• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.1053-1063
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• 2011

We discuss the perturbation theory of "left" and "right" Browder operators, which come somewhere between Browder operators and semi Browder operators.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.493-507
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• 1995

The Hartman-Stampacchia variational inequality was first suggested and studied by Hartman and Stampacchia [8] in finite dimensional spaces during the time establishing the base of variational inequality theory in 1960s [4]. Then it was generalized by Lions et al. [6], [9], [10], Browder [3] and others to the case of infinite dimensional inequality [3], [9], [10], and the results concerning this variational inequality have been applied to many important problems, i.e., mechanics, control theory, game theory, differential equations, optimizations, mathematical economics [1], [2], [6], [9], [10]. Recently, the Browder-Hartman-Stampaccnia variational inequality was extended to the case of set-valued monotone mappings in reflexive Banach sapces by Shih-Tan [11] and Chang [5], and under different conditions, they proved some existence theorems of solutions of this variational inequality.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.853-869
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• 2018

We give a new characterization of Browder's theorem through equality between the pseudo B-Weyl spectrum and the generalized Drazin spectrum. Also, we will give conditions under which pseudo B-Fredholm and pseudo B-Weyl spectrum introduced in [9] and [25] become stable under commuting Riesz perturbations.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.81-87
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• 1992

Wellknown invariance of domain theorems are Brower's invariance of domain theorem for continuous mappings defined on a finite dimensional space and Schauder-Leray's invariance of domain theorem for the class of mappings I+C defined on a infinite dimensional Banach space with I the identity and C compact. The two classical invariance of domain theorems were proved by applying the homotopy invariance of Brower's degree and Leray-Schauder's degree respectively. Degree theory for some class of mappings is a useful tool for mapping theorems. And mapping theorems (or surjectivity theorems of mappings) are closely related with invariance of domain theorems for mappings. In[4, 5], Browder and Petryshyn constructed a multi-valued degree theory for A-proper mappings. From this degree Petryshyn [9] obtained some invariance of domain theorems for locally A-proper mappings. Recently Browder [6] has developed a degree theory for demicontinuous mapings of type ( $S_{+}$) from a reflexive Banach space X to its dual $X^{*}$. By applying this degree we obtain some invariance of domain theorems for demicontinuous mappings of type ( $S_{+}$). ( $S_{+}$).

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.1027-1040
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• 2012

The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum $S{\oplus}T$, where S and T are bounded linear operators acting on a Banach space X. Among other results, we prove that if both T and S possesses property ($gb$) and if ${\Pi}(T){\subset}{\sigma}_a(S)$, ${\PI}(S){\subset}{\sigma}_a(T)$, then $S{\oplus}T$ possesses property ($gb$) if and only if ${\sigma}_{SBF^-_+}(S{\oplus}T)={\sigma}_{SBF^-_+}(S){\cup}{\sigma}_{SBF^-_+}(T)$. Moreover, we prove that if T and S both satisfies generalized Browder's theorem, then $S{\oplus}T$ satis es generalized Browder's theorem if and only if ${\sigma}_{BW}(S{\oplus}T)={\sigma}_{BW}(S){\cup}{\sigma}_{BW}(T)$.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.77-95
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• 2012

For a bounded linear operator T we prove the following assertions: (a) If T is algebraically (p, k)-quasihyponormal, then T is a-isoloid, polaroid, reguloid and a-polaroid. (b) If $T^*$ is algebraically (p, k)-quasihyponormal, then a-Weyl's theorem holds for f(T) for every $f{\in}Hol({\sigma}T))$, where $Hol({\sigma}(T))$ is the space of all functions that analytic in an open neighborhoods of ${\sigma}(T)$ of T. (c) If $T^*$ is algebraically (p, k)-quasihyponormal, then generalized a-Weyl's theorem holds for f(T) for every $f{\in}Hol({\sigma}T))$. (d) If T is a (p, k)-quasihyponormal operator, then the spectral mapping theorem holds for semi-B-essential approximate point spectrum $\sigma_{SBF_+^-}(T)$, and for left Drazin spectrum ${\sigma}_{lD}(T)$ for every $f{\in}Hol({\sigma}T))$.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.83-89
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• 1993

Applications of the classical Knaster-Kuratowski-Mazurkiewicz theorem [KKM] and the fixed point theory of multifunctions defined on convex subsets of topological vector spaces have been greatly improved by adopting the concept of convex spaces due to Lassonde[L]. Recently, this concept has been extended to pseudo-convex spaces, contractible spaces, or spaces having certain families of contractible subsets by Horvath[H1-4]. In the present paper we give a far-reaching generalization of the best approximation theorem of Ky Fan[F1, 2] to pseudo-metric spaces and improved versions of the well-known fixed point theorems due to Brouwer [B] and Schauder [S] for spaces having certain families of contractible subsets. Our basic tool is a generalized Fan-Browder type fixed point theorem in our previous works [P3, 4].

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.831-850
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• 2023

Our long-standing Metatheorem in Ordered Fixed Point Theory is applied to some well-known order theoretic fixed point theorems. In the first half of this article, we introduce extended versions of the Zermelo fixed point theorem, Zorn's lemma, and the Caristi fixed point theorem based on the Brøndsted-Jachymski principle and our 2023 Metatheorem. We show some of their applications to other fixed point theorems or theorems on the existence of maximal elements in partially ordered sets. In the second half, we collect and improve order theoretic fixed point theorems in the collection of Howard-Rubin in 1991 and others. In fact, we improve or extend several ordering principles or fixed point theorems due to Brézis-Browder, Brøndsted, Knaster-Tarski, Tarski-Kantorovitch, Turinici, Granas-Horvath, Jachymski, and others.