Inversely, let m ∈ E − ( m 1, m 2 ) ∩ E + ( m 1, m 2 ) ; then R ( m 1, m, m 2 ) = m and so m is the unique ( m 1, m 2 ) -stabilizable mean.
Let A M 1 ̃, p 1 be the covering transformation group of the cover p 1 : M ̃ 1 → M 1 and A M 2 ̃, p 2 the covering transformation group of the cover p 2 : M ̃ 2 → M 2. For every α ∈ A M ̃ 1, p 1, let f× be the unique covering transformation in A M 2 ̃, p 2 that satisfies f ̃ α = f × f ̃.
Proof (1) Take 0 < M < h, where h is as in Lemma 2.1, and let v = v M be the unique positive solution of (2.4).
Let S be a non-empty subset of a Riemannian manifold M, which is a geodesic η-invex set with respect to η : M × M → T M, and let x and u be two arbitrary points of S. Let γ : [ 0, 1 ] → M be the unique geodesic such that γ ( 0 ) = u, γ ′ ( 0 ) = η ( x, u ), γ ( t ) ∈ S, for all t ∈ [ 0, 1 ].
Then Z G ( m ) is a reductive R -group and its radical contains a unique maximal split torus S ( m ) of G. (2) Let S is a maximal split torus of G, and let m ( S ) be the unique maximal subalgebra of Lie algebra Lie ( S ) which is an AD subalgebra of g (see Remark 6.4).
Then, the Green's operator G is defined as G : C ∞ ( ∧ l M ) → H ⊥ ∩ C ∞ ( ∧ l M ) by assigning G u) be the unique element of H ⊥ ∩ C ∞ ( ∧ l M ) satisfying Poisson's equation ΔG u) = u - H u), where H is the harmonic projection operator that maps C∞ ∧ l M) onto H so that H u) is the harmonic part of u [[7, 8], for more properties of these operators].
Let { ν n, m } be the unique solution of the system of equations { ν − A B ν = f n, m, ν − B B ν = g n, m, ν − S T ν = t n, m, ν − T T ν = r n, m, where A, B, S, T : X → X satisfy the following conditions: (d1) The pairs { A, S } and { B, T } are compatible of type (B), (d2) A 2 = B 2 = S 2 = T 2 = I, where I denotes the identity mapping, and.
Let { ν n, m } be the unique solution of the system of equations { F ν − A B ν = f n, m, F ν − B B ν = g n, m, F ν − S T ν = t n, m, F ν − T T ν = r n, m, where F, A, B, S, T : X → X satisfy the following conditions: (d1) The pairs { A, S } and { B, T } are compatible of type (B), (d2) A 2 = B 2 = S 2 = T 2 = I, where I denotes the identity mapping, and.
For a fixed ε > 0, let the process u ε = { u ε ( t, x ) ; 0 ≤ t ≤ T, x ∈ R m } be the unique solution to equation (11).
Let (u=u_{varepsilon}in W_{1}(M,T)) be the unique weak solution of the problem ((P_{varepsilon})).
