Dini derivative

About: Dini derivative is a research topic. Over the lifetime, 196 publications have been published within this topic receiving 3130 citations.

Stability Theory by Liapunov’s Direct Method

Abstract: I. Elements of Stability Theory.- 1. A First Glance at Stability Concepts.- 2. Various Definitions of Stability and Attractivity.- 3. Auxiliary Functions.- 4. Stability and Partial Stability.- 5. Instability.- 6. Asymptotic Stability.- 7. Converse Theorems.- 8. Bibliographical Note.- II. Simple Topics in Stability Theory.- 1. Theorems of E.A. Barbashin and N.N. Krasovski for Autonomous and Periodic Systems.- 2. A Theorem of V.M. Matrosov on Asymptotic Stability.- 3. Introduction to the Comparison Method.- 4. Total Stability.- 5. The Frequency Method for Stability of Control Systems.- 6. Non-Differentiable Liapunov Functions.- 7. Bibliographical Note.- III. Stability of a Mechanical Equilibrium.- 1. Introduction.- 2. The Lagrange-Dirichlet Theorem and Its Variants.- 3. Inversion of the Lagrange-Dirichlet Theorem Using Auxiliary Functions.- 4. Inversion of the Lagrange-Dirichlet Theorem Using the First Approximation.- 5. Mechanical Equilibrium in the Presence of Dissipative Forces.- 6. Mechanical Equilibrium in the Presence of Gyroscopic Forces.- 7. Bibliographical Note.- IV. Stability in the Presence of First Integrals.- 1. Introduction.- 2. General Hypotheses.- 3. How to Construct Liapunov Functions.- 4. Eliminating Part of the Variables.- 5. Stability of Stationary Motions.- 6. Stability of a Betatron.- 7. Construction of Positive Definite Functions.- 8. Bibliographical Note.- V. Instability.- 1. Introduction.- 2. Definitions and General Hypotheses.- 3. Fundamental Proposition.- 4. Sectors.- 5. Expellers.- 6. Example of an Equation of N Order.- 7. Instability of the Betatron.- 8. Example of an Equation of Third Order.- 9. Exercises.- 10. Bibliographical Notes.- VI. A Survey of Qualitative Concepts.- 1. Introduction.- 2. A View of Stability and Attractivity Concepts.- 3. Qualitative Concepts in General.- 4. Equivalence Theorems for Qualitative Concepts.- 5. A Tentative Classification of Concepts.- 6. Weak Attractivity, Boundedness, Ultimate Boundedness.- 7. Asymptotic Stability.- 8. Bibliographical Note.- VII. Attractivity for Autonomous Equations.- 1. Introduction.- 2. General Hypotheses.- 3. The Invariance Principle.- 4. An Attractivity and a Weak Attractivity Theorem.- 5. Attraction of a Particle by a Fixed Center.- 6. A Class of Nonlinear Electrical Networks.- 7. The Ecological Problem of Interacting Populations.- 8. Bibliographical Note.- VIII. Attractivity for Non Autonomous Equations.- 1. Introduction, General Hypotheses.- 2. The Families of Auxiliary Functions.- 3. Another Asymptotic Stability Theorem.- 4. Extensions of the Invariance Principle and Related Questions.- 5. The Invariance Principle for Asymptotically Autonomous and Related Equations.- 6. Dissipative Periodic Systems.- 7. Bibliographical Note.- IX. The Comparison Method.- 1. Introduction.- 2. Differential Inequalities.- 3. A Vectorial Comparison Equation in Stability Theory.- 4. Stability of Composite Systems.- 5. An Example from Economics.- 6. A General Comparison Principle.- 7. Bibliographical Note.- Appendix I. DINI Derivatives and Monotonic Functions.- 1. The Dini Derivatives.- 2. Continuous Monotonic Functions.- 3. The Derivative of a Monotonic Function.- 4. Dini Derivative of a Function along the Solutions of a Differential Equation.- Appendix II. The Equations of Mechanical Systems.- Appendix III. Limit Sets.- List of Examples.- Author Index.

On error bounds for lower semicontinuous functions

Abstract: We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f.

Advanced Analysis: on the Real Line

Abstract: 0 Preliminaries.- 0.1 Lebesgue Measure.- 0.2 The Lebesgue Integral.- 0.3 Vitali Covering Theorem.- 0.4 Baire Category Theorem and Baire Class Functions.- 1 Monotone Functions.- 1.1 Continuity Properties.- 1.2 Differentiability Properties.- 1.3 Reconstruction of f from f?.- 1.4 Series of Monotone Functions.- Exercises.- 2 Density and Approximate Continuity.- 2.1 Preliminaries and Definitions.- 2.2 The Lebesgue Density Theorem.- 2.3 Approximate Continuity.- 2.4 Approximate Continuity and Integrability.- 2.5 Further Results on Approximate Continuity.- 2.6 Sierpinski's Theorem.- 2.7 The Darboux Property and the Density Topology.- Exercises.- 3 Dini Derivatives.- 3.1 Preliminaries and Definitions.- 3.2 Simple Properties of Derivatives.- 3.3 Ruziewicz's Example.- 3.4 Further Properties of Derivatives.- 3.5 The Denjoy-Saks-Young Theorem.- 3.6 Measurability of Dini Derivatives.- 3.7 Dini Derivatives and Convex Functions.- Exercises.- 4 Approximate Derivatives.- 4.1 Definitions.- 4.2 Measurability of Approximate Derivatives.- 4.3 Analogue of the Denjoy-Saks-Young Theorem.- 4.4 Category Results for Approximate Derivatives.- 4.5 Other Properties of Approximate Derivatives.- Exercises.- 5 Additional Results on Derivatives.- 5.1 Derivatives.- 5.2 Derivates.- 5.3 Approximate Derivatives.- 5.4 The Denjoy Property.- 5.5 Metrically Dense.- 6 Bounded Variation.- 6.1 Bounded Variation of Finite Intervals.- 6.2 Stieltjes Integral.- 6.3 The Space BV[a,b].- BVloc and L1loc.- 6.5 Additional Remarks on Fubini's Theorem.- Exercises.- 7 Absolute Continuity.- 7.1 Absolute Continuity.- 7.2 Rectifiable Curves.- Exercises.- 8 Cantor Sets and Singular Functions.- 8.1 The Cantor Ternary Set and Function.- 8.2 Hausdorff Measure.- 8.3 Generalized Cantor Sets-Part I.- 8.4 Generalized Cantor Sets-Part II.- 8.5 Cantor-like Sets.- 8.6 Strictly Increasing Singular Functions.- Exercises.- 9 Spaces of BV and AC Functions.- 9.1 Convergence in Variation.- 9.2 Convergence in Length.- 9.3 Norms on AC.- 9.4 Norms on BV.- 10 Metric Separability.- Exercises.

From scalar to vector optimization

Abstract: Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem ϕ(x) → min, x ∈ ℝ m , are given. These conditions work with arbitrary functions ϕ: ℝ m → ℝ, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if ϕ is of class $$\mathcal{C}^{1,1}$$ (i.e., differentiable with locally Lipschitz derivative). Further, considering $$\mathcal{C}^{1,1}$$ functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmaki, Krizek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.