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Variational inclusion problems in Hadamard manifolds. (English) Zbl 1395.49013

A complete simply connected Riemannian manifold \((M,g)\) of non-positive sectional curvature is called a Hadamard manifold. If \(M\) is a Hadamard manifold , then the set of all single-valued vector fields \(A:M\to TM\) on a \(M\) is \(\Omega(M)\), and the set of all set-valued vector fields \(B:M\rightrightarrows2^{TM}\) such that \(B(x)\subset T_xM\) for all \(x\in\{x;\;B(x)\neq\varnothing\}\) is denoted by \(\chi(M)\). A vector field \(A\in\Omega(M)\) is said to be monotone if \(g(A(x),\exp_x^{-1}y)\leq g(A(y),-\exp_y^{-1}x)\) for all \(x,y\in M\), and a vector field \(B\in\chi(M)\) is said to be monotone if \(g(u,\exp_x^{-1}y)\leq g(v,-\exp_y^{-1}x)\) for all \(u\in B(x)\) and \(v\in B(y)\). In a Hilbert space \(H\), the variational inclusion problem is to find \(x\in C\) such that \(x\in(A+B)^{-1}(0)\) is a zero of \(A+B\), where \(A\) is a monotone operator and \(B\) is a maximal monotone operator. For a set-valued maximal monotone operator \(B:H\to2^H\) and \(\lambda>0\), the resolvent of order \(\lambda\) associated to \(B\) is a single-valued mapping \(J_\lambda^B:H\to H\) defined by \(J_\lambda^B(x)=(I+\lambda B)^{-1}(x)\). The set of zeros \(B^{-1}(0)\) of \(B\) coincides with the fixed point set of \(J_\lambda^B\). In [J. Math. Anal. Appl. 75, 287–292 (1980; Zbl 0437.47047)], S. Reich showed that \(J_\lambda^B(x)\) converges strongly to the metric projection onto \(A^{-1}(0)\) for every \(x\in H\).
In this paper, the authors consider variational inclusion problems in the setting of Hadamard manifolds. If \(M\) is a Hadamard manifold, \(A\in\Omega(M)\), and \(B\in\chi(M)\), then the variational inclusion problem is to find \(\bar x\in M\) such that \((A+B)(\bar x)=0\). The authors present the algorithm for computing the approximate solutions of the problem. They show that for a nonempty bounded, closed, and convex subset \(C\) of a Hadamard manifold \(M\) with constant curvature, a monotone and continuous vector field \(A\in\Omega(M)\), and a maximal monotone vector field \(B\in\chi(M)\) on \(C\), the authors create the computer program to construct a sequence \(\{x_n\}\), \(x_n\in C\), converging to a solution \(\bar x\) of the variational inclusion problem.

MSC:

49J53 Set-valued and variational analysis
49J20 Existence theories for optimal control problems involving partial differential equations
54C60 Set-valued maps in general topology

Citations:

Zbl 0437.47047
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