A class of nonlinear mixed ordered inclusion problems for ordered (α A , λ)-ANODM set-valued mappings with strong comparison mapping A Abstract The purpose of this paper is to introduce and study a new class of nonlinear mixed ordered inclusion problems in ordered Banach spaces and to obtain an existence theorem and a comparability theorem of the resolvent operator. Further, by using fixed point theory and the resolvent operator, the authors constructed and studied an approximation algorithm for this kind of problems, and they show the relation between the first valued x 0 and the solution of the problems. The results obtained seem to be general in nature. MSC: 49J40; 47H06Keywords: a new class of nonlinear mixed ordered inclusion problems; ordered (α A , λ ω )-ANODM set-valued mapping; strong comparison mapping; ordered Banach spaces; convergence Proof This directly follows from Lemma ., Lemma ., and Lemma .. Lemma . []Let X be a real ordered Banach space with norm · , a zero θ , a normal cone P, a normal constant N of P and a partial ordered relation ≤ defined by the cone P, and the operator ⊕ be a XOR operator. If A is a strong comparison mapping, and M : X →  X is a λ-ordered monotone mapping with respect to J A M,λ , then the resolvent operator J A M,λ : X → X is a comparison mapping. Lemma . []Let X be a real ordered Banach space with norm · , a zero θ , a normal cone P, a normal constant N of P and a partial ordered relation ≤ defined by the cone P, and the operator ⊕ be a XOR operator. If A is a strong comparison mapping, and M : X →  X is a λ-ordered monotone mapping with respect to J A M,λ , then the resolvent operator J A ωM, λ ω : X → X is a comparison mapping.Proof This directly follows from Lemma ., Lemma ., and Lemma .. Lemma . []Let X be a real ordered Banach space with norm · , a zero θ , a normal cone P, a normal constant N of P and a partial ordered relation ≤ defined by the cone P. If A is a γ -ordered non-extended mapping, and M : X →  X is a (α A , λ)-ANODM mapping, which is a α A -non-ordinary difference mapping with respect to A and λ-ordered monotone mapping with respect to J A M,λ , then for the resolvent operator J A M,λ : X → X, the following relation holds:where α A λ > .

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. Preliminaries and a New Class of GNOVI StructuresLet 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 R be real set, let H be Hilbert space with an inner product ⟨⋅, ⋅⟩, a norm ‖ ⋅ ‖, and a zero element , let nonempty closed convex subsets C ⊆ H be a cone, let be a normal constant of C, and let relation ≤ defined by a normal cone C

A general nonlinear framework for an Ishikawa-hybrid proximal point algorithm using the notion of A, η -accretive is developed. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding to A, η -accretive mapping due to Lan-Cho-Verma in Banach space. The result that sequence {x n } generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem with the convergence rate θ is proved.

In this paper, a new class of general set-valued parametric ordered variationalBanach spaces. Then, by using fixed point theory and the resolvent operator associated with (α, λ)-NODSM 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

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 (A, η)-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 (A, η)-accretive mapping due to Lan et al. The result that the sequence {x n } ∞ n=1 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. MSC: 49J40; 47H06