Implicit Function Theorem Research Articles

The paper deals with a solution of a stationary problem with unknown boundaries for the Navier–Stokes equations corresponding to the slow rigid rotation of a viscous two-phase drop consisting of compressible and incompressible embedded fluids. In this case, the internal fluid is incompressible. It is bounded by a closed interface that does not intersect the outer free surface. It is assumed that the compressible fluid is barotropic. Surface tension forces act at the boundaries. The existence of a family of equilibrium figures close to embedded balls is proved. The proof is carried out in Hölder spaces by means of the implicit function theorem.

Abstract We study the action of the nonlinear mapping G[z] between real or complex Banach spaces in the vicinity of a given curve with respect to possible linearization, emerging patterns of level sets, as well as existing solutions of $$G[z]=0$$ G [ z ] = 0 . The results represent local generalizations of the standard implicit or inverse function theorem and of Newton’s Lemma, considering the order of approximation needed to obtain solutions of $$G[z]=0$$ G [ z ] = 0 . The main technical tool is given by Jordan chains with increasing rank, used to obtain an Ansatz, appropriate for transformation of the nonlinear system to its linear part. The family of linear mappings is restricted to the case of an isolated singularity. Geometrically, the Jordan chains define a generalized cone around the given curve, composed of approximate solutions of order 2k with k denoting the maximal rank of Jordan chains needed to ensure k-surjectivity of the linear family. Along these lines, the zero set of G[z] in the cone is calculated immediately, agreeing up to the order of $$k-1$$ k - 1 with the given approximation. Hence, the results may also be interpreted as a version of Tougeron’s implicit function theorem in Banach spaces, essentially restricted to the arc case of a single variable. Finally, by considering a left shift of the Jordan chains, the Ansatz can be modified in a systematic way to obtain a sequence of refined versions of linearization theorems and Newton Lemmas in Banach spaces.

Abstract In the local gluing one glues local neighborhoods around the critical point of the stable and unstable manifolds to gradient flow lines defined on a finite-time interval $$[-T,T]$$ [ - T , T ] for large T. If the Riemannian metric around the critical point is locally Euclidean, the local gluing map can be written down explicitly. In the non-Euclidean case the construction of the local gluing map requires an intricate version of the implicit function theorem. In this paper we explain a functional analytic approach how the local gluing map can be defined. For that we are working on infinite dimensional path spaces and also interpret stable and unstable manifolds as submanifolds of path spaces. The advantage of this approach is that similar functional analytical techniques can as well be generalized to infinite dimensional versions of Morse theory, for example Floer theory. A crucial ingredient is the Newton-Picard map. We work out an abstract version of it which does not involve troublesome quadratic estimates.

We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type ( ( A ( x / ϵ ) + B ( x / ϵ ) ) u ′ ( x ) + c ( x , u ( x ) ) ) ′ = d ( x , u ( x ) ) for x ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0. Here A ∈ L ∞ ( R ; M n ) is 1-periodic, B ∈ L ∞ ( R ; M n ) ∩ L 1 ( R ; M n ) , A ( y ) and A ( y ) + B ( y ) are positive definite uniformly with respect to y, c ( x , ⋅ ) , d ( x , ⋅ ) ∈ C 1 ( R n ; R n ) and c ( ⋅ , u ) , d ( ⋅ , u ) ∈ L ∞ ( ( 0 , 1 ) ; R n ) . For small ϵ > 0 , we show the existence of weak solutions u = u ϵ as well as their local uniqueness for ‖ u − u 0 ‖ ∞ ≈ 0 , where u = u 0 is a given non-degenerate weak solution to the homogenized problem ( ( ∫ 0 1 A ( y ) − 1 d y ) − 1 u ′ ( x ) + c ( x , u ( x ) ) ) ′ = d ( x , u ( x ) ) for x ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0. Further, we show that ‖ u ϵ − u 0 ‖ ∞ → 0 and if c ( ⋅ , u 0 ( ⋅ ) ) ∈ W 1 , ∞ ( ( 0 , 1 ) ; R n ) , then ‖ u ϵ − u 0 ‖ ∞ = O ( ϵ ) for ϵ → 0 . Moreover, all these statements are true, roughly speaking, uniformly with respect to the localized defects B. The main tool of the proofs is an abstract result of implicit function theorem type which has been tailored for applications to nonlinear singular perturbation and homogenization problems.

We study properties of the central path underlying a nonlinear semidefinite optimization problem, called an NSDP for short. The latest radical work on this topic was contributed by Yamashita and Yabe [Yamashita H, Yabe H (2012) Local and superlinear convergence of a primal-dual interior point method for nonlinear semidefinite programming. Mathematical Programming 132(1-2):1–30]: they proved that the Jacobian of a certain equation system derived from the Karush–Kuhn–Tucker (KKT) conditions of the NSDP is nonsingular at a KKT point under the second-order sufficient condition (SOSC), the strict complementarity condition (SC), and the nondegeneracy condition (NC). This yields uniqueness and existence of the central path through the implicit function theorem. In this paper, we consider the following three assumptions on a KKT point: the strong SOSC, the SC, and the Mangasarian–Fromovitz constraint qualification. Under the absence of the NC, the Lagrange multiplier set is not necessarily a singleton, and the nonsingularity of the above-mentioned Jacobian is no longer valid. Nonetheless, we establish that the central path exists uniquely, and moreover prove that the dual component of the path converges to the so-called analytic center of the Lagrange multiplier set. As another notable result, we clarify a region around the central path where Newton’s equations relevant to primal-dual interior-point methods are uniquely solvable. Funding: This work was supported by the Japan Society for the Promotion of Science [Grant-in-Aid for Young Scientists 20K19748 and Grant-in-Aid for Scientific Research (C) 20H04145]. The Japan Society for the Promotion of Science [Grant-in-Aid for Scientific Research (C) 25K15008].

Estimating Remaining Useful Life (RUL) is crucial in modern Prognostic and Health Management (PHM) systems providing valuable information for planning the maintenance strategy of critical components in complex systems such as aircraft engines. Deep Learning (DL) models have shown great performance in the accurate prediction of RUL, building hierarchical representations by the stacking of multiple explicit neural layers. In the current research paper, we follow a different approach presenting a Deep Equilibrium Model (DEM) that effectively captures the spatial and temporal information of the sequential sensor. The DEM, which incorporates convolutional layers and a novel dual-input interconnection mechanism to capture sensor information effectively, estimates the degradation representation implicitly as the equilibrium solution of an equation, rather than explicitly computing it through multiple layer passes. The convergence representation of the DEM is estimated by a fixed-point equation solver while the computation of the gradients in the backward pass is made using the Implicit Function Theorem (IFT). The Monte Carlo Dropout (MCD) technique under calibration is the final key component of the framework that enhances regularization and performance providing a confidence interval for each prediction, contributing to a more robust and reliable outcome. Simulation experiments on the widely used NASA Turbofan Jet Engine Data Set show consistent improvements, with the proposed framework offering a competitive alternative for RUL prediction under diverse conditions.

In this paper, we consider the existence of real transmission eigenvalues in a stratified medium by applying a modified version of the implicit function theorem through perturbation analysis. The Neumann-Dirichlet operator is defined and two useful lemmas are provided. We transform the existence problem of transmission eigenvalues into its zero eigenvalues of the Neumann-Dirichlet operator and prove the analyticity of the eigenvalues of the Neumann-Dirichlet operator on parameter k by the first lemma. Subsequently, relying on the introduced Sturm-Liouville problem and the second lemma, the continuity on another parameter ϵ of the eigenvalues of the Neumann-Dirichlet operator in the context of a stratified medium is proved. Eventually, we illustrate that the transmission eigenvalues exist under the perturbation method of the implicit function theorem.

A continuous point of a trajectory for an ordinary differential equation can be viewed as a special impulsive point; i.e., the pulsed proportional change rate and the instantaneous increment for the prey and predator populations can be taken as 0. By considering the variation multiple pulse intervention effects (i.e., several indefinite continuous points are regarded as impulsive points), an impulsive predator–prey model for characterizing chemical and biological control processes at different fixed times is first proposed. Our modeling approach can describe all possible realistic situations, and all of the traditional models are some special cases of our model. Due to the complexity of our modeling approach, it is essential to examine the dynamical properties of the periodic solutions using new methods. For example, we investigate the permanence of the system by constructing two uniform lower impulsive comparison systems, indicating the mathematical (or biological) essence of the permanence of our system; furthermore, the existence and global attractiveness of the pest-present periodic solution is analyzed by constructing an impulsive comparison system for a norm V(t), which has not been addressed to date. Based on the implicit function theorem, the bifurcation of the pest-present periodic solution of the system is investigated under certain conditions, which is more rigorous than the corresponding traditional proving method. In addition, by employing the variational method, the eigenvalues of the Jacobian matrix at the fixed point corresponding to the pest-free periodic solution are determined, resulting in a sufficient condition for its local stability, and the threshold condition for the global attractiveness of the pest-free periodic solution is provided in terms of an indicator Ra. Finally, the sensitivity of indicator Ra and bifurcations with respect to several key parameters are determined through numerical simulations, and then the switch-like transitions among two coexisting attractors show that varying dosages of insecticide applications and the numbers of natural enemies released are crucial.

Exploring threshold dynamics in a spatially heterogeneous ecosystem with memory-based diffusion and hunting cooperation on predators

We extend Clarke’s local inversion theorem for Sobolev mappings. We use this result to find a general implicit function theorem for continuous locally Lipschitz mapping in the first variable and satisfying just a topological condition in the second variable. An application to control systems is given.

In the context of normed space, Banach's fixed point theorem for mapping is studied in this paper. This idea is generalized in Banach's classical fixed-point theory. Fixed point theory explains many situations where maps provide great answers through an amazing combination of mathematical analysis. Picard- Lendell's theorem, Picard's theorem, implicit function theorem, and other results are created by other mathematicians later using this fixed-point theorem. We have come up with ideas that Banach's theorem can be used to easily deduce many well-known fixed-point theorems. Extending the Banach contraction principle to include metric space with modular spaces has been included in some recent research, the aim of study proves some properties of Banach space.

We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform vorticity distribution in this inner phase and they have a vortex sheet on its surface. Our construction relies on a perturbative approach around an explicit spherical solution, given by Hill’s vortex enclosed by a spherical vortex sheet. The construction is sensitive to the Weber numbers describing the flow. At critical Weber numbers, we perform a bifurcation analysis utilizing the Crandall–Rabinowitz theorem in Sobolev spaces on the 2-sphere. Away from these critical numbers, our construction relies on the implicit function theorem. Our results imply that the model containing surface tension is richer than the ordinary one-phase Euler equations, in the sense that for the latter, Hill’s spherical vortex is unique (modulo translations) among all axisymmetric simply connected uniform vortices of a given circulation.

<p>This paper mainly studied the stochastic stability and the design of state feedback controllers for nonlinear singular continuous semi-Markov jump systems under false data injection attacks. Based on the Lyapunov function and the implicit function theorem, a basic stochastic stability condition of the system was given to ensure that the nonlinear singular semi-Markov jump system under attack was regular, impulse-free, unique, and stochastically stable. On this basis, the stochastic admissible linear matrix inequality conditions of the system were obtained by using the singular value decomposition of the matrix and Schur's complement lemma. To design the state feedback controller, based on the upper and lower bounds of the time-varying transition probability of the semi-Markov jump system and the singular value decomposition method, the stochastic stable linear matrix inequality condition of the closed-loop system under the false data injection attack was established. Finally, the validity and feasibility of the results were verified by numerical examples.</p>

A common approach to singular perturbation and homogenization II: Semilinear elliptic systems

Abstract We prove that the sublinear equation with an obstacle { u ¨ + u α = p ( t ) , 0 < α < 1 , u ≥ 0 , u ( t 0 ) = 0 ⇒ u ˙ ( t 0 + ) = − u ˙ ( t 0 − ) has infinitely many bounded solutions for non-periodic forcing p by the implicit function theorem and the non-periodic twist maps’ theory established by Kunze and Ortega.

Abstract Nonlinear multi‐point constraints are essential in modeling various engineering problems, for example, joints undergoing large rotations or coupling of different element types in finite element analysis. They can be handled using Lagrange multipliers, the penalty method, or master‐slave elimination. The latter satisfies the constraints exactly and reduces the dimension of the resulting system of equations, which is particularly advantageous when a large number of constraints have to be considered. However, the existing schemes in literature are limited to linear constraints. Therefore, the authors introduced an extension of the method to arbitrary nonlinear constraints. A mathematically rigorous derivation of this new method is presented. The starting point is the optimization problem with constraints. It is transformed into a modified optimization problem without constraints using the implicit function theorem. For this, an appropriate selection of slave degrees of freedom (dofs) is crucial, ensuring that the Jacobian constraints meets specific conditions. This implies the consideration of three aspects: The handling of redundant constraints, the automatic selection of slave dofs, and the change of slave dofs in the context of large deformations. Additionally, it allows for the combination of the new method with existing constraint methods.

In this paper, a heterogeneous single population model with memory effect and nonlinear boundary condition is investigated. By virtue of the implicit function theorem and Lyapunov-Schmidt reduction, spatially nonconstant positive steady state solutions appear from two trivial solutions, respectively. Through the distribution of the eigenvalues and bifurcation analysis, sufficient conditions for the occurrence of the Hopf bifurcation associated with one spatially nonconstant positive steady state are obtained. Results about the stability associated with the other one are also yielded. It is found that with the interaction of memory delay, spatial heterogeneity and nonlinear boundary condition, the Hopf bifurcation will happen in such diffusive model. To be specific, the memory delay could induce a single stability switch from stability to instability. As contrast to the diffusive model only with memory effect, the stability switch is independent of memory delay. The obtained results are applied to some specific models, from which we can find that the theoretical analysis is verified and the results enrich the existing ones.

Abstract The phase-field system is a nonlinear model that has significant applications in material sciences. In this paper, we are concerned with the uniqueness of determining the nonlinear energy potential in a phase-field system consisting of Cahn–Hilliard and Allen–Cahn equations. This system finds widespread applications in the development of alloys engineered to withstand extreme temperatures and pressures. The goal is to reconstruct the nonlinear energy potential through the measurements of concentration fields. We establish the local well-posedness of the phase-field system based on the implicit function theorem in Banach spaces. Both of the uniqueness results for recovering time-independent and time-dependent energy potential functions are provided through the higher order linearization technique.

A diffusive phytoplankton–zooplankton–fish model with toxin and Crowley–Martin functional response is considered. By the global bifurcation theorem and the fixed point index theory, the existence and multiplicity of coexistence states are discussed. Secondly, we investigate the bifurcation from a double eigenvalue by virtue of Lyapunov–Schmidt procedure and implicit function theorem. Then, the effect of large k, which measures the magnitude of interference among zooplankton, is studied by means of the combination of the perturbation theory and topological degree theory. The results indicate that if k is large enough, this system has only a unique asymptotically stable coexistence state provided that toxin is properly small and the maximal growth rates of phytoplankton and fish are suitably large. Furthermore, the extinction and permanence of the time-dependent system are determined by virtue of the comparison principle. Finally, we make some numerical simulations to validate and complement the theoretical analysis and exhibit the critical role of toxin, spatial diffusion and magnitude of interference among zooplankton in the dynamics. The findings suggest that the spatiotemporal dynamics of the systems with toxin and Crowley–Martin functional response are richer and more complex, and toxin and spatial diffusion have significant effects on the coexistence of phytoplankton–zooplankton–fish species.

This paper is concerned with the tracking control problem of nonlinear time delay systems. For time delay systems, we consider nonaffine pure-feedback structure. Additionally, the case where the time delay systems are subject to input saturation and unknown states is also taken into account. To begin with, a mean value theorem and an implicit function theorem are exploited to transform the systems into an affine form. Afterwards, a state observer is developed to estimate the unknown state variables and an extreme learning machine (ELM) is used to approximate the unknown functions. The output of the command filter replaces the virtual control signal'derivative, thereby circumventing the problem of ‘explosion of complexity’. Meanwhile, compensation signals are introduced to eliminate the filtering errors in a dynamic surface control (DSC). A Lyapunov–Krasovskii functional is designed to mitigate the unknown constant state time delay. During the controller design process, combining a prescribed performance control (PPC) with a command filter control, the tracking error can converge to a predetermined bound. Additionally, the effect of input saturation is eliminated with the aid of an auxiliary system. Finally, the effectiveness of such controller is further proved by taking an electromechanical system as an application object.