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We establish fairly general sufficient conditions for a locally compact group (a Baire topological group) to admit partitions into finitely many congruent μ -thick (everywhere of second category) subsets.

We study a particular class of transition kernels that stems from differentiating Markov kernels in the weak sense. Sufficient conditions are established for this type of kernels to admit a Jordan-type decomposition. The decomposition is explicitly constructed.

Sufficient optimality criteria are derived for a control problem under generalized invexity. A Mond-Weir type dual to the control problem is proposed and various duality theorems are validated under generalized invexity assumptions on functionals appearing in the problems. It is pointed out that these results can be applied to the control problem with free boundary conditions and have linkage with results for nonlinear programming problems in the presence of inequality and equality constraints already established in the literature.

In this paper we present a new method of determining Koebe domains. We apply this method by giving a new proof of the well-known theorem of A. W. Goodman concerning the Koebe domain for the class T of typically real functions. We applied also the method to determine Koebe sets for classes of the special type, i.e. for T M,g = {ƒ ∈ T : ƒ(Δ) ⊂ M g (Δ)}, g ∈ T ∩ S , M > 1, where Δ = { z ∈ C : | z | < 1} and T , S stand for the classes of typically real functions and univalent functions respectively. In particular, we find the Koebe domains for the class T M of all typically real and bounded functions, and for the class T ( M ) of all typically real functions with ranges in a given strip.

The purpose of this paper is to prove the existence of a solution of the following periodic boundary value problem in the presence of an upper solution β and a lower solution α with β ≤ α , where ƒ( t , u , v ) satisfies one side Lipschitz condition.

Under a class of generalized (Type I, F , ρ )-convexity assumptions, the author formulates the sufficient conditions about Pareto efficiency and Geoffrion proper efficiency and gets the dual results between multiobjective programming and its Wolfe dual.

In this paper, we introduce a new class of general mixed vector F -implicit complementarity problems and general mixed vector F -implicit variational inequality problems, and study the equivalence between of them under certain assumptions in Banach spaces. We also derive some new existence theorems of solutions for the general mixed vector F -implicit complementarity problems and the general mixed vector F -implicit variational inequality problems by using the FKKM theorem under some suitable assumptions without monotonicity. Moreover, we establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution mapping of the general mixed vector F -implicit variational inequality problems.

Modelling of singularities given by discontinuous functions or Schwartz distributions by means of generalized functions of Colombeau has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions that model singularities given by distributions with singular point support. Moreover, we evaluate various products of singularity-modelling generalized functions whenever the result admits an associated distribution.

It is known that a Fréchet space 𝔽 can be realized as a projective limit of a sequence of Banach spaces . The space K c (𝔽) of all compact, convex subsets of a Fréchet space, 𝔽, is realized as a projective limit of the semilinear metric spaces . Using the notion of Hukuhara derivative for maps with values in K c (𝔽), we prove the local and global existence theorems for an initial value problem associated with a set differential equation.

This paper concerns the existence and multiplicity of positive solutions for a class of nonresonant m -point boundary-value problem of second-order differential equations Lx = λw ( t )ƒ( t , x ), 0 < t < 1. The interesting points are that Lx ≔ – x ″ + ρqx and w is L p -integrable for some 1 ≤ p ≤ +∞. The arguments are based upon fixed point theorems in a cone and Hölder's inequality. The nonexistence of positive solution is also studied. In addition, some examples are included to demonstrate the main results.

A nonlinear multiobjective programming problem is considered. Weak, strong and strict converse duality theorems are established under generalized second order ( F, α, ρ, d )-convexity for second order Mangasarian type and general Mond-Weir type vector duals.