• 대한수학회보
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• 제49권6호
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• pp.1131-1145
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• 2012

The purpose of this paper is to use an appropriate variational framework to discuss the boundary value problem with p-Laplacian type operators $$\{({\alpha}(t,x^{\Delta}(t)))^{\Delta}-a(t){\phi}_p(x^{\sigma}(t))+f({\sigma}(t),x^{\sigma}(t))=0,\;{\Delta}-a.e.\;t{\in}I\\x^{\sigma}(0)=0,\\{\beta}_1x^{\sigma}(1)+{\beta}_2x^{\Delta}({\sigma}(1))=0,$$ where ${\beta}_1$, ${\beta}_2$ > 0, $I=[0,1]^{k^2}$, ${\alpha}({\cdot},x({\cdot}))$ is an operator of $p$-Laplacian type, $\mathbb{T}$ is a time scale. Some sufficient conditions for the existence of constant-sign solutions are obtained.

• 대한수학회지
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• 제41권6호
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• pp.977-994
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• 2004

A pair of quasi-definite linear functionals {u$_{0}$, u$_1$} is a generalized $\Delta$-coherent pair if monic orthogonal polynomials (equation omitted) relative to u$_{0}$ and u$_1$, respectively, satisfy a relation (equation omitted) where $\sigma$$_{n}$ and T$_{n}$ are arbitrary constants and $\Delta$p = p($\chi$＋1) - p($\chi$) is the difference operator. We show that if {u$_{0}$, u$_1$} is a generalized $\Delta$-coherent pair, then u$_{0}$ and u$_{1}$ must be discrete-semiclassical linear functionals. We also find conditions under which either u$_{0}$ or u$_1$ is discrete-classical.ete-classical.

• Kyungpook Mathematical Journal
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• 제55권4호
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• pp.953-967
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• 2015

In this article, by making use of a linear multiplier fractional differential operator $D^{{\delta},m}_{\lambda}$, we introduce a new subclass of spiral-like functions. The main object is to provide some subordination results for functions in this class. We also find sufficient conditions for a function to be in the class and derive Fekete-$Szeg{\ddot{o}}$ inequalities.

• Journal of applied mathematics & informatics
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• 제39권3_4호
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• pp.487-496
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• 2021

In this paper we show that if A ∈ L(X) and B ∈ L(Y), X and Y complex Banach spaces, then A ⊕ B ∈ L(X ⊕ Y) is subscalar if and only if both A and B are subscalar. We also prove that if A, Q ∈ L(X) satisfies AQ = QA and Q^p = 0 for some nonnegative integer p, then A has property (C) (resp. property (𝛽)) if and only if so does A + Q (resp. property (𝛽)). Finally, we show that A ∈ L(X, Y) and B, C ∈ L(Y, X) satisfying operator equation ABA = ACA and BA ∈ L(X) is subscalar with property (𝛿) then both Lat(BA) and Lat(AC) are non-trivial.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제5권1호
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• pp.23-32
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• 1998

In the present paper the author studies the decomposition property ($\delta$) of the bounded linear operators.

• 대한수학회지
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• 제42권1호
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• pp.111-127
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• 2005

Let B and S be the unit ball and the unit sphere in $\mathbb{C}^n$, respectively. Let ${\sigma}$ be the normalized Lebesgue measure on S. Define the Privalov spaces $N^P(B)\;(1\;<\;p\;<\;{\infty})$ by $$N^P(B)\;=\;\{\;f\;{\in}\;H(B) : \sup_{0 where H(B) is the space of all holomorphic functions in B. Let ${\varphi}$ be a holomorphic self-map of B. Let ${\mu}$ denote the pull-back measure ${\sigma}o({\varphi}^{\ast})^{-1}$. In this paper, we prove that the composition operator $C_{\varphi}$ is metrically bounded on $N^P$(B) if and only if ${\mu}(S(\zeta,\delta)){\le}C{\delta}^n$ for some constant C and $C_{\varphi}$ is metrically compact on $N^P(B)$ if and only if ${\mu}(S(\zeta,\delta))=o({\delta}^n)$ as ${\delta}\;{\downarrow}\;0$ uniformly in ${\zeta}\;\in\;S. Our results are an analogous results for Mac Cluer's Carleson-measure criterion for the boundedness or compactness of $C_{\varphi}$ on the Hardy spaces $H^P(B)$.

• Nonlinear Functional Analysis and Applications
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• 제27권2호
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• pp.349-361
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• 2022

In this paper, we study the following nonlinear problem: $$\{-\Delta_{p}^{3}(x)u\;=\;{\lambda}V_{1}(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u\;=\;{\Delta}u\;{\Delta}^{2}u\;=\;0\;on\;{\partial}\Omega, $$ under adequate conditions on the exponent functions p, q and the weight function V₁. We prove the existence and nonexistence of eigenvalues for p(x)-triharmonic problem with Navier boundary value conditions on a bounded domain in ℝ^N. Our technique is based on variational approaches and the theory of variable exponent Lebesgue spaces.

• 충청수학회지
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• 제24권1호
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• pp.19-34
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• 2011

A bounded linear operator T on a complex Banach space X is said to have property (I) provided that T has Bishop's property (${\beta}$) and there exists an integer p > 0 such that for a closed subset F of ${\mathbb{C}}$ ${X_T}(F)={E_T}(F)=\bigcap_{{\lambda}{\in}{\mathbb{C}}{\backslash}F}(T-{\lambda})^PX$ for all closed sets $F{\subseteq}{\mathbb{C}}$, where $X_T$(F) denote the analytic spectral subspace and $E_T$(F) denote the algebraic spectral subspace of T. Easy examples are provided by normal operators and hyponormal operators in Hilbert spaces, and more generally, generalized scalar operators and subscalar operators in Banach spaces. In this paper, we prove that if T has property (I), then the quasi-nilpotent part $H_0$(T) of T is given by $$KerT^P=\{x{\in}X:r_T(x)=0\}={\bigcap_{{\lambda}{\neq}0}(T-{\lambda})^PX$$ for all sufficiently large integers p, where ${r_T(x)}=lim\;sup_{n{\rightarrow}{\infty}}{\parallel}T^nx{\parallel}^{\frac{1}{n}}$. We also prove that if T has property (I) and the spectrum ${\sigma}$(T) is finite, then T is algebraic. Finally, we prove that if $T{\in}L$(X) has property (I) and has decomposition property (${\delta}$) then T has a non-trivial invariant closed linear subspace.

• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.185-220
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• 2016

We study the Gauss and Poisson semigroups connected with the Riemann-Liouville operator defined on the half plane. Next, we establish a principle of maximum for the singular partial differential operator $${\Delta}_{\alpha}={\frac{{\partial}^2}{{\partial}r^2}+{\frac{2{\alpha}+1}{r}{\frac{\partial}{{\partial}r}}+{\frac{{\partial}^2}{{\partial}x^2}}+{\frac{{\partial}^2}{{\partial}t^2}}};\;(r,x,t){\in}]0,+{\infty}[{\times}{\mathbb{R}}{\times}]0,+{\infty}[$$. Later, we define the Littlewood-Paley g-function and using the principle of maximum, we prove that for every $p{\in}]1,+{\infty}[$, there exists a positive constant $C_p$ such that for every $f{\in}L^p(d{\nu}_{\alpha})$, $${\frac{1}{C_p}}{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}{\leqslant}{\parallel}g(f){\parallel}_{p,{\nu}_{\alpha}}{\leqslant}C_p{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}$$.

• Journal of applied mathematics & informatics
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• 제41권2호
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• pp.273-286
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• 2023

In this paper we show that if T ∈ L(X) and S ∈ L(X) is a Riesz operator commuting with T and X_S(F) ∈ Lat(S), where F = {0} or F ⊆ ℂ ⧵ {0} is closed then T|X_S(F) and T|X_T(F) + S|X_S(F) share the local spectral properties such as SVEP, Dunford's property (C), Bishop's property (𝛽), decomopsition property (𝛿) and decomposability. As a corollary, if T ∈ L(X) and Q ∈ L(X) is a quasinilpotent operator commuting with T then T is Riesz if and only if T + Q is Riesz. We also study some spectral properties of Riesz operators acting on Banach spaces. We show that if T, S ∈ L(X) such that TS = ST, and Y ∈ Lat(S) is a hyperinvarinat subspace of X for which 𝜎(S|Y ) = {0} then 𝜎_*(T|Y + S|Y ) = 𝜎_*(T|Y ) for 𝜎_* ∈ {𝜎, 𝜎_loc, 𝜎_sur, 𝜎_ap}. Finally, we show that if T ∈ L(X) and S ∈ L(Y ) on the Banach spaces X and Y and T is similar to S then T is Riesz if and only if S is Riesz.