MONOTONE OPERATORS ON BUSEMANN SPACES

Transcription

1 MONOTONE OPERATORS ON BUSEMANN SPACES David Ariza-Ruiz Genaro López-Acedo Universidad de Sevilla Departamento de Análisis Matemático V Workshop in Metric Fixed Point Theory and Applications Noviembre, 2011 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

2 We have divided this presentation in two parts. PART I. How to define monotone operators in geodesic spaces? (Genaro) PART II. Fixed point results in Busemann spaces (David) (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

3 Part I DEFINITION OF MONOTONE OPERATORS (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

4 Summary of Part I 2 Why definition in metric spaces? 1 Classic Theory Definition Relationship with other class of operators Motivation Bibliography 3 Previous work Riemannian manifolds Hyperbolic spaces 4 Our setting: Busemann spaces Basic Tools (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

5 Classic Theory (DEFINITION) Classic Theory Definition Let H be a Hilbert space and let A : D H 2 H be a set-valued operator. If for any x,y D and for each u A(x) and v A(y) we say that 1 A is Monotone, if 0 u v,x y ; 2 A is α-strongly monotone, α > 0, if α x y 2 u v,x y ; 3 A is Maximal monotone, if T is monotone and its graph is maximal. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

6 Classic Theory (DEFINITION) Classic Theory Definition Let H be a Hilbert space and let A : D H 2 H be a set-valued operator. If for any x,y D and for each u A(x) and v A(y) we say that 1 A is Monotone, if 0 u v,x y ; 2 A is α-strongly monotone, α > 0, if α x y 2 u v,x y ; 3 A is Maximal monotone, if T is monotone and its graph is maximal. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

7 Classic Theory (DEFINITION) Classic Theory Definition Let H be a Hilbert space and let A : D H 2 H be a set-valued operator. If for any x,y D and for each u A(x) and v A(y) we say that 1 A is Monotone, if 0 u v,x y ; 2 A is α-strongly monotone, α > 0, if α x y 2 u v,x y ; 3 A is Maximal monotone, if T is monotone and its graph is maximal. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

8 Classic Theory (DEFINITION) Classic Theory Definition Let H be a Hilbert space and let A : D H 2 H be a set-valued operator. If for any x,y D and for each u A(x) and v A(y) we say that 1 A is Monotone, if 0 u v,x y ; 2 A is α-strongly monotone, α > 0, if α x y 2 u v,x y ; 3 A is Maximal monotone, if T is monotone and its graph is maximal. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

9 Classic Theory Classic Theory (ACCRETIVE OPERATORS) Relationship with other class of operators Let H be a Hilbert space and let A : D H 2 H be a set-valued operator. If for any x,y D and for each u A(x) and v A(y) we say that 1 A is Accretive if for all r 0 x y x y + r(u v) ; 2 A is α-strongly accretive, α > 0, if for all r 0 3 A is m-accretive, If is accretive and R(I + A) = H. (1 + αr) x y x y + r(u v). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

10 Classic Theory Classic Theory (ACCRETIVE OPERATORS) Relationship with other class of operators Let H be a Hilbert space and let A : D H 2 H be a set-valued operator. If for any x,y D and for each u A(x) and v A(y) we say that 1 A is Accretive if for all r 0 x y x y + r(u v) ; 2 A is α-strongly accretive, α > 0, if for all r 0 3 A is m-accretive, If is accretive and R(I + A) = H. (1 + αr) x y x y + r(u v). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

11 Classic Theory Classic Theory (ACCRETIVE OPERATORS) Relationship with other class of operators Let H be a Hilbert space and let A : D H 2 H be a set-valued operator. If for any x,y D and for each u A(x) and v A(y) we say that 1 A is Accretive if for all r 0 x y x y + r(u v) ; 2 A is α-strongly accretive, α > 0, if for all r 0 3 A is m-accretive, If is accretive and R(I + A) = H. (1 + αr) x y x y + r(u v). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

12 Classic Theory Classic Theory (ACCRETIVE OPERATORS) Relationship with other class of operators Let H be a Hilbert space and let A : D H 2 H be a set-valued operator. If for any x,y D and for each u A(x) and v A(y) we say that 1 A is Accretive if for all r 0 x y x y + r(u v) ; 2 A is α-strongly accretive, α > 0, if for all r 0 3 A is m-accretive, If is accretive and R(I + A) = H. (1 + αr) x y x y + r(u v). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

13 Classic Theory Classic Theory (THE RESOLVENT) Relationship with other class of operators Let H be a Hilbert space and let A : H 2 H be a proper set-valued operator and D = R(I + A). The resolvent of A is J A = (I + A) 1 If T = J A D, the following hold 1 A = T 1 I. 2 A is monotone if and only if T is firmly non expansive. 3 A is maximally monotone if and only if T is firmly non expansive and D = H. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

14 Classic Theory Classic Theory (THE RESOLVENT) Relationship with other class of operators Let H be a Hilbert space and let A : H 2 H be a proper set-valued operator and D = R(I + A). The resolvent of A is J A = (I + A) 1 If T = J A D, the following hold 1 A = T 1 I. 2 A is monotone if and only if T is firmly non expansive. 3 A is maximally monotone if and only if T is firmly non expansive and D = H. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

15 Classic Theory Classic Theory (THE RESOLVENT) Relationship with other class of operators Let H be a Hilbert space and let A : H 2 H be a proper set-valued operator and D = R(I + A). The resolvent of A is J A = (I + A) 1 If T = J A D, the following hold 1 A = T 1 I. 2 A is monotone if and only if T is firmly non expansive. 3 A is maximally monotone if and only if T is firmly non expansive and D = H. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

16 Classic Theory Classic Theory (THE RESOLVENT) Relationship with other class of operators Let H be a Hilbert space and let A : H 2 H be a proper set-valued operator and D = R(I + A). The resolvent of A is J A = (I + A) 1 If T = J A D, the following hold 1 A = T 1 I. 2 A is monotone if and only if T is firmly non expansive. 3 A is maximally monotone if and only if T is firmly non expansive and D = H. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

17 Classic Theory Motivation Classic Theory (ABSTRACT CAUCHY PROBLEM) Reasonable operators A H H such that the abstract Cauchy problem d dt x(t) Ax(t), for L 1 a.e. t 0 x(0) = x 0 D(A), has a unique solution for each x 0 D(A). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

18 Classic Theory Motivation Classic Theory (ABSTRACT CAUCHY PROBLEM) Let H be a Hilbert space and let A H H be a maximal monotone operator. then for each u 0 D(A) the curve u(0) := u 0, ( ) nu0 u(t) := lim J tn n for t > 0 has the following properties: (P 1 ) u(t) D(A) for each t > 0; (P 2 ) u is Lipschitz with Lipschitz constant Au 0 ; (P 3 ) 0 du dt + Au(t) for a.e. t 0; (P 4 ) u(0) = u 0 ; (P 5 ) u is differentiable from the right for each t 0 and we have du dt + + Au(t) = 0 for all t 0; (P 6 ) The function 0 t A(t) is continuous from the right and the function 0 t A(t) is non-increasing; (P 7 ) If two functions u and ũ have properties (P 1 ), (P 2 ), and (P 3 ) above, then u(t) ũ(t) u(0) ũ(0) for all t 0; (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

19 Classic Theory Motivation Classic Theory (ABSTRACT CAUCHY PROBLEM) Let H be a Hilbert space and let A H H be a maximal monotone operator. then for each u 0 D(A) the curve u(0) := u 0, ( ) nu0 u(t) := lim J tn n for t > 0 has the following properties: (P 1 ) u(t) D(A) for each t > 0; (P 2 ) u is Lipschitz with Lipschitz constant Au 0 ; (P 3 ) 0 du dt + Au(t) for a.e. t 0; (P 4 ) u(0) = u 0 ; (P 5 ) u is differentiable from the right for each t 0 and we have du dt + + Au(t) = 0 for all t 0; (P 6 ) The function 0 t A(t) is continuous from the right and the function 0 t A(t) is non-increasing; (P 7 ) If two functions u and ũ have properties (P 1 ), (P 2 ), and (P 3 ) above, then u(t) ũ(t) u(0) ũ(0) for all t 0; (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

20 Classic Theory Motivation Classic Theory (SUBPOTENTIAL OPERATORS) Let H be a Hilbert space and f : H R a lower semi-continuous convex, proper function. The subdifferential of f at x is defined as f (x) = {u H : u,y x f (y) f (x), y H}. The f is a maximal monotone operator also called subpotencial maximal operator and satisfies 0 f (x 0 ) f (x 0 ) f (x) for all x H (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

21 Classic Theory Motivation Classic Theory (SUBPOTENTIAL OPERATORS) Let H be a Hilbert space and f : H R a lower semi-continuous convex, proper function. The subdifferential of f at x is defined as f (x) = {u H : u,y x f (y) f (x), y H}. The f is a maximal monotone operator also called subpotencial maximal operator and satisfies 0 f (x 0 ) f (x 0 ) f (x) for all x H (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

22 Classic Theory Motivation Classic Theory (SUBPOTENTIAL OPERATORS) Let H be a Hilbert space and f : H R a lower semi-continuous convex, proper function. The subdifferential of f at x is defined as f (x) = {u H : u,y x f (y) f (x), y H}. The f is a maximal monotone operator also called subpotencial maximal operator and satisfies 0 f (x 0 ) f (x 0 ) f (x) for all x H (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

23 Classic Theory Classic Theory (REFERENCES) Bibliography BREZIS, H., Operateurs maximaux monotones et semigropus des contractions dans les espaces de Hilbert, (1973), GARCÍA FALSET, J., Operadores no lineales: una aproximacin a la teora del punto fijo y la acretividad, (2010), BARBU, V., Nonlinear Differentia equations of monotone types in Banach spaces, (2010), BAUSCHKE, H. AND COMBETTES, P., Convex analysis and monotone theory in Hilbert spaces spaces, (2011), (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

24 Classic Theory Classic Theory (REFERENCES) Bibliography BREZIS, H., Operateurs maximaux monotones et semigropus des contractions dans les espaces de Hilbert, (1973), GARCÍA FALSET, J., Operadores no lineales: una aproximacin a la teora del punto fijo y la acretividad, (2010), BARBU, V., Nonlinear Differentia equations of monotone types in Banach spaces, (2010), BAUSCHKE, H. AND COMBETTES, P., Convex analysis and monotone theory in Hilbert spaces spaces, (2011), (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

25 Classic Theory Classic Theory (REFERENCES) Bibliography BREZIS, H., Operateurs maximaux monotones et semigropus des contractions dans les espaces de Hilbert, (1973), GARCÍA FALSET, J., Operadores no lineales: una aproximacin a la teora del punto fijo y la acretividad, (2010), BARBU, V., Nonlinear Differentia equations of monotone types in Banach spaces, (2010), BAUSCHKE, H. AND COMBETTES, P., Convex analysis and monotone theory in Hilbert spaces spaces, (2011), (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

26 Classic Theory Classic Theory (REFERENCES) Bibliography BREZIS, H., Operateurs maximaux monotones et semigropus des contractions dans les espaces de Hilbert, (1973), GARCÍA FALSET, J., Operadores no lineales: una aproximacin a la teora del punto fijo y la acretividad, (2010), BARBU, V., Nonlinear Differentia equations of monotone types in Banach spaces, (2010), BAUSCHKE, H. AND COMBETTES, P., Convex analysis and monotone theory in Hilbert spaces spaces, (2011), (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

27 Why definition in metric spaces? Why definition in metric spaces? Development of the theory of gradient flows in various metric settings. L. AMBROSIO, N. GIGLI, G. SAVARÉ, Gradient flows in metric spaces and in the space of probability measure, (2005), U.F. MAYER, Gradient flowson nonpositively curved metric space and harmonic maps, Comm. Anal. Geom., 6 (1998), Nonconvex optimization problems. RAPCSAK, Sectional curvature in nonlinear optimization, J. Global Optim., 40 (2008), Variational inequalities. S.Z. NEMETH, Variational inqualities on Hadamard manifolds, Nonlinear Anal., 52, , (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

28 Why definition in metric spaces? Why definition in metric spaces? Development of the theory of gradient flows in various metric settings. L. AMBROSIO, N. GIGLI, G. SAVARÉ, Gradient flows in metric spaces and in the space of probability measure, (2005), U.F. MAYER, Gradient flowson nonpositively curved metric space and harmonic maps, Comm. Anal. Geom., 6 (1998), Nonconvex optimization problems. RAPCSAK, Sectional curvature in nonlinear optimization, J. Global Optim., 40 (2008), Variational inequalities. S.Z. NEMETH, Variational inqualities on Hadamard manifolds, Nonlinear Anal., 52, , (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

29 Why definition in metric spaces? Why definition in metric spaces? Development of the theory of gradient flows in various metric settings. L. AMBROSIO, N. GIGLI, G. SAVARÉ, Gradient flows in metric spaces and in the space of probability measure, (2005), U.F. MAYER, Gradient flowson nonpositively curved metric space and harmonic maps, Comm. Anal. Geom., 6 (1998), Nonconvex optimization problems. RAPCSAK, Sectional curvature in nonlinear optimization, J. Global Optim., 40 (2008), Variational inequalities. S.Z. NEMETH, Variational inqualities on Hadamard manifolds, Nonlinear Anal., 52, , (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

30 Riemannian manifolds VICTORIA PHD Previous work Riemannian manifolds Monotone operators. C.LI, G. LÓPEZ, V. Martín-Márquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. Lond. Math. Soc., 79(3) (2009), Resolvent and Firmly nonexpansive mappings C.LI, G. LÓPEZ, V. MARTÍN-MÁRQUEZ, J. WANG, Resolvents of set valued monotone vector fields in Hadamard manifolds, Set-Valued Var. Anal., 19(3) (2011), Accretive operators C.LI, G. LÓPEZ, V. MARTÍN-MÁRQUEZ, J. WANG, Monotone and accretive operators in Riemannian manifolds, J. Optim. Theory Appl., 146(3) (2010), fondosdigitales.us.es/tesis/autores/1223/ (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

31 Previous work HYPERBOLIC SPACES Previous work Hyperbolic spaces The Hilbert ball. I. SHAFRIR, Coacretive operators and firmly nonexpansive mappings in the Hilbert ball, Nonlinear Analysis 18(6), (1992) CAT(0) spaces. I. STOJKOVIC, Approximation for convex functionals on non-positivily curved spaces and the Trotter-Kato formula, Advances in Calculus of Variations 5, (2012) M. BA CÁK, The proximal point algorithm in metric spaces, Israel J. of Mathematics To appear (2012) (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

32 Previous work HYPERBOLIC SPACES Previous work Hyperbolic spaces The Hilbert ball. I. SHAFRIR, Coacretive operators and firmly nonexpansive mappings in the Hilbert ball, Nonlinear Analysis 18(6), (1992) CAT(0) spaces. I. STOJKOVIC, Approximation for convex functionals on non-positivily curved spaces and the Trotter-Kato formula, Advances in Calculus of Variations 5, (2012) M. BA CÁK, The proximal point algorithm in metric spaces, Israel J. of Mathematics To appear (2012) (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

33 Our setting: Busemann spaces Our setting: Busemann spaces A geodesic space (X,d) is a Busemann space if for any two geodesics γ,γ : [0,1] X, the map D γ,γ : [0,1] [0,1] R, D γ,γ (s,t) = d(γ(s),γ(t)) is convex. A normed space is a Busemann space if and only if it is strictly convex. CAT(0) spaces and some simply connected Finsler manifolds with nonpositive flag curvature are Busemann spaces. In A. PAPADOPOULOS, Metric spaces, convexity and nonpositive curvature (2005), and M. BRIDSON, A. HAEFLIGER, Metric spaces of non-positive curvature (1999), one can find more examples of Busemann spaces. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

34 Our setting: Busemann spaces Our setting: Busemann spaces A geodesic space (X,d) is a Busemann space if for any two geodesics γ,γ : [0,1] X, the map D γ,γ : [0,1] [0,1] R, D γ,γ (s,t) = d(γ(s),γ(t)) is convex. A normed space is a Busemann space if and only if it is strictly convex. CAT(0) spaces and some simply connected Finsler manifolds with nonpositive flag curvature are Busemann spaces. In A. PAPADOPOULOS, Metric spaces, convexity and nonpositive curvature (2005), and M. BRIDSON, A. HAEFLIGER, Metric spaces of non-positive curvature (1999), one can find more examples of Busemann spaces. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

35 Our setting: Busemann spaces Our setting: Busemann spaces A geodesic space (X,d) is a Busemann space if for any two geodesics γ,γ : [0,1] X, the map D γ,γ : [0,1] [0,1] R, D γ,γ (s,t) = d(γ(s),γ(t)) is convex. A normed space is a Busemann space if and only if it is strictly convex. CAT(0) spaces and some simply connected Finsler manifolds with nonpositive flag curvature are Busemann spaces. In A. PAPADOPOULOS, Metric spaces, convexity and nonpositive curvature (2005), and M. BRIDSON, A. HAEFLIGER, Metric spaces of non-positive curvature (1999), one can find more examples of Busemann spaces. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

36 Our setting: Busemann spaces Our setting: Busemann spaces A geodesic space (X,d) is a Busemann space if for any two geodesics γ,γ : [0,1] X, the map D γ,γ : [0,1] [0,1] R, D γ,γ (s,t) = d(γ(s),γ(t)) is convex. A normed space is a Busemann space if and only if it is strictly convex. CAT(0) spaces and some simply connected Finsler manifolds with nonpositive flag curvature are Busemann spaces. In A. PAPADOPOULOS, Metric spaces, convexity and nonpositive curvature (2005), and M. BRIDSON, A. HAEFLIGER, Metric spaces of non-positive curvature (1999), one can find more examples of Busemann spaces. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

37 Basic tools DEFINITION OF ANGLE Our setting: Busemann spaces Basic Tools Let (X,d) be a metric space and let γ : [0,l] X and γ : [0,l ] X be two geodesics with γ(0) = γ (0). The Alexandrov angle is the number γ,γ [0,π] defined by (γ,γ ) := limsup t,t 0 γ(0) (γ(t),γ (t )) = lim sup ε 0 0<t,t <ε where γ(0) (γ(t),γ (t )) denotes to the comparison angle in R 2. γ(0) (γ(t),γ (t )), If the limit lim t,t 0 γ(0)(γ(t),γ (t )) exists, then we say the angle exists in the strict sense. If (X,d) is uniquely geodesic, p x and p y, then the angle between the geodesic segments [p,x] and [p,y] may be denoted p (x,y). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

38 Basic tools Our setting: Busemann spaces Basic Tools THE 0-CONE OVER A METRIC SPACE Given a metric space X, the 0-cone over X is the metric space C 0 X which is defined as the quotient set given by [0, ) and the equivalence relation (t,x) (t,x ) if (t = t = 0) or (t = t > 0 and x = x ). The equivalence class of (t,x) is denoted tx. The class of (0,x) is denoted 0 and is called the vertex of the cone, or the cone point. Let d π (x,x ) := min { π,d(x,x ) }. the distance between two points z = tx and z = t x in X is defined so that d(z,z ) = t if z = 0 and so that if t,t > 0. cos ( 0 (z,z ) ) = d π (x,x ) d(z,z ) 2 = t 2 + t 2 2tt cos ( d π (x,x ) ). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

39 Basic tools Our setting: Busemann spaces Basic Tools THE SPACE OF THE DIRECTIONS and THE TANGENT CONE γ γ p (γ,γ ) = 0 defines an equivalence relation on the set of non-trivial geodesics issuing from p, and, moreover, induces a metric on the set of equivalence classes. The resulting metric spaces is calle the space of direction at p and is denoted S p (X). Note that two geodesics segments issuing from p may have the same direction but intersect only at p. C 0 S p (X), the 0-cone over S p (X), is called the tangent cone at p. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

40 Basic tools WEAK CONVERGENCE Our setting: Busemann spaces Basic Tools -convergence. T.C. LIM, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), Weak convergence. J. JOST, Equilibrium maps between metric spaces metric spaces, Cal. Var. Partial Diff. Equa., 2(1994), E.N. SOSOV, Analogs of weak convergence in special metric spaces, Russian Mathematics, 148 (2004), R. ESPÍNOLA-GARCÍA, A. FERNÁNDEZ LEÓN, CAT(k)-spaces, weak convergence and fixed points, J. Math. Anal. Appl., 353 (2009), (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

41 Basic tools WEAK CONVERGENCE Our setting: Busemann spaces Basic Tools -convergence. T.C. LIM, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), Weak convergence. J. JOST, Equilibrium maps between metric spaces metric spaces, Cal. Var. Partial Diff. Equa., 2(1994), E.N. SOSOV, Analogs of weak convergence in special metric spaces, Russian Mathematics, 148 (2004), R. ESPÍNOLA-GARCÍA, A. FERNÁNDEZ LEÓN, CAT(k)-spaces, weak convergence and fixed points, J. Math. Anal. Appl., 353 (2009), (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

42 Basic tools FIXED POINT RESULTS Our setting: Busemann spaces Basic Tools Firmly Nonexpansive mappings. D. ARIZA-RUIZ, L. LEUSTEAN, G. LÓPEZ-ACEDO Firmly nonexpansive mappings in classes of geodesic spaces, (Accepted Trans. Am. Math. Soc.) arxiv.org/abs/ Compact mappings. D. ARIZA-RUIZ, G. LÓPEZ-ACEDO, Schauder theorem in Busemann spaces, (Preprint). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

43 Part II FIXED POINT THEOREMS IN BUSEMANN SPACES (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

44 Summary of Part II 5 Preliminaries Preliminaries 6 Topological Fixed Point Results in Busemann spaces Schauder Fixed Point Theorem A direct proof of Schauder FPT 7 Metric Fixed point Theorems (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

45 Preliminaries Preliminaries Preliminaries A metric space (X,d) is said to be a (uniquely) geodesic space if every two distinct points x and y of X are joined by a (unique) geodesic, that is, a map γ : [0,l] R X such that γ(0) = x, γ(l) = y, and d(γ(s),γ(t)) = s t for all s,t [0,l]. The image γ([0,l]) of such a geodesic forms a geodesic segment which joins x and y, and it will be denoted [x,y]. Notice that in a (uniquely) geodesic space (X,d), given two distinct points x,y X, a point z in X belongs to the (unique) segment [x,y] if and only if there exists t [0,1] such that d(x,z) = t d(x,y) and d(y,z) = (1 t)d(x,y). We shall write z = (1 t)x t y for simplicity. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

46 Preliminaries Preliminaries Preliminaries A metric space (X,d) is said to be a (uniquely) geodesic space if every two distinct points x and y of X are joined by a (unique) geodesic, that is, a map γ : [0,l] R X such that γ(0) = x, γ(l) = y, and d(γ(s),γ(t)) = s t for all s,t [0,l]. The image γ([0,l]) of such a geodesic forms a geodesic segment which joins x and y, and it will be denoted [x,y]. Notice that in a (uniquely) geodesic space (X,d), given two distinct points x,y X, a point z in X belongs to the (unique) segment [x,y] if and only if there exists t [0,1] such that d(x,z) = t d(x,y) and d(y,z) = (1 t)d(x,y). We shall write z = (1 t)x t y for simplicity. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

47 Preliminaries Preliminaries Preliminaries A metric space (X,d) is said to be a (uniquely) geodesic space if every two distinct points x and y of X are joined by a (unique) geodesic, that is, a map γ : [0,l] R X such that γ(0) = x, γ(l) = y, and d(γ(s),γ(t)) = s t for all s,t [0,l]. The image γ([0,l]) of such a geodesic forms a geodesic segment which joins x and y, and it will be denoted [x,y]. Notice that in a (uniquely) geodesic space (X,d), given two distinct points x,y X, a point z in X belongs to the (unique) segment [x,y] if and only if there exists t [0,1] such that d(x,z) = t d(x,y) and d(y,z) = (1 t)d(x,y). We shall write z = (1 t)x t y for simplicity. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

48 Busemann spaces Preliminaries Preliminaries A geodesic space (X,d) is a Busemann space if d ( (1 t)x t y,(1 t)z t w ) (1 t)d(x,z) + t d(y,w), ( ) for all x,y,z,w X and t [0,1]. Recall that a geodesic space (X,d) is said to have convex metric if for all x,y,z,w X and t [0,1]. d ( (1 t)x t y,z ) (1 t)d(x,z) + t d(y,z), We note that if (X,d) is an arbitrary geodesic space, the convexity of its metric does not imply that (X,d) is a Busemann space, since any Banach space has convex metric and some of these Banach spaces are not Busemann spaces (to be precise, non-strictly convex Banach spaces). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

49 Busemann spaces Preliminaries Preliminaries A geodesic space (X,d) is a Busemann space if d ( (1 t)x t y,(1 t)z t w ) (1 t)d(x,z) + t d(y,w), ( ) for all x,y,z,w X and t [0,1]. Recall that a geodesic space (X,d) is said to have convex metric if for all x,y,z,w X and t [0,1]. d ( (1 t)x t y,z ) (1 t)d(x,z) + t d(y,z), We note that if (X,d) is an arbitrary geodesic space, the convexity of its metric does not imply that (X,d) is a Busemann space, since any Banach space has convex metric and some of these Banach spaces are not Busemann spaces (to be precise, non-strictly convex Banach spaces). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

50 Busemann spaces Preliminaries Preliminaries A geodesic space (X,d) is a Busemann space if d ( (1 t)x t y,(1 t)z t w ) (1 t)d(x,z) + t d(y,w), ( ) for all x,y,z,w X and t [0,1]. Recall that a geodesic space (X,d) is said to have convex metric if for all x,y,z,w X and t [0,1]. d ( (1 t)x t y,z ) (1 t)d(x,z) + t d(y,z), We note that if (X,d) is an arbitrary geodesic space, the convexity of its metric does not imply that (X,d) is a Busemann space, since any Banach space has convex metric and some of these Banach spaces are not Busemann spaces (to be precise, non-strictly convex Banach spaces). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

51 Topological Fixed Point Results in Busemann spaces Schauder Fixed Point Theorem Topological Fixed Point Results in Busemann spaces I Schauder FPT Let K a nonempty, closed, and convex subset of a Busemman space (X,d). Then, any continuous T : K K with T(K) compact has at least a fixed point in K. In C. HORVATH, A note on metric spaces with continuous midpoints, Annals of the Academy of Romanian Scientists Series of Mathematics and its Applications, 1 (2009), Schauder FPT is proved (unclearly) using a fixed point result for compactly nullhomotopic self-map into an absolute neighborhood retract. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

52 Topological Fixed Point Results in Busemann spaces Schauder Fixed Point Theorem Topological Fixed Point Results in Busemann spaces I Schauder FPT Let K a nonempty, closed, and convex subset of a Busemman space (X,d). Then, any continuous T : K K with T(K) compact has at least a fixed point in K. In C. HORVATH, A note on metric spaces with continuous midpoints, Annals of the Academy of Romanian Scientists Series of Mathematics and its Applications, 1 (2009), Schauder FPT is proved (unclearly) using a fixed point result for compactly nullhomotopic self-map into an absolute neighborhood retract. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

53 Topological Fixed Point Results in Busemann spaces Schauder Fixed Point Theorem Topological Fixed Point Results in Busemann spaces II In C.P. NICULESCU, I. ROVENTA, Schauder fixed point theorem in spaces with global nonpositive curvature, Fixed Point Theory Appl. (2009), Art. ID , 8 pp. Schauder FPT is proved in CAT(0) spaces with the additional hypothesis The closed convex hull of every finite subset is compact. The idea used to prove Schauder FPT is the same that in the setting of Banach spaces: A Schauder like Approximation Method. Next, we give a direct proof of Schauder FPT in Busemann spaces (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

54 Topological Fixed Point Results in Busemann spaces Schauder Fixed Point Theorem Topological Fixed Point Results in Busemann spaces II In C.P. NICULESCU, I. ROVENTA, Schauder fixed point theorem in spaces with global nonpositive curvature, Fixed Point Theory Appl. (2009), Art. ID , 8 pp. Schauder FPT is proved in CAT(0) spaces with the additional hypothesis The closed convex hull of every finite subset is compact. The idea used to prove Schauder FPT is the same that in the setting of Banach spaces: A Schauder like Approximation Method. Next, we give a direct proof of Schauder FPT in Busemann spaces (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

55 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT A direct proof of Schauder FPT Let {x 1,...,x m } a finite nonempty subset of a uniquely geodesic space (X,d). We set where D 1 = {x 1 } and, for any 2 j m, D({x i } m i=1 ) := m D i i=1 D j := { [x j,z] : z D j 1 }. Notice that any element y j D j D({x i } m i=1 ) can be written in the form y j = (1 t j )x j t j y j 1, where t j [0,1] and y j 1 D j 1. Lemma 1. Let (X,d) be a Busemann space, x X and r > 0. Assume that {x 1,...,x k } is a nonempty finite subset of X. If x i B(x,r) for all i = 1,...,k, then D({x i } k i=1 ) B(x,r). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

56 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

57 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

58 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

59 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

60 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

61 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

62 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

63 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

64 Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT (STEPS) A direct proof of Schauder FPT 1 There exists {x 1,...,x m } K such that T(K) 2 We define G : K 2 K as G(x) := T(K) B(x, 1 n ). m B(x i, 1 n ). 3 Prove that there exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) i=1 k G( x i ). 4 Then, there exists x T ( D({ x i } k i=1 )) such that x i B(x, 1 n ) for all i = 1,...,k. There exists x n D({ x i } k i=1 ) such that x = T(x n ). 5 By Lemma 1, D({ x i } k i=1 ) B(x, 1 n ). = d(x n,t(x n )) < 1 n. 6 There exists a subsequence {T(x nj )} j N converges to p K. Clearly, the subsequence {x nj } j N is also convergent to p. Since T is continuous, we conclude that p is a fixed point of T. i=1 (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

65 The third step Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT There exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) In its proof, by contradiction, we use KKM Lemma. k G( x i ). KKM Lemma Let C 0,...,C n be closed subsets of the standard n-dimensional n. Suppose that for every non-empty finite subset J = {j 1,...,j k } of {1,...,n}, we have that J j J C j, where J := e j1,...,e jk is the k-simplex. Then, n i=0 C i /0. i=1 We define inductively the map H : n D({x i } m i=1 ) and we prove that H is continuous (only here we have used that X is a Busemann space) (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

66 The third step Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT There exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) In its proof, by contradiction, we use KKM Lemma. k G( x i ). KKM Lemma Let C 0,...,C n be closed subsets of the standard n-dimensional n. Suppose that for every non-empty finite subset J = {j 1,...,j k } of {1,...,n}, we have that J j J C j, where J := e j1,...,e jk is the k-simplex. Then, n i=0 C i /0. i=1 We define inductively the map H : n D({x i } m i=1 ) and we prove that H is continuous (only here we have used that X is a Busemann space) (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

67 The third step Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT There exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) In its proof, by contradiction, we use KKM Lemma. k G( x i ). KKM Lemma Let C 0,...,C n be closed subsets of the standard n-dimensional n. Suppose that for every non-empty finite subset J = {j 1,...,j k } of {1,...,n}, we have that J j J C j, where J := e j1,...,e jk is the k-simplex. Then, n i=0 C i /0. i=1 We define inductively the map H : n D({x i } m i=1 ) and we prove that H is continuous (only here we have used that X is a Busemann space) (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

68 The third step Topological Fixed Point Results in Busemann spaces A direct proof of Schauder FPT There exists { x 1,..., x k } in K such that T ( D({ x i } k i=1 )) In its proof, by contradiction, we use KKM Lemma. k G( x i ). KKM Lemma Let C 0,...,C n be closed subsets of the standard n-dimensional n. Suppose that for every non-empty finite subset J = {j 1,...,j k } of {1,...,n}, we have that J j J C j, where J := e j1,...,e jk is the k-simplex. Then, n i=0 C i /0. i=1 We define inductively the map H : n D({x i } m i=1 ) and we prove that H is continuous (only here we have used that X is a Busemann space) (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

69 Metric Fixed point Theorems UC-Busemann spaces Recall that a Busemann space (X,d) is uniformly convex if for any r > 0 and any ε (0,2] there exists η(r,ε) (0,1] such that for all a,x,y X, d(a,x) r d(a,y) r d(x,y) ε r = d(a, 1 2 x 1 2 y) ( 1 η(r,ε) ) r. η : (0, ) (0, 2] (0, 1] is called a modulus of uniform convexity. We shall refer (X,d) as a UC-Busemann space when it is a uniformly convex Busemann space with a decreasing modulus of uniform convexity with respect to r (for a fixed ε). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

70 Metric Fixed point Theorems UC-Busemann spaces Recall that a Busemann space (X,d) is uniformly convex if for any r > 0 and any ε (0,2] there exists η(r,ε) (0,1] such that for all a,x,y X, d(a,x) r d(a,y) r d(x,y) ε r = d(a, 1 2 x 1 2 y) ( 1 η(r,ε) ) r. η : (0, ) (0, 2] (0, 1] is called a modulus of uniform convexity. We shall refer (X,d) as a UC-Busemann space when it is a uniformly convex Busemann space with a decreasing modulus of uniform convexity with respect to r (for a fixed ε). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

71 Metric Fixed point Theorems UC-Busemann spaces Recall that a Busemann space (X,d) is uniformly convex if for any r > 0 and any ε (0,2] there exists η(r,ε) (0,1] such that for all a,x,y X, d(a,x) r d(a,y) r d(x,y) ε r = d(a, 1 2 x 1 2 y) ( 1 η(r,ε) ) r. η : (0, ) (0, 2] (0, 1] is called a modulus of uniform convexity. We shall refer (X,d) as a UC-Busemann space when it is a uniformly convex Busemann space with a decreasing modulus of uniform convexity with respect to r (for a fixed ε). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

72 Metric Fixed point Theorems Firmly nonexpansive mappings in Busemann spaces Definition 2. Given λ (0,1), we say that T is λ-firmly nonexpansive if d(tx,ty) d((1 λ)x λtx,(1 λ)y λty) for all x,y C. If the above condition holds for all λ (0,1), then T is said to be firmly nonexpansive. Remark 3. Any λ-firmly nonexpansive mapping is nonexpansive, by ( ). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

73 Metric Fixed point Theorems Firmly nonexpansive mappings in Busemann spaces Definition 2. Given λ (0,1), we say that T is λ-firmly nonexpansive if d(tx,ty) d((1 λ)x λtx,(1 λ)y λty) for all x,y C. If the above condition holds for all λ (0,1), then T is said to be firmly nonexpansive. Remark 3. Any λ-firmly nonexpansive mapping is nonexpansive, by ( ). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

74 Metric Fixed point Theorems Firmly nonexpansive mappings in Busemann spaces Definition 2. Given λ (0,1), we say that T is λ-firmly nonexpansive if d(tx,ty) d((1 λ)x λtx,(1 λ)y λty) for all x,y C. If the above condition holds for all λ (0,1), then T is said to be firmly nonexpansive. Remark 3. Any λ-firmly nonexpansive mapping is nonexpansive, by ( ). (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

75 Metric Fixed point Theorems Firmly nonexpansive mappings A generalization of Smarzewski s FPT Let (X,d) be a UC-Busemann space, C = p k=1 C k be a union of nonempty closed convex subsets C k of X, and T : C C be λ-firmly nonexpansive for some λ (0,1). The following are equivalent: (i) T has bounded orbits. (ii) T has fixed points. (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33

76 Metric Fixed point Theorems Thank you for your attention! (Universidad de Sevilla) Monotone Operators on Busemann spaces November 18, / 33