mapping and is an α-weak-non-ordinary difference mapping with respect to A, then inclusion problem (1.1) has a solution if and only if
Proof This directly follows from (1.1), Lemma 2.4, and the definition of the resolvent operator of . □
Theorem 4.2 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in the X, and let ≤ be an ordered relation defined by the cone P. Let be two single-valued β, ξ ordered compression mappings, respectively, let A be a γ non-extended strongly comparison mapping, , , let M be a -weak-non-ordinary difference mapping with respect to A and be a λ-order different weak-comparison mapping with respect to and for . Then is an ordered -weak-ANODD mapping with respect to . And if and
(where , , and they are constants), then a sequence converges strongly to solution of problem (1.1), which is generated by the following algorithm:
For any given , let , set
thus the following holds:
Proof Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, let ≤ be an ordered relation defined by the cone P. By using Lemma 3.4-Lemma 3.6 and Theorem 3.7, is an ordered -weak-ANODD mapping with respect to .
For any , let , let be an ordered -weak-ANODD mapping with respect to , , and the comparability of , we know that by Lemma 2.2. Further, we can obtain a sequence , and (where ). Using Lemma 2.4 and (2) in Lemma 3.6, we have
By Definition 2.1(ii), we obtain
where . Hence, for any , we have
It follows from condition (4.1) that and , as , and so is a Cauchy sequence in the complete space X. Let as . By the conditions, we can have
We know that is a solution of inclusion problem (1.1). It is follows that () from Lemma 2.3, and (4.1)
holds. This completes the proof. □
Remark 4.3 Though the method of solving the problem by the resolvent operator is the same as that in [31–37], or [47] for the nonlinear inclusion problem, but the character of an ordered -ANODD set-valued mapping is different from the one of an -accretive mapping [31–37], or an -maximal monotone mapping [47].
References
Hassouni A, Moudafi A: A perturbed algorithms for variational inequalities. J. Math. Anal. Appl. 2001, 185: 706–712.
Chang SS, Zhou HY: On variational inequalities for fuzzy mappings. Fuzzy Sets Syst. 1989, 32: 359–367. 10.1016/0165-0114(89)90268-6
Chang SS, Huang NJ: Generalized complementarity problem for fuzzy mappings. Fuzzy Sets Syst. 1993, 55: 227–234. 10.1016/0165-0114(93)90135-5
Ding XP, Jong YP: A new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings. J. Comput. Appl. Math. 2002, 138: 243–257. 10.1016/S0377-0427(01)00379-X
Jin MM: Generalized nonlinear implicit quasi-variational inclusions with relaxed monotone mappings. Adv. Nonlinear Var. Inequal. 2004, 7(2):173–181.
Li HG:Iterative algorithm for a new class of generalized nonlinear fuzzy set-valued variational inclusions involving -monotone mappings. Adv. Nonlinear Var. Inequal. 2007, 10(1):89–100.
Chang SS, Huang NJ: Generalized random multivalued quasi-complementarity problem. Indian J. Math. 1993, 33: 305–320.
Ahmad R, Bazán FF: An iterative algorithm for random generalized nonlinear mixed variational inclusions for random fuzzy mappings. Appl. Math. Comput. 2005, 167: 1400–1411. 10.1016/j.amc.2004.08.025
Chang SS: Variational Inequality and Complementarity Problem Theory with Applications. Shanghai Sci. Technol., Shanghai; 1991.
Cho YJ, Huang NJ, Kang SM: Random generalized set-valued strongly nonlinear implicit quasi-variational inequalities. J. Inequal. Appl. 2000, 5: 515–531.
Huang NJ, Fang YP: Generalized m -accretive mappings in Banach spaces. J. Sichuan Univ. Eng. Sci. Ed. 2001, 38(4):591–592.
Huang NJ, Fang YP: A new class of general variational inclusions involving maximal η -monotone mappings. Publ. Math. (Debr.) 2003, 62(1–2):83–98.
Ding XP, Luo CL: Perturbed proximal point algorithms for generalized quasi-variational-like inclusions. J. Comput. Appl. Math. 2000, 210: 153–165.
Fang YP, Huang NJ: H -Monotone operator and resolvent operator technique for variational inclusions. Appl. Math. Comput. 2003, 145: 795–803. 10.1016/S0096-3003(03)00275-3
Fang YP, Huang NJ, Thompson HB:A new system of variational inclusions with -monotone operators in Hilbert spaces. Comput. Math. Appl. 2005, 49: 365–374. 10.1016/j.camwa.2004.04.037
Lan HY, Cho YJ, Verma RU:Nonlinear relaxed cocoercive inclusions involving -accretive mappings in Banach spaces. Comput. Math. Appl. 2006, 51: 1529–1538. 10.1016/j.camwa.2005.11.036
Amann H: On the number of solutions of nonlinear equations in ordered Banach space. J. Funct. Anal. 1972, 11: 346–384. 10.1016/0022-1236(72)90074-2
Du YH: Fixed points of increasing operators in ordered Banach spaces and applications. Appl. Anal. 1990, 38: 1–20. 10.1080/00036819008839957
Ge DJ, Lakshmikantham V: Couple fixed points of nonlinear operators with applications. Nonlinear Anal. TMA 1987, 3811: 623–632.
Ge DJ: Fixed points of mixed monotone operators with applications. Appl. Anal. 1988, 31: 215–224. 10.1080/00036818808839825
Li HG: Approximation solution for generalized nonlinear ordered variational inequality and ordered equation in ordered Banach space. Nonlinear Anal. Forum 2008, 13(2):205–214.
Li H-g: Approximation solution for a new class of general nonlinear ordered variational inequalities and ordered equations in ordered Banach space. Nonlinear Anal. Forum 2009, 14: 89–97.
Li H-g: Nonlinear inclusion problem for ordered RME set-valued mappings in ordered Hilbert space. Nonlinear Funct. Anal. Appl. 2011, 16(1):1–8.
Li H-g: Nonlinear inclusion problem involving - NODM set-valued mappings in ordered Hilbert space. Appl. Math. Lett. 2012, 25: 1384–1388. 10.1016/j.aml.2011.12.007
Li H-g: Sensitivity analysis for general nonlinear ordered parametric variational inequality with restricted-accretive mapping in ordered Banach space. Nonlinear Funct. Anal. Appl. 2011, 17(1):109–118.
Schaefer HH: Banach Lattices and Positive Operators. Springer, Berlin; 1974.
Lan HY, Cho YJ, Verma RU:On nonlinear relaxed cocoercive inclusions involving -accretive mappings in Banach spaces. Comput. Math. Appl. 2006, 3151: 1529–1538.
Li HG:Iterative algorithm for a new class of generalized nonlinear fuzzy set-valued variational inclusions involving -monotone mappings. Adv. Nonlinear Var. Inequal. 2007, 10(1):89–100.
Li HG, Xu AJ, Jin MM:A hybrid proximal point three-step algorithm for nonlinear set-valued quasi-variational inclusions system involving -accretive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 635382 10.1155/2010/635382
Li HG, Xu AJ, Jin MM:A Ishikawa-hybrid proximal point algorithm for nonlinear set-valued inclusions problem based on -accretive framework. Fixed Point Theory Appl. 2010., 2010: Article ID 501293 10.1155/2010/501293
Lan HY, Cho YJ, Verma RU: On solution sensitivity of generalized relaxed cocoercive implicit quasivariational inclusions with A -monotone mappings. J. Comput. Anal. Appl. 2006, 8: 75–87.
Lan HY, Cho YJ, Huang N-j: Stability of iterative procedures for a class of generalized nonlinear quasi-variational-like inclusions involving maximal η -monotone mappings. 6. Fixed Point Theory Appl. 2007, 107–116.
Lan HY, Kim JH, Cho YJ: On a new system of nonlinear A -monotone multivalued variational inclusions. J. Math. Anal. Appl. 2007, 327: 481–493. 10.1016/j.jmaa.2005.11.067
Cho YJ, Lan HY: A new class of generalized nonlinear multi-valued quasi-variational-like-inclusions with H -monotone mappings. Math. Inequal. Appl. 2007, 10: 389–401.
Lan HY, Kang JI, Cho YJ:Nonlinear -monotone operator inclusion systems involving non-monotone set-valued mappings. Taiwan. J. Math. 2007, 11: 683–701.
Cho YJ, Qin XL, Shang MJ, Su YF:Generalized nonlinear variational inclusions involving -monotone mappings in Hilbert spaces. Fixed Point Theory Appl. 2007., 2007: Article ID 29653
Cho YJ, Lan HY:Generalized nonlinear random -accretive equations with random relaxed cocoercive mappings in Banach spaces. Comput. Math. Appl. 2008, 55: 2173–2182. 10.1016/j.camwa.2007.09.002
Cho YJ, Qin X: Systems of generalized nonlinear variational inequalities and its projection methods. Nonlinear Anal. 2008, 69: 4443–4451. 10.1016/j.na.2007.11.001
Alimohammady M, Balooee J, Cho YJ, Roohi M:A new system of nonlinear fuzzy variational inclusions involving -accretive mappings in uniformly smooth Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 806727 10.1155/2010/806727
Alimohammady M, Balooee J, Cho YJ, Roohi M: Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities. Nonlinear Anal. 2010, 73: 3907–3923. 10.1016/j.na.2010.08.022
Alimohammady M, Balooee J, Cho YJ, Roohi M: New perturbed finite step iterative algorithms for a system of extended generalized nonlinear mixed-quasi variational inclusions. Comput. Math. Appl. 2010, 60: 2953–2970. 10.1016/j.camwa.2010.09.055
Yao Y, Cho YJ, Liou Y: Iterative algorithms for variational inclusions, mixed equilibrium problems and fixed point problems approach to optimization problems. Cent. Eur. J. Math. 2011, 9: 640–656. 10.2478/s11533-011-0021-3
Yao Y, Cho YJ, Liou Y: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 2011, 212: 242–250. 10.1016/j.ejor.2011.01.042
Li HG, Xu AJ, Jin MM: A hybrid proximal point three-step algorithm for nonlinear set-valued quasi-variational inclusions system involving (A, η )-accretive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 635382 10.1155/2010/635382
Li HG, Xu AJ, Jin MM: An Ishikawa-hybrid proximal point algorithm for nonlinear set-valued inclusions problem based on -accretive framework. Fixed Point Theory Appl. 2010., 2010: Article ID 501293 10.1155/2010/501293
Pan XB, Li HG, Xu AJ: The over-relaxed a-proximal point algorithm for general nonlinear mixed set-valued inclusion framework. Fixed Point Theory Appl. 2011., 2011: Article ID 840978 10.1155/2011/840978
Verma RU:A hybrid proximal point algorithm based on the -maximal monotonicity framework. Appl. Math. Lett. 2008, 21: 142–147. 10.1016/j.aml.2007.02.017
Li HG, Qiu M: Ishikawa-hybrid proximal point algorithm for NSVI system. Fixed Point Theory Appl. 2012., 2012: Article ID 13663 10.1186/1687-1812-2012-195
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11201512) and the Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, H.G., Qiu, D. & Zou, Y. Characterizations of weak-ANODD set-valued mappings with applications to an approximate solution of GNMOQV inclusions involving ⊕ operator in ordered Banach spaces. Fixed Point Theory Appl 2013, 241 (2013). https://doi.org/10.1186/1687-1812-2013-241
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-241