Invariant monotone vector fields on Riemannian manifolds

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1 Nonlinear Analsis 70 (2009) Invariant monotone vector fields on Riemannian manifolds A. Barani, M.R. Pouraevali Department of Mathematics, Universit of Isfahan, P.O.Box , Isfahan, Iran Received 7 Jul 2007; accepted 25 Februar 2008 Abstract Various concepts of invariant monotone vector fields on Riemannian manifolds are introduced. Some examples of invariant monotone vector fields are given. Several notions of invexities for functions on Riemannian manifolds are defined their relations with invariant monotone vector fields are studied. c 2008 Elsevier Ltd. All rights reserved. Kewords: Generalized invex functions; Monotone vector fields; Invariant monotone vector fields; Riemannian manifolds. Introduction The theor of variational inequalit has proven its applicabilit in man fields of mathematics phsics. On the other h monotonicit has plaed a ver important role in the stud of the existence of solutions of variational inequalit problems. It is also well known that generalized monotonicit is a powerful tool in the stud of the existence the sensitivit analsis of solutions for variational inclusion complementarit problems. The convexit of a real valued function is equivalent to the monotonicit of the corresponding gradient function; see [3,5,9]. The relation between generalized convexit of functions generalized monotone operators has been investigated b man authors; for example see [4,7,6,8]. In the nonsmooth case the relationships between monotonicit convexit of subdifferential operators as an application of the approximate Mean Value Theorem are studied in Section of [4]. In [] Hanson introduced the concept of invexit which was a generalization of convexit. Generalized invexit its relation with generalized invex monotonicit has been investigated in [6,23]. The notion of invariant monotonicit as a generalization of monotonicit was studied in [24]. In general a manifold is not a linear space extensions of concepts techniques from linear spaces to Riemannian manifolds are natural. Rapscák [20] Udriste [22] considered a generalization of convexit called geodesic convexit extended man results of convex analsis optimization theor to Riemannian manifolds. The also studied the monotonicit of the gradient of a geodesic convex function. The interested reader is referred to the books [20,22] references therein. The concept of monotone vector field on Riemannian manifolds which was a generalization of monotone operator was introduced in [5] b Németh. Since then numerous articles have appeared in the literature reflecting further generalizations applications in this categor; for example see [6,0]. Corresponding author. Tel.: ; fax: addresses: barani@math.ui.ac.ir (A. Barani), poura@math.ui.ac.ir (M.R. Pouraevali) X/$ - see front matter c 2008 Elsevier Ltd. All rights reserved. doi:0.06/j.na

2 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) In [7] the notion of invex function on Riemannian manifolds was introduced in [2] invex functions preinvex functions on invex subsets of Riemannian manifolds are studied. In this paper various concepts of generalized invexities for functions on Riemannian manifolds are introduced. Then, invariant monotone vector fields generalized invariant monotone vector fields on Riemannian manifolds are defined their relations with generalized invexities are investigated. The organization of the paper is as follows. In Section 2 some concepts facts from Riemannian geometr are collected. In Section 3 we recall the notion of invexit introduce the concept of strong invexit on Riemannian manifolds. Section 4 is devoted to the notions of monotone invariant monotone vector fields. Then the relation between invexit monotonicit is studied. Finall in Section 5 we introduce pseudoinvexit invariant pseudomonotonicit of functions on Riemannian manifolds. 2. Preliminaries In this section, we recall some definitions known results concerning Riemannian manifolds which will be used throughout the paper. We refer the reader to [2,3] for the stard material of differential geometr. Throughout this paper M is a C smooth manifold modelled on a Hilbert space H, either finite dimensional or infinite dimensional, endowed with a Riemannian metric g p (.,.) on the tangent space T p M = H. Therefore, we have a smooth assignment of an inner product to each tangent space. It is usual to write g p (v, w) = v, w p for all v, w T p M. Thus, M is now a Riemannian manifold. The corresponding norm is denoted b p. Let us recall that the length of a piecewise C curve γ : [a, b] M is defined b L(γ ) := b a γ (t) γ (t) dt. Minimizing this length functional on the set of all piecewise C curves joining p, q M, we obtain a distance function (p, q) d(p, q). Then d is a distance which induces the original topolog on M. Given a manifold M, denote b X(M) the space of all vector fields over M. The metric induces a map f grad f X(M) which associates with each f its gradient via the rule d f, X = d f (X) for each X X(M). On ever Riemannian manifold there exists exactl one covariant derivation called the Levi-Civita connection, denoted b X Y for an vector fields X, Y on M. We also recall that a geodesic is a C smooth path γ whose tangent is parallel along the path γ, that is, γ satisfies the equation dγ (t)/dt dγ (t)/dt = 0. An path γ joining p q in M such that L(γ ) = d(p, q) is a geodesic, it is called a minimal geodesic. The existence theorem for ordinar differential equations implies that for ever v T M there exist an open interval J(v) containing 0 a unique geodesic γ v : J(v) M with dγ (0)/dt = v. This implies that there is an open neighborhood T M of the submanifold M of T M such that for ever v T M the geodesic γ v (t) is defined for t < 2. The exponential mapping exp : T M M is then defined as exp(v) = J v () the restriction of exp to a fiber T p M in T M is denoted b exp p for ever p M. Since (T p M, p ) is a Hilbert space, there is a linear isometric identification between this space its dual (T p M, p ). If γ : [a, b] M is a geodesic then for each t, t 2 [a, b] the Levi-Civita connection induces an isometr P t 2 t,γ : T γ (t )M T γ (t2 )M, the so called parallel translation along γ from γ (t ) to γ (t 2 ). We also recall that a simpl connected complete Riemannian manifold of nonpositive sectional curvature is called a Cartan Hadamard manifold. If f is a differentiable map from manifold M to manifold N, we shall denote b d f x the differential of f at x. 3. Invexit on Riemannian manifolds In this section, motivated b [7], the notion of strong invexit for functions is introduced. Let M be a Riemannian manifold η : M M T M be a function such that for ever x, M, η(x, ) T M. Definition 3.. Let M be a Riemannian manifold f : M R be a differentiable function; then, f is said to be: (i) Invex with respect to η on M if f (x) f () d f (η(x, )) for all x, M. ()

3 852 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) If the inequalit () is strict for x, then f is said to be strictl invex with respect to η. (ii) Strongl invex with respect to η on M if there exists a constant α > 0 such that for ever x, M we have f (x) f () d f (η(x, )) + α η(x, ) 2. Example 3.. Let M be a Riemannian manifold f : M R be a differentiable function such that for ever M, d f 0. Suppose that the function η : M M T M is defined b η(x, ) := ( f (x) f ()) d f 2 d f. Then, for ever x, M we have ( f (x) f ()) d f, η(x, ) = d f, d f 2 d f ( f (x) f ()) = d f 2 d f, d f ( f (x) f ()) = d f 2 d f 2 = f (x) f (). Therefore, f is an invex function with respect to η. The following definitions are introduced in [7]. Definition 3.2. Let M be a Riemannian manifold α x, : [0, ] M be a curve on M such that α x, (0) = α x, () = x. Then, α x, is said to possess the propert (P) with respect to, x M if α x, (s)(t s) = η(α x,(t), α x, (s)), for all s, t [0, ]. Definition 3.3. Let M be a Riemannian manifold. Then, the function η : M M T M is said to be integrable if for ever x, M there exists at least one curve α x, possessing the propert (P) with respect to, x M. For an example of an integrable map η, see [7, p. 305]. Remark 3.. Let M be a Riemannian manifold function η : M M T M be integrable. Then, we have η(x, ) = η(α x, (), α x, (0)) = α x, (0). In the case where α x, is a geodesic, then, η(, α x, (s)) = sα x, (s) = s Ps 0,α x, [α x, (0)] = s Ps 0,α x, [η(x, )] η(α x, (), α x, (s)) = ( s)α x, (s) = ( s)ps 0,α x, [η(x, )]. 4. Monotonicit on Riemannian manifolds Let us recall the definition of a monotone vector field on a Riemannian manifold; see [0,5] for the details. Definition 4.. Let M be a Riemannian manifold X be a vector field on M. Then, X is said to be monotone on M if for ever x, M we have α x, (0), P0,α x, [X (x)] X () 0, where α x, is a geodesic joining x. (2)

4 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) Motivated b Yang et al. [24] we generalize the notion of invariant monotonicit to Riemannian manifolds. Definition 4.2. Let M be a Riemannian manifold X be a vector field on M; then: (i) X is said to be invariant monotone on M with respect to η if for ever x, M one has X (), η(x, ) + X (x), η(, x) x 0. (3) If the inequalit (3) is strict for x, then X is said to be strictl invariant monotone with respect to η. (ii) X is said to be strongl invariant monotone on M with respect to η if there exists an α > 0 such that for ever, x M one has X (), η(x, ) + X (x), η(, x) x α( η(x, ) 2 + η(, x) 2 x ). It should be noted that on ever Cartan Hadamard manifold M each monotone vector field X is an invariant monotone vector field with respect to η(x, ) := exp (x). Indeed, for ever x, M, α x, (t) := exp (t exp (x)), for all t [0, ], is the unique geodesic joining to x α x, (0) := exp (x). B (2) the propert P 0,α x, [exp (x)] = exp () of parallel translation we have x exp Therefore, (x), P0,α x, [X (x)] X () = P0,α x, [exp (x)], X (x) x exp (x), X () exp x (), X (x) x + exp (x), X () 0. = exp x (), X (x) x exp (x), X () 0. (4) Considering η(x, ) = exp (x) for ever x, M, we conclude that X is an invariant monotone vector field on M with respect to η. Despite the above fact, in the following example we show that on ever Cartan Hadamard manifold M there exists a strictl invariant monotone vector field X with respect to an η : M M T M which is not monotone. Example 4.. Let M be a Cartan Hadamard manifold x M. Then, the distance function z d(x, z) is smooth on M \ {x}. We denote the partial derivatives of the function d : M M R b d(x,) d(x,). Suppose that x l := d(x, ) = exp (x) x. If γ (t) := exp (t exp (x)/ exp (x) x) for all t [0, l] is the unique minimizing geodesic joining to x, then, b [, p. 347], we have the following antismmetr properties: [ ] d(x, ) P0,γ l d(x, ) =. (5) On the other h, b [2, p. 08] we have d(x, ) = γ (l), d(x, ) = d(x, ) =. x Since γ (0) = l exp (x), b Eq. (6) we can write d(x, ) = γ (l) = P l 0,γ [γ (0)] (6) (7)

5 854 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) [ ] = P0,γ l l exp (x) = l Pl 0,γ [exp (x)] = l exp x (). (8) Now, we define the function η vector field X; η(x, ) := l exp (x) X () := d(x, ). The vector field X is invariant monotone with respect to η. Indeed, b utilizing (8) we get d(x, ) X (x), η(, x) x =, l exp x () x = l l exp x (), exp x () Similarl, Hence, X (), η(x, ) = l 2. = exp x () 2 x = l 2. X (x), η(, x) x + X (), η(x, ) < 0. We show that the vector field X is not monotone. B (5) (7) we get [ ] γ (0), P,γ 0 [X (x)] X () = γ (0), P,γ 0 d(x, ) = γ (0), P 0,γ = γ (0), P,γ 0 = 2 γ (0), P,γ 0 = 2 P0,γ [γ (0)], x [ d(x, ) [ d(x, ) [ ] d(x, ) d(x, ) d(x, ) d(x, ) = 2, x = 2 d(x, ) 2 x = 2 < 0. ( ] P,γ 0 ] + P,γ 0 x ) d(x, ) [ ] d(x, ) [ ] d(x, ) The following example shows that there exists a strictl invariant monotone vector field on ever Riemannian manifold.

6 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) Example 4.2. Let M be a Riemannian manifold f : M R be a differentiable function such that for ever M, d f 0. Suppose that for ever x, M the function η : M M T M is defined b η(x, ) := 2 d f 2 d f. Then, d f is a strictl invariant monotone vector field with respect to η. Indeed, for ever x, M with x we have d f, η(x, ) = d f, 2 d f 2 d f Similarl, we have = 2 d f 2 d f, d f = 2 d f 2 d f 2 = 2. (9) d f x, η(, x) x = 2. (0) Therefore, d f, η(x, ) + d f x, η(, x) x < 0. Let S be a convex subset of a finite dimensional Cartan Hadamard manifold M x M. Then, there exists a unique point p S (x) such that for each S, d(x, p S (x)) d(x, ). The point p S (x) is called the projection of x onto S; see [9, p. 262]. Now, we give an example of a strictl invariant monotone vector field which is not of gradient tpe. Example 4.3. Let M be a Cartan Hadamard manifold of constant sectional curvature K 0. Suppose that x, M, x, r > 0. Moreover, assume that B x := B(x, r) B := B(, r) are two closed balls such that B x B = φ. Let x := P Bx (x) := P B () be projections of x on B x B, respectivel. Suppose that l := exp x. Let γ (t) := exp (t exp x / exp x ), for all t [0, l], be the normalized geodesic joining to x. Assume that J is a Jacobi field along γ normal to γ w(t) is a parallel field along γ with w(t) γ (t) = γ, w(t) γ (t) = 0 for ever t [0, l]. We define the function η : M M T M b η(x, ) := P 0 l,γ [ w(l)]. We also define the vector field X as follows: X () := P 0 l,γ [J(l)]. B [8, p. 2] we have J(l) = sinh(l K ) w(l). K () B the definitions of X η utilizing () we get X (), η(x, ) = Pl,γ 0 [J(l)], P0 l,γ [ w(l)] = J(l), w(l) x = sinh(l K ) w(l), w(l) K x

7 856 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) = sinh(l K ) K w(l), w(l) x = sinh(l K ) K < 0. Therefore, X (), η(x, ) < 0. Similarl, let l be the length of the normalized geodesic β joining x to defined b β(t) := exp x (t exp x / exp x x ), for all t [0, l ]. Then, we have η(, x) = P 0 l,β [ w (l )], X (x) = P 0 l,β [J (l )]. Hence, b (3) (4) we have X (), η(x, ) < 0. B appling (2) (5) we get X (), η(x, ) + X (), η(x, ) < 0. Therefore, X is strictl invariant monotone with respect to η. (2) (3) (4) (5) Proposition 4.. Let M be a Riemannian manifold f : M R be a differentiable function. If f is (strongl, strictl) invex with respect to η, then d f is (strongl, strictl) invariant monotone with respect to η on M. Proof. We prove onl the result when f is strongl invex. The proofs for other cases are similar. Let f be strongl invex on M x, M. Since f is strongl invex, we have f (x) f () d f (η(x, )) + α η(x, ) 2, f () f (x) d f x (η(, x)) + α η(, x) 2 x. B adding these two inequalities we get 0 d f (η(x, )) + d f x (η(, x)) + α( η(x, ) 2 + η(, x) 2 x ). We need the following proposition from [7]. Proposition 4.2. Let M be a Riemannian manifold f : M R be a differentiable function. Suppose that the function η : M M T M is integrable. Then, f is an invex function on M if onl if g x, (t) = f (α x, (t)) is convex on [0, ] for some α x,. Theorem 4.. Let M be a Riemannian manifold f : M R be a differentiable function. Suppose that the function η : M M T M is integrable. If d f is invariant monotone on M with respect to η, then f is an invex function on M. Proof. Let x, M. Since η is integrable, there exists at least one curve α x, possessing the Propert (P) such that α x, (0) = α x, () = x. If f is not invex on M, then, b Proposition 4.2, f (α x, (t)) is not convex on [0, ]. Hence there exists a λ (0, ) such that f (α x, (λ)) > λf (x) + ( λ) f (),

8 or A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) λ[ f (α x, (λ)) f (x)] > ( λ)[ f (α x, (λ)) f ()]. B the Mean Value Theorem there exist λ, λ 2 (0, ), 0 < λ 2 < λ < λ < λ(λ )d f αx, (λ )(α x, (λ )) + ( λ)λd f αx, (λ 2 )(α x, (λ 2)) > 0. This is equivalent to d f αx, (λ )(α x, (λ )) + d f αx, (λ 2 )(α x, (λ 2)) > 0. B the condition (P) we have Therefore, (λ λ 2 ) η(α x,(λ ), α x, (λ 2 )) = α x, (λ 2), (λ λ 2 ) η(α x,(λ 2 ), α x, (λ )) = α x, (λ ). d f αx, (λ )(η(α x, (λ 2 ), α x, (λ ))) + d f αx, (λ 2 )(η(α x, (λ ), α x, (λ 2 ))) > 0. This contradicts the invariant monotonicit of f. Theorem 4.2. Let M be a Riemannian manifold such that there is a geodesic between ever two points let f : M R be a differentiable function. Suppose that the function η : M M T M is integrable. If d f is strongl invariant monotone on M with respect to η, then f is a strongl invex function on M. Proof. Let x, M. Then, there exists a geodesic α x, : [0, ] M such that α x, (0) =, α x, () = x. Let z =: α x, ( 2 ); then, b the Mean Value Theorem there exist t, t 2 (0, ) such that 0 < t 2 < 2 < t < we have f (x) f (z) = 2 d f u(α x, (t )), (6) f (z) f () = 2 d f v(α x, (t 2)), (7) where u := α x, (t ), v := α x, (t 2 ). Since d f is strongl invariant monotone, then, d f u (η(, u)) + d f (η(u, )) α( η(, u) 2 u + η(u, ) 2 ). (8) Since condition (P) holds, then, b Remark 3. utilizing the parallel translation, we have η(, α x, (t )) = t P t 0,α x, [η(x, )], η(α x, (t ), ) = t η(x, ). B combining (8) (20) we have d f u ( t P t 0,α x, [η(x, )]) + d f (t [η(x, )]) α( t P t 0,α x, [η(x, )] 2 u + t [η(x, )] 2 ). This is equivalent to (9) (20) d f u (P t 0,α x, [η(x, )]) + d f ([η(x, )]) 2t α [η(x, )] 2. (2) Since P t 0,α x, [η(x, )] = α x, (t ) we have 2 d f u(α x, (t )) 2 d f ([η(x, )]) + t α [η(x, )] 2. (22)

9 858 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) In a similar wa we get 2 d f v(α x, (t 2)) 2 d f ([η(x, )]) + t 2 α [η(x, )] 2. (23) From (6), (7), (22) (23) we conclude that f (x) f (z) 2 d f (η(x, )) + t α [η(x, )] 2, f (z) f () 2 d f (η(x, )) + t 2 α [η(x, )] 2. B adding these two inequalities we obtain f (x) f () d f (η(x, )) + α(t + t 2 ) η(x, ) 2 d f (η(x, )) + 2 α η(x, ) 2. Therefore, f is strongl invex. 5. Pseudoinvexit invariant pseudomonotonicit Definition 5.. Let M be a Riemannian manifold f : M R be a differentiable function; then: (i) f is said to be pseudoinvex with respect to η on M if for ever x, M, one has d f (η(x, )) 0 f (x) f (). (ii) f is said to be strictl pseudoinvex with respect to η on M if for ever x, M with x one has d f (η(x, )) 0 f (x) > f (). (iii) f is said to be strongl pseudoinvex with respect to η on M if there exists a constant α > 0 such that for ever x, M, one has d f (η(x, )) 0 f (x) f () + α η(x, ) 2. Definition 5.2. Let M be a Riemannian manifold X be a vector field on M; then: (i) X said to be invariant pseudomonotone on M with respect to η if for ever x, M one has X (x), η(, x) x 0 X (), η(x, ) 0. (ii) X is said to be strictl invariant pseudomonotone on M with respect to η if for ever x, M with x one has X (x), η(, x) x 0 X (), η(x, ) < 0. (iii) X is said to be strongl invariant pseudomonotone on M with respect to η if there exists a constant α > 0 such that for ever x, M one has X (x), η(, x) x 0 X (), η(x, ) α η(x, ) 2. The following theorem is a generalization of Theorem 4. in [24] to Riemannian manifolds. Theorem 5.. Let M be a Riemannian manifold f : M R be a differentiable function. Suppose that the function η : M M T M is integrable. Then, f is strictl pseudoinvex with respect to η on M if onl if d f is strictl invariant pseudomonotone on M with respect to η.

10 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) Proof. Suppose that f is strictl pseudoinvex with respect to η on M. Let x, M, x be such that d f (η(x, )) 0. Then, b the strict pseudoinvexit of f we have f (x) > f (). We need to show that d f x (η(, x)) < 0. Assume that d f x (η(, x)) 0. Again utilizing the strict pseudoinvexit of f with respect to η ields f () f (x), which contradicts (24). Conversel, suppose that d f is strictl invariant pseudomonotone on M. Let x, M, x be such that d f (η(x, )) 0. Since η is integrable, there exists at least one curve α x, possessing the Propert (P) such that α x, (0) = α x, () = x. We need to show that f (x) > f (). Assume f (x) f (), B the Mean Value Theorem there exists λ (0, ) such that (24) (25) (26) f (x) f () = d f αx, (λ)(α x, (λ)). Now, (26) (27) implies that d f αx, (λ)(α x, (λ)) 0. Hence, b the Propert (P) we have d f αx, (λ)(α x, (λ)) = d f α x, (λ) [ ] λ η(, α x,(λ)) (27) = λ d f α x, (λ)[η(, α x, (λ))] 0. Therefore, d f αx, (λ)(η(, α x, (λ))) 0. Since d f is strictl invariant pseudomonotone, from (28) we have d f (η(α x, (λ), )) < 0, b the Propert (P) we conclude that d f (λα x, (0)) = λd f (η(x, )) < 0. Therefore, d f (η(x, )) < 0 which contradicts (25). Hence, f is strictl pseudoinvex with respect to η on M. (28) Theorem 5.2. Let M be a Riemannian manifold such that there is a geodesic between ever two points let f : M R be a differentiable function. Suppose that the function η : M M T M is integrable. If f is strongl invariant pseudomonotone on M with respect to η, then f is strongl pseudoinvex with respect to η on M. Proof. Suppose that d f η((x, )) 0. (29) Let x, M. Then, there exists a geodesic α x, : [0, ] M such that α x, (0) =, α x, () = x. Let z := α x, ( 2 ); then, b the Mean Value Theorem, there exist t, t 2 (0, ) such that 0 < t 2 < 2 < t < f (x) f (z) = 2 d f u(α x, (t )) (30) f (z) f () = 2 d f v(α x, (t 2)), (3)

11 860 A. Barani, M.R. Pouraevali / Nonlinear Analsis 70 (2009) where u := α x, (t ), v := α x, (t 2 ). B the propert (P) equalities (30) (3) we obtain f (x) f (z) = 2t d f u η((, u)), f (z) f () = 2t 2 d f v η((, v)). The Propert (P) (29) impl that (32) (33) 0 d f (η(x, )) = t d f (η(u, )) = t 2 d f (η(v, )). (34) Since d f is strongl invariant pseudomonotone on M with respect to η, b utilizing (34) Propert (P) we have d f u (η(, u)) α η(, u) 2 u = α t P t 0,α x, [η(x, )] 2 u = αt 2 Pt 0,α x, [η(x, )] 2 u = αt 2 η(x, ) 2, (35) d f v (η(, v)) α η(, v) 2 v = α t P t 2 0,α x, [η(x, )] 2 v = αt 2 Pt 2 0,α x, [η(x, )] 2 v = αt 2 2 η(x, ) 2. Now, from (32) (35) we conclude that f (x) f (z) α 2 t η(x, ) 2. (36) (37) Similarl, b using (33) (36) we have f (z) f () α 2 t 2 η(x, ) 2. (38) B adding (37) (38) we obtain f (x) f () α 2 (t + t 2 ) η(x, ) 2 α 4 η(x, ) 2. Therefore, f is strongl pseudoinvex with respect to η on M. Acknowledgement This research was partiall supported b the Center of Excellence for Mathematics, Universit of Isfahan, Isfahan, Iran. References [] D. Azagra, J. Ferrera, F. López-Mesas, Nonsmooth analsis Hamilton Jacobi equations on Riemannian manifolds, Journal of Functional Analsis 220 (2005) [2] A. Barani, M.R. Pouraevali, Invex sets preinvex functions on Riemannian manifolds, Journal of Mathematical Analsis Applications 328 (2007)

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