• 대한수학회보
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• 제39권4호
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• pp.561-569
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• 2002

In this paper we prove the existence and uniqueness of a mild solution of a functional differential equation of Sobolev type with nonlocal condition using the semigroup theory and the Banach fixed point principle.

• 충청수학회지
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• 제23권4호
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• pp.731-739
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• 2010

In this paper, we prove stability of the reciprocal difference functional equation $$r\(\frac{{\sum}_{i=1}^{m}x_i}{m}\)-r\(\sum_{i=1}^{m}x_i\)=\frac{(m-1){\prod}_{i=1}^{m}r(x_i)}{{\sum}_{i=1}^{m}{\prod}_{k{\neq}i,1{\leq}k{\leq}m}r(x_k)$$ and the reciprocal adjoint functional equation $$r\(\frac{{\sum}_{i=1}^{m}x_i}{m}\)+r\(\sum_{i=1}^{m}x_i\)=\frac{(m+1){\prod}_{i=1}^{m}r(x_i)}{{\sum}_{i=1}^{m}{\prod}_{k{\neq}i,1{\leq}k{\leq}m}r(x_k)$$ in m-variables. Stability of the reciprocal difference functional equation and the reciprocal adjoint functional equation in two variables were proved by K. Ravi, J. M. Rassias and B. V. Senthil Kumar [13]. We extend their result to m-variables in similar types.

• 충청수학회지
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• 제17권1호
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• pp.29-45
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• 2004

We prove the Cauchy-Rassias stability of cyclic functional equations in Banach modules over a unital $C^*$-algebra.

• 충청수학회지
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• 제20권3호
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• pp.269-277
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• 2007

It is shown that $f:\mathbb{R}{\rightarrow}\mathbb{R}$ satisfies the following functional inequalities (0.1) ${\mid}f(x)+f(y){\mid}{\leq}{\mid}f(x+y){\mid}$, (0.2) ${\mid}f(x)+f(y){\mid}{\leq}{\mid}2f(\frac{x+y}{2}){\mid}$, (0.3) ${\mid}f(x)+f(y)-2f(\frac{x-y}{2}){\mid}{\leq}{\mid}2f(\frac{x+y}{2}){\mid}$, respectively, then the function $f:\mathbb{R}{\rightarrow}\mathbb{R}$ satisfies the Cauchy functional equation, the Jensen functional equation and the Jensen quadratic functional equation, respectively.

• 대한수학회보
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• 제45권1호
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• pp.111-118
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• 2008

Using the Hyers-Ulam-Rassias stability method, we investigate isomorphisms in quasi-Banach algebras and derivations on quasi-Banach algebras associated with the Cauchy-Jensen functional equation $$2f(\frac{x+y}{2}+z)$$=f(x)+f(y)+2f(z), which was introduced and investigated in [2, 17]. The concept of Hyers-Ulam-Rassias stability originated from the Th. M. Rassias' stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. Furthermore, isometries and isometric isomorphisms in quasi-Banach algebras are studied.

• 대한수학회보
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• 제43권4호
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• pp.703-709
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• 2006

Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0 and $$(0.1)\;f(\frac {x+y} 2+z)+f(\frac {x+y} 2-z)+f(\frac {x-y} 2+z)+f(\frac {x-y} 2-z)=f(x)+f(y)+4f(z)$$ for all x, y, z ${\in}$X, then the mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Cauchy-Rassias stability of the functional equation (0.1) in Banach spaces.

• 대한수학회지
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• 제43권2호
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• pp.323-356
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• 2006

Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$mn_{mn-2}C_{k-2}f(\frac {x_1+...+x_{mn}} {mn})$$ $(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1<... if and only if the mapping $f : X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation $(\ddagger)$ in Banach modules over a unital $C^*-algebra$. Let A and B be unital $C^*-algebra$ or Lie $JC^*-algebra$. As an application, we show that every almost homomorphism h : $A{\rightarrow}B$ of A into B is a homomorphism when $h(2^d{\mu}y) = h(2^d{\mu})h(y)\;or\;h(2^d{\mu}\;o\;y)=h(2^d{\mu})\;o\;h(y)$ for all unitaries ${\mu}{\in}A,\;all\;y{\in}A$, and d = 0,1,2,..., and that every almost linear almost multiplicative mapping $h:\;A{\rightarrow}B$ is a homomorphism when h(2x)=2h(x) for all $x{\in}A$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*-algebras$ or in Lie $JC^*-algebras$, and of Lie $JC^*-algebra$ derivations in Lie $JC^*-algebras$.

• 충청수학회지
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• 제26권3호
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• pp.631-639
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• 2013

We investigate the existence of a global attractor for the Cauchy problem $$x^{\prime}(t)=Ax(t)+F(x(t)),\;x(0)=x_0{\in}X$$ on a Banach space X according to the remark in You and Yuan's paper.

• 대한수학회보
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• 제47권5호
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• pp.987-996
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• 2010

In this paper, we investigate the Cauchy-Rassias stability in Banach spaces and also the Cauchy-Rassias stability using the alternative fixed point for the functional equation: $$f(\frac{sx+ty}{2}+rz)+f(\frac{sx+ty}{2}-rz)+f(\frac{sx-ty}{2}+rz)+f(\frac{sx-ty}{2}-rz)=s^2f(x)+t^2f(y)+4r^2f(z)$$ for any fixed nonzero integers s, t, r with $r\;{\neq}\;{\pm}1$.

• 충청수학회지
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• 제28권3호
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• pp.365-375
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• 2015

Of concern is the existence, uniqueness, and continuous dependence of a mild solution of a nonlocal Cauchy problem for a semilinear mixed Volterra-Fredholm functional integrodifferential equation. Our analysis is based on the theory of a strongly continuous semigroup of operators and the Banach fixed point theorem.