On maximal monotonicity of bifunctions on Hadamard manifolds
Assume that \(M\) is an \(n\)-dimensional Hadamard manifold. The \(n\)-dimensional tangent space of \(M\) at \(x,\) and the tangent bundle of \(M\)\ are, respectively, denoted by \(T_{x}M,\) and \(TM=\bigcup_{x\in M}T_{x}M\). Also, the geodesic \(\gamma=\gamma_{\nu}(\cdot,x)\) is determined by its position \(x\) and velocity \(\nu\) at \(x\). In addition, the exponential map \(\exp:T_{x}M\rightarrow M\) is defined by \(\exp_{x}\nu=\gamma_{\nu}(1,x)\), for each \(x\in M\). A set \(\Omega \subset M\) is called convex if \(\gamma:\left[ a,b\right] \rightarrow M\) is a geodesic such that \(x=\gamma (a)\in \Omega\) and \(y=\gamma (b)\in\Omega\), then \[ \gamma ((1-t)a+tb)\in \Omega, \ \ \ \forall t\in \left[ 0,1\right] . \] The convex hull of \(C\subset M\) is denoted by \(\operatorname{conv}(C)\). Let \(\Omega\subset M\) be a convex set. A function \(f:\Omega\rightarrow\mathbb{R}\) is called convex if for each geodesic segment \(\gamma:[a,b]\rightarrow\Omega\), with \(a<b\), the composition \(f\circ \gamma:[a,b]\rightarrow\mathbb{R}\) is convex. A vector \(s\in T_{p}M\) is said to be a subgradient of \(f\) at \(p\in\Omega\) if \[ f(q)\geq f(p)+\left \langle s,\exp_{p}^{-1}q\right \rangle ,\ \ \ \ \forall q\in \Omega. \] Let \(\chi (M)\) be the set of all set-valued vector fields \(A:M\rightarrow2^{TM}\) such that \(A(x)\subset T_{x}M\), for each \(x\in M\), and the domain \(\mathcal{D}(A)\) of \(A\) is closed and convex. A vector field \(T\in \chi (M)\) is said to be monotone if the following condition holds \[ \left \langle u,\exp_{x}^{-1}y\right \rangle \leq \left \langle v,-\exp_{y}^{-1}x\right \rangle ,\ \ \ \ \forall x,y\in \mathcal{D}(T) ,\ \forall u\in T(x)\ \text{ and }\ \forall v\in T(y). \] Also, \(T\in\) \(\chi (M)\) is called maximal monotone if it is monotone and the following implication holds for any \(x\in \mathcal{D}(T)\) and \(u\in T_{x}M\): \[ \left \langle u,\exp_{x}^{-1}y\right \rangle \leq \left \langle v,-\exp_{y}^{-1}x\right \rangle ,\ \ \ \ \forall y\in \mathcal{D}(T)\ \text{ and }\ \forall v\in T(y)\implies u\in T(x). \] Given a nonempty, closed and convex subset \(\Omega\) of \(M\), by the term ``bifunction'' we understand any function \(F:\Omega \times \Omega \rightarrow \mathbb{R}\) such that \(F(x,x)=0\) for all \(x\in \Omega\). A bifunction \(F\) is called monotone if \[ F(x,y)+F(y,x)\leq0,\ \ \ \ \forall x,y\in \Omega. \] Given any bifunction \(F:\Omega \times \Omega \rightarrow \mathbb{R}\), one defines the vector filed \(A^{F}:M\rightarrow2^{TM}\) by \[ A^{F}(x)=\left \{ \begin{array} [c]{cc} \left \{ u\in T_{x}M:F(x,y)\geq \left \langle u,\exp_{x}^{-1}y\right \rangle ,\ \ \forall y\in \Omega \right \} , & \text{if }x\in \Omega,\\ \emptyset, & \text{if }x\notin \Omega. \end{array} \right. \] Following the works of Hadjisavvas in the spirit of the Banach spaces, a monotone bifunction \(F\) is called maximal monotone if \(A^{F}\) is maximal monotone. Given a \(T\in\) \(\chi (M)\), one can define the bifunction \(G_{T}:D(T)\times D(T)\rightarrow \mathbb{R}\) by \[ G_{T}(x,y)=\sup_{u\in T(x)}\left \langle u,\exp_{x}^{-1}y\right \rangle . \] The main results of the paper under review are: Theorem 1. Let \(F:\Omega \times \Omega \rightarrow \mathbb{R}\) be a continuous monotone bifunction. Then, \(A^{F}\)\ is maximal monotone. Theorem 2. If \(T\in\) \(\chi (M)\) is monotone, then \(G_{T}\) is monotone. Moreover, if \(T\) is maximal monotone, then \(G_{T}\) is maximal monotone and \(A^{G_{T}}=T\). Theorem 3. Suppose that \(T\in\chi (M)\), \(D(T)=M,\) and \(T(x)\) is a closed and convex set. If \(G_{T}\) is maximal monotone, then \(T\) is maximal monotone. Also, the authors studied the existence of solutions to equilibrium problems in Hadamard manifolds obtained by perturbing the equilibrium bifunction.
scientific article; zbMATH DE number 6990996
scientific article