Implicit Function Theorem Research Articles

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

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type∂xi(aijαβ(x/ε)∂xjuβ(x)+biα(x,u(x)))=bα(x,u(x)) for x∈Ω. For small ε>0 we prove existence of weak solutions u=uε as well as their local uniqueness for ‖u−u0‖∞≈0, where u0 is a given non-degenerate weak solution to the homogenized boundary value problem, and we estimate the rate of convergence to zero of ‖uε−u0‖∞ for ε→0. Our assumptions are, roughly speaking, as follows: The functions aijαβ are bounded, measurable and Z2-periodic, the functions biα(⋅,u) and bα(⋅,u) are bounded and measurable, the functions biα(x,⋅) and bα(x,⋅) are C1-smooth, and Ω is a bounded Lipschitz domain in R2. Neither global solution uniqueness is supposed nor growth restrictions of biα(x,⋅) or bα(x,⋅) nor higher regularity of u0, and cross-diffusion is allowed. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence, local uniqueness and error estimates for singularly perturbed problems and for homogenization problems.

Existence of bounded solutions for a non-periodic sublinear equation with an obstacle*

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.

Dynamics of a single population model with memory and nonlinear boundary condition

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.

A uniqueness theory on determining the nonlinear energy potential in phase-field system

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.

The effects of toxin and mutual interference among zooplankton on a diffusive plankton–fish model with Crowley–Martin functional response

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.

Adaptive output feedback command filter control based on extreme learning machine for nonaffine time delay systems with input saturation

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.

Well-posedness of the stationary and slowly traveling wave problems for the free boundary incompressible Navier-Stokes equations

We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a general phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The setting of our result is a horizontally-infinite fluid of finite depth with a flat, rigid bottom and a free boundary top. A constant gravitational field acts normal to bottom, and the free boundary experiences surface tension. In addition to these gravity-capillary effects, we allow for applied stress tensors to act on the free surface region and applied forces to act in the bulk. These are posited to be in either stationary or traveling form.In the absence of any applied stress or force, the system reverts to a quiescent equilibrium; in contrast, when such sources of stress or force are present, stationary or traveling waves are generated. We develop a small data well-posedness theory for this problem by proving that there exists a neighborhood of the origin in stress, force, and wave speed data-space in which we obtain the existence and uniqueness of stationary and traveling wave solutions that depend continuously on the stress-force data, wave speed, and other physical parameters. To the best of our knowledge, this is the first proof of well-posedness of the solitary stationary wave problem and the first continuous embedding of the stationary wave problem into the traveling wave problem. Our techniques are based on vector-valued harmonic analysis, a novel method of indirect symbol calculus, and the implicit function theorem.

Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

Learnable Graph Matching: A Practical Paradigm for Data Association.

Data association is at the core of many computer vision tasks, e.g., multiple object tracking, image matching, and point cloud registration. however, current data association solutions have some defects: they mostly ignore the intra-view context information; besides, they either train deep association models in an end-to-end way and hardly utilize the advantage of optimization-based assignment methods, or only use an off-the-shelf neural network to extract features. In this paper, we propose a general learnable graph matching method to address these issues. Especially, we model the intra-view relationships as an undirected graph. Then data association turns into a general graph matching problem between graphs. Furthermore, to make optimization end-to-end differentiable, we relax the original graph matching problem into continuous quadratic programming and then incorporate training into a deep graph neural network with KKT conditions and implicit function theorem. In MOT task, our method achieves state-of-the-art performance on several MOT datasets. For image matching, our method outperforms state-of-the-art methods on a popular indoor dataset, ScanNet. For point cloud registration, we also achieve competitive results.

Augmented Lagrangian method for nonlinear circular conic programs: a local convergence analysis

In this paper, we analyse a local convergence of augmented Lagrangian method (ALM) for a class of nonlinear circular conic optimization problems. In light of the singular value decomposition, the Debreu theorem and the implicit function theorem, we prove that the sequence generated by ALM converges to a local minimizer in the linear convergence rate under the constraint nondegeneracy condition and the strong second-order sufficient condition, in which the ratio constant is proportional to 1 / τ , where τ is the associated penalty parameter with a given lower threshold. As a byproduct, we also derive explicit expressions of critical cone and its affine hull for the given nonlinear circular conic program.

An implicit function theorem for the stream calculus

In the context of the stream calculus, we present an Implicit Function Theorem (IFT) for polynomial systems, and discuss its relations with the classical IFT from calculus. In particular, we demonstrate the advantages of the stream IFT from a computational point of view, and provide a few example applications where its use turns out to be valuable.

Balanced and Aeppli parameters for the heterotic moduli

In this paper, we fix the complex structure and explore the moduli space of the heterotic system by considering two different yet “dual” deformation paths starting from a Kähler solution. They correspond to deformation along the Bott–Chern cohomology class and the Aeppli cohomology class, respectively. Using the implicit function theorem, we prove the stability of the existence of heterotic solutions under these two deformations and hence establish an initial step in constructing local moduli coordinates around a Kähler solution.

Stability of Minima in Constrained Optimization Problems and Implicit Function Theorem

Stability of Minima in Constrained Optimization Problems and Implicit Function Theorem

Existence, uniqueness, and asymptotic behaviors of ground state solutions of Kirchhoff‐type equation with fourth‐order dispersion

In this paper, we focus on the following Schrödinger–Kirchhoff‐type problem with fourth‐order dispersion: where are constants and . We make use of Nehari manifold technique together with concentration‐compactness principle to prove that the above equation has at least a ground state solution for if , 6, and 7, and for if . Moreover, we also investigate the asymptotic behaviors of ground state solutions when some coefficients tend to zero. Among them, a uniqueness result about ground state solutions is obtained by implicit function theorem, and a blow‐up result is established by Pohozaev identity if dimension .

Continuous/smooth transonic solutions to the quasi-one-dimensional steady relativistic Euler equations

This paper establishes the existence and uniqueness of continuous/smooth Meyer type transonic solutions to the quasi-one-dimensional steady isentropic relativistic Euler equations. We use the high-order Taylor’s expansion of the velocity near the sonic point to overcome the difficulties of applying the implicit function theorem caused by the sonic degeneracy. It is verified that the flow in the De Laval nozzle accelerates from subsonic upstream and continuously/smoothly pass through the sonic surface at the throat and then becomes supersonic downstream. Moreover, the flow may have a zero or positive acceleration at the sonic surface, which depends on the geometric property of the nozzle at the throat.

Orbital stability of the black soliton for the quintic Gross–Pitaevskii equation

In this work, a proof of the orbital stability of the black soliton solution of the quintic Gross–Pitaevskii equation in one spatial dimension is obtained. We first build and show explicitly black and dark soliton solutions and we prove that the corresponding Ginzburg–Landau energy is coercive around them by using some orthogonality conditions related to perturbations of the black and dark solitons. The existence of suitable perturbations around black and dark solitons satisfying the required orthogonality conditions is deduced from an implicit function theorem. In fact, these perturbations involve dark solitons with sufficiently small speeds and some proportionality factors arising from the explicit expression of their spatial derivative.

On the computation of the gradient in implicit neural networks

Implicit neural networks and the related deep equilibrium models are investigated. To train these networks, the gradient of the corresponding loss function should be computed. Bypassing the implicit function theorem, we develop an explicit representation of this quantity, which leads to an easily accessible computational algorithm. The theoretical findings are also supported by numerical simulations.

Implicit Function Theorem for Nonlinear Time-Delay Systems With Algebraic Constraints

Implicit Function Theorem for Nonlinear Time-Delay Systems With Algebraic Constraints

On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equations

We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a Taylor series, then the solution can be locally expressed as a fractional power series. We use, in the real field, the method of majorants and the analytic version of the implicit-function theorem, in a way that circumvents difficulties associated to fractional calculus. Some corollaries on continuity are derived, with computational examples for illustration, and a discussion on fractional partial differential equations is included with a case study and counterexamples. Open problems are raised at the end.

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under [Formula: see text]-Gevrey assumptions on the residual equation, we establish [Formula: see text]-Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.