An implicit function theorem: Comment

In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:R ⁿ ×R ^m R ⁿ, withF(x ⁰,y ⁰)=0, that requires neither differentiability ofF nor nonsingularity of _x F(x ⁰,y ⁰). In the proof, the local one-to-one condition forF(·,y):A R ⁿ R ⁿ for ally B is consciously or unconsciously treated as implying thatF(·,y) mapsA one-to-one ontoF(A, y) for ally B, and the proof is not perfect. A proof can be given directly, and the theorem is shown to be the strongest, in the sense that the condition is truly if and only if.     

An implicit function theorem for Banach spaces and some applications

We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach–Lie groups, as well as Weil’s local rigidity for homomorphisms from finitely generated groups to Banach–Lie groups.      

An application for an implicit function theorem of Craven and Nashed: Continuum limits of lattice differential equations

We deal with the continuum limit of lattice differential equations and show how an implicit function theorem of Craven and Nashed can be used in order to continuate solutions of the resulting partial differential equation to solutions of the original spatially discrete system.     

Global solution branches and a topological implicit function theorem

We show some abstract (purely set-topological) principles which allow to prove the existence of global solution branches. The results apply either for the locally compact situation and then allow to prove global bifurcation results of Rabinowitz type, or they apply for a locally connected situation and allow to prove global branches of arbitrarily small perturbations without any compactness hypotheses. As two applications, we obtain a generalization of the Rabinowitz theorem for bifurcation from an interval and an implicit function type theorem for nondifferentiable functions.     

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C², or even C¹, or even Fréchet-differentiable.     

Generalized implicit function theorem and its application to parametric optimal control problems

Many problems in physics and mathematics may be reduced to solving equations depending on a parameter. The justification of the existence of solutions to the equations and the sensitivity analysis may be conducted based on Implicit Function Theorem (IFT) under certain regularity assumptions. Provided the regularity assumptions do not hold, generalizations of IFT are needed in order to study solutions to the equations. The paper focuses on a particular generalization of IFT which is then applied to a parametric linear time-optimal control problem.     

Floquet theorem for linear implicit nonautonomous difference systems

The aim of this paper is to develop the Floquet theory for linear implicit difference systems (LIDS). It is proved that any index-1 LIDS can be transformed into its Kronecker normal form. Then the Floquet theorem on the representation of the fundamental matrix of index-1 periodic LIDS has been established. As an immediate consequence, the Lyapunov reduction theorem is proved. Some applications of the obtained results are discussed.     

Differentiability of implicit functions: Beyond the implicit function theorem

The implicit function theorem (IFT) can be used to deduce the differentiability of an implicit mapping S:u?y$S:u?y$ given by the equation e(y,u)=0$e(y,u)=0$. 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 _ey(y,u)$e_y(y,u)$. 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 ∇y.     

Busy periods of discrete-time queues using the Lagrange implicit function theorem

In this paper, we study the busy-period distribution for discrete-time queues assuming a Bernoulli arrival process, with arbitrary service-time and batch-arrival distributions. We derive explicit analytic formulas for these distributions using the Lagrange Implicit Function Theorem applied to probability generating functions. The convenient coefficient operator notation used in these formulas leads to a computationally efficient method for obtaining the distributions in their entirety from these analytic formulas.     

Normal coderivative for multifunctions and implicit function theorems

In the framework of the theory of normal coderivative for multifunctions, new implicit function theorems are obtained. The main tools of the proofs are the Ekeland variational principle, a nonsmooth version of Fermat's rule, a sum rule, and the differential estimate for marginal functions established by B.S. Mordukhovich and Y. Shao [B.S. Mordukhovich, Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996) 1235-1280].     

Critical cases of stability. Converse implicit function theorem for dynamical systems with cosymmetry

Stability criteria of boundary equilibria for dynamical systems in the three critical cases, (n, k)=(3, 0), (2, 1), and (1, 1), are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 572–578, April, 1998.The author wishes to thank V. I. Yudovich for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01791 and by the International Science Foundation under grant NRQ000.     

Implicit function theorem and its application to a Ulam’s problem for exact differential equations

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.     

The Lagrange inversion theorem in the smooth case

The classical Lagrange inversion theorem is a concrete, explicit form of the implicit function theorem for real analytic functions. An explicit construction shows that the formula is not true for all merely smooth functions. The authors modify the Lagrange formula by replacing the smooth function by its Maclaurin polynomials. The resulting modified Lagrange series is, in analogy to the Maclaurin polynomials, an approximation to the solution function accurate to o(x^N) as x→0.     

An implicit function theorem for directionally differentiable functions

A new implicit function theorem for a class of nonsmooth functions is proved. It is used to improve the directional implicit function theorem of Demidova and Demyanov (Ref. 1).     

An implicit function theorem for algebraically closed fields

Let F be an algebraically closed field of characteristic 0, and let f : F ⁿ → F. We show that f is implicitly definable by a system of polynomial equations if and only if f is a special kind of piecewise rational function.     

The ABC theorem for higher-dimensional function fields

We generalize the ABC theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non-singular in codimension one. We also obtain an upper bound for the minimal order sequence of Wronskians over such function fields of positive characteristic.

     

Analogs of the Kolmogorov-Seliverstov-Plessner theorem for nonorthogonal function systems

For arbitrary function systems, the growth of partial sums is estimated depending on the growth of the corresponding Lebesgue functions. We prove analogs of the Kolmogorov-Seliverstov-Plessner convergence theorem for trigonometric series and the Kaczmarz convergence theorem for orthogonal series for arbitrary (nonorthogonal) function systems as well as for orthogonal-like and generalized orthogonal-like systems. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 87–101, January, 2000.     

Theorems on the representation of nonlinear mapping families and implicit function theorems

We derive a result on local representations of a nonlinear mapping family in a neighborhood of a 2-regular singularity. Theorems of this kind are special forms of implicit function theorems and contain many very important results of local nonlinear analysis. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 57–68, January, 2000.     

An existence theorem for systems of boundary value problems

We prove an existence theorem for , , in , using the shooting method. The function is supposed to be asymptotically linear.