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S. Z. Németh

Ph.D. thesis

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Our dissertation presents some geometrical results concerning the Minty-Browder monotonicity. In the first chapter different computational methods are presented for determining the domains of monotonicity of an operator $f:\,{\bf R}^n\to{\bf R}^n$. Most of them are connected to a new notion introduced by us the scalar derivative. In the 2-dimensional case the connection of the scalar derivative with the complex derivative and Kasner circle is outlined.

In the second chapter the notion of Minty-Browder monotonicity is extended to Riemannian manifolds, introducing the monotone vector fields. The monotone vector fields of gradient type were analyzed by C. Udriste and T. Rapcsák in [34], [35] and [31]. However the nongradient case was not considered yet. We deal mostly with this general case.

Connections with first variations of the length of a geodesic, closed geodesics, one parameter expansive transformation groups, integral submanifolds, auto-parallel submanifolds, the Lie derivative, conformal vector fields, projection maps, nonexpansive maps, singularities of vector fields are exhibited. Throughout the dissertation many examples are given, and different computation methods are presented for determinining the domains of monotonicity of a vector field. Most of them are connected with a new notion introduced by us the geodesic scalar derivative. We generalized the notion of monotonicity, introducing the pseudomonotone, strictly pseudomonotone and quasimonotone vector fields. Connections with the corresponding types of convexities are presented.


The Minty-Browder monotonicity notion was an important achievement of nonlinear analysis. It throws a new light on many known results and becomes a good instrument in various application oriented new investigations. 

Convex functionals in Hilbert spaces have as gradients monotone operators. This result was extended to Riemannian manifolds by C. Udriste [34], [35]. He proved that the gradient of a (geodesic) convex functional [31] on a Riemannian manifold is monotone. T. Rapcsák [31] gives a more explicit form of this result. Hovewer, there are nongradient type monotone vector fields which (as far as we know) has not been studied yet.

Recent developments of convex optimization problems on Riemannian manifolds [31] focus the attention to nongradient type monotone vector fields, since in the Hilbert case in the analysis and solution of complementary problems, variational inequalities, and evolution equations it is commonly assumed that the defining map is monotone (and not necessarily of gradient type).

The main purpose of our thesis is to present global results on the geometrical aspects of the Minty-Browder monotonicity. We consider that this notion has a strong geometrical meaning, which has not been analysed yet.

The thesis is structured on chapters, sections and subsections. The first chapter is devoted to the euclidean case. It contains three sections. The first two deals with first order characterizations of monotone maps through the scalar derivative, a new notion introduced by us. In the two dimensional case the connection of scalar derivatives with the complex derivative and the Kasner circle is also encountered.

The third section analyses the connection of scalar derivative with conformity. It is proved that globally scalar differentiability of a map (which can be identified with a vector field) implies the conformity of the one parameter transformation group generated by this map (vector field). Locally, the connection is given using the conformal derivative, a new notion introduced by us. In this section the well-known connection between monotone operators and nonexpansive semigroups [40] is also presented, from a more geometrical point of view.

The second chapter is devoted to the Riemannian case. It contains six sections. In the first section the notion of geodesic monotone vector fields is introduced [35], which generalize the monotone operators. First order characterization of geodesic monotone vector fields are given, using the geodesic scalar derivatives, a new notion introduced by us. Different computational formulae are presented for this derivative. The constant curvature case is analysed separately, since it has important peculiarities. We present the connection of geodesic monotone vector fields with the first variation of lengths of geodesics and the existence of closed geodesics. For Hadamard manifolds (complete, simply connected Riemannian manifolds of nonpositive sectional curvature) a class of geodesic monotone vector fields is given which generalize the position vector fields [26]. It is proved that there are monotone vector fields on Hadamard manifolds, which are not gradient fields.

In the second section we analyse the connection of geodesic monotone vector fields with one parameter expansive transformation groups. We introduce the Killing monotone vector fields, which are the vector fields generated by such transformation groups. We proved that the notion of Killing monotone vector fields and geodesic monotone vector fields coincide. For brevity's sake they are called monotone vector fields. These results are generalizations of the results of section 1.3.2. We also analyse the connection of the Lie derivative with monotonicity and (geodesic) scalar derivatives. It is proved that a vector field is conformal if and only if it is (geodesic) scalar differentiable, generalizing the corresponding result of section 1.3.1.

The third and fourth section deals with complementary vector fields of maps on Hadamard manifolds. We define the notion of complementary vector fields and $\lambda$-complementary vector fields of maps on Hadamard manifolds. It is proved that for $\lambda$ increasing the $\lambda$-complementary vector fields of projection maps to closed (geodesic) convex sets [31] are monotone. It is also proved that the complementary vector fields of nonexpansive maps are monotone. As a particular case composition of projection maps to closed convex sets are considered. These results are generalizations of some results of [38].

In the fifth section different types of generalized monotone vector fields are introduced. It is showed that for gradient fields, these generalized monotonicity properties correspond to generalized convexity properties of the underlying functions. These results extend some results of [11].

In the sixth section we prove that the zeros of a monotone vector field, defined in a convex open subset of an Hadamard manifold, form a convex set.

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