Volume 8, Issue 1

Using arguments which apply equally well to the study of Brownian motion and Feynman path integrals, the relationship between two expressions which arise in derivative asset pricing theory is examined. Detailed explanations are given for some of the key points in the theory of Henstock integrals in function spaces.

The aim of this paper is to provide an explicit formula for solutions of the following system of delay difference equations F n +1 = A n F n + B n F γ n + ƒ n , n ∈ ℕ, where γ n = kα n , α n = [ n / k ] (the symbol [ x ] stands for entire part of the real number x and k is a fixed positive integer). ( A n ), ( B n ), n ∈ ℕ, are sequences of square matrices of order m , (ƒ n ) is a sequence of vectors from . From this formula conditions for the stability and asymptotic stability of solutions are derived.

In this paper we investigate the existence and controllability of mild solutions to the first order semilinear evolution inclusions in Banach spaces with nonlocal conditions. We shall rely of a fixed point theorem for condensing maps due to Martelli.

Results on products of Schwartz distributions are obtained when they have coinciding point singularities and only sums of the products exist in the distribution space. These results follow the pattern of a well-know distributional product published by Jan Mikusiński in 1966, and are named Mikusiński type products. The formulas are derived as the distributions are embedded in Colombeau algebra of generalized functions. This algebra possesses optimal properties regarding the distributional multiplication, and its notion of “association” allows one to obtain the results in terms of distributions.

We consider a mathematical model which describes the frictional contact between a deformable body and an obstacle, say a foundation. The body is assumed to be linear elastic and the contact is modeled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. The novelty consists here in the fact that we consider a slip dependent coefficient of friction and a quasistatic process. We present two alternative yet equivalent formulations of the problem and establish existence and uniqueness results. The proofs are based on a new result obtained in [Motreanu and Sofonea, Abstr. Appl. Anal. 4: 255–279, 1999] in the study of evolutionary variational inequalities.

We apply the Kurzweil-Henstock integral setting to prove a Fredholm Alternative-type result for the integral equation x ( t ) – K ∫ [ a, b ] α ( t, s ) x ( s ) ds = ƒ ( t ), t ∈ [ a, b ], where x and ƒ are Kurzweil integrable functions (possibly highly oscillating) defined on a compact interval [ a, b ] of the real line with values on Banach spaces. An application is given.

The cyclic group labelled family of α -projection operators implicitly present in [Ungar, J. Pure Appl. Math. 15: 301–304, 1984] is used as in [Ben Cheikh, Matematiche (Catania) 52: 365–378, 1997]–[Ben Cheikh, Appl. Math. Inform. 4: 30–53, 2000], [Kwaśniewski, Adv. Appl. Clifford Algebras 9: 41–54, 1999] for investigation of decomposition of functions with respect to the cyclic group of order n . Series of new identities thus arising are demonstrated and new perspectives for further investigation are indicated as for example in the case of q -extended special polynomials. The paper constitutes an example of the application of the method of projections introduced in [Muldoon, Ungar, Math. Mag. 69: 3–14, 1996]; see also references [Ben Cheikh, Matematiche (Catania) 52: 365–378, 1997]–[Ben Cheikh, Appl. Math. Inform. 4: 30–53, 2000].

The thermo-elastic plate system of equations is analysed. The sufficient conditions of existence, uniqueness and continuity dependence on initial data of the Cauchy problem solutions for differential-operational equation of mixed type (a part of the equation of hyperbolic type, and a part of parabolic type) are given in this paper. If the operational coefficients are suitably chosen, the investigated equation can be used to obtain a differential equation describing vibrations of a plate — the modified Germain-Lagrange equation of hyperbolic type. Moreover, in order to define the temperature field, one can use a three-dimensional equation of thermal conductivity (a parabolic equation).

We consider the stationary flow of a heat conducting Power Law shear thinning fluid in a bounded domain in . We present an elementary proof of existence of at least one weak solution.