A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces.

*(English)*Zbl 1215.65119The essential aim of the present paper is to consider a new iterative scheme to study the approximate solvability problem for a system of generalized nonlinear variational inequalities in the framework of uniformly smooth and strictly convex Banach spaces by using the generalized projection approach. Here the authors consider some special cases of the problem (the so-called system of generalized nonlinear variational inequalities problem, SGNIVP): find \(x^*, y^*, z^* \in K\) (a nonempty closed convex subset of real strictly convex Banach space) such that for all \(x \in K\) (\(S,T,U: K^3 \rightarrow X^*\) are the nonlinear mappings)

(1)\(\langle S(y^*, z^*, x^*), x-x^*\rangle \geq 0\), \(\langle T(z^*, x^*, y^*), x-y^*\rangle \geq 0\), \(\langle U(x^*, y^*, z^*), x-z^*\rangle \geq 0\).

(I) If \(U=0\) and \((S,T)\) are bifurcations from \(K^2\) to \(X^*\) (the dual space) with \(S(y,x) = \varrho T_1(y,x) + Jx - Jy\) (\(J: E \rightarrow 2 ^{E^*}\) is the normalized duality mapping) and \(T(x,y) = \eta T_2 (x,y) + Jy - Jx\), where \(T_i: K^2 \rightarrow X^*\) is a mapping, \(i = 1,2\), then the problem (1) reduces to find \(x^*, y^* \in K\) such that for all \(x \in X\):

(2) \(\langle \varrho T_1 (y^*, x^*)+Jx^* - Jy^*, x-x^* \rangle \geq 0\), \(\langle \eta T_2 (x^*, y^*)+Jy^* - Jx^*, x-y^* \rangle \geq 0\), \(\forall x \in K\);

where \(\varrho\) and \(\eta\) are two positive constants.

(II) If \(S\) and \(T\) both are univariate mappings then the problem (2) reduces to the (SGNVIP). Find \(x^*, y^* \in K\) such that for all \(x \in X\);

(3) \(\langle \varrho T_1 (y^*) + Jx^* - Jy^*, x-x^*\rangle \geq 0\), \(\langle \eta T_2 (x^*) + Jy^* - Jx^*, x-y^*\rangle \geq 0\).

Main result: If \(X\) is a real smooth and strictly convex Banach space with Kadec-Klee property, \(K\) is a nonempty closed and convex subset of \(X\) with \(\theta \in K\) and \(S,T,U,: K^3 \rightarrow X^*\) be continuous mappings satisfying the condition: There exist a compact subset \(C \subset X^*\) and constants \(\varrho > 0\), \(\eta > 0\), \(\xi > 0\) such that \((J- \varrho S) (K^3) \cup (J- \eta T) (K^3) \cup (J- \xi U)(K^3) \cup C\), where \(J(x,y,z) = Jz\), \(\forall (x,y,z) \in K^3\) and \(\langle S(x,y,z), J^{-1}(Jz- \varrho S (x,y,z)\rangle \geq 0\), \(\langle T(x,y,z), J^{-1}(Jz- \eta T (x,y,z)\rangle \geq 0\) \(\langle U(x,y,z)\), \(J^{-1}(Jz- \eta U (x,y,z)\rangle \geq 0\), then problem (1) has a solution \( (x^*, y^*, z^*) \in K^3\) and \( x_n \rightarrow x^*\), \(y_n \rightarrow y^*\), \(z_n \rightarrow z^*\), and the sequences converge strongly to a unique solution \([x^*, y^*, z^*] \in K^3\).

The problem (1) is a more general system of generalized nonlinear variational inequality problem, which includes many kinds of well-known systems of variational inequalities as its special case.

(1)\(\langle S(y^*, z^*, x^*), x-x^*\rangle \geq 0\), \(\langle T(z^*, x^*, y^*), x-y^*\rangle \geq 0\), \(\langle U(x^*, y^*, z^*), x-z^*\rangle \geq 0\).

(I) If \(U=0\) and \((S,T)\) are bifurcations from \(K^2\) to \(X^*\) (the dual space) with \(S(y,x) = \varrho T_1(y,x) + Jx - Jy\) (\(J: E \rightarrow 2 ^{E^*}\) is the normalized duality mapping) and \(T(x,y) = \eta T_2 (x,y) + Jy - Jx\), where \(T_i: K^2 \rightarrow X^*\) is a mapping, \(i = 1,2\), then the problem (1) reduces to find \(x^*, y^* \in K\) such that for all \(x \in X\):

(2) \(\langle \varrho T_1 (y^*, x^*)+Jx^* - Jy^*, x-x^* \rangle \geq 0\), \(\langle \eta T_2 (x^*, y^*)+Jy^* - Jx^*, x-y^* \rangle \geq 0\), \(\forall x \in K\);

where \(\varrho\) and \(\eta\) are two positive constants.

(II) If \(S\) and \(T\) both are univariate mappings then the problem (2) reduces to the (SGNVIP). Find \(x^*, y^* \in K\) such that for all \(x \in X\);

(3) \(\langle \varrho T_1 (y^*) + Jx^* - Jy^*, x-x^*\rangle \geq 0\), \(\langle \eta T_2 (x^*) + Jy^* - Jx^*, x-y^*\rangle \geq 0\).

Main result: If \(X\) is a real smooth and strictly convex Banach space with Kadec-Klee property, \(K\) is a nonempty closed and convex subset of \(X\) with \(\theta \in K\) and \(S,T,U,: K^3 \rightarrow X^*\) be continuous mappings satisfying the condition: There exist a compact subset \(C \subset X^*\) and constants \(\varrho > 0\), \(\eta > 0\), \(\xi > 0\) such that \((J- \varrho S) (K^3) \cup (J- \eta T) (K^3) \cup (J- \xi U)(K^3) \cup C\), where \(J(x,y,z) = Jz\), \(\forall (x,y,z) \in K^3\) and \(\langle S(x,y,z), J^{-1}(Jz- \varrho S (x,y,z)\rangle \geq 0\), \(\langle T(x,y,z), J^{-1}(Jz- \eta T (x,y,z)\rangle \geq 0\) \(\langle U(x,y,z)\), \(J^{-1}(Jz- \eta U (x,y,z)\rangle \geq 0\), then problem (1) has a solution \( (x^*, y^*, z^*) \in K^3\) and \( x_n \rightarrow x^*\), \(y_n \rightarrow y^*\), \(z_n \rightarrow z^*\), and the sequences converge strongly to a unique solution \([x^*, y^*, z^*] \in K^3\).

The problem (1) is a more general system of generalized nonlinear variational inequality problem, which includes many kinds of well-known systems of variational inequalities as its special case.

Reviewer: Jan Lovíšek (Bratislava)

##### MSC:

65K15 | Numerical methods for variational inequalities and related problems |

49J40 | Variational inequalities |

49M25 | Discrete approximations in optimal control |

##### Keywords:

generalized projection mapping; system of generalized nonlinear variational inequalities; uniformly smooth Banach space; Lyapunov functional; normalized duality mapping
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\textit{S. S. Chang} et al., Appl. Math. Comput. 217, No. 16, 6830--6837 (2011; Zbl 1215.65119)

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