تجزیه و تحلیل حساسیت برای پارامتری غیر خطی تعمیم یافته (A، η، M) (A، η، M) سیستم ورود اپراتور یکنواخت حداکثری با اپراتور نوع اجباری مشترک آرام

By using Lim’s inequalities, Nadler’s results, the new parametric resolvent operator technique associated with (A,η,m)(A,η,m)-maximal monotone operators, in this paper, the existence theorem for a new class of generalized nonlinear parametric (A,η,m)(A,η,m)-maximal monotone operator inclusion systems with relaxed cocoercive type operators in Hilbert spaces is analyzed and established. Our results generalize sensitivity analysis results of other recent works on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions and nonlinear mixed quasi-variational inclusion systems in Hilbert spaces.

For i=1,2i=1,2, let XiXi be real Hilbert space, ΛiΛi be nonempty open subset of XiXi in which the parameter λiλi take values, and let S,E:X1×Λ1→2X1S,E:X1×Λ1→2X1 and T,G:X2×Λ2→2X2T,G:X2×Λ2→2X2 be multi-valued operators, and gi,pi:Xi×Λi→Xigi,pi:Xi×Λi→Xi, N1:X1×X2×Λ1→X1N1:X1×X2×Λ1→X1 and N2:X1×X2×Λ2→X2N2:X1×X2×Λ2→X2 be single-valued operators. Suppose that for i=1,2i=1,2, View the MathML sourceAi:Xi→Xi, View the MathML sourceηi:Xi×Xi→Xi and View the MathML sourceMi:Xi×Xi×Λi→2Xi are any nonlinear operators such that for all (ϖ,λ1)∈X1×Λ1(ϖ,λ1)∈X1×Λ1, M1(⋅,ϖ,λ1):X1→2X1M1(⋅,ϖ,λ1):X1→2X1 is an (A1,η1,m1)(A1,η1,m1)-maximal monotone operator with (g1−p1)λ1(X1)∩dom(M1(⋅,ϖ,λ1))≠0̸(g1−p1)λ1(X1)∩dom(M1(⋅,ϖ,λ1))≠0̸ and for all (w,λ2)∈X2×Λ2(w,λ2)∈X2×Λ2, M2(⋅,w,λ2):X2→2X2M2(⋅,w,λ2):X2→2X2 is an (A2,η2,m2)(A2,η2,m2)-maximal monotone operator with (g2−p2)λ2(X2)∩dom(M2(⋅,w,λ2))≠0̸(g2−p2)λ2(X2)∩dom(M2(⋅,w,λ2))≠0̸, respectively, where (gi−pi)λi(v)=(gi−pi)(v(λi),λi)(gi−pi)λi(v)=(gi−pi)(v(λi),λi) for λi∈Λiλi∈Λi and v(λi)∈Xiv(λi)∈Xi. Throughout this paper, unless otherwise stated, we shall consider the following generalized nonlinear parametric (A,η,m)(A,η,m)-maximal monotone operator inclusion systems: For each fixed View the MathML sourceλi∈Λi(i=1,2), find (u(λ1),v(λ2))∈X1×X2(u(λ1),v(λ2))∈X1×X2 such that x(λ1)∈Sλ1(u)x(λ1)∈Sλ1(u), y(λ2)∈Tλ2(v)y(λ2)∈Tλ2(v), z∈Eλ1(u)z∈Eλ1(u), ω(λ2)∈Gλ2(v)ω(λ2)∈Gλ2(v) and equation(1.1) View the MathML source{0∈N1(u(λ1),y(λ2),λ1)+Mλ11((g1−p1)λ1(u),z),0∈N2(x(λ1),v(λ2),λ2)+Mλ22((g2−p2)λ2(v),ω), Turn MathJax on where View the MathML sourceMλii(u,v)=Mi(u(λi),v(λi),λi) for all View the MathML source(u,v,λi)∈Xi×Xi×Λi,i=1,2. Example 1.1. If E:X1×Λ1→X1E:X1×Λ1→X1 and G:X2×Λ2→X2G:X2×Λ2→X2 are single-valued operators, then for each fixed View the MathML sourceλi∈Λi(i=1,2), the problem (1.1) reduces to finding (u(λ1),v(λ2))∈X1×X2(u(λ1),v(λ2))∈X1×X2 such that x(λ1)∈Sλ1(u)x(λ1)∈Sλ1(u), y(λ2)∈Tλ2(v)y(λ2)∈Tλ2(v) and equation(1.2) View the MathML source{0∈N1(u(λ1),y(λ2),λ1)+Mλ11((g1−p1)λ1(u),Eλ1(u)),0∈N2(x(λ1),v(λ2),λ2)+Mλ22((g2−p2)λ2(v),Gλ2(v)). Turn MathJax on The problem (1.2) is called a system of general nonlinear parametric mixed quasi-variational-like inclusions, which was studied by us [1] when gi−pi=E=G≡Igi−pi=E=G≡I for i=1,2i=1,2, the identity operator. Further, the problem (1.2) was introduced and studied by Agarwal and Verma [2] if E=G≡IE=G≡I, gi−pi=0gi−pi=0 for i=1,2i=1,2, and operators u(λ1)=uu(λ1)=u and x(λ1)=xx(λ1)=x for all λ1∈Λ1λ1∈Λ1, v(λ2)=vv(λ2)=v and y(λ2)=yy(λ2)=y for all λ2∈Λ2λ2∈Λ2, and Ni(u,v,λi)=Ni(u,v)Ni(u,v,λi)=Ni(u,v) for all (u,v,λi)∈X1×X2×Λi(u,v,λi)∈X1×X2×Λi and i=1,2i=1,2 in (1.2). Example 1.2. If S:X1×Λ1→X1S:X1×Λ1→X1 and T:X2×Λ2→X2T:X2×Λ2→X2 are single-valued operators, then for each fixed View the MathML sourceλi∈Λi(i=1,2), the problem (1.2) reduces to finding (u(λ1),v(λ2))∈X1×X2(u(λ1),v(λ2))∈X1×X2 such that equation(1.3) View the MathML source{0∈N1(u(λ1),Tλ2(v),λ1)+Mλ11((g1−p1)λ1(u),Eλ1(u)),0∈N2(Sλ1(u),v(λ2),λ2)+Mλ22((g2−p2)λ2(v),Gλ2(v)). Turn MathJax on Example 1.3. If X1=X2=XX1=X2=X, View the MathML sourcegi−pi=I(i=1,2), View the MathML sourceMλ11(u,v)=Mλ11(u) for all (u,v,λ1)∈X×X×Λ1(u,v,λ1)∈X×X×Λ1 and View the MathML sourceMλ22(u,v)=Mλ22(u) for all (u,v,λ2)∈X×X×Λ2(u,v,λ2)∈X×X×Λ2, then the problem (1.3) is equivalent to the following system of generalized nonlinear parametric mixed quasi-variational inclusions: find (u(λ1),v(λ2))∈X×X(u(λ1),v(λ2))∈X×X such that View the MathML source{0∈N1(u(λ1),Tλ2(v),λ1)+Mλ11(u),0∈N2(Sλ1(u),v(λ2),λ2)+Mλ22(v), Turn MathJax on which was studied by Jeong and Kim [3] when View the MathML sourceMλii is HH-accretive operators and NiNi is a special case for i=1,2i=1,2. We remark that for appropriate and suitable choices of NiNi, MiMi, SS, TT, EE, GG, gigi, pipi, AiAi, ηiηi and XiXi for i=1,2i=1,2, it is easy to see that the problem (1.1) is a generalized version of some problems, which includes a number (systems) of (parametric) quasi-variational inclusions, (parametric) generalized quasi-variational inclusions, (parametric) quasi-variational inequalities, (parametric) implicit quasi-variational inequalities studied by many authors as special cases, see, for example, [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] and [23] and the references therein. Further, the study of such types of problems is motivated by an increasing interest in the sensitivity analysis (or existence) of solutions for variational inclusion problems involving strongly monotone and relaxed cocoercive mappings under suitable second order and regularity assumptions have been carried out based on the general resolvent operator techniques by several researchers, see, for example, [1], [2], [3], [4], [5], [6], [8], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] and [23] and the references therein. Moreover, it also generalizes the theory of multi-valued maximal monotone mappings and provides a general framework to examine convex programming and other variational inclusion problems. General resolvent operator techniques have been in use for a while and are being applied to a broad range of problems arising from model equilibria problems in economics, optimization and control theory, operations research, transportation network modelling, and mathematical programming. Example 1.4 [24]. Let XX be a real Hilbert space, and M:dom(M)⊂X→XM:dom(M)⊂X→X be an operator on XX such that MM is monotone and R(I+M)=XR(I+M)=X, Then based on the Yosida approximation View the MathML sourceMρ=1ρ(I−(I+ρM)−1), for each given u0∈dom(M)u0∈dom(M), there exists exactly one continuous function u:[0,1)→Xu:[0,1)→X such that the following first-order evolution equation: equation(1.4) View the MathML source{u′(t)+Mu(t)=0,0<t<∞,u(0)=u0, Turn MathJax on where the derivative u′(t)u′(t) exists in the sense of weak convergence, that is, View the MathML sourceu(t+h)−u(t)h⇀u′(t)as h→0. Turn MathJax on holds for all t∈(0,∞)t∈(0,∞). As Verma [20] pointed out “Among significant applications of the notion of AA-maximal (m)(m)-relaxed monotonicity, it seems that one can generalize the Yosida approximation and apply it to the solvability of evolution equations of the form (1.4)”. It is well known that it is easier to solve the Yosida approximate evolution equation than Eq. (1.4). Further, regarding the results of Refs. [25], [26] and [27], we know that the studied problem can be used to extend some duality results in nonsmooth optimization literature. On the other hand, in [15], the author introduced a new concept of (A,η)(A,η)-monotone operators, which generalizes the (H,η)(H,η)-monotonicity and AA-monotonicity in Hilbert spaces and other existing monotone operators as special cases, and some properties of (A,η)(A,η)-monotone operators were studied and the resolvent operator associated with (A,η)(A,η)-monotone operators was defined. At the same time, Verma [19] introduced the notion of (A,η)(A,η)-maximal monotonicity (also referred to as (A,η)(A,η)-monotonicity) in the context of studying sensitivity analysis for quasi-variational inclusions using the resolvent operator technique. For other related works, we refer to [2], [6], [13], [17] and [19] and the references therein. Very recently, by using the resolvent operator technique of (K,η)(K,η)-monotone mappings and the property of a fixed point set of multi-valued contractive mappings, Ding and Wang [6] studied the behavior and sensitivity analysis for a new system of parametric generalized mixed quasi-variational inclusions involving (K,η)(K,η)-monotone mappings under the assumption that set-values mapping GG is αGαG-strongly monotone and Lipschitz continuous in the first argument and other suitable conditions. However, we note that in 2001, Liu and Li [16] showed that the Lipschitz continuous multi-valued operator cannot be monotone, that is, all the monotone and Lipschitz continuous multi-valued mappings are single-valued mappings indeed. Inspired and motivated by the above works, the purpose of this paper is to analyze and establish the existence theorem for a new class of generalized nonlinear parametric (A,η,m)(A,η,m)-maximal monotone operator inclusion systems with relaxed cocoercive type operators based upon using some results provided by Lim [28] and Nadler [29], and the new parametric resolvent operator technique associated with (A,η,m)(A,η,m)-maximal monotone operators in Hilbert spaces. Our results generalize sensitivity analysis results of other recent works on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions and nonlinear mixed quasi-variational inclusion systems in Hilbert spaces.