• Journal of Applied and Pure Mathematics
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• 제5권1_2호
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• pp.69-93
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• 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.

• Communications of the Korean Mathematical Society
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• 제9권2호
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• pp.331-336
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• 1994

Let $P_{0}$ be a $\delta$-ring (a ring closed with respect to the forming of countable intersections) of subsets of a nonempty set $\Omega$. Let X and Y be Banach spaces and L(X, Y) the Banach space of all bounded linear operators from X to Y. A set function m : $P_{0}$ longrightarrow L(X, Y) is called an operator valued measure countably additive in the strong operator topology if for every x $\epsilon$ X the set function E longrightarrow m(E)x is a countably additive vector measure. From now on, m will denote an operator valued measure countably additive in the strong operator topology.(omitted)

• East Asian mathematical journal
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• 제17권1호
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• pp.71-85
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• 2001

The strong convergences of solutions to a nonlinear Volterra equation are studied in Banach space. As an application, a nonlinear heat flow in a homogeneous bar of unit length of a material with memory is investigated.

• Korean Journal of Mathematics
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• 제4권2호
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• pp.125-133
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• 1996

We show that if X is a reflexive Banach space with a uniformly G$\hat{a}$teaux differentiable norm, then the convergence of semigroups acting on Banach spaces $X_n$ implies the convergence of resolvents of generators of semigroups.

• Honam Mathematical Journal
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• 제20권1호
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• pp.111-115
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• 1998

In this note, we give an economical Kato's decomposition for a regular operator on a Banach space.

• Journal of the Korean Mathematical Society
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• 제48권5호
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• pp.1043-1052
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• 2011

We introduce a new norm, called the $N^p$-norm (1 $\leq$ p < ${\infty}$ on the space $N^p$(V,W) where V and W are abstract operator spaces. By proving some fundamental properties of the space $N^p$(V,W), we also discover that if W is complete, then the space $N^p$(V,W) is also a Banach space with respect to this norm for 1 $\leq$ p < ${\infty}$.

• Bulletin of the Korean Mathematical Society
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• 제48권6호
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• pp.1129-1135
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• 2011

In this note, we prove that every subscalar operator with finite spectrum is algebraic. In particular, a quasi-nilpotent subscala operator is nilpotent. We also prove that every subscalar operator with property (${\delta}$) on a Banach space of dimension greater than 1 has a nontrivial invariant closed linear subspace.

• Bulletin of the Korean Mathematical Society
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• 제54권2호
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• pp.443-454
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• 2017

We study a notion of q-frequent hypercyclicity of linear maps between the Banach algebras consisting of operators on a separable infinite dimensional Banach space. We derive a sufficient condition for a linear map to be q-frequently hypercyclic in the strong operator topology. Some properties are investigated regarding q-frequently hypercyclic subspaces as shown in [5], [6] and [7]. Finally, we study q-frequent hypercyclicity of tensor products and direct sums of operators.

• Communications of the Korean Mathematical Society
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• 제15권1호
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• pp.77-90
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• 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.

• Korean Journal of Mathematics
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• 제25권3호
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• pp.359-377
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• 2017

In this paper, we consider a system of generalized nonlinear ordered variational inclusions in real ordered Banach spaces and define an iterative algorithm for a solution of our problems. By using the resolvent operator techniques to prove an existence result for the solution of the system of generalized nonlinear ordered variational inclusions and discuss convergence of sequences suggested by the algorithms.