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Convergence of a continuation method under majorant conditions
Shwet Nisha,P. K. Parida,Chandni Kumari 강원경기수학회 2019 한국수학논문집 Vol.27 No.4
The paper is devoted to study local convergence of a continuation method under the assumption of majorant conditions. The method is used to approximate a zero of an operator in Banach space and is of third order. It is seen that the famous Kantorovich-type and Smale-type conditions are special cases of our majorant conditions. This infers that our result is a generalized one in comparison to results based on Kantorovich-type and Smale-type conditions. Finally a number of numerical examples have been computed to show applicability of the convergence analysis.
LOCAL CONVERGENCE OF THE GAUSS-NEWTON METHOD FOR INJECTIVE-OVERDETERMINED SYSTEMS
Amat, Sergio,Argyros, Ioannis Konstantinos,Magrenan, Angel Alberto Korean Mathematical Society 2014 대한수학회지 Vol.51 No.5
We present, under a weak majorant condition, a local convergence analysis for the Gauss-Newton method for injective-overdetermined systems of equations in a Hilbert space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than in earlier studies such us [10, 11, 13, 14, 18]. Special cases and numerical examples are also included in this study.
LOCAL CONVERGENCE OF THE GAUSS-NEWTON METHOD FOR INJECTIVE-OVERDETERMINED SYSTEMS
Sergio Amat,Ioannis Konstantinos Argyros,Ángel Alberto Magreñán 대한수학회 2014 대한수학회지 Vol.51 No.5
We present, under a weak majorant condition, a local conver- gence analysis for the Gauss-Newton method for injective-overdetermined systems of equations in a Hilbert space setting. Our results provide un- der the same information a larger radius of convergence and tighter er- ror estimates on the distances involved than in earlier studies such us [10, 11, 13, 14, 18]. Special cases and numerical examples are also in- cluded in this study.
BLOCH-TYPE SPACES ON THE UPPER HALF-PLANE
Fu, Xi,Zhang, Junding Korean Mathematical Society 2017 대한수학회보 Vol.54 No.4
We define Bloch-type spaces of ${\mathcal{C}}^1({\mathbb{H}})$ on the upper half plane H and characterize them in terms of weighted Lipschitz functions. We also discuss the boundedness of a composition operator ${\mathcal{C}}_{\phi}$ acting between two Bloch spaces. These obtained results generalize the corresponding known ones to the setting of upper half plane.
Bloch-type spaces on the upper half-plane
Xi Fu,Junding Zhang 대한수학회 2017 대한수학회보 Vol.54 No.4
We define Bloch-type spaces of $\mathcal{C}^1(\IH)$ on the upper half plane $\mathbb{H}$ and characterize them in terms of weighted Lipschitz functions. We also discuss the boundedness of a composition operator $C_\phi$ acting between two Bloch spaces. These obtained results generalize the corresponding known ones to the setting of upper half plane.
OSCILLATION CRITERIA OF DIFFERENTIAL EQUATIONS OF SECOND ORDER
Kim, Rae Joong The Kangwon-Kyungki Mathematical Society 2011 한국수학논문집 Vol.19 No.3
We give sufficient conditions that the homogeneous differential equations : for $t{\geq}t_0$(> 0), $$x^{{\prime}{\prime}}(t)+q(t)x^{\prime}(t)+p(t)x(t)=0,\\x^{{\prime}{\prime}}(t)+q(t)x^{\prime}(t)+F(t,x({\phi}(t)))=0$$, are oscillatory where $0{\leq}{\phi}(t)$, 0 < ${\phi}^{\prime}(t)$, $\lim_{t\to{\infty}}{\phi}(t)={\infty}$. and $F(t,u){\cdot}sgn$ $u{\leq}p(t)|u|$. We obtain comparison theorems.
DISCUSSION ON THE ANALYTIC SOLUTIONS OF THE SECOND-ORDER ITERATED DIFFERENTIAL EQUATION
Liu, HanZe,Li, WenRong Korean Mathematical Society 2006 대한수학회보 Vol.43 No.4
This paper is concerned with a second-order iterated differential equation of the form $c_0x'(Z)+c_1x'(z)+c_2x(z)=x(az+bx(z))+h(z)$ with the distinctive feature that the argument of the unknown function depends on the state. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained.
OSCILLATION AND NONOSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIONS OF SECOND ORDER
Kim, RakJoong The Kangwon-Kyungki Mathematical Society 2011 한국수학논문집 Vol.19 No.4
We give necessary and sufficient conditions such that the homogeneous differential equations of the type: $$(r(t)x^{\prime}(t))^{\prime}+q(t)x^{\prime}(t)+p(t)x(t)=0$$ are nonoscillatory where $r(t)$ > 0 for $t{\in}I=[{\alpha},{\infty})$, ${\alpha}$ > 0. Under the suitable conditions we show that the above equation is nonoscillatory if and only if for ${\gamma}$ > 0, $$(r(t)x^{\prime}(t))^{\prime}+q(t)x^{\prime}(t)+p(t)x(t-{\gamma})=0$$ is nonoscillatory. We obtain several comparison theorems.
INVERTIBLE AND ISOMETRIC COMPOSITION OPERATORS ON VECTOR-VALUED HARDY SPACES
Sharma, S.D.,Bhand, Udhey Korean Mathematical Society 2004 대한수학회보 Vol.41 No.3
Invertible and isometric composition operators acting on vector-valued Hardy space $H^2$(E) are characterized.
Argyros, Ioannis Konstantinos,Silva, Gilson do Nascimento Korean Mathematical Society 2019 대한수학회지 Vol.56 No.2
The aim of this paper is to extend the applicability of Gauss-Newton method for solving underdetermined nonlinear least squares problems in cases not covered before. The novelty of the paper is the introduction of a restricted convergence domain. We find a more precise location where the Gauss-Newton iterates lie than in earlier studies. Consequently the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the local as well the semilocal convergence of Gauss-Newton method. The new developmentes are obtained under the same computational cost as in earlier studies, since the new Lipschitz constants are special cases of the constants used before. Numerical examples further justify the theoretical results.