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

By using ordered fixed point theory, we set up a new class of GNOVI structures (general nonlinear ordered variational inclusions) with (γ G , λ)-weak-GRDmappings, 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.


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][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]).For related work, we refer the reader to  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.

Preliminaries and a New Class of GNOVI Structures
∧, ∨, and ⊕ are called AND, OR, and XOR operations, respectively; then (H, ∨, ∧, ≤) is an ordered lattice [35].Definition 1.An ordered Hilbert space H with an inner product ⟨⋅, ⋅⟩ is said to be a positive Hilbert space (denoted by H  ) with a partially ordered relation ≤, if  ≥ ,  ≥  for any ,  ∈ H; (1) then ⟨, ⟩ ≥ 0 holds, or H with an inner product ⟨⋅, ⋅⟩ is said to be a nonpositive Hilbert space (denoted by H  ) with a partially ordered relation ≤.
Theorem 3 (see [8,9,16]).Let H be an ordered Hilbert space and let ≤ be a partial ordered relation in H; then the following conclusions hold: Theorem 4 (see [22]).If H  is a positive Hilbert space and ≤ is a partial ordered relation in H  , then the inequalities, It is worth noting that (1)-( 5) metric inequalities in Theorem 4 are failure in nonpositive Hilbert space H  , for example, H NC 2 .Definition 5. Let H  be a real positive Hilbert space, and let  : H  × H  → H  be a mapping.The mapping  : H × H → H is said to be ordered Lipschitz continuous mapping with constants (, ]); if  ∝ V and  ∝ , then (, ) ∝ (, V) and there exist constants , ] > 0 such that (2) Definition 6.Let H  be a real positive Hilbert space, let  : H  → CB(H  ) be a set-valued mapping, and let  : H  → H  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 V  ∈ (()) and V  ∈ (()) such that  ∝ V  ,  ∝ V  , and V  ∝ V  , where V  and V  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  > 0 such that, for any ,  ∈ H  , there exist V  ∈ (()), V  ∈ (()), (V  − V  ) ∝  −  holds, where V  and V  are said to be -elements, respectively.(3)  is said to be an ordered rectangular mapping, if, for each ,  ∈ H  , and any V  ∈ () and any (4)  is said to be a   -ordered rectangular mapping with respect to ; if there exists a constant   ≥ 0, for any ,  ∈ H  , there exist V  ∈ (()) and V  ∈ (()) such that holds, where V  and V  are said to be   -elements, respectively.(5) A weak comparison mapping  with respect to  is said to be a (  , )-weak-GRD mapping with respect to , if  is a   -ordered rectangular and -weak ordered different comparison mapping with respect to  and ( + )(H  ) = H  for  > 0, and there exist V  ∈ (()) and V  ∈ (()) such that V  and V  are (  , )-elements, respectively.
Remark 7 (see [9]).Let H  be a real positive Hilbert space, let  : H  → H  be a single-valued mapping, and let  : H  → CB(H  ) 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 H  be a real positive Hilbert space with normal constant , and let  be a strong comparison and -ordered compressed mapping.Let  : H  → (H  ) be an   -weak ordered rectangular set-valued mapping and  is an identical mapping.Let mapping  ,  = ( + ) −1 : H  → 2 H  be an inverse mapping of ( + ).
Let R be real set, and let H  be a real positive Hilbert space with normal constant , a norm ‖ ⋅ ‖, an inner product ⟨⋅, ⋅⟩, and zero .Let  : H  → CB(H  ) and (H  ) = {V | V ∈ (H  )} be two set-valued mappings, and let  : H  → H  and  : H  × H  → H  be two single-valued nonlinear ordered compression mappings.We consider the following structures.
For  > 0 and any  ∈ R, find  ∈ H  such that which is called a new class of general nonlinear ordered variational inclusion structures (GNOVI structures) in positive Hilbert spaces.

Existence Theorem of the Solution for GNOVI Structures
In this section, by using Definition then there exists a solution  * of GNOVI structures (5), which is a fixed point of  ,  (() + (/)(, ())).
Proof.Let H  be a positive Hilbert space with an inner product ⟨⋅, ⋅⟩ and a normal constant , let  be a strong comparison and -ordered compression mapping, and let () = {V | V ∈ ()} : H  → CB(H  ) ( > 0) be a (  , )weak-GRD set-valued mapping with respect to  ,  .
Based on Theorem 11, we can develop a new Ishikawa iterative sequence for solving problem (5) as follows.
Algorithm 14.Let R be real set, and let H  be a real positive Hilbert space with normal constant .

Remark 15.
For a suitable choice of the mappings , , , ,   ,   , , and  and space H  , then Algorithm 14 can be degenerated to known the algorithms in [9].
Proof.Let R, H  , , ,  be the same as in Theorem 11.If (13) holds then ( 7) is true.
There is one more point; we prove (ii).
Remark 17.For a suitable choice of the mappings , , , , , we can obtain known results [9,22] as special cases of Theorem 11.