_{Nonlinear operator 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. }

_{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. Download PDF Abstract: We develop a theoretical analysis for special neural network architectures, termed operator recurrent neural networks, for approximating nonlinear functions whose inputs are linear operators. Such functions commonly arise in solution algorithms for inverse boundary value problems. Traditional neural networks …A "linear" function usually means one who's graph is a straight line, or that involves no powers higher than 1. And yet, many sources will tell you that the Fourier transform is a "linear transform". Both the discrete and continuous Fourier transforms fundamentally involve the sine and cosine functions. These functions are about as non -linear ...1.2.4 A spatial derivative d h du dx = lim e!0 ux +ehx ux e = dh dx 1.2.5 A functional Let J : H1(W) !R be J[u] = W 1 2 u2 x + 1 2 u2 dx. Then d hJ = lim e!0 W h 1 2 u 2 x+ 1 2 u 2 +euh+euxhx + 1 2e 2h2 x + 1 2eh 2 1 2 u 2 1 2 u 2 i dx e d hJ = W [uh+u xh ] dx Note: it's routine in inﬁnite-dimensional optimization problems to exchange integration and Gateaux differ-Martin R.H.: Nonlinear Operators and Differential Equations in Banach Spaces. Robert E Krieger Publishing Co, Florida (1987) Google Scholar Obrecht E.: Evolution operators for higher order abstract parabolic equations. Czech. Math. J. 36, 210-222 (1986) MathSciNet MATH Google Scholar Peng Y., Xiang X.:bility of an explicit timestepping method assuming a potentially highly nonlinear dynamics with indeterminant nonlinear coupling. In other words, the framework developed here can also be used to determine whether operator splitting, and/or implicit timestepping, might be necessary given a nonlinear system of equations.In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input.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. To illustrate the concept of linear systems representing nonlinear evolution in original coordinates we show the evolution of the respective eigenfunctions in Fig. 2.The linear combination of the linearly evolving eigenfunctions fully describes all trajectories of the nonlinear system from Example 2.1.This highlights the globality of the Koopman …Sep 1, 2022 · Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural operators such as deep operator networks (DeepONets) provide a new simulation paradigm in science and engineering. An antilinear operator satisfies the following two properties: (1) (2) where is the complex conjugate of . See also Antiunitary, Linear Operator Explore with Wolfram|Alpha. More things to try: [3,8) expand (x^2 + 1)(x^2 - 1)(x+1)^3 (n,k)-Turan graph edge count; ReferencesPositive nonlinear operators associated with isoperimetric eigenvalue problems in algebra. Positive nonlinear systems: Some results and applications. Random products of contractions. Functional-differential equations with non-Volterra operator. Searching for periodic orbits in a four dimensional symplectic map.This 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. On Non-Linear operators for Geometric Deep Learning Grégoire Sergeant-Perthuis, Jakob Maier, Joan Bruna, Edouard Oyallon; Momentum Aggregation for Private Non-convex ERM Hoang Tran, Ashok Cutkosky; Learning in Congestion Games with Bandit Feedback Qiwen Cui, Zhihan Xiong, Maryam Fazel, Simon S. DuIn this section, numerical examples of both linear and non-linear differential equations solved by Khan-Green's fixed point iterative method are shown as proof. In addition, the examples were also computed by Picard-Green's method to show comparisons of the outcomes for both methods to reveal the high accuracy of Khan-Green's method.The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral …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 there is a (finite) number , called the operator norm (a similar assertion is also true for arbitrary normed spaces). The continuous linear operators from into form a subspace of which is a Banach space with respect to .Definition 2.2.1. Let F be a nonlinear operator defined on a subset D of a linear space X with values in a linear space Y, i.e., F ∈ ( D, Y) and let x, y be two points of D. A linear operator from X into Y, denoted [ x, y ], which satisfies the condition. is called a divided difference of F at the points x and y. In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the … Sign with adobe. When this is done, the nonlinear operator framework is used, and the model is converted to a big-M model when the problem is solved. We will however do the modelling by hand here, to illustrate the underlying problems. The big-M model for this case is easy to derive, and is easily seen to be given by the following model.Tracking of reference signals is addressed in the context of a class of nonlinear controlled systems modelled by r-th-order functional differential equations, encompassing inter alia systems with unknown "control direction" and dead-zone input effects. A control structure is developed which ensures that, for every member of the underlying system class and every admissible reference signal ...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.A neural network can approximate a continuous function using a nonlinear basis that is computed on-the-fly based on different activation functions in the form of sigmoids, tanh, or other non-polynomial activation functions [9]. A less known result is that a neural network can also approximate nonlinear continuous operators [6].It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a …Here, we develop a non-Hermitian linear response theory that considers the dynamical response of a Hermitian system to a non-Hermitian probe, and we can also relate such a dynamical response to ... 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 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. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from ...Pierre-Louis Lions. Pierre-Louis Lions ( French: [ljɔ̃ːs]; [1] born 11 August 1956) is a French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Medal and the 1991 Prize of the Philip Morris tobacco and cigarette company.Therefore, nonlinear spectral theory studies eigenfunctions of nonlinear operators, as for instance the p-Laplacian. In this thesis, nonlinear spectral theory is further developed using ...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 ...Linear system. In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator . Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control ...1 Answer Sorted by: 1 If linear, such an operator would be unbounded. Unbounded linear operators defined on a complete normed space do exist, if one takes the axiom of …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 In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are useful tools in classical mechanics.Operators are even more important in quantum mechanics, where they … This relationship between DMD and the Koopman operator has motivated an effort to machine-learn Koopman eigenfunctions from data in order to linearize nonlinear dynamical systems globally on the ..."Nonlinear analysis is a quite young area in mathematical sciences, and it has grown tremendously in the last thirty years. … In addition, all methods discussed in this book are illustrated by carefully chosen examples from applied mathematics, physics, engineering and geometry. … Overall, the book presents a unified approach, and is an ...2 Eigenvectors of nonlinear operators We give here a brief introduction to the broad topic of eigenvectors of nonlinear oper-ators. More details are provided in relation to the variational setting. We would like to extend the linear eigenvalue problem Lu= u; given a matrix L, to a generalized problem, given a bounded nonlinear operator T:Abstract. A new unified theory and methodology is presented to characterize and model long-term memory effects of microwave components by extending the poly-harmonic distortion (PHD) model to ...Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types ...This 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.Therefore, nonlinear spectral theory studies eigenfunctions of nonlinear operators, as for instance the p-Laplacian. In this thesis, nonlinear spectral theory is further developed using ...Uncover the Useful Interactions of Fixed Point Theory with Topological StructuresNonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications is the first book to tackle the topological fixed point theory for block operator ...Let us now consider the uniformly elliptic operator Lgiven by Lu ¸ n i;j 1 B jp a ijB iuq ; (5.1) and the respective PDE Lu f in ; (5.2) where P Rn is open and fP L2p q . It is straightforward, and will be apparent from the proof how to extend the regularity theory to operators that contain lower-order terms. We de ne a weak solution as the ...We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace's equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L... Bath and body works oak lawn. Origin of rock chalk jayhawk. However, many interesting learning tasks entail learning operators, i.e., mappings be-tween an in nite-dimensional input Banach space and (possibly) an in nite-dimensional output space. A prototypical example in scienti c computing is provided by nonlinear operators that map the initial datum into the (time series of) solution of a nonlinear time-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 there is a (finite) number , called the operator norm (a similar assertion is also true for arbitrary normed spaces). The continuous linear operators from into form a subspace of which is …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. Keywords: Nonlinear Dynamics, Operator Theory, Observer Design 1. INTRODUCTION In this pap r we develop new appr ach for bserver synth sis for nonlinear systems based on K opman operator theoretic framework. Observer design for nonli ear systems is an ex- tensively researched area, se Misawa and Hedrick (1989); Ni- jmeijer …Hello, I am trying to solve an equation using complex numbers. However I get the following error: ufl.algorithms.check_arities.ArityMismatch: Applying nonlinear operator Real to expression depending on form argument v_1. It would seem that this is a classical problem but I have not found a satisfactory solution yet. Here's a MWE.Kerr electro-optic effect. The Kerr electro-optic effect, or DC Kerr effect, is the special case in which a slowly varying external electric field is applied by, for instance, a voltage on electrodes across the sample material. Under this influence, the sample becomes birefringent, with different indices of refraction for light polarized parallel to or perpendicular to the applied field.Operator regression approaches have been successful in learning nonlinear operators for complex PDEs directly from observations; however, in many real-world applications, collecting the required ...A linear operator is a function that maps one vector onto other vectors. They can be represented by matrices, which can be thought of as coordinate representations of linear operators (Hjortso & Wolenski, 2008). Therefore, any n x m matrix is an example of a linear operator. An example of an operator that isn't linear: Gα = α 2.The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn ...The proposed nonlinear approach employs a differential Teager energy operator and the energy separation algorithm to obtain formant AM and FM modulations from filtered speech recordings. A new speech measure is proposed based on parameterization of the autocorrelation envelope of the AM response.Free linear first order differential equations calculator - solve ordinary linear first order differential equations step-by-step.This reference text, now in its second edition, offers a modern unifying presentation of three basic areas of nonlinear analysis: convex analysis, monotone operator theory, and the fixed point theory of nonexpansive operators. Taking a unique comprehensive approach, the theory is developed from the ground up, with the rich connections and ... …. Extraction of the gravitational potential and high-frequency wave perturbation properties of nonlinear (3 + 1)-dimensional Vakhnenko-Parkes equation via novel approach. Li Yan, Haci Mehmet Baskonus, Carlo Cattani, Wei Gao. Version of Record online: 23 September 2022.The Adomian decomposition method (ADM) is a semi-analytical method for solving ordinary and partial nonlinear differential equations.The method was developed from the 1970s to the 1990s by George Adomian, chair of the Center for Applied Mathematics at the University of Georgia. It is further extensible to stochastic systems by using the Ito integral. The aim of this method is towards a unified ...This article concerns the improvement of digital image quality using mathematical tools such as nonlinear partial differential operators. In this paper, to …Operator norm. In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm of a linear map is the maximum factor by which it ...The di erential operator is called the Laplacian. The wave equation. For u: R1+d!R or C, u= 0; where = @2 0 + : ... It turns out that many important and interesting PDEs are nonlinear. Let us see a few key examples from Geometry and Physics. To relate with the previously listed fundamental PDEs, the type of each nonlinear PDE (elliptic ...In this article we prove a new uniform boundedness principle for monotone, positively homogeneous, subadditive, and Lipschitz mappings defined on a suitable cone of functions (Theorem 2 ). This result is applicable to several classes of classically nonlinear operators (Examples 4 and 5 and Remarks 7 and 8 ). Let be a nonempty set.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.6.4.1 Non-Linear Perturbations of Linear VIEs 231 6.4.2 Hammerstein VIEs with Convolution Kernels 233 6.5 Exercises and Research Problems 236 6.6 Notes 238 7 Cordial Volterra Integral Equations 241 7.1 Cordial Volterra Integral Operators 241 7.1.1 Basic Properties of Cordial Volterra Integral Operators 242 Nonlinear operator, In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation.The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can ..., 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 …, Wanting a filter means the input has some frequencies you regard as noise, and want them attenuated. A filter attenuates and phase shifts the input. It pretty much makes sense to talk about filters only in terms of linear circuits. (What can happen to frequencies after they pass through a nonlinear operator?), 1. 2. We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular ..., 1. 2. We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a …, Dynamic mode decomposition ( DMD) is a dimensionality reduction algorithm developed by Peter J. Schmid and Joern Sesterhenn in 2008. [1] [2] Given a time series of data, DMD computes a set of modes each of which is associated with a fixed oscillation frequency and decay/growth rate. For linear systems in particular, these modes and frequencies ..., H. Gaevskii, K. Gregor, and K. Zakharias, Nonlinear Operator Equations and Operator Differential Equations [Russian translation], Mir, Moscow (1978). Google Scholar S. L. Sobolev, Applications of Functional Analysis to Mathematical Physics [in Russian], Leningrad State Univ. (1950)., is assumed to be either a fully nonlinear operator satisfying the assumptions in Section 1.1 or a quasilinear divergence-form operator of the form (1.4). Our main theorem is the following comparison principle: Theorem 1 (Theorem 3.1 and Theorem 3.24). Let ube a viscosity subsolution and va viscosity supersolution of (1.1) on Q= (0;T] for some T>0. , Examples of nonlinear operators: √(f + g) is not equal to √f + √g inverse (f + g) = 1/(f + g) is not equal to 1/f + 1/g Cautionary note: When trying to determine the result of operations with operators that include partial derivatives, always. using a function as a “place holder”. For example, what is, Airline Operating Expenses - Airline operating costs range from airport space to fuel costs. Learn about airline operating expenses on this page. Advertisement Running an airline is unavoidably expensive. One of the most basic costs is the..., Operator Norm. The operator norm of a linear operator is the largest value by which stretches an element of , It is necessary for and to be normed vector spaces. The operator norm of a composition is controlled by the norms of the operators, When is given by a matrix, say , then is the square root of the largest eigenvalue of the symmetric ..., The Koopman operator \({\cal K}\) induces a linear system on the space of all measurement functions g, trading finite-dimensional nonlinear dynamics in (2) for infinite-dimensional linear dynamics ..., MONOTONICITY IN THE THEORY OF ALMOST PERIODIC SOLUTIONS OF NONLINEAR OPERATOR EQUATIONS ... [18] R. I. Kačurovskiĭ 1968 Nonlinear monotone operators in Banach ..., H. Gaevskii, K. Gregor, and K. Zakharias, Nonlinear Operator Equations and Operator Differential Equations [Russian translation], Mir, Moscow (1978). Google Scholar S. L. Sobolev, Applications of Functional Analysis to Mathematical Physics [in Russian], Leningrad State Univ. (1950)., Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. NMI, 2021. paper \n. Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. \n \n \n. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. SA, 2021. paper \n, Sep 9, 2022 · In this paper, the numerical radius of nonlinear operators in Hilbert spaces is studied. First, the relationship between the spectral radius and the numerical radius of nonlinear operators is given. Then, the famous inequality $$\\frac{1}{2}\\Vert T\\Vert \\le w(T)\\le \\Vert T\\Vert $$ 1 2 ‖ T ‖ ≤ w ( T ) ≤ ‖ T ‖ and inclusion $$\\sigma (A^{-1}B)\\subseteq \\frac{\\overline{W(B ... , Aug 11, 2015 · Uncover the Useful Interactions of Fixed Point Theory with Topological StructuresNonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications is the first book to tackle the topological fixed point theory for block operator ... , The Koopman operator \({\cal K}\) induces a linear system on the space of all measurement functions g, trading finite-dimensional nonlinear dynamics in (2) for infinite-dimensional linear dynamics ..., Koopman operator is a linear but an infinite-dimensional operator that governs the time evolution of system outputs in a linear fashion. e exploit this property to synthesize an observer form which enables the use of Luenberger/Kalman-like linear observers for nonlinear estimation., 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. , In the comments it was mentioned that the square root is nonlinear and operator monotone, but it is non-convex. The motivation is me trying to generalise of a certain optimization problem from measures with values in $[0, \infty)$ to tensor-valued measures, that is, measures having as values real symmetric positive definite matrices., The convection-diffusion equation is a combination of the diffusion and convection ( advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the ..., In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi-Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted [X, Y] . Conceptually, the Lie bracket [X, Y] is the derivative of Y ..., The 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 ..., Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized. We review the properties of subdifferentials as maximally monotone operators and, in tandem, investigate those of proximity operators as resolvents. In ..., Session Overview. This session consists of an imaginary dialog written by Prof. Haynes Miller and performed in his 18.03 class in spring 2010. It takes the form of a debate between Linn E. R. representing linear first order ODE’s and Chao S. doing the same for first order nonlinear ODE’s., which is similar to the example on the doc page for BilevelJuMP about Non Linear Models. (I changed it slightly to match my use case that is lower-objective depending also on x) This can be solved using BilevelJuMP.jl as explained on that page, with the following code: using BilevelJuMP,Ipopt model = BilevelModel (Ipopt.Optimizer, mode ..., 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 ..., DeepONet can learn continuous nonlinear operators between input and output , so that it can be used to approximate various explicit and implicit mapping functions like Laplace transform and PDEs, which are the most common but difficult mathematical relationships to investigate in various dynamic systems. To find effective nonlinear operators ..., 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 ... , In this work, we consider parametrized and nonlinear partial differential equations of the general form (1) u t + N [u; λ] = 0, x ∈ Ω, t ∈ [0, T], where u (t, x) denotes the latent (hidden) solution, N [⋅; λ] is a nonlinear operator parametrized by λ, and Ω is a subset of R D. This setup encapsulates a wide range of problems in ..., the operator system (CSOS) involved in the nonlinear operators Si and Tj, which aims to locate the point x 2Hsuch that ˆ 0 = x Six, 8i 2â1, 0 = x Tjx, 8j 2â2. (3) Example 1. When â1 and â2 are single point sets, i.e., p k = 1 for k = 1,2, and S1 and T1 are separately denoted as S and T, one has the following special nonlinear operator ..., ℘ψ(t) represents the bounded linear diﬀerential operator in t, i.e. it is possible to ﬁnd number α>0 such that ℘ψ (t)≤ αψ , Ξψ(t) indicates the nonlinear diﬀerential operator of general type in t and is Lipschitz continuous having β>0 satisfying |Ξψ(t)−Ξ (t)|≤β|ψ − | and Φ(t) stands for the source term.}