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Ekeland variational principle

with

generalizations

and

variants

CONTENTS

Introduction 1

Chapter I. Ekeland variational principle 3

1 Ekeland priciple in complete metric spaces 5

Several applications of Ekeland principle 30

2 Ekeland principle versus equivalent theorems 63

A. Ekeland principle versus the Drop theorem and the Flower

Petal theorem 63

Support functional and support point for closed convex sets 71

The surjectivity of nonlinear operators 76

B. Ekeland principle versus Caristi - Kirk theorem 81

C. Ekeland principle versus Takahashi theorem, Caristi - Kirk

theorem and Oettli - Théra theorem 83

D. Siegel theorem 88

3 Ekeland principle in Banach spaces 95

- derivative 105

4 Palais - Smale type conditions 113

(PS)c 114

(PS)c, F 117

Critical points and weak solutions for an elliptic type equation 120

5 Subdifferential calculus 125

A. Subderivative 125

B. Superderivative 141

C. Directional derivative 141

D. - subderivative and - superderivative 147

E. The property (H) and the Banach space D (X) 149

F. Clarke derivative 153

G. Clarke subderivative 156

Clarke subderivative 156

Strict derivative of Hadamard type 162

Rules of subdifferential calculus 166

H. Critical points for nondifferentiable functionals 178

I. Mathematical programming 185

6 Variational theorems of minimax type in Banach spaces 187

Minimax theorem I 187

Minimax theorem II 197

Mountain pass theorem and Saddle point theorem for locally

Lipschitz functions 211

Minimax theorem III 218

7 Variational theorems of minimax type on Finsler manifolds 225

Minimax theorem I 229

Minimax theorem II (strong form) 234

Ljusternik - Schnirelmann theorem 245

Minimax theorem III (relaxed boundary conditions) 249

Relaxed Mountain pass and Saddle point theorems 255

Critical orbits of the real periodic functionals 259

8 Ekeland principle for vector - valued functions and minimal

points in product spaces 271

Preliminaries of affine geometry 271

Minimal point theorem I 275

Minimal point theorem II 279

Minimal point theorem III 281

Minimal point theorem IV 288

9 Other variants of Ekeland principle 295

A. Vectorial variant of Ekeland principle 295

B. Vectorial variant of Ekeland principle 305

C. Vectorial variant of Ekeland principle 310

D. Vectorial generalization of the Ekeland and Caristi - Kirk

theorems 314

E. Multivalued variant of Ekeland principle 321

F. Multivalued variants of Ekeland principle 328

G. Ekeland principle in sequentially complete locally convex

spaces 339

H. Ekeland principle in locally complete locally convex spaces 353

Local completeness 354

Two l-complete variants of Ekeland principle 355

The l-complete versions of Ekeland principle face to the l-complete

version of Daneš drop theorem 358

I. Ekeland principle in uniform spaces 361

Introduction 362

Notations 362

Uniform space 363

The topology of a uniform space 364

Pseudo-distance 368

Family of quasi-distances 369

Ekeland principle in uniform spaces (families of quasi-distances) 371

Ekeland principle in uniform spaces (generalized families of quasi-

distances) 377

Generalized family of quasi-distances 377

Minimal point theorem 378

J. Ekeland principle face to the w-distance 382

10. Ekeland principle and Caristi - Kirk theorem in

probabilistic metric spaces 391

A. Menger PM-space 391

Distribution function 391

t-norm 392

Menger PM-space 392

B. Caristi - Kirk theorem in probabilistic metric spaces 396

C. Ekeland principle in probabilistic metric spaces 398

Chapter II. Smooth perturbed variational principles 403

1. Borwein - Preiss principle 405

2. Deville - Godefroy - Zizler principle 417

3. Ghoussoub - Maurey theorems 427

4. Generalization and unification of the Ekeland and Borwein -

Preiss principles 445

Annex 459

Differential manifolds 459

Finsler manifolds 477

Riemann manifolds 482

Bibliography 487

Additional bibliography (with brief presentations) 495

Index 521

INTRODUCTION

In the frame of Variational Calculus, the elementary proposition

“If : X R , X real normed space, has in x0 a local minimum point (hence in particular a global minimum point) and it is Gâteaux differentiable at x0 , then (x 0) = 0” (x0 is critical point)

is called variational principle ([30], 1; w comes from weak).

This is the reason why the neighbour propositions, for instance those in which X is replaced by a metric space, or in which the statement “(x0) = 0” is replaced by “, > 0 any“ etc, are called variational principles (perturbed). The adjective “perturbed” is imposed by the fact that not the function is minimized, but a function of the form + (Ekeland), or of more general form + (Borwein - Preiss), or of even more general form + f (Deville- Godefroy - Zizler), f having some given properties (the second term is the perturbation function).

The Ekeland principle is a perturbed variational principle discovered in 1972 ([30], 1) and nowadays, after more than 30 years, it was proved to be, and this monograph will strongly support this, the foundation of the modern Variational Calculus (see, for instance, the minimax theorems in Banach spaces or in the Finsler manifolds, in which the key step of demonstration is made by the application of the Ekeland principle).

As referring the applications, these are numerous and various: the geometry of Banach spaces, nonlinear analysis, differential equations and partial differential equations, global analysis, probabilistic analysis, differential geometry, fixed point theorems, nonlinear semigroups, dynamical systems, optimization, mathematical programming, optimal control.

Finally, we cannot close this INTRODUCTION without the confession of Ekeland ([30], 4):

“The grandfather of these all is the celebrated 1961 theorem of Bishop and Phelps that the set of continuous linear functionals on a Banach space E, which attain their maximum on a prescribed non void closed convex bounded subset X E, is norm-dense in E*”.

This Bishop-Phelps theorem can be found in 2.35, Chapter I, 2.

*

* *

All the normed spaces that appear in this monograph are real.

Let f : X ( , + ] be given. The domain of f , dom f , is

dom f {x X : f ( x ) < +}.

f is proper dom f .

N will denote the set of integer numbers 1.

Both for the functions and for the sequences will be used the terms increasing and strictly increasing, decreasing and strictly decreasing instead of nondecreasing and increasing respectively nonincreasing and decreasing.

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