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Soonmo Jung - One of the best experts on this subject based on the ideXlab platform.
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a fixed point approach to the stability of an n dimensional mixed type additive and quadratic functional Equation
2012Co-Authors: Yanghi Lee, Soonmo JungAbstract:and Applied Analysis 3 Moreover, they also investigated the Hyers-Ulam-Rassias stability of 1.3 by using the direct method see 18 . Indeed, they tried to approximate the even and odd parts of each solution of a perturbed inequality by the even and odd parts of an “exact” solution of 1.3 , respectively. In Theorems 3.1 and 3.3 of this paper, we will apply the fixed point method and prove the Hyers-Ulam-Rassias stability of the n-dimensional mixed-type additive and quadratic functional Equation. The advantage of this paper, in comparison with 18 , is to approximate each solution of a perturbed inequality by an “exact” solution of 1.3 , and we obtain sharper estimations in consequence of this advantage. Throughout this paper, let V be a real or complex vector space, Y a Banach space, and n an integer larger than 1. 2. Preliminaries Let X be a nonempty set. A function d : X2 → 0,∞ is called a generalized metric on X if and only if d satisfies the following: M1 d x, y 0 if and only if x y; M2 d x, y d y, x for all x, y ∈ X; M3 d x, z ≤ d x, y d y, z for all x, y, z ∈ X. We remark that the only difference between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to 19 . Theorem 2.1. Let X, d be a complete generalized metric space. Assume that Λ : X → X is a strict contraction with the Lipschitz constant L < 1. If there exists a nonnegative integer n0 such that d Λ0 1x,Λn0x < ∞ for some x ∈ X, then the following statements are true. i The sequence {Λnx} converges to a fixed point x∗ of Λ. ii x∗ is the unique fixed point of Λ in X∗ {y ∈ X | d Λ0x, y < ∞}. iii If y ∈ X∗, then d ( y, x∗ ) ≤ 1 1 − L ( Λy, y ) . 2.1 In 1991, Baker applied the fixed point method to prove the Hyers-Ulam stability of a nonlinear functional Equation see 20 . Thereafter, Radu noticed that many theorems concerning the Hyers-Ulam stability of various functional Equations follow from the fixed point alternative Theorem 2.1 . Indeed, he applied the fixed point method to prove the existence of a solution of the inequality 1.1 and investigated the Hyers-Ulam stability of the additive Cauchy Equation see 21 and also 22–26 . For a somewhat different fixed point approach to stability of functional Equations, see 27, 28 . 4 Abstract and Applied Analysis 3. Hyers-Ulam-Rassias Stability Let V be a real or complex vector space and let Y be a Banach space. For a given function f : V → Y , we use the following abbreviation: Df x1, x2, . . . , xn : 2f ⎛ ⎝ n ∑
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hyers ulam rassias stability of functional Equations in nonlinear analysis
2011Co-Authors: Soonmo JungAbstract:-1. Introduction. -2. Additive Cauchy Equation (Behavior of additive functions, Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Stability on a restricted domain, Method of invariant means, Fixed point method, Composite functional congruences, Pexider Equation, Remarks). -3. Generalized Additive Cauchy Equations (Functional Equation f(ax+by)=af(x)+bf(y), Additive Cauchy Equations of general form, Functional Equation f(x+y)2=(f(x)+f(y))2). -4. Hossza"'s Functional Equation (Stability in the sense of Borelli, Hyers-Ulam stability, Generalized Hossza"'s Equation is not stable on the unit interval, Hossza"'s functional Equation of Pexider type). -5. Homogeneous Functional Equation(Homogeneous Equation between Banach algebras, Superstability on a restricted domain, Homogeneous Equation between vector spaces, Homogeneous Equation of Pexide type). -6. Linear Functional Equations (A system for linear functions, Functional Equation f(x+cy)=f(x)+cf(y), Stability for other Equations).-7. Jensen's Functional Equation (Hyers-Ulam-Rassias stability, Stability on a restriced domain, Fixed point method, Lobacevskii s functional Equation). -8. Quadratic Functional Equations (Hyers-Ulam-Rassias stability, Stability on a restricted domain, Fixedpoint method, Quadratic functional Equation of other type, Quadratic functional Equation of Pexider type). -9. Exponential Functional Equations (Superstability, Stability in the sense of Ger, Stability on a restricted domain, Exponential functional Equation of other type). -10. Multiplicative Functional Equations (Superstability, delta-multiplicative functional, Theory of AMNM algebras, Functional Equation f(xy)= f(x)y, Functional Equation f(x+y)= f(x)f(y)f(1/x+1/y)). -11. Logarithmic Functional Equations (Functional Equation f(xy)= yf(x), Superstability of Equation f(xy)= yf(x), Functional Equation of Heuvers). -12. Trigonometric Functional Equations (Cosine functional Equation, Sine functional Equation, Trigonometric Equations with two unknowns, Butler-Rassias functional Equation, Remarks). -13. Isometric Functional Equation (Hyers-Ulam stability, Stability on a restricted domain, Fixed point method, Wigner Equation). -14. Miscellaneous (Associativity Equation, Equation of multiplicative derivation, Gamma functional Equation). -Bibliography. -Index.
Jae-young Chung - One of the best experts on this subject based on the ideXlab platform.
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stability of a conditional Cauchy Equation on a set of measure zero
2014Co-Authors: Jae-young ChungAbstract:In this paper, we prove the Hyers–Ulam stability theorem when \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set \({\Gamma \subset \mathbb{R}^{2}}\) of measure \({m(\Gamma) = 0}\), which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy the Pexider Equation $$f(x + y) - g(x) - h(y) = 0$$ in \({\Gamma}\), then the Equation holds for all \({x, y \in \mathbb{R}}\). Using our method of construction of the set, we can find a set \({\Gamma \subset \mathbb{R}^{2n}}\) of 2n-dimensional measure 0 and obtain the above result for the functions \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\).
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stability of a conditional Cauchy Equation
2012Co-Authors: Jae-young ChungAbstract:Let \({\mathbb R}\) be the set of real numbers, \({f : \mathbb {R} \to \mathbb {R}}\), \({\epsilon \ge 0}\) and d > 0. We denote by {(x1, y1), (x2, y2), (x3, y3), . . .} a countable dense subset of \({\mathbb {R}^2}\) and let $$U_d:=\bigcup\nolimits_{j=1}^{\infty} \{(x, y)\in \mathbb {R}^2:\,|x|+|y| > d,\, |x-x_j| < 1,\, |y-y_j| < 2^{-j}\}.$$ We consider the Hyers-Ulam stability of the conditional Cauchy functional inequality $$|f(x+y)-f(x)-f(y)|\le \epsilon$$ for all \({(x, y) \in U_d}\) .
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hyers ulam rassias stability of Cauchy Equation in the space of schwartz distributions
2004Co-Authors: Jae-young ChungAbstract:Abstract We reformulate and prove the Hyers–Ulam–Rassias stability of Cauchy Equation in the space of Schwartz tempered distributions and Fourier hyperfunctions.
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The stability of Cauchy Equations in the space of Schwartz distributions
2004Co-Authors: Jae-young Chung, Soon-yeong Chung, Han KimAbstract:Abstract We reformulate the stability theorem of D.H. Hyers in the Schwartz tempered distributions and prove that every ϵ -additive tempered distribution can be approximated by a linear function. We also consider the superstability of the Cauchy Equation f ( x + y )= f ( x ) f ( y ).
Yanghi Lee - One of the best experts on this subject based on the ideXlab platform.
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a fixed point approach to the stability of an n dimensional mixed type additive and quadratic functional Equation
2012Co-Authors: Yanghi Lee, Soonmo JungAbstract:and Applied Analysis 3 Moreover, they also investigated the Hyers-Ulam-Rassias stability of 1.3 by using the direct method see 18 . Indeed, they tried to approximate the even and odd parts of each solution of a perturbed inequality by the even and odd parts of an “exact” solution of 1.3 , respectively. In Theorems 3.1 and 3.3 of this paper, we will apply the fixed point method and prove the Hyers-Ulam-Rassias stability of the n-dimensional mixed-type additive and quadratic functional Equation. The advantage of this paper, in comparison with 18 , is to approximate each solution of a perturbed inequality by an “exact” solution of 1.3 , and we obtain sharper estimations in consequence of this advantage. Throughout this paper, let V be a real or complex vector space, Y a Banach space, and n an integer larger than 1. 2. Preliminaries Let X be a nonempty set. A function d : X2 → 0,∞ is called a generalized metric on X if and only if d satisfies the following: M1 d x, y 0 if and only if x y; M2 d x, y d y, x for all x, y ∈ X; M3 d x, z ≤ d x, y d y, z for all x, y, z ∈ X. We remark that the only difference between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to 19 . Theorem 2.1. Let X, d be a complete generalized metric space. Assume that Λ : X → X is a strict contraction with the Lipschitz constant L < 1. If there exists a nonnegative integer n0 such that d Λ0 1x,Λn0x < ∞ for some x ∈ X, then the following statements are true. i The sequence {Λnx} converges to a fixed point x∗ of Λ. ii x∗ is the unique fixed point of Λ in X∗ {y ∈ X | d Λ0x, y < ∞}. iii If y ∈ X∗, then d ( y, x∗ ) ≤ 1 1 − L ( Λy, y ) . 2.1 In 1991, Baker applied the fixed point method to prove the Hyers-Ulam stability of a nonlinear functional Equation see 20 . Thereafter, Radu noticed that many theorems concerning the Hyers-Ulam stability of various functional Equations follow from the fixed point alternative Theorem 2.1 . Indeed, he applied the fixed point method to prove the existence of a solution of the inequality 1.1 and investigated the Hyers-Ulam stability of the additive Cauchy Equation see 21 and also 22–26 . For a somewhat different fixed point approach to stability of functional Equations, see 27, 28 . 4 Abstract and Applied Analysis 3. Hyers-Ulam-Rassias Stability Let V be a real or complex vector space and let Y be a Banach space. For a given function f : V → Y , we use the following abbreviation: Df x1, x2, . . . , xn : 2f ⎛ ⎝ n ∑
Amy L Oldenburg - One of the best experts on this subject based on the ideXlab platform.
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inversion of displacement fields to quantify the magnetic particle distribution in homogeneous elastic media from magnetomotive ultrasound
2019Co-Authors: Diwash Thapa, Benjamin E Levy, Daniel L Marks, Amy L OldenburgAbstract:: Magnetomotive ultrasound (MMUS) contrasts superparamagnetic iron-oxide nanoparticles (SPIOs) that undergo submicrometer-scale displacements in response to a magnetic gradient force applied to an imaging sample. Typically, MMUS signals are defined in a way that is proportional to the medium displacement, rendering an indirect measure of the density distribution of SPIOs embedded within. Displacement-based MMUS, however, suffers from 'halo effects' that extend into regions without SPIOs due to their inherent mechanical coupling with the medium. To reduce such effects and to provide a more accurate representation of the SPIO density distribution, we propose a model-based inversion of MMUS displacement fields by reconstructing the body force distribution. Displacement fields are modelled using the static Navier-Cauchy Equation for linear, homogeneous, and isotropic media, and the body force fields are, in turn, reconstructed by minimizing a regularized least-squares error functional between the modelled and the measured displacement fields. This reconstruction, when performed on displacement fields of two tissue-mimicking phantoms with cuboidal SPIO-laden inclusions, improved the range of errors in measured heights and widths of the inclusions from 54%-282% pre-inversion to-15%-20%. Likewise, the post-inversion contrast to noise ratios (CNRs) of the images were significantly larger than displacement-derived CNRs alone (p = 0.0078, Wilcoxon signed rank test). Qualitatively, it was found that inversion ameliorates halo effects and increases overall detectability of the inclusion. These findings highlight the utility of model-based inversion as a tool for both signal processing and accurate characterization of the number density distribution of SPIOs in magnetomotive imaging.
Marcin Balcerowski - One of the best experts on this subject based on the ideXlab platform.
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on a modification of some conditional Cauchy Equation
2018Co-Authors: Marcin BalcerowskiAbstract:We show connections between generalized versions of some conditional Cauchy Equation and its basic form. As a consequence we obtain the solution of the generalized Equations in some classes of regular functions.