Nonlinear operator

Elliptic operator; Hyperbolic partial differential equation; Parabolic partial differential equation; PDEs of second order (for fuller discussion) References External links "Elliptic partial differential equation", Encyclopedia of Mathematics, EMS Press, 2001 ...

We study an irregular double obstacle problem with Orlicz growth over a nonsmooth bounded domain. We establish a global Calderón-Zygmund estimate by proving that the gradient of the solution to such a nonlinear elliptic problem is as integrable as both the nonhomogeneous term in divergence form and the gradient of the associated double obstacles. We also investigate minimal regularity ...No headers. An important aspect of linear systems is that the solutions obey the Principle of Superposition, that is, for the superposition of different oscillatory modes, the amplitudes add linearly.The linearly-damped linear oscillator is an example of a linear system in that it involves only linear operators, that is, it can be written in the operator form (appendix \(19.6.2\))E mail: [email protected]. The purpose of the "Journal of Applied Functional Analysis" (JAFA) is to publish high quality original research articles, survey articles and book reviews from all subareas of Applied Functional Analysis in the broadest form plus from its applications and its connections to other topics of Mathematical Sciences.Non‐Linear Filters Pixels in filter range combined by some non‐linear function Simplest examples of nonlinear filters: Min and Max filters Before filtering After filtering Step Edge (shifted to right) Narrow Pulse (removed) Linear Ramp (shifted to right) Effect of Minimum filterThis article concerns the improvement of digital image quality using mathematical tools such as nonlinear partial differential operators. In this paper, to perform smoothing on digital images, we propose to use the p(x)-Laplacian operator. Its smoothing power plays a main role in the restoration process. This enables us to dynamically process certain areas of an image.

The concept of coupled fixed point for nonlinear operators was considered first by Opoitsev (see [25-27]) and then, in 1987, by Guo and Lakshmikantham (see ) in connection with coupled quasisolutions of an initial value problem for ordinary differential equations.The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful ...also referred to as the Gateaux derivative, or R-operator (R-op). Visu-ally, if frepresents a curve or surface in Rn, then the directional derivatives represent tangent vectors to the curve or surface. JAX provides the jacfwdfunction for computing directional derivatives. Behind the scenes, directional derivatives are computed using a procedureOffer details. Description. In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation ...The basic results for nonlinear operators are given. These results include nonlinear versions of classical uniform boundedness theorem and Hahn-Banach theorem. Furthermore, the mappings from a metrizable space into another normed space can fall in some normed spaces by defining suitable norms.It is widely recognized that the Koopman operator provides a powerful tool to learn dynamical models of nonlinear systems from data [18, 28,30,37]. Based on the equivalence between contraction and ...: This paper examines the existence of weak solutions for a nonlinear boundary value problem of p ( x ) -Kirchhoff type involving the p ( x ) -Kirchhoff type triharmonic operator and perturbed external source terms. We establish our results by using a Fredholm-type result for a couple of nonlinear operators, in the framework of variable exponent Sobolev spaces.The Fredholm Integral Operator, denoted by K, is de ned as on functions f2C([a;b]) as Kf:= Z b a k(x;y)f(y)dy where k is an F.I.E. kernel. Note that Kis a linear operator. The F.I.E. is then written f= g+ Kf which can also be written Tf= g+ Kf using the xed point equation Tf= f. Note that Tf 0 = g+ Kf 0 T 2f 0 = T(Tf

Nov 21, 2021 · Generalized Inversion of Nonlinear Operators. Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore-Penrose inverse, widely used in physics ... contradiction to show that a function is nonlinear. Example 2. The function f(x) = sin x is nonlinear. To prove this fact, we'll use contradiction: assume f is linear, in which case 2sin x = sin(2x) 8x. Since this must hold for all x, it certainly must hold in the particular case x = p 2. Then 2sin p 2 = sin 2p 2 which leads to the ridiculous ...the nonlinear system (1.2) from the linear stability of the linearized equation. This prompts ... In this paper, k · k is an operator norm or l2 norm. | · | will in general denote a sup norm. For a ∈ Zd, we use |a| for the l2 norm. The dimension d will be fixed and p > d 2. Let A be aLet us consider some simple applications of systems and how to set up the equations. Example 3.1.2 3.1. 2. First, we consider salt and brine tanks, but this time water flows from one to the other and back. We again consider that the tanks are evenly mixed. Figure 3.1.1 3.1. 1: A closed system of two brine tanks.2023. 4. 14. ... EXISTENCE OF FIXED POINT FOR NONLINEAR OPERATOR IN PARTIALLY ORDERED METRIC SPACES. Advances in Differential Equations and Control Processes ...To overcome the above issue, this article proposed an improved nonlinear tuna swarm optimization algorithm based on Circle chaos map and levy flight operator (CLTSO). In order to compare it with some advanced heuristic algorithms, the performance of CLTSO is tested with unimodal functions, multimodal functions, and some CEC2014 benchmark functions.

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Jun 6, 2020 · Y. Kobayashi, "Difference approximation of Gauchy problems for quasi-dissipative operators and generation of nonlinear semigroups" J. Math. Soc. Japan, 27 : 4 (1975) pp. 640–665 [6] Y. Konishi, "On the uniform convergence of a finite difference scheme for a nonlinear heat equation" Proc. Japan. We propose a theoretical model that approximates the solution operator of parametric PDEs, and prove a theorem of its universal approximation capability in the nonlinear operator space. Based on the theoretical framework, for practical application, we introduce the point-based NNs as the backbone to approximate the solution operator of ...Beyond deep learning approaches, operator-valued kernel methods (38, 39) have also been demonstrated as a powerful tool for learning nonlinear operators, and they can naturally be generalized to neural networks acting on function spaces , but their applicability is generally limited due to their computational cost.DeepONet: Learning nonlinear operators The source code for the paper L. Lu, P. Jin, G. Pang, Z. Zhang, & G. E. Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.

Another important application of nonlinear approximation lies in the solu-tion of operator equations. Most notable, of course, are the adaptive nite element methods for elliptic equations (see Babu ska and Suri (1994)) as well as the emerging nonlinear wavelet methods in the same subject (see Dahmen (1997)).The LSTM-RNN representation of the nonlinear integral operator allows us to turn a system of nonlinear integro-differential equations into a system of ordinary differential equations for which many efficient solvers are available. Furthermore, because the use of LSTM-RNN representation of the nonlinear integral operator in an IDE eliminates the ...If an operator is not linear, it is said to be nonlinear. 1So, operators are function-valued functions of functions... 2Here, I am being very sloppy with what kind of functions can be input for an operator, i.e. I am ignoring domain issues. E mail: [email protected]. The purpose of the "Journal of Applied Functional Analysis" (JAFA) is to publish high quality original research articles, survey articles and book reviews from all subareas of Applied Functional Analysis in the broadest form plus from its applications and its connections to other topics of Mathematical Sciences.Generalized Inversion of Nonlinear Operators. Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore-Penrose inverse, widely used in physics ...The state space H endowed with the inner product 〈.,.〉, and the corresponding norm ∥.∥, v(t) is a scaler valued control.The dynamic A is an unbounded operator with domain D(A) ⊂ H and generates a semigroup of contractions (S(t)) t≥ ;0 on H. N is a nonlinear operator from H into H which is dissipative, such that N(0) = ;0, and B …By adding more Fourier blocks, FNO is capable of approximating complex nonlinear operators. In the 3rd part, the output is projected to the original space using a decoder. The Fourier neural operator has the same layer structure as the GKN, except that its integral operator \(\kappa _\theta \) is void of any dependence on the function a. That ...This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation: (*) $$ - \\Delta u + b(x)u = f(x, u), x \\in \\mathbb{R}^n .$$ The functionf is assumed to be "superlinear". A special case is the power nonlinearityf(x, z)=∥z∥ s−1 z where 1<s<(n+2)(n ...Jacobian matrix and determinant. In vector calculus, the Jacobian matrix ( / dʒəˈkoʊbiən /, [1] [2] [3] / dʒɪ -, jɪ -/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the ...which is a nonlinear matrix equation and appears in the area of transport of a beam of particles. For the Riccati equation $$\begin {aligned} XBX+XA-DX-C=0, \end {aligned}$$ (1.2) many authors study its solutions. An interesting question is found that solutions of the Riccati equation are closely related to the invariant subspace of some operator.The generalized modular string averaging procedure and its applications to iterative methods for solving various nonlinear operator theory problems. Article. Full-text available. May 2023;

This model-agnostic framework pairs a BED scheme that actively selects data for quantifying extreme events with an ensemble of deep neural operators that approximate infinite-dimensional nonlinear ...

Nonlinear optics. Structure of KTP crystal, viewed down b axis, used in second harmonic generation. Nonlinear optics ( NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light.To see how complex-valued constraints can be used in a more advanced setting, we solve the covariance estimation problem from the sedumi manual. The problem is to find a positive-definite Hermitian Toeplitz matrix Z such that the Frobenious norm of P-Z is minimized (P is a given complex matrix.). The matrix P isUniversal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems ... the necessary and sufficient condition is that it is not a polynomial; 3) the capability of approximating nonlinear functionals defined on some compact set of a Banach space and nonlinear operators ...Based on the engaged senses, multimedia can be five main types: text, images, audio, video and animations. Multimedia can also divide into linear and nonlinear categories depending on whether the user has navigational control.Beyond deep learning approaches, operator-valued kernel methods (38, 39) have also been demonstrated as a powerful tool for learning nonlinear operators, and they can naturally be generalized to neural networks acting on function spaces , but their applicability is generally limited due to their computational cost.Recently Koopman operator has become a promising data-driven tool to facilitate real-time control for unknown nonlinear systems. It maps nonlinear systems into equivalent linear systems in embedding space, ready for real-time linear control methods. However, designing an appropriate Koopman embedding function remains a challenging task. Furthermore, most Koopman-based algorithms only consider ...Some of the results in Sect. 3.2 have been extended to the case of homogeneous and subadditive operators in and applied for the construction of ISS Lyapunov functions for infinite networks of input-to-state stable systems with homogeneous and subadditive gain operators. Nevertheless, in the nonlinear case many problems …In the field of nonlinearity, nonlinear effects as a function of the Lévy index ... where \(\hat{D}\) is a differential operator that demonstrates diffraction in a linear medium and \(\hat{N}\) is a nonlinear operator that shows waveguide nonlinearities effect on soliton propagation. These operators areAbout this book. The greatest mathematicians, such as Archimedes, Newton, and Gauss, always united theory and applications in equal measure. Felix Klein There exists the remarkable possibility that one can master a subject mathemati­ cally, without really understanding its essence. Albert Einstein Don't give us numbers: give us insight!Choices of observations and local subset drawn from the global state for the local analysis are determined by the newly implemented modules. Since the observation operator is implemented in an independent computation module outside of the LETKF system, it is flexible in using any kind of nonlinear operator.

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Dec 1, 2000 · Nonlinear Operator Theory 1369 Observe that Example 2 is not possible for Y = R, since any continuous surjective map F : X R, with dimX > 2, has the property that F~ ( {y}) is unbounded for any y R. EXAMPLE 3. Let X = Y == R and F (x) = arctana;. Then, F is locally invertible, but has none of the remaining properties. Then use the Definition operator to assign the Find function for the same variables. Then outside of the Solve Block, evaluate the vector or individual variables to see the solutions. I like Solve Blocks because they can be used to solve both linear and nonlinear systems of equations. A linear system is one in which the variables are all raised ...2019. 11. 8. ... Therefore, a nonlinear formulation of the gyrokinetic Coulomb collision operator is crucial to adequately describe the dynamics in the periphery ...The name PRONTO stands for PRojection Operator based Netwon's method for Trajectory Optimization . The method is also known, for short, as the projection operator approach. Interesting applications that have been tackled with this method include, e.g., computing minimum-time trajectories for a race car, exploiting the energy from the ...The purpose of this paper is to study and analyze two different kinds of extragradient-viscosity-type iterative methods for finding a common element of the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous operator and the set of fixed points of a demicontractive mapping in real Hilbert spaces. Although the problem can be translated to a common ...The operator is anti-Hermitian, as shown for i.e., for arbitrary and though is Hermitian (being the position operator) Therefore, we see and are Hermitian adjoints, i.e., Note neither of these operators is Hermitian, i.e., ˆ† 1 2 d a d 1 ˆ 2 d a dnonlinear operators for the study of the spectrum of the nonlinear operator one needs to approach by another way. This paper is proposed a new approach for the study of the spectrum of con-tinuous nonlinear operators in the Banach spaces. Really here we find the first eigenvalue of the nonlinear continuous operator in Banach space and this shows LEARNING LOW-DIMENSIONAL NONLINEAR STRUCTURES FROM HIGH-DIMENSIONAL NOISY DATA: AN INTEGRAL OPERATOR APPROACH BY XIUCAI DING1 AND RONG MA2 1Department of Statistics, University of California, [email protected] 2Department of Statistics, Stanford [email protected] We propose a kernel-spectral embedding …In this article, we study the JF iterative algorithm to approximate the fixed points of a non-linear operator that satisfies condition (E) in uniformly convex Banach spaces. Further, some weak and strong convergence results are presented for the same operator using the JF iterative algorithm. We also demonstrate that the JF iterative algorithm is weakly w2G-stable with respect to almost ...Solving Linear Differential Equations. For finding the solution of such linear differential equations, we determine a function of the independent variable let us say M (x), which is known as the Integrating factor (I.F). Multiplying both sides of equation (1) with the integrating factor M (x) we get; M (x)dy/dx + M (x)Py = QM (x) ….. ….

The concept of coupled fixed point for nonlinear operators was considered first by Opoitsev (see [25-27]) and then, in 1987, by Guo and Lakshmikantham (see ) in connection with coupled quasisolutions of an initial value problem for ordinary differential equations.have applied ADM for solving multi-term linear and non-linear diffusion-wave equations. Con- ... where f is a given function, L and N are linear and nonlinear operators respectively. It isThe operator is anti-Hermitian, as shown for i.e., for arbitrary and though is Hermitian (being the position operator) Therefore, we see and are Hermitian adjoints, i.e., Note neither of these operators is Hermitian, i.e., ˆ† 1 2 d a d 1 ˆ 2 d a dThe Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is a relativistic wave equation, related to the Schrödinger equation.It is second-order in space and time and manifestly Lorentz-covariant.It is a quantized version of the relativistic energy-momentum relation = + ().Its solutions include a quantum scalar or pseudoscalar field, a field ...$\begingroup$ @Alex Indeed, I should say the derivative at a point is a linear transformation. The analog of the derivative function from one dimensional calculus is a linear transformation-valued map on some subset of $\mathbb{R}^n$.2. Generalized conditional gradient method for nonlinear inverse problems In this section, the generalized conditional gradient method described in the previous section will be applied to problem (1), (2). First, we state conditions on the nonlinear operator K which ensure that theorem 1.2 can be applied. Secondly, we will have a closer look at the Non‐Linear Filters Pixels in filter range combined by some non‐linear function Simplest examples of nonlinear filters: Min and Max filters Before filtering After filtering Step Edge (shifted to right) Narrow Pulse (removed) Linear Ramp (shifted to right) Effect of Minimum filterworks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator,This article concerns with the existence of multiple positive solutions for the following logarithmic Schrödinger equation. { − ϵ 2 Δ u + V ( x) u = u log u 2, in R N, u ∈ H 1 ( R N), where ϵ > 0, N ≥ 1 and V is a continuous function with a global minimum. Using variational method, we prove that for small enough ϵ > 0, the "shape" of ...The Koopman operator is a linear operator that describes the evolution of scalar observables (i.e., measurement functions of the states) in an infinitedimensional Hilbert space. This operator theoretic point of view lifts the dynamics of a finite-dimensional nonlinear system to an infinite-dimensional function space where the evolution of the … Nonlinear operator, While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator [5]. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators ... , Our module consists of multiple variants of the Koopman neural operator (KNO), a kind of mesh-independent neural-network-based PDE solvers developed following dynamic system theory. ... and non-linear PDEs. All variants are validated by mesh-independent and long-term prediction experiments implemented on representative PDEs (e.g., the Navier ..., the nonlinear continuous operators acting in the Banach spaces is investigated. For the study, this question is used a different approach that allows the study-ing of all eigenvalues of the nonlinear operator relative to another nonlinear operator. Here shows that in nonlinear operators case is necessary to seek the, Abstract. In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O (1/ N) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on ..., Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision.It is a formal theory for handling image structures at different scales, by representing an image as a one-parameter family of smoothed images, the scale-space ..., Title: Monotone Operators in Banach Space and Nonlinear Partial Differential Eq uations Author: R.E. Showalter Created Date: 11/13/2001 2:49:12 PM, Gamma correction or gamma is a nonlinear operation used to encode and decode luminance or tristimulus values in video or still image systems. ... On some operating systems running the X Window System, one can set the gamma correction factor (applied to the existing gamma value) ..., The basic results for nonlinear operators are given. These results include nonlinear versions of classical uniform boundedness theorem and Hahn-Banach theorem. Furthermore, the mappings from a metrizable space into another normed space can fall in some normed spaces by defining suitable norms., Edward Rówiński. The foundations of non-linear quantum mechanics are based on six postulates and five propositions. On a first quantised level, these approaches are built on non-linear differential operators, non-linear eigenvalue equations, and the notion of non-linear observables and non-linear states. The present theory predicts that the ..., Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find ..., While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator [5]. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators ..., GRG Nonlinear: Use if your worksheet model is nonlinear and smooth. In general terms, a smooth model is one in which a graph of the equation used doesn't show sharp edges or breaks. ... Use the int (integer) operator when you need a constraint, such as total employees, to be an integer value instead of a real number (that is, a number with a ..., These networks, introduced in Lu et al. , rely on a universal approximation theorem for operators (Chen & Chen, 1995), stating that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. The DeepOnet can then be used to learn the mapping between the terminal function of a PDE and its solution., Galerkin method. method of moments. A method for finding the approximate solution of an operator equation in the form of a linear combination of the elements of a given linearly independent system. Let $ F $ be a non-linear operator, with domain of definition in a Banach space $ X $ and range of values in a Banach space $ Y $., where L is a linear second-order operator, R is a linear first-order operator, N is the nonlinear operator and N ½yxðÞis the nonlinear convolution term which is defined by: N yxðÞ ¼ f y ;y 0;y 00;:: ;y ðÞn gy;y 0;y 00;: ;y ðÞn According to the variational iteration method, we can construct a correction functional as follows:, Neural Operator: Graph Kernel Network for Partial Differential Equations. zongyi-li/graph-pde; Markov Neural Operators for Learning Chaotic Systems. DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators, Nonlinear equations of evolution: Fluid flows and gas dynamics. 21. Nonlinear PDE's and nonlinear functional analysis. 22. Free boundary value problems: Variational inequalities. ... The theory of singular integral operators, which started in the 1930's in connection with PDE's, has become, through the Calderon Zygmund, A broad range of topics is covered, such as nonlinear functional analysis, linear operator theory, theory of operator algebras, applications including the solvability of some partial differential equations, and special spaces that contain Banach spaces and topological vector spaces., Splitting Algorithms for the Sum of Two Nonlinear Operators. P. L. Lions and ..., Compared to other neural networks to learn functions, it aims at the problem of learning nonlinear operators. However, it can be challenging by using the original model to learn nonlinear operators for high-dimensional stochastic problems. We propose a new multi-resolution autoencoder DeepONet model referred to as MultiAuto-DeepONet to deal ..., Leggett, RW, Williams, LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673-688 (1979) Article MATH MathSciNet Google Scholar Download references. Acknowledgements. The authors express their sincere thanks to the referees for the careful and details reading of the manuscript and very ..., The attention operator in either \eqref{eq:fourier-p} or \eqref{eq:attn-simple} is a nonlinear operator with respect to both its input and the trainable parameters. How can we bridge it to something like a Galerkin or Petrov-Galerkin projection (which are linear)? ... In Artificial Intelligence and Machine Learning for Multi-Domain Operations ..., However, the nonlinear operator \(N_{4} \left( \tau \right)\) (Eq. ) contains a fractional-order term, which represents an important feature of the turbulent orifice equation Eq. in the model of the VHCS containing CBVs. This existence makes the solutions of Eq. can not conform to the form of Eq. ., The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis . , The non-linear operator does not have "a" one period, it has a period range. The linear operator however does have one period. So the equality statement of the period needs some elaboration. The solution you cite has period $2\pi$, this need not be a period in the period range of the non-linear operator., We show that the knowledge of the Dirichlet--to--Neumann map for a nonlinear magnetic Schr\"odinger operator on the boundary of a compact complex manifold, equipped with a K\"ahler metric and ..., Aug 1, 2022 · In the NKNs, the iterative kernel integration is also assumed to be layer-independent. Differs from the GKNs where the next layer representation is defined via a nonlinear operator, the increment of each layer network representation is defined as a nonlinear operator in the NKNs. , where D_r = d_{33} E_{z}^2, d_{33} is the nonlinear coefficient, and E z is the z-component of the electric field.. With the frequency-domain analysis of the beam, only one particular frequency can be analyzed at one instance.(In other words, only one frequency can be analyzed with the Helmholtz equation.) Hence, the model sets up two interfaces and couples the two physics., On Non-Linear operators for Geometric Deep Learning. Grégoire Sergeant-Perthuis (LML), Jakob Maier, Joan Bruna (CIMS), Edouard Oyallon (ISIR) This work studies operators mapping vector and scalar fields defined over a manifold , and which commute with its group of diffeomorphisms . We prove that in the case of scalar fields , those operators ..., A linear operator between Banach spaces is continuous if and only if it is bounded, that is, the image of every bounded set in is bounded in , or equivalently, if …, I have implemented a simple parameter continuation scheme to find the stationary solutions of a nonlinear problem at different parameter values. However, my scheme cannot handle bifurcations - it fails to find solutions near turning points. I therefore need to implement a more sophisticated continuation scheme, but am unsure how to proceed., nonlinear operator theory. The nonlinear operator theory and their applications to PDE’s problems are presented in details, including classical variational optimization problems in Newtonian and non-Newtonian mechanics and uid. An emerging appli-cation of optimizations include the imaging and signal analysis and the classi cation and machine ..., Abstract. A stochastic forcing of a non-linear singular/degenerated parabolic problem with random growth conditions is proposed in the framework of Orlicz Lebesgue and Sobolev spaces with variable random exponents. We give a result of existence and uniqueness of the solution, for additive and multiplicative problems.