Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 9217091, 7 pages
http://dx.doi.org/10.1155/2016/9217091

Research Article

Stability for a New Class of GNOVI with ()-Weak-GRD Mappings in Positive Hilbert Spaces

Hong Gang Li,¹ Yongqin Yang,² Mao Ming Jin,³ and Qinghua Zhang¹

¹School of Science, Chongqing University of Posts and Telecommunications, Chongqing, Nan’an 400056, China
²School of Science, Chongqing Jiaotong University, Chongqing, Nan’an 400074, China
³Institute of Nonlinear Analysis Research, Changjiang Normal University, Chongqing, Fuling 400803, China

Received 9 March 2016; Accepted 7 August 2016

Academic Editor: Ben T. Nohara

Copyright © 2016 Hong Gang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By using ordered fixed point theory, we set up a new class of GNOVI structures (general nonlinear ordered variational inclusions) with -weak-GRD mappings, discuss an existence theorem of solution, consider a perturbed Ishikawa iterative algorithm and the convergence of iterative sequences generated by the algorithm, and show the stability of algorithm for GNOVI structures in positive Hilbert spaces. The results in the instrument are obtained.

1. Introduction

Stability for variational inequality or general nonlinear ordered variational inclusions problems are of course powerful tools to deal with the problems occurring in control, nonlinear programming, economics, engineering sciences and optimization, and so forth. In recent years, there are some achievements in terms of systems of inequalities [1], weak vector variational inequality [2], differential mixed variational inequalities [3], and so forth. Moreover, Jin [4] studied the stability for strong nonlinear quasi-variational inclusion involving H-accretive operators in 2006. After that the authors investigated some the stability problems of perturbed Ishikawa iterative algorithms for nonlinear variational inclusion problems involving -accretive mappings [5, 6].

On the other hand, in 1972, Amann [7] had the number of solutions of nonlinear equations in ordered Banach spaces. Focusing on the work done related to the fixed points of nonlinear increasing operators in ordered Banach spaces, it is worth mentioning that work done by Du [8] is quite interesting and applicable in pure and applied sciences. From 2008, the authors have some results with regard to the approximation algorithm, the approximation solution for a variety of generalized nonlinear ordered variational inequalities, ordered equations and inclusions, and sensitivity analysis for a class of parametric variational inclusions in ordered Banach spaces (see [7–22]). For related work, we refer the reader to [1–36] and the references therein.

Taking into account the importance of above-mentioned research works, in this paper, a new class of generalized nonlinear ordered variational inclusion structures, GNOVI structures, are introduced in positive Hilbert spaces. By using the resolvent operator for -weak-GRD set-valued mappings and fixed point theory, an existence theorem of solution for the GNOVI frameworks is established, a perturbed Ishikawa iterative algorithm is suggested, and the stability and the convergence of iterative sequences generated by the algorithm are discussed in positive Hilbert spaces. In this field, the results in the instrument are obtained.

2. Preliminaries and a New Class of GNOVI Structures

Let us recall the following results and concepts for research stability for a new class of GNOVI with -weak-GRD mappings in positive Hilbert spaces.

Let be real set, let be Hilbert space with an inner product , a norm , and a zero element , let nonempty closed convex subsets be a cone, let be a normal constant of , and let relation defined by a normal cone be a partial ordered relation in ; then formats an ordered Hilbert space for the ordered relation and and are said to be compared to each other (denoted by ) for if or holds in . If express the least upper bound of a binary set and express the greatest lower bound of a binary set on the partial ordered relation for any , and exist, and some binary operators can be defined as follows:

(i).(ii).(iii).

, , and are called AND, OR, and XOR operations, respectively; then is an ordered lattice [35].

Definition 1. An ordered Hilbert space with an inner product is said to be a positive Hilbert space (denoted by ) with a partially ordered relation , if

then holds, or with an inner product is said to be a nonpositive Hilbert space (denoted by ) with a partially ordered relation .

As an example, let be closed convex subsets and let defined by a normal cone be a partial ordered relation in (denoted by ); it is clear that is a positive Hilbert space with the partially ordered relation . However, when letting , then is closed convex subsets. Obviously, is a nonpositive Hilbert space with because for , (denoted by ).

The following results and structural relationships are achievements gained by some folks in ordered Banach spaces (see [8, 9, 16–28, 30–35]), and they are as same as right in positive Hilbert space .

Theorem 2 (see [9, 35]). Let be real set, let be an ordered Hilbert space, and let be an ordered lattice; then the following relations hold:

(1)If and can be compared, then (2).(3)If , then .(4)Let be real; then .(5) exists and .(6) exists and .(7)If , and can be compared to each other, then .(8)Let exist, and if and , then .(9)If , and can be compared to each other, then .(10) for .

For arbitrariness, , and .

Theorem 3 (see [8, 9, 16]). Let be an ordered Hilbert space and let be a partial ordered relation in ; then the following conclusions hold:

(i)If , then (1) and exist, (2) , and (3) .(ii)If, for any natural number , and , then .

Theorem 4 (see [22]). If is a positive Hilbert space and is a partial ordered relation in , then the inequalities,

(1)if  , then ,(2)if  , then ,(3)if  , then ,(4)if  , then ,(5)if  , then ,

hold for .

It is worth noting that metric inequalities in Theorem 4 are failure in nonpositive Hilbert space , for example, .

Definition 5. Let be a real positive Hilbert space, and let be a mapping. The mapping is said to be ordered Lipschitz continuous mapping with constants ; if and , then and there exist constants such that

Definition 6. Let be a real positive Hilbert space, let be a set-valued mapping, and let be a strong comparison and -ordered compressed mapping.

(1) is said to be a weak comparison mapping with respect to ; if, for any , , then there exist and such that , , and , where and are said to be weak-comparison elements, respectively.(2) with respect to is said to be a -weak ordered different comparison mapping with respect to ; if there exists a constant such that, for any , there exist , , holds, where and are said to be -elements, respectively.(3) is said to be an ordered rectangular mapping, if, for each , and any and any such that holds.(4) is said to be a -ordered rectangular mapping with respect to ; if there exists a constant , for any , there exist and such that

 holds, where and are said to be -elements, respectively.(5)A weak comparison mapping with respect to is said to be a -weak- mapping with respect to , if is a -ordered rectangular and -weak ordered different comparison mapping with respect to and for , and there exist and such that and are -elements, respectively.

Remark 7 (see [9]). Let be a real positive Hilbert space, let be a single-valued mapping, and let be a set-valued mapping; then one has the following:

(i)If (identical mapping), then a -ordered rectangular mapping must be ordered rectangularly in [15].(ii)An ordered RME mapping must be -weak-GRD in [15].(iii)A -ordered monotone mapping must be -weak ordered different comparison [22].

Theorem 8 (see [22]). Let be a real positive Hilbert space with normal constant , and let be a strong comparison and -ordered compressed mapping. Let be an -weak ordered rectangular set-valued mapping and is an identical mapping. Let mapping be an inverse mapping of .
If , , and is a -weak-GRD set-valued mapping with respect to , then the resolvent operator of is a single-valued comparison, and

for and , which are , , and -elements, respectively.

Let be real set, and let be a real positive Hilbert space with normal constant , a norm , an inner product , and zero . Let and be two set-valued mappings, and let and be two single-valued nonlinear ordered compression mappings. We consider the following structures.

For and any , find such that

which is called a new class of general nonlinear ordered variational inclusion structures (GNOVI structures) in positive Hilbert spaces.

Remark 9. (i) If , , and , then problem (5) becomes the ordered variational inequality , which was studied by Li [9].
(ii) If and , then problem (5) becomes the ordered variational inequality , which was studied by Li [10].
(iii) If , , and , then problem (5) becomes the ordered variational inequality , which was studied by Li et al. [20].

3. Existence Theorem of the Solution for GNOVI Structures

In this section, by using Definition 1 and Theorems 2–4 and 8, we study a new class of general nonlinear ordered variational inclusion structures in positive Hilbert spaces.

Lemma 10. Let be a real positive Hilbert space with normal constant , let be a strong comparison and -ordered compressed mapping, and let be a -weak ordered GRD set-valued mapping with respect to . Let and be two single-valued nonlinear mappings. Then inclusion problem (5) has a solution if and only if in .

Proof. For , take notice of the fact that if and only if ; this directly follows from the definition of and problem (5).

Theorem 11. Let be real set, and let be a real positive Hilbert space with an inner product and a normal constant . Let be a strong comparison and -ordered compressed mapping, let be an -ordered rectangular and -weak-GRD set-valued mapping with respect to , and let and be , , and -elements, respectively. Let be an ordered Lipschitz continuous mapping with constants , and let be single-valued nonlinear -ordered compression mapping. If and are compared to each other and and satisfy

then there exists a solution of GNOVI structures (5), which is a fixed point of .

Proof. Let be a positive Hilbert space with an inner product and a normal constant , let be a strong comparison and -ordered compression mapping, and let be a -weak-GRD set-valued mapping with respect to .

Since and by condition (6), we have

By Theorems 4 and 8 in [9] and the conditions, if , then for , and

It follows that has a fixed point , which is a solution for GNOVI (5), from Lemma 10 and for (7).

4. Stability of Algorithm for GNOVI Structures

Definition 12. Let be a self-mapping, , and let define an iteration procedure which yields a sequence of points in . Suppose that and converge to a fixed point of . Let and let . If implies that , then the iteration procedure defined by is said to be -stable or stable with respect to .

Lemma 13 (see [36]). Let be a nonnegative real sequence and let be a real sequence in such that . If there exists a positive integer such that

where for all and , then .

Based on Theorem 11, we can develop a new Ishikawa iterative sequence for solving problem (5) as follows.

Algorithm 14. Let be real set, and let be a real positive Hilbert space with normal constant . Let and be two sequences such that and . Let and be two sequences in introduced to take into account possible inexact computation, where and . For any given , the perturbed Ishikawa iterative sequence is defined by

Let be any sequence in and define by

where , and

Remark 15. For a suitable choice of the mappings , , , , , , , and and space , then Algorithm 14 can be degenerated to known the algorithms in [9].

Theorem 16. Let be the same as in Theorem 11, and let and be two sequences such that and . Let and be two sequences in introduced to take into account possible inexact computation, where and . If condition

holds, then one has the following:

(i)If , then sequence generated by (11) converges strongly to , and is a unique solution of problem (5).(ii)Moreover, if , then if and only if , where is defined by (12); that is, sequence generated by (11) is -stable.

Proof. Let be the same as in Theorem 11. If (13) holds then (7) is true.
In the first place, we show that (i) is right.
Let be a unique solution of problem (5); then we have

From (11), (14), and (9) and Theorems 2, 4, and 11, it follows that

where

Similarly, we can prove that

It follows that

from (15), (16), and (17) and Theorems 2 and 4.
By assumption (13), we have and deduce

for (18) and Theorem 11, and and .
Let

then (20) can be written as

It follows from Lemma 13 and that , and so converges strongly to unique solution of problem (5).
There is one more point; we prove (ii).
Let for . By (12) and Theorems 2 and 4 and (14), we obtain

As in the proof of inequality (19) and Theorem 11, we have

Since , by (23), we have

Suppose that ; we have for , Theorem 3, and .
Conversely, if , then, by (14) and , we get

From (12) and Theorem 11, it follows that

we can have

Sequence generated by (11) is -stable. This completes the proof.

Remark 17. For a suitable choice of the mappings , we can obtain known results [9, 22] as special cases of Theorem 11.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

The main idea of this paper was proposed by Hong Gang Li. Hong Gang Li, Yongqin Yang, Mao Ming Jin, and Qinghua Zhang prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

Acknowledgments

The authors acknowledge the support of the National Natural Science Foundation of China (no. 61472056) and the Educational Science Foundation of Chongqing (KJ1400426).

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