• Journal of the Korean Mathematical Society
• /
• v.42 no.5
• /
• pp.1031-1056
• /
• 2005

In this paper, we define the conditional first variation over Wiener paths in abstract Wiener space and investigate its properties. Using these properties, we also investigate relationships among first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transforms of functions in a Banach algebra which is equivalent to the Fresnel class. Finally, we provide another method evaluating the Fourier-Feynman transform for the product of a function in the Banach algebra with n linear factors.

• Journal of applied mathematics & informatics
• /
• v.17 no.1_2_3
• /
• pp.567-572
• /
• 2005

A Banach space X is a Hilbert space if and only if $$E{\mid}x=Y{\mid}{\le}2$$ for all x$\in$ X and all X-valued Bochner integrable functions Y satisfying EY = 0 and ${\mid}x-Y{\mid}{\le}2$ a.e.

• Journal of Applied and Pure Mathematics
• /
• v.5 no.1_2
• /
• pp.69-93
• /
• 2023

Here we give multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ^N, N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a parametrized Gudermannian sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

• Journal of the Chungcheong Mathematical Society
• /
• v.8 no.1
• /
• pp.147-152
• /
• 1995

In this paper, we study the properties of the McShane and Henstock integrals for the case in which the function has values in a Banach space.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.4
• /
• pp.779-789
• /
• 2011

In this paper we study the Banach space $L^1$(G) of real valued measurable functions which are integrable with respect to a vector measure G in the sense of D. R. Lewis. First, we investigate conditions for a scalarly integrable function f which guarantee $f{\in}L^1$(G). Next, we give a sufficient condition for a sequence to converge in $L^1$(G). Moreover, for two vector measures F and G with values in the same Banach space, when F can be written as the integral of a function $f{\in}L^1$(G), we show that certain properties of G are inherited to F; for instance, relative compactness or convexity of the range of vector measure. Finally, we give some examples of $L^1$(G) related to the approximation property.

• Journal of the Chungcheong Mathematical Society
• /
• v.35 no.2
• /
• pp.161-170
• /
• 2022

It is presented a Banach space of functions of bounded mean oscillation BMO type.

• Communications of the Korean Mathematical Society
• /
• v.20 no.3
• /
• pp.505-509
• /
• 2005

We show that every $\varepsilon-approximate$ Jordan functional on a Banach algebra A is continuous. From this result we obtain that every $\varepsilon-approximate$ Jordan mapping from A into a continuous function space C(S) is continuous and it's norm less than or equal $1+\varepsilon$ where S is a compact Hausdorff space. This is a generalization of Jarosz's result [3, Proposition 5.5].

• Journal of applied mathematics & informatics
• /
• v.4 no.1
• /
• pp.83-108
• /
• 1997

Chebysheff-Halley methods are probably the best known cubically convergent iterative procedures for solving nonlinear equa-tions. These methods however require an evaluation of the second Frechet-derivative at each step which means a number of function eval-uations proportional to the cube of the dimension of the space. To re-duce the computational cost we replace the second Frechet derivative with a fixed bounded bilinear operator. Using the majorant method and Newton-Kantorovich type hypotheses we provide sufficient condi-tions for the convergence of our method to a locally unique solution of a nonlinear equation in Banach space. Our method is shown to be faster than Newton's method under the same computational cost. Finally we apply our results to solve nonlinear integral equations appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field.

• Journal of the Korean Mathematical Society
• /
• v.41 no.6
• /
• pp.995-1005
• /
• 2004

The aim of this paper is to prove the stability in the sense of Hyers-Ulam- Rassias of the Banach space valued differentialequation y' = λy, where λ is a complex constant. That is, suppose f is a Banach space valued strongly differentiable function on an open interval. If f is an approximate solution of the equation y' = λy, then there exists an exact solution of the equation near to f.

• Communications of the Korean Mathematical Society
• /
• v.15 no.1
• /
• pp.77-90
• /
• 2000

In this paper, we will establish the existence theorem of the operator valued function space integral over paths in abstract Wiener space under the general conditions rather than the known conditions.