Suppose thatF:D⊂Rn×Rm→Rn, withF(x0,y0)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ∂1F(x0,y0) is nonsingular. We strengthen this theorem by remo...
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Suppose thatF:D⊂R
n×Rm→Rn, withF(x
0,y
0)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ∂1
F(x
0,y
0) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.
In this note, we discuss a generalization of the well-known implicit function theorem to the time-delay case. We show that the latter problem is closely related to the bicausal changes of coordinates of time-delay sys...
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In this note, we discuss a generalization of the well-known implicit function theorem to the time-delay case. We show that the latter problem is closely related to the bicausal changes of coordinates of time-delay systems [Califano and Moog (2014), Califano and Moog (2017)]. An iterative algorithm is proposed to check the conditions and to construct the desired bicausal change of coordinates for the proposed implicit function theorem. Moreover, we show that our results can be applied to delayed differential-algebraic equations to reduce their indices and to get their solutions. Some numerical examples are given to illustrate our results.
We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y）...
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We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y） ＋ h（x, y）y＇ =0.
In this paper, we prove an implicit function theorem and we study the regularity of the functionimplicitly defined. The implicit function theorem had already been proved in homogeneous Lie groups by Franchi, Serapion...
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In this paper, we prove an implicit function theorem and we study the regularity of the functionimplicitly defined. The implicit function theorem had already been proved in homogeneous Lie groups by Franchi, Serapioni and Serra Cassano, while the regularity problem of the functionimplicitly defined was still open even in the simplest Lie group.
We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type epsilon(2)u '' = f(x, u, epsilon u', epsilon), 0 < x < 1, with ...
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We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type epsilon(2)u '' = f(x, u, epsilon u', epsilon), 0 < x < 1, with Dirichlet and Neumann boundary conditions. For that we assume that there is given a family of approximate solutions which satisfy the differential equation and the boundary conditions with certain low accuracy. Moreover, we show that, if this accuracy is high, then the closeness of the approximate solution to the exact solution is correspondingly high. The main tool of the proofs is a generalized implicit function theorem which is close to those of Fife and Greenlee (Uspechi Mat. Nauk 24 (1974), 103-130) and of Magnus (Proc. Royal Soc. Edinburgh 136A (2006), 559-583). Finally we show how to construct approximate solutions under certain natural conditions.
This paper provides an example of the implicit function theorem to the accuracy of global positioning system (GPS) navigation. The implicit function theorem allows one to approximate the timing accuracy required by th...
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This paper provides an example of the implicit function theorem to the accuracy of global positioning system (GPS) navigation. The implicit function theorem allows one to approximate the timing accuracy required by the GPS navigation system to locate a user within a certain degree of precision.
In this article, we present explicit estimates of size of the domain on which the implicit function theorem and the inverse functiontheorem are valid. For maps that are twice continuously differentiable, these estima...
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In this article, we present explicit estimates of size of the domain on which the implicit function theorem and the inverse functiontheorem are valid. For maps that are twice continuously differentiable, these estimates depend upon the magnitude of the first-order derivatives evaluated at the point of interest, and a bound on the second-order derivatives over a region of interest. One of the key contributions of this article is that the estimates presented require minimal numerical computation. In particular, these estimates are arrived at without any intermediate optimization procedures. We then present three applications in optimization and systems and control theory where the computation of such bounds turns out to be important. First, in electrical networks, the power flow operations can be written as quadratically constrained quadratic programs, and we utilize our bounds to compute the size of permissible power variations to ensure stable operations of the power system network. Second, the robustness margin of positive-definite solutions to the algebraic Riccati equation (frequently encountered in control problems) subject to perturbations in the system matrices is computed with the aid of our bounds. Finally, we employ these bounds to provide quantitative estimates of the size of the domains for feedback linearization of discrete-time control systems.
Let H : R-m x R-n --> R-n be a locally Lipschitz function in a neighborhood of ((y) over bar, (x) over bar) and H((y) over bar, (x) over bar) = 0 for some (y) over bar epsilon R-m and (x) over bar epsilon R-n. The ...
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Let H : R-m x R-n --> R-n be a locally Lipschitz function in a neighborhood of ((y) over bar, (x) over bar) and H((y) over bar, (x) over bar) = 0 for some (y) over bar epsilon R-m and (x) over bar epsilon R-n. The implicit function theorem in the sense of Clarke (Pacific J. Math. 64 (1976) 97;Optimization and Nonsmooth Analysis, Wiley, New York, 1983) says that if pi(x)partial derivativeH((y) over bar, (x) over bar) is of maximal rank, then there exist a neighborhood Y of (y) over bar and a Lipschitz function G((.)) : Y --> R-n such that G((y) over bar) = (x) over bar and for every y in Y, H(y,G(y)) = 0. In this paper, we shall further show that if H has a superlinear (quadratic) approximate property at ((y) over bar, (x) over bar), then G has a superlinear (quadratic) approximate property at (y) over bar. This result is useful in designing Newton's methods for nonsmooth equations. (C) 2001 Elsevier Science B.V. All rights reserved.
We consider the quasi-linear eigenvalue problem -Lambda(p)u = lambda g(u) subject to Dirichlet boundary conditions on a bounded open set Omega, where g is a locally Lipschitz continuous function. Imposing no further c...
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We consider the quasi-linear eigenvalue problem -Lambda(p)u = lambda g(u) subject to Dirichlet boundary conditions on a bounded open set Omega, where g is a locally Lipschitz continuous function. Imposing no further conditions on Omega or g, we show that for lambda near zero the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions parameterized by lambda depends continuously on the parameter. (C) 2011 Elsevier Ltd. All rights reserved.
The implicit function theorem (IFT) can be used to deduce the differentiability of an implicit mapping S : u -> y given by the equation e(y, u) = 0. However, the IFT is not applicable when different norms are neces...
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The implicit function theorem (IFT) can be used to deduce the differentiability of an implicit mapping S : u -> y given by the equation e(y, u) = 0. However, the IFT is not applicable when different norms are necessary for the differentiation of e w.r.t. y and the invertibility of the partial derivative e(y) (y, u). We prove theorems ensuring the (twice) differentiability of the mapping S which can be applied in this case. We highlight the application of our results to quasilinear partial differential equations whose principal part depends nonlinearly on the gradient of the state del y. (C) 2014 Elsevier Inc. All rights reserved.
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