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Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with mappings in ordered Banach spaces
Fixed Point Theory and Applications volume 2014, Article number: 122 (2014)
Abstract
In this paper, a new class of general set-valued parametric ordered variational inclusions, , with mappings is studied in ordered Banach spaces. Then, by using fixed point theory and the resolvent operator associated with set-valued mappings, an existence theorem and a sensitivity analysis of the solution set for this kind of parametric variational inclusion is proved and discussed in ordered Banach spaces. The obtained results seem to be general in nature.
MSC:49J40, 47H06.
1 Introduction
Generalized nonlinear ordered variational inequalities and inclusions (ordered equation) have wide applications in many fields including, for example, mathematics, physics, optimization and control, nonlinear programming, economics, and engineering sciences etc. In recent years, nonlinear mapping fixed point theory and applications have been extensively studied in ordered Banach spaces [1–3]. 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 [4]. 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 in ordered Banach spaces [5]. By using the resolvent operator associated with an set-valued mapping, the author introduced and studied a class of nonlinear inclusion problems for ordered MR set-valued mappings 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 [6]. In 2012, the author introduced and studied a class of nonlinear inclusion problems, to find such that ( is a set-valued mapping) for ordered set-valued mappings, and he then, applying the resolvent operator associated with set-valued mappings, established the existence theorem on the solvability and a general algorithm applied to the approximation solvability of this class of nonlinear inclusion problems, based on the existence theorem and the new model in ordered Hilbert space [7]. For Banach spaces, the author made a 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 [8]. In this field, the obtained results seem to be general in nature. In 2013, the author introduced and studied characterizations of ordered -weak- set-valued mappings, which was applied to finding an approximate solution for a new class of general nonlinear mixed-order quasi-variational inclusions involving the ⊕ operator in ordered Banach spaces [9], and, applying the matrix analysis and the vector-valued mapping fixed point analysis method, he introduced and studied a new class of generalized nonlinear mixed-order variational inequalities systems with ordered B-restricted-accretive mappings for ordered Lipschitz continuous mappings in ordered Banach spaces [10].
On the other hand, as everyone knows, the sensitivity analysis for a class of general nonlinear variational inequalities (inclusions) has wide applications to many fields. In 1999, Noor and Noor have studied a sensitivity analysis for strongly nonlinear quasi-variational inclusions [11]. From 2000, Agarwal et al. have discussed a sensitivity analysis for strongly nonlinear quasi-variations in Hilbert spaces by using the resolvent operator technique [12]; furthermore, Bi et al. [13], Lan et al. [14, 15], Dong et al. [16], Jin [17], Verma [18], Li et al. [9], and Li [19] have shown the existence of solutions and made a sensitivity analysis for a class of nonlinear variational inclusions involving generalized nonlinear mappings in Banach spaces, respectively. Recently, it has become of the highest interest that we are studying a new class of nonlinear ordered inclusion problems for ordered set-valued mappings and a sensitivity analysis of the solution set for this kind of parametric variational inclusions in ordered Banach spaces by using the resolvent operator technique [20] associated with ordered set-valued mappings. For details, we refer the reader to [1–35] and the references therein.
Let X be a real ordered Banach space with a norm , zero θ, and a partial ordering relation ≤ defined by the normal cone P, and a normal constant N of P [4]. Let Ω be a nonempty open subset of X and we have the parametric . Let (), be a single-valued mapping and be a set-valued mapping. We consider the following problem:
Find () such that
and the solution of the inclusion problem (1.1) is continuous from Ω and X.
Problem (1.1) is called a nonlinear generalized set-valued parametric ordered variational inclusions for ordered set-valued mappings in ordered Banach spaces.
Remark 1.1 When mapping M is single-valued and , then the problem (1.1) reduces to problem (2.1) in [4].
When the mapping is set-valued, then the problem (1.1) reduces to problem (1.1) in [7].
Inspired and motivated by recent research work in this field, in this paper, a new class of nonlinear generalized parametric ordered variational inclusions with mappings is studied in ordered Banach spaces. Then, by using the resolvent operator associated with set-valued mappings, an existence theorem of this class of nonlinear inclusions is established, and a sensitivity analysis of the solution set for this kind of parametric variational inclusions is proved and discussed in ordered Banach spaces. The obtained results 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 ordering 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 ordering 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 If , then and exist, , and .
Proof If , then or . Let , then and , and . It follows that and exist, and . , then . □
Lemma 2.3 If for any natural number n, , and (), then .
Proof If for any natural number n, and (), then or for any natural number n. Since P is a nonempty closed convex subsets of X, we have or . Therefore, . □
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 ordering relation in X;
-
(2)
;
-
(3)
;
-
(4)
;
-
(5)
let λ be a 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 [4] and Lemma 2.3 in [5], and (8)-(13) directly follow from (1)-(8). □
Definition 2.5 Let X be a real ordered Banach space, let Ω be a nonempty open subset of X in which the parametric ρ takes values, let (), be a single-valued mapping and be a set-valued mapping and be a nonempty closed subset in X.
-
(1)
A set-valued mapping M is said to be a comparison mapping, if for any , , and if , then for any and any , ().
-
(2)
A set-valued mapping M is said to be a comparison mapping with respect to g, if for any , , and if , then for any and any , ().
-
(3)
A comparison mapping M is said to be an α-non-ordinary difference mapping, if there exists a constant , for each , , and such that
-
(4)
A comparison mapping M is said to be λ-ordered strongly monotonic increase mapping, if for there exists a constant such that
-
(5)
A comparison mapping M is said to be a mapping, if M is a α-non-ordinary difference and λ-ordered strongly monotone increasing mapping, and for .
Obviously, if M is a comparison mapping, then ().
Definition 2.6 [4]
Let X be a real ordered Banach space, P be a normal cone with a normal constant N in X; a mapping is said to be β-ordered compression, if A is comparison, and there exists a constant such that
Definition 2.7 [4]
Let X be a real ordered Banach space. A mapping is said to be a restricted-accretive mapping with constants , if A is a comparison, and there exist two constants such that for arbitrary ,
holds, where I is the identity mapping on X.
Definition 2.8 Let X be a real ordered Banach space, let Ω be a nonempty open subset of X in which the parametric ρ takes values, let (). is said to be a comparison element when, if then for any .
Lemma 2.9 Let . If M is a α-non-ordinary difference mapping, then an inverse mapping of is a single-valued mapping (), where I is the identity mapping on X.
Proof Let , and x and y be two elements in . It follows that and , and
Since M is a α-non-ordinary difference mapping, we have
and from Lemma 2.4. Also, holds. Thus is a single-valued mapping. The proof is completed. □
Definition 2.10 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in the X, let be a α-non-ordinary difference mapping. The resolvent operator of the is defined by
where is a constant.
3 Existence theorem of the solution
In this section, we will show an existence theorem on the solvability of this class of nonlinear inclusion problems (1.1).
Theorem 3.1 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in X, let ≤ be an ordering relation defined by the cone P. If is an α-non-ordinary difference mapping, then the inclusion problem (1.1) has a solution x if and only if in X.
Proof This directly follows from the definition of the resolvent operator of . □
Theorem 3.2 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in the X, let ≤ be an ordering relation defined by the cone P, the operator ⊕ be a operator. If is an mapping with respect to , then the resolvent operator is a comparison mapping.
Proof Since is an α-non-ordinary difference mapping and a comparison mapping with respect to so that . For any , let , and and . Setting
by using the λ-order strongly monotonicity of M, we have
and if then , and if then . Therefore for Lemma 2.4. □
Theorem 3.3 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in X, let ≤ be an ordering relation defined by the cone P. Let be a set-valued mapping with respect to . If , then for the resolvent operator , the following relation holds:
Proof Let be a set-valued mapping with respect to . For , let , and , then for . Since is an mapping with respect to the , the following relation holds by Lemma 2.4 and the condition :
It follows that and from the condition . The proof is completed. □
Theorem 3.4 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in the X, let ≤ be an ordering relation defined by the cone P. Let be an set-valued mapping with respect to the first argument and be a γ-ordered compression and an 1-ordered strongly monotonic increase with respect to the first argument and , and for M with respect to the first argument and for M with respect to the second argument be two restricted-accretive mappings with constants and , respectively, and . Suppose that for any
and
hold. For any parametric , for the nonlinear parametric inclusion problem (1.1) there exists a solution .
Proof Let X be a real ordered Banach space, let P be a normal cone with normal constant N in the X, let ≤ be an ordering relation defined by the cone P, let Ω be a nonempty open subset of X in which the parametric ρ takes values, let , and for any given and for . If , and setting
where , by (3.1) and the λ-ordered strongly monotonicity of M,
by , the comparability of , and the 1-ordered monotonic increase of , it follows from that . Using (3.3), (3.5), Lemma 2.4, Theorem 3.3, and , from the conditions that for M with respect to the first argument and for M with respect to the second argument are two restricted-accretive mappings with constants and , respectively, it follows that
and, by Definition 2.1(2), we obtain
where . It follows from the condition (3.4) that , and has a fixed point and the is a solution of the generalized nonlinear ordered parametric equation
Further, satisfies the generalized nonlinear ordered parametric equation
Then for the nonlinear parametric inclusion problems (1.1) there exists a solution for any parametric . This completes the proof. □
Remark 3.5 Though the method of solving problem by the resolvent operator is the same as in [20, 25–28] and [34] for the nonlinear inclusion problem, the character of the ordered set-valued mapping is different from the one of the -accretive mapping [25], the -monotone mapping [26], the -monotone mapping [27] and the monotone mapping [34].
4 Sensitivity analysis of the solution
Theorem 4.1 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in the X, let ≤ be an ordering relation defined by the cone P. Let be a set-valued mapping and be a γ-ordered compression, continuous and 1-ordered monotonic increase of with respect to first argument , and , and for M with respect to first argument and for M with respect to second argument be two restricted-accretive mappings with constants and , respectively, and . Suppose that for any
and
hold; if the solution of the nonlinear parametric inclusion problem (1.1) is a comparison element, which is said to be a comparison solution of the nonlinear parametric inclusion problem (1.1), then , a comparison solution, is continuous on Ω.
Proof For any given , by Theorem 3.4, let be a comparison solution, and and satisfy parametric problem (1.1), then for any , we have
By the condition that M, g, , and are comparisons for each other and by Lemma 2.4, we have
Further, and are two restricted-accretive mappings with constants and , respectively, so that from Lemma 2.4 and Theorem 3.3, , and from (3.6), it follows that
where for the condition (4.1), and
Combining (4.4), (4.5), and (4.6), and by using Lemma 2.4, we get
Therefore,
It follows that
By Lemma 2.4, , and continuity of g with respect to the first argument , we have
and
which implies that the solution of problem (1.1) is continuous at . This completes the proof. □
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The main idea of this paper was proposed by HGL, and HGL and LPL etc. prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Li, H.G., Li, L.P., Zheng, J.M. et al. Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with mappings in ordered Banach spaces. Fixed Point Theory Appl 2014, 122 (2014). https://doi.org/10.1186/1687-1812-2014-122
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DOI: https://doi.org/10.1186/1687-1812-2014-122