ON GENERALIZED PROJECTIONS OF ANALYTIC SETS

THE LSRS PROPERTY AND THE CONVERGENCE THEOREM OF HENSTOCK-KURZWEIL INTEGRAL

Chapter 13 - The Henstock—Kurzweil Integral

Controlled convergence theorems for infinite dimension Henstock integrals of fuzzy valued functions based on weak equi-integrability

The concept of the GAP-integral was introduced by the authors [GANGULY, D. K.-MUKHERJEE, R.: The generalized approximate Perron integral, Math. Slovaca 58 (2008), 31-42]. In this paper we prove the controlled convergence theorem for the GAP-integral and deduce other convergence theorems as corollaries.

The Controlled Convergence Theorem for the Approximately Continuous Integral of Burkill

In this paper the approximate solution of a self-adjoint linear differential system by a non-orthogonal finite sum is examined in its most general form. In obtaining the coefficients, the variational concept is based on minimizing the “energy norm” of the error function of the approximation. This systematic treatment is shown to encompass the usual Rayleigh-Ritz and Galerkin methods which arise in certain boundary value problems with homogeneous boundary conditions and the Trefftz method which arises in certain boundary value problems with homogeneous differential equations. Using theorems proviously proven by Ho, it is then shown that the approximation is improved in the “mean” when an additional term is introduced (“stepwise mean convergence”). Conditions on the uniqueness of the solution and the proper prescription of the boundary conditions follow directly from the form of the equation describing the selfadjoint operation. The method is exemplified in the paper by application to the problem in antiplane elastic shear deformation of a finite crack. A surprisingly elegant closed form solution is obtained.

We provide a justification of the Reissner–Mindlin plate theory, using linear three-dimensional elasticity as framework and Γ-convergence as technical tool. Essential to our developments is the selection of a transversely isotropic material class whose stored energy depends on (first and) second gradients of the displacement field. Our choices of a candidate Γ-limit and a scaling law of the basic energy functional in terms of a thinness parameter are guided by mechanical and formal arguments that our variational convergence theorem is meant to validate mathematically.

The controlled convergence theorems for the strong Henstock integrals of fuzzy-number-valued functions

In this paper, a controlled convergence theorem is proved for \(n\)-dimensional strong variational Banach-valued integrals, also referred herein as Banach-valued Multiple Integrals. The methods used in the proof for one dimensional case given in [15], in which linearization was used, cannot be applied for the higher dimensional case. Instead, we follow the ideas in [17, Chapter 5, Section 21; 4; 18].

is to evaluate a double integral involving generalized I-function of two variables.

In this paper, using the properties of the strong Henstock integrals of fuzzy-number-valued functions and controlled convergence theorem, we prove the existence theorem for the discontinuous fuzzy system x′ = \(\tilde f(t,x)\) in fuzzy number space, where f is strong fuzzy Henstock integrable.

By using the strong fuzzy Henstock integral and its controlled convergence theorem, we generalized the existence theorems of solution for initial problems of fuzzy discontinuous integral equation.

We generalized the existence theorems and the continuous dependence of a solution on parameters for initial problems of fuzzy discontinuous differential equation by the strong fuzzy Henstock integral and its controlled convergence theorem.

In this paper, we generalized the existence theorems of Caratheodory solution for initial problems of fuzzy discontinuous differential equation by the definition of the \(\omega -ACG^{*}\) for a fuzzy-number-valued function and the nonabsolute fuzzy integral and its controlled convergence theorem.

A Short Proof of the Controlled Convergence Theorem for Henstock Integrals