• Journal of applied mathematics & informatics
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• v.6 no.3
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• pp.867-872
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• 1999

In this paper we study the stability theorems for nonlinear Fredholm-Volterra integral equations system.

• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.791-802
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• 2000

In 1968, Cameron and Storvick introduce the definition and the theories of the operator-valued function space integral. Since then, the stability theorems of the integral was developed by Johnson, Skoug, Chang etc [1, 2, 4, 5]. Recently, the authors establish the existence theorem of the operator-valued function space [8]. In this paper, we will prove the stability theorems of the operator-valued function space integral over paths in abstract Wiener space $C_0(B)$.

• East Asian mathematical journal
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• v.17 no.1
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• pp.33-45
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• 2001

In this paper an improved version of a fixed point theorem of the present author [3] in Banach algebras is obtained under the weaker conditions with a different method and using measure of non-compactness. The newly developed fixed point theorem is further-applied to certain nonlinear integral equations of mixed type for proving the existence theorems and stability of the solution in Banach algebras.

• Journal of the Korean Mathematical Society
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• v.35 no.4
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• pp.999-1018
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• 1998

In this paper, we prove stability theorems for the operator-valued Feynman integral of certain functionals involving some Borel measures on (0, t) as a bounded linear operator from L$_1$(R) to $C_{0}$(R).

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.515-536
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• 2012

This work focuses on the general decay stability of nonlinear stochastic differential equations with unbounded delay. A Razumikhin-type theorem is first established to obtain the moment stability but without almost sure stability. Then an improved edition is presented to derive not only the moment stability but also the almost sure stability, while existing Razumikhin-type theorems aim at only the moment stability. By virtue of the $M$-matrix techniques, we further develop the aforementioned Razumikhin-type theorems to be easily implementable. Two examples are given for illustration.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.3
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• pp.531-541
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• 2012

Feynman's operational calculus for noncommuting operators was studied by means of measures on the time inteval. And various stability theorems for Feynman's operational calculus were investigated. In this paper we see the time-dependent stability properties when the operator-valued functions take their values in a separable Hilbert space.

• Korean Journal of Mathematics
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• v.23 no.3
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• pp.393-399
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• 2015

W. Park [J. Math. Anal. Appl. 376 (2011) 193-202] proved the Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces. But there are serious problems in the control functions given in all theorems of the paper. In this paper, we correct the statements of these results and prove the corrected theorems. Moreover, we prove the superstability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces under the original given conditions.

• The Pure and Applied Mathematics
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• v.30 no.4
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• pp.407-416
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• 2023

In this paper, we introduce the concept of C^*-algebra-valued quasi b-metric space and prove some existence and uniqueness theorems. Furthermore, we prove the Hyers-Ulam stability results for fixed point problems via C^*-algebra-valued extended quasi b-metric space.

• The Pure and Applied Mathematics
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• v.8 no.1
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• pp.71-78
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• 2001

In 1984, Johnson[A bounded convergence theorem for the Feynman in-tegral, J, Math. Phys, 25(1984), 1323-1326] proved a bounded convergence theorem for hte Feynman integral. This is the first stability theorem of the Feynman integral as an $L(L_2 (\mathbb{R}^N), L_2(\mathbb{R}^{N}))$ theory. Johnson and Lapidus [Generalized Dyson series, generalized Feynman digrams, the Feynman integral and Feynmans operational calculus. Mem, Amer, Math, Soc. 62(1986), no 351] studied stability theorems for the Feynman integral as an $L(L_2 (\mathbb{R}^N), L_2(\mathbb{R}^{N}))$ theory for the functional with arbitrary Borel measure. These papers treat functionals which involve only a single integral. In this paper, we obtain the stability theorems for the Feynman integral as an $L(L_1 (\mathbb{R}^N), L_{\infty}(\mathbb{R}^{N}))$theory for the functionals which involve double integral with some Borel measures.

• Communications of the Korean Mathematical Society
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• v.15 no.1
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• pp.45-50
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• 2000

The modified Hyers-Ulam-Rassias Stability Of the generalized form g(x+p) : $\phi$(x)g(x) for the Gamma functional equation shall be proved. As a consequence we obtain the stability theorems for the gamma functional equation.