A General Duality Scheme for Nonconvex Minimization Problems with a Strict Inequality Constraint

We establish a duality formula for the problem Minimize f(x)+g(x) for h(x)+k(x)<0 where g, k are extended-real-valued convex functions and f, h belong to the class of functions that can be written as the lower envelope of an arbitrary family of convex functions. Applications in d.c. and Lipschitzian optimization are given.     

A topological invariant for pairs of maps

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝⁿ, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.     

Periodic solutions of singular second order differential equations: Upper and lower functions

In this paper, we continue the study of the periodic problem for the second-order equation u^′′+f(u)u^′+g(u)=h(t,u), where h is a Carathéodory function and f,g are continuous functions on (0,+∞) which may have singularities at zero. Both attractive and repulsive singularities are considered. The method relies on a novel technique of construction of lower and upper functions. As an application, we obtain new sufficient conditions for the existence of periodic solutions to the Rayleigh–Plesset equation.     

A generalized mixed type of quartic–cubic–quadratic–additive functional equations

We determine the general solution of the functional equation f(x + ky) + f(x-ky) = g(x + y) + g(x-y) + h(x) + h(y) for fixed integers with k ≠ 0; ±1 without assuming any regularity conditions for the unknown functions f, g, h, and0020[(h)\tilde] \tilde{h} . The method used for solving these functional equations is elementary but it exploits an important result due to Hosszú. The solution of this functional equation can also be obtained in groups of certain type by using two important results due to Székelyhidi.     

First-Order Necessary Conditions of Optimality for Strongly Monotone Nonlinear Control Problems

Necessary conditions for the optimality of a pair (y*, u*) with respect to the cost functional g(y) + h(u) subject to AyBu + f are given in terms of generalized gradients. Here, g is locally Lipschitz, h is convex, A is a maximal strongly monotone operator, and B is linear. Two examples of applications of our necessary conditions to nonlinear partial differential equations of elliptic type are presented.     

A fixed point approach to stability of a quadratic equation

Using the fixed point alternative theorem we establish the orthogonal stability of the quadratic functional equation of Pexider type f (x+y)+g(x−y) = h(x)+k(y), where f, g, h, k are mappings from a symmetric orthogonality space to a Banach space, by orthogonal additive mappings under a necessary and sufficient condition on f.     

Berezin Transform, Mellin Transform and Toeplitz Operators

We consider functions of the form f₁[`(g)]<sub>1</sub>+h{f_1\bar g_1+h} in the range of the Berezin transform B, where f <sub>1</sub> and g <sub>1</sub> are holomorphic on the unit disk \mathbb D{\mathbb D}, and h is either harmonic or of the form f<sub>2</sub>[`(g)]₂{f_2\bar g_2} for some holomorphic functions f ₂ and g ₂ on \mathbb D{\mathbb D}. First, by using the Mellin transform, we complement Ahern’s Theorem (Ahern in J Funct Anal 215:206–216, 2004) by proving that if u ? L¹{u\in L^1} and B(u) is harmonic, then u is harmonic. Secondly, we extend Ahern’s Theorem when h is harmonic, and give very precise relations between f ₁ and f ₂, g ₁ and g ₂ when h=f₂[`(g)]₂{h=f_2\bar g_2} and g ₂(z) = z ⁿ with n ≥ 1. Finally, some applications of our results to the theory of Toeplitz operators are discussed.     

Products of Toeplitz Operators on the Polydisk

This paper studies products of Toeplitz operators on the Hardy space of the polydisk. We show that T _f T _g = 0 if and only if T _f T _g is a finite rank if and only if T _f or T _g is zero. The product T _f T _g is still a Toeplitz operator if and only if there is a h $ \in $ L $ \infty $ (T ⁿ ) such that T _f T _g - T _h is a finite rank operator. We also show that there are no compact simi-commutators with symbols pluriharmonic on the polydisk. Submitted: October 5, 2000     

Hölder Continuity of Roots of Complex and p-adic Polynomials

Let f be a polynomial with coefficients in an algebraically closed, valued field. We show a refinement of the principle of continuity of roots, namely, that each root α of f is locally Hölder continuous of order 1/μ as a function of the coefficients of f, where μ is the root multiplicity of α. This is derived as a consequence of a principle that could be called continuity of factors, namely, that if f = gh is a factorisation with (g, h) = 1, then the coefficients of g and h are locally Lipschitz continuous as functions of the coefficients of f. The proofs are elementary and of an algebraic nature.     

Two-generated groups acting on trees

We study 2-generated subgroups of groups that act on simplicial trees. We show that any generating pair {g,h}\{{g},h\} of such a subgroup is Nielsen-equivalent to a pair {f,s}\{f,s\} where either powers of f and s or powers of f and sfs^-1sfs^{-1} have a common fixed point if the subgroup ág,h?\langle {g},h\rangle is freely indecomposable. Analogous results are obtained for generating pairs of fundamental groups of graphs of groups. Some simple applications are given.     

Second-order optimality conditions for minimizing a max-function

We study Chaney's and Ben-Tal-Zowe's second-order directional derivatives with applications in minimization problem for max-functions of the formh(x): = max {f(x, τ); τ ∈T},where T is a compact metricspace. We improve Kawasaki's result on necessary condition for such functions in the minimization problem.     

Sensitivity analysis of composite piecewise smooth equations

This paper is a contribution to the sensitivity analysis of piecewise smooth equations. A piecewise smooth function is a Lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible B-derivative at this point. We emphasise the role of functions of the typef=g °h whereg is piecewise smooth andh is smooth and present verifiable conditions which ensure that the functionf=g ° is a Lipschitzian homeomorphism near a given point for every sufficiently close toh with respect to theC ¹-topology. Revised version of part of the paper “Sensitivity analysis and Newton’s method for composite piecewise smooth equations”.     

A variational problem for pullback metrics

Let (M, g) and (N, h) be Riemannian manifolds without boundary and let f : M → N be a smooth map. Let ||f^*h||{\|f^*h\|} denote the norm of the pullback metric of h by f. In this paper, we consider the functional F(f) = ò_M ||f^*h||² dv_g{{\Phi (f) = \int_M \|f^*h\|^2 dv_g}}. We prove the existence of minimizers of the functional Φ in each 3-homotopy class of maps, where maps f ₁ and f ₂ are 3-homotopic if they are homotopic on the three dimensional skeltons of a triangulation of M. Furthermore, we give a monotonicity formula and a Bochner type formula.     

Comparison semigroups and algebras of transformations

We characterize algebras of transformations on a set under the operations of composition and the pointwise switching function defined as follows: (f,g)[h,k](x)=h(x) if f(x)=g(x), and k(x) otherwise. The resulting algebras are both semigroups and comparison algebras in the sense of Kennison. The same characterization holds for partial transformations under composition and a suitable generalisation of the quaternary operation in which agreement of f,g includes cases where neither is defined. When a zero element is added (modelling the empty function), the resulting signature is rich enough to encompass many operations on semigroups of partial transformations previously considered, including set difference and intersection, restrictive product, and a functional analog of union. When an identity element is also added (modelling the identity function), further domain-related operations can be captured.     

Invertibility of the Sum of Idempotents

We study necessary and sufficient conditions for the invertibility of the sum f+g when f and g are idempotents in a unital ring or bounded linear operators in Hilbert or Banach spaces. We describe the relation between the invertibility of f+g and f m g.     

Conjugation of injections by permutations

Let Ω be a countably infinite set, Inj(Ω) the monoid of all injective endomaps of Ω, and Sym(Ω) the group of all permutations of Ω. Also, let f,g,h∈Inj(Ω) be any three maps, each having at least one infinite cycle. (For instance, this holds if f,g,h∈Inj(Ω)∖Sym(Ω).) We show that there are permutations a,b∈Sym(Ω) such that h=afa ⁻¹ bgb ⁻¹ if and only if |Ω∖(Ω)f|+|Ω∖(Ω)g|=|Ω∖(Ω)h|. We also prove a generalization of this statement that holds for infinite sets Ω that are not necessarily countable.     

The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems

The DC programming and its DC algorithm (DCA) address the problem of minimizing a function f=g−h (with g,h being lower semicontinuous proper convex functions on R ⁿ ) on the whole space. Based on local optimality conditions and DC duality, DCA was successfully applied to a lot of different and various nondifferentiable nonconvex optimization problems to which it quite often gave global solutions and proved to be more robust and more efficient than related standard methods, especially in the large scale setting. The computational efficiency of DCA suggests to us a deeper and more complete study on DC programming, using the special class of DC programs (when either g or h is polyhedral convex) called polyhedral DC programs. The DC duality is investigated in an easier way, which is more convenient to the study of optimality conditions. New practical results on local optimality are presented. We emphasize regularization techniques in DC programming in order to construct suitable equivalent DC programs to nondifferentiable nonconvex optimization problems and new significant questions which have to be answered. A deeper insight into DCA is introduced which really sheds new light on DCA and could partly explain its efficiency. Finally DC models of real world nonconvex optimization are reported.     

Solutions of linear iterative functional equations behaving like the forcing term in the equation

Summary. We are dealing with those continuous solutions j \varphi of the functional equation¶¶j°f=g·j+h \varphi\circ f=g\cdot \varphi+h ¶that are asymptotically comparable at the origin (the fixed point of f) with the function h. Connections with a linear iterative functional inequality of second order are also mentioned.     

A generalization of the cosine-sine functional equation on groups

The functional equation f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y) is solved where f, g, h are complex functions defined on a group.