Solving Yosida inclusion problem in Hadamard manifold
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Arabian Journal of Mathematics
Mohammad Dilshad
Solving Yosida inclusion problem in Hadamard manifold
Received: 2 October 2018 / Accepted: 26 June 2019 © The Author(s) 2019
Abstract We consider a Yosida inclusion problem in the setting of Hadamard manifolds. We study Korpelevich-type algorithm for computing the approximate solution of Yosida inclusion problem. The resolvent and Yosida approximation operator of a monotone vector field and their properties are used to prove that the sequence generated by the proposed algorithm converges to the solution of Yosida inclusion problem. An application to our problem and algorithm is presented to solve variational inequalities in Hadamard manifolds. Mathematics Subject Classification
49J53 · 47J20 · 48C06
1 Introduction Variational inequalities introduced by Hartman and Stampacchia have been studied in different spaces, namely Hilbert spaces, Banach spaces, see for example [2,6,7,15,23]. There are various problems in applied sciences which can be formulated as variational inequalities or boundary value problems on manifolds. Therefore, the extensions of the concepts and techniques of the theory of variational inequalities and related topics from Euclidean spaces to Riemannian or Hadamard manifolds are natural and interesting but not easy. Németh introduced the concept of variational inequalities on Hadamard manifold: Find x ∈ K such that F(x), exp−1 x y ≥ 0, ∀ y ∈ K , where K is nonempty closed, convex subset of Hadamard manifold M. F : K → T M is a vector field, that is F(x) ∈ Tx M for each x ∈ K and exp−1 is the inverse of exponential mapping. Németh generalized some basic existence and uniqueness results of the classical theory of variational inequality from Euclidean space to Hadamard manifold which is simply connected complete Riemannian manifold with nonpositive sectional curvature. Li et al. [12] studied the variational inequality problem on Riemannian manifolds. Fang and Chen [8] proved the convergence of projection algorithm to estimate the solution of set-valued variational inequalities on Hadamard manifolds. Noor et al. [17] studied Two-steps methods to solve variational inequalities in Hadamard manifolds. An important generalization of variational inequalities is variational inclusion. The inclusion problem 0 ∈ B(x) for set-valued monotone operator B on Hilbert space H is formulated as mathematical model of many problems arising in operation research, economics, physics, etc. It is well known that set-valued monotone operator can be regularized into a single-valued monotone operator by the process known as the Yosida approximation. Yosida approximation is a tool to solve a variational inclusion problem using nonexpansive resolvent operator. Due to the fact that the zeros of maximal monotone operator are the fixed point sets of resolvent operator, the resolvent associated with a set-valued maximal monotone operator plays an M. Dilshad (B) Faculty of Science, Department of Mathematics, University of Tabuk, Tabuk 71491, Kingdom of Saudi Arabia E-mail: mdils
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