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Journal articles on the topic 'Operator equations, Nonlinear'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Burýšková, Věra, and Slavomír Burýšek. "On solvability of nonlinear operator equations and eigenvalues of homogeneous operators." Mathematica Bohemica 121, no. 3 (1996): 301–14. http://dx.doi.org/10.21136/mb.1996.125984.

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2

Takači, Djurdjica. "Nonlinear operator differential equations." Nonlinear Analysis: Theory, Methods & Applications 30, no. 1 (December 1997): 47–52. http://dx.doi.org/10.1016/s0362-546x(97)00404-5.

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3

Ryazantseva, Irina P. "Simplification method for nonlinear equations of monotone type in Banach space." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 23, no. 2 (June 30, 2021): 185–92. http://dx.doi.org/10.15507/2079-6900.23.202102.185-192.

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Abstract. In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

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4

Binh, Tran Quoc, and Nguyen Minh Chuong. "APPROXIMATION OF NONLINEAR OPERATOR EQUATIONS*." Numerical Functional Analysis and Optimization 22, no. 7-8 (November 30, 2001): 831–44. http://dx.doi.org/10.1081/nfa-100108311.

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5

Marchenko, V. A. "Nonlinear equations and operator algebras." Physica D: Nonlinear Phenomena 28, no. 1-2 (September 1987): 227. http://dx.doi.org/10.1016/0167-2789(87)90152-7.

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6

Motsa, S. S. "On the Optimal Auxiliary Linear Operator for the Spectral Homotopy Analysis Method Solution of Nonlinear Ordinary Differential Equations." Mathematical Problems in Engineering 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/697845.

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The purpose of this study is to identify the auxiliary linear operator that gives the best convergence and accuracy in the implementation of the spectral homotopy analysis method (SHAM) in the solution of nonlinear ordinary differential equations. The auxiliary linear operator is an essential element of the homotopy analysis method (HAM) algorithm that strongly influences the convergence of the method. In this work we introduce new procedures of defining the auxiliary linear operators and compare solutions generated using the new linear operators with solutions obtained using well-known linear operators. The applicability and validity of the proposed linear operators is tested on four highly nonlinear ordinary differential equations with fluid mechanics applications that have recently been reported in the literature. The results from the study reveal that the new linear operators give better results than the previously used linear operators. The identification of the optimal linear operator will direct future research on further applications of HAM-based methods in solving complicated nonlinear differential equations.

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7

Argyros, Ioannis K., Santhosh George, and P. Jidesh. "Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/754154.

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We present a new iterative method which does not involve inversion of the operators for obtaining an approximate solution for the nonlinear ill-posed operator equationF(x)=y. The proposed method is a modified form of Tikhonov gradient (TIGRA) method considered by Ramlau (2003). The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005). The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper.

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8

Hofmann, Bernd, and Robert Plato. "On ill-posedness concepts, stable solvability and saturation." Journal of Inverse and Ill-posed Problems 26, no. 2 (April 1, 2018): 287–97. http://dx.doi.org/10.1515/jiip-2017-0090.

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AbstractWe consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces. It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.

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9

Çavuş, Abdullah, Djavvat Khadjiev, and Seda Öztürk. "On periodic solutions to nonlinear differential equations in Banach spaces." Filomat 30, no. 4 (2016): 1069–76. http://dx.doi.org/10.2298/fil1604069c.

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Let A denote the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a complex Banach space H. In this work, an analog of the resolvent operator which is called quasi-resolvent operator and denoted by R? is defined for points of the spectrum, some equivalent conditions for compactness of the quasi-resolvent operators R? are given. Then using these, some theorems on existence of periodic solutions to the non-linear equations ?(A)x = f (x) are given, where ?(A) is a polynomial of A with complex coefficients and f is a continuous mapping of H into itself.

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10

ARAI, ASAO. "HEISENBERG OPERATORS, INVARIANT DOMAINS AND HEISENBERG EQUATIONS OF MOTION." Reviews in Mathematical Physics 19, no. 10 (November 2007): 1045–69. http://dx.doi.org/10.1142/s0129055x07003206.

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An abstract operator theory is developed on operators of the form AH(t) := eitHAe-itH, t ∈ ℝ, with H a self-adjoint operator and A a linear operator on a Hilbert space (in the context of quantum mechanics, AH(t) is called the Heisenberg operator of A with respect to H). The following aspects are discussed: (i) integral equations for AH(t) for a general class of A; (ii) a sufficient condition for D(A), the domain of A, to be left invariant by e-itH for all t ∈ ℝ; (iii) a mathematically rigorous formulation of the Heisenberg equation of motion in quantum mechanics and the uniqueness of its solutions; (iv) invariant domains in the case where H is an abstract version of Schrödinger and Dirac operators; (v) applications to Schrödinger operators with matrix-valued potentials and Dirac operators.

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11

Emmrich, Etienne. "External Approximation of Nonlinear Operator Equations." Numerical Functional Analysis and Optimization 30, no. 5-6 (June 30, 2009): 486–98. http://dx.doi.org/10.1080/01630560902987329.

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12

Chidume, C. E., and H. Zegeye. "Approximation methods for nonlinear operator equations." Proceedings of the American Mathematical Society 131, no. 8 (November 13, 2002): 2467–78. http://dx.doi.org/10.1090/s0002-9939-02-06769-2.

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13

Chronopoulos, A. T., and Z. Zlatev. "Iterative methods for nonlinear operator equations." Applied Mathematics and Computation 51, no. 2-3 (October 1992): 167–80. http://dx.doi.org/10.1016/0096-3003(92)90072-9.

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14

Petrov, Vladilen V. "SOLUTION OF NON-LINEAR PROBLEMS OF STRUCTURAL MECHANICS BY METHOD OF STEEPEST DESCENT." International Journal for Computational Civil and Structural Engineering 13, no. 3 (September 11, 2017): 103–11. http://dx.doi.org/10.22337/1524-5845-2017-13-3-103-111.

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The algorithm of application of the method of steepest descent to the solution of problems of structural mechanics and solid mechanics, described by nonlinear differential equations. For application of this method to nonlinear operators are described by a sequence of linear operators in incremental form, unlimited and complex linear operator, in line with the idea of L. V. Kantorovich is limited to a simple linear unbounded operator.

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15

Gavrilyuk, I. P. "Approximation of the Operator Exponential and Applications." Computational Methods in Applied Mathematics 7, no. 4 (2007): 294–320. http://dx.doi.org/10.2478/cmam-2007-0019.

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AbstractA review of the exponentially convergent approximations to the operator exponential is given. The applications to inhomogeneous parabolic and elliptic equations, nonlinear parabolic equations, tensor-product approximations of multidimensional solution operators as well as to parabolic problems with time dependent coefficients and boundary conditions are discussed.

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16

Wei, Li, Liling Duan, and Haiyun Zhou. "Solution of Nonlinear Elliptic Boundary Value Problems and Its Iterative Construction." Abstract and Applied Analysis 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/210325.

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We study a kind of nonlinear elliptic boundary value problems with generalizedp-Laplacian operator. The unique solution is proved to be existing and the relationship between this solution and the zero point of a suitably defined nonlinear maximal monotone operator is investigated. Moreover, an iterative scheme is constructed to be strongly convergent to the unique solution. The work done in this paper is meaningful since it combines the knowledge of ranges for nonlinear operators, zero point of nonlinear operators, iterative schemes, and boundary value problems together. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work.

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17

Kopach, M. I., A. F. Obshta, and B. A. Shuvar. "Two-sided inequalities with nonmonotone sublinear operators." Carpathian Mathematical Publications 7, no. 1 (July 3, 2015): 78–82. http://dx.doi.org/10.15330/cmp.7.1.78-82.

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18

LATRACH, KHALID. "COMPACTNESS RESULTS FOR TRANSPORT EQUATIONS AND APPLICATIONS." Mathematical Models and Methods in Applied Sciences 11, no. 07 (October 2001): 1181–202. http://dx.doi.org/10.1142/s021820250100129x.

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The goal of this paper is to give a systematic analysis of compactness properties for transport equations with general boundary conditions where an abstract boundary operator relates the incoming and outgoing fluxes. The analysis involves two parameters: The velocity measure and the collision operator. Hence, for a large class of (velocity) measures and under appropriate assumptions on scattering operators compactness results are obtained. Using the positivity (in the lattice sense) and the comparison arguments by Dodds–Fremlin, their converses are derived, and necessary conditions for some remainder term of the Dyson–Phillips expansion to be compact are given. Our results are independent of the properties of the boundary operators and play a crucial role in the understanding of the time asymptotic structure of evolution transport problems. Also, although solutions of transport equations propagate singularities (due to the hyperbolic nature of the operator), they bring the regularity in the variable space (regardless of the boundary operator). We end the paper by applying the obtained results to discuss the existence of solutions to a nonlinear boundary value problem and to describe in detail the various essential spectra of transport operators with abstract boundary conditions.

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19

Frumin, Leonid L. "Linear least squares method in nonlinear parametric inverse problems." Journal of Inverse and Ill-posed Problems 28, no. 2 (April 1, 2020): 307–12. http://dx.doi.org/10.1515/jiip-2019-0009.

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AbstractA generalization of the linear least squares method to a wide class of parametric nonlinear inverse problems is presented. The approach is based on the consideration of the operator equations, with the selected function of parameters as the solution. The generalization is based on the two mandatory conditions: the operator equations are linear for the estimated parameters and the operators have discrete approximations. Not requiring use of iterations, this approach is well suited for hardware implementation and also for constructing the first approximation for the nonlinear least squares method. The examples of parametric problems, including the problem of estimation of parameters of some higher transcendental functions, are presented.

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20

LAZURCHAK, I. I., V. L. MAKAROV, and D. SYTNYK. "Two-Sided Approximations For Nonlinear Operator Equations." Computational Methods in Applied Mathematics 8, no. 4 (2008): 386–92. http://dx.doi.org/10.2478/cmam-2008-0028.

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AbstractWe propose an analytical-numerical method for nonlinear operator equations which converges exponentially and provides two-sided approximations. A numerical example confirms the theoretical results

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21

Kiguradze, I. "On solvability conditions for nonlinear operator equations." Mathematical and Computer Modelling 48, no. 11-12 (December 2008): 1914–24. http://dx.doi.org/10.1016/j.mcm.2007.05.019.

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22

Bu, Yucheng, and Yujun Dong. "Existence of solutions for nonlinear operator equations." Discrete & Continuous Dynamical Systems - A 39, no. 8 (2019): 4429–41. http://dx.doi.org/10.3934/dcds.2019180.

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23

Jones, John. "Nonlinear operator equations and applications to modelling." ACM SIGSIM Simulation Digest 20, no. 4 (April 1990): 135–42. http://dx.doi.org/10.1145/99637.99655.

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24

Sang, Yanbin. "Sign-Changing Solutions for Nonlinear Operator Equations." Journal of Operators 2014 (March 6, 2014): 1–6. http://dx.doi.org/10.1155/2014/379056.

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The existence of six solutions for nonlinear operator equations is obtained by using the topological degree and fixed point index theory. These six solutions are all nonzero. Two of them are positive, the other two are negative, and the fifth and sixth ones are both sign-changing solutions. Furthermore, the theoretical results are applied to elliptic partial differential equations.

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25

Potra, Florian A. "Monotone iterative methods for nonlinear operator equations∗)." Numerical Functional Analysis and Optimization 9, no. 7-8 (January 1987): 809–43. http://dx.doi.org/10.1080/01630568708816262.

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26

Karlsen, Kenneth Hvistendahl, and Nils Henrik Risebro. "Corrected Operator Splitting for Nonlinear Parabolic Equations." SIAM Journal on Numerical Analysis 37, no. 3 (January 2000): 980–1003. http://dx.doi.org/10.1137/s0036142997320978.

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27

HAYASHI, NAKAO, and PAVEL I. NAUMKIN. "SCATTERING OPERATOR FOR NONLINEAR KLEIN–GORDON EQUATIONS." Communications in Contemporary Mathematics 11, no. 05 (October 2009): 771–81. http://dx.doi.org/10.1142/s0219199709003582.

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We prove the existence of the scattering operator in [Formula: see text] in the neighborhood of the origin for the nonlinear Klein–Gordon equation with a power nonlinearity [Formula: see text] where [Formula: see text]μ ∈ C, n=1,2.

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28

Tanana, V. P. "On approximate solution of nonlinear operator equations." Siberian Mathematical Journal 39, no. 5 (October 1998): 1017–25. http://dx.doi.org/10.1007/bf02672925.

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29

Agarwal, Ravi P., Donal O’Regan, and Veli Shakhmurov. "Higher order nonlinear degenerate differential operator equations." Applied Mathematics Letters 22, no. 10 (October 2009): 1556–61. http://dx.doi.org/10.1016/j.aml.2009.04.004.

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30

Bao, Jiguang. "Fully Nonlinear Elliptic Equations on General Domains." Canadian Journal of Mathematics 54, no. 6 (December 1, 2002): 1121–41. http://dx.doi.org/10.4153/cjm-2002-042-9.

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AbstractBy means of the Pucci operator, we construct a function u0, which plays an essential role in our considerations, and give the existence and regularity theorems for the bounded viscosity solutions of the generalized Dirichlet problems of second order fully nonlinear elliptic equations on the general bounded domains, which may be irregular. The approximation method, the accretive operator technique and the Caffarelli's perturbation theory are used.

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31

Jafari, Hossein, and Hassan Kamil Jassim. "Approximate Solution for Nonlinear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operator." Journal of Zankoy Sulaimani - Part A 18, no. 1 (August 30, 2015): 127–32. http://dx.doi.org/10.17656/jzs.10456.

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32

Holtby, Derek W. "HIGHER-ORDER ESTIMATES FOR FULLY NONLINEAR DIFFERENCE EQUATIONS. II." Proceedings of the Edinburgh Mathematical Society 44, no. 1 (February 2001): 87–102. http://dx.doi.org/10.1017/s0013091598000200.

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AbstractThe purpose of this work is to establish a priori $C^{2,\alpha}$ estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finite difference operator $\F$ is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the results for the special case that the operator does not depend explicitly upon the independent variables (the so-called frozen case) established in part I to approach the general case of explicit dependence upon the independent variables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger and an especially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author’s knowledge, is novel. In this paper we establish the desired Hölder estimate in the large, that is, on the entire mesh $n$-plane. In a subsequent paper a truly interior estimate will be established in a mesh $n$-box.AMS 2000 Mathematics subject classification: Primary 35J60; 35J15; 39A12. Secondary 39A70; 39A10; 65N06; 65N22; 65N12

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33

Shakhno, Stepan, and Halyna Yarmola. "Differential-difference method with approximation of the inverse operator." Physico-mathematical modelling and informational technologies, no. 33 (September 6, 2021): 186–90. http://dx.doi.org/10.15407/fmmit2021.33.186.

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The problem of finding an approximate solution of a nonlinear equation with operator decomposition is considered. For equations of this type, a nonlinear operator can be represented as the sum of two operators – differentiable and nondifferentiable. For numerical solving such an equation, a differential-difference method, which contains the sum of the derivative of the differentiable part and the divided difference of the nondifferentiable part of the nonlinear operator, is proposed. Also, the proposed iterative process does not require finding the inverse operator. Instead of inverting the operator, its one-step approximation is used. The analysis of the local convergence of the method under the Lipschitz condition for the first-order divided differences and the bounded second derivative is carried out and the order of convergence is established.

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34

KONOPELCHENKO, B. G. "ON OPERATOR REPRESENTATION OF INTEGRABLE EQUATIONS." Modern Physics Letters A 03, no. 18 (December 1988): 1807–11. http://dx.doi.org/10.1142/s0217732388002178.

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35

GESZTESY, F., and H. HOLDEN. "TRACE FORMULAS AND CONSERVATION LAWS FOR NONLINEAR EVOLUTION EQUATIONS." Reviews in Mathematical Physics 06, no. 01 (February 1994): 51–95. http://dx.doi.org/10.1142/s0129055x94000055.

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New trace formulas for linear operators associated with Lax pairs or zero-curvature representations of completely integrable nonlinear evolution equations and their relation to (polynomial) conservation laws are established. We particularly study the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the sine–Gordon equation, and the infinite Toda lattice though our methods apply to any element of the AKNS–ZS class. In the KdV context, we especially extend the range of validity of the infinite sequence of conservation laws to certain long-range situations in which the underlying one-dimensional Schrödinger operator has infinitely many (negative) eigenvalues accumulating at zero. We also generalize inequalities on moments of the eigenvalues of Schrödinger operators to this long-range setting. Moreover, our contour integration approach naturally leads to higher-order Levinson-type theorems for Schrödinger operators on the line.

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36

Medina, Rigoberto. "Existence and Boundedness of Solutions for Nonlinear Volterra Difference Equations in Banach Spaces." Abstract and Applied Analysis 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/1319049.

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We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated.

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37

He, Jia Wei, and Yong Zhou. "Stability analysis for discrete time abstract fractional differential equations." Fractional Calculus and Applied Analysis 24, no. 1 (January 29, 2021): 307–23. http://dx.doi.org/10.1515/fca-2021-0013.

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Abstract In this paper, we consider a discrete-time fractional model of abstract form involving the Riemann-Liouville-like difference operator. On account of the C 0-semigroups generated by a closed linear operator A and based on a distinguished class of sequences of operators, we show the existence of stable solutions for the nonlinear Cauchy problem by means of fixed point technique and the compact method. Moreover, we also establish the Ulam-Hyers-Rassias stability of the proposed problem. Two examples are presented to explain the main results.

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38

Gasiński, Leszek, and Nikolaos S. Papageorgiou. "Nonhomogeneous Nonlinear Dirichlet Problems with ap-Superlinear Reaction." Abstract and Applied Analysis 2012 (2012): 1–28. http://dx.doi.org/10.1155/2012/918271.

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We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reactionf(z,ζ), whose primitivef(z,ζ)isp-superlinear near±∞, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like thep-Laplacian, the(p,q)-Laplacian, and thep-generalized mean curvature operator.

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39

Galaktionov, Victor A., Enzo Mitidieri, and Stanislav I. Pohozaev. "Capacity Induced by a Nonlinear Operator and Applications." gmj 15, no. 3 (September 2008): 501–16. http://dx.doi.org/10.1515/gmj.2008.501.

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Abstract We consider the application of the concept of nonlinear capacity induced by nonlinear operators to blow-up problems for various types of nonlinear partial differential equations involving equations with nonlocal nonlinearities.

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40

Kravvaritis, Dimitrios. "Nonlinear random equations with maximal monotone operators in Banach spaces." Mathematical Proceedings of the Cambridge Philosophical Society 98, no. 3 (November 1985): 529–32. http://dx.doi.org/10.1017/s0305004100063726.

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Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such thatRandom equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

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41

Singh, Sukhjit, Eulalia Martínez, Abhimanyu Kumar, and D. K. Gupta. "Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations." Mathematics 8, no. 3 (March 9, 2020): 384. http://dx.doi.org/10.3390/math8030384.

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In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Fréchet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Hölder condition; however, these operators verify the omega condition established. Then, the semilocal convergence balls are obtained and the R-order of convergence and error bounds can be obtained by following thee main theorem. Finally, we perform a numerical experience by solving a nonlinear Hammerstein integral equations in order to show the applicability of the theoretical results by obtaining the existence and uniqueness balls.

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42

Esposito, Giampiero. "A parametrix for quantum gravity?" International Journal of Geometric Methods in Modern Physics 13, no. 05 (April 21, 2016): 1650060. http://dx.doi.org/10.1142/s0219887816500602.

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In the 60s, DeWitt discovered that the advanced and retarded Green functions of the wave operator on metric perturbations in the de Donder gauge make it possible to define classical Poisson brackets on the space of functionals that are invariant under the action of the full diffeomorphism group of spacetime. He therefore tried to exploit this property to define invariant commutators for the quantized gravitational field, but the operator counterpart of such classical Poisson brackets turned out to be a hard task. On the other hand, in the mathematical literature, it is by now clear that, rather than inverting exactly an hyperbolic (or elliptic) operator, it is more convenient to build a quasi-inverse, i.e. an inverse operator up to an operator of lower order which plays the role of regularizing operator. This approximate inverse, the parametrix, which is, strictly, a distribution, makes it possible to solve inhomogeneous hyperbolic (or elliptic) equations. We here suggest that such a construction might be exploited in canonical quantum gravity provided one understands what is the counterpart of classical smoothing operators in the quantization procedure. We begin with the simplest case, i.e. fundamental solution and parametrix for the linear, scalar wave operator; the next step are tensor wave equations, again for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear Einstein equations are studied, relying upon the well-established Choquet-Bruhat construction, according to which the fifth derivatives of solutions of a nonlinear hyperbolic system solve a linear hyperbolic system. The latter is solved by means of Kirchhoff-type formulas, while the former fifth-order equations can be solved by means of well-established parametrix techniques for elliptic operators. But then the metric components that solve the vacuum Einstein equations can be obtained by convolution of such a parametrix with Kirchhoff-type formulas. Some basic functional equations for the parametrix are also obtained, that help in studying classical and quantum version of the Jacobi identity.

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43

Galewski, Marek. "The Existence of Solutions for Nonlinear Operator Equations." gmj 15, no. 1 (March 2008): 45–52. http://dx.doi.org/10.1515/gmj.2008.45.

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Abstract We provide the existence results for a nonlinear operator equation Λ*Φ′ (Λ𝑥) = 𝐹′(𝑥), in case 𝐹 – Φ is not necessarily convex. We introduce the dual variational method which is based on finding global minima of primal and dual action functionals on certain nonlinear subsets of their domains and on investigating relations between the minima obtained. The solution is a limit of a minimizng sequence whose existence and convergence are proved. The application for the non-convex Dirichlet problem with P.D.E. is given.

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44

Zhukovskaya, Zukhra Tagirovna, and Sergey Evgenyevich Zhukovskiy. "ON EQUATIONS GENERATED BY NONLINEAR NILPOTENT MAPPINGS." Tambov University Reports. Series: Natural and Technical Sciences, no. 124 (2018): 637–42. http://dx.doi.org/10.20310/1810-0198-2018-23-124-637-642.

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A generalization of a nilpotent linear operator concept is proposed for nonlinear mapping acting from R^2 to R^2. The properties of nonlinear nilpotent mappings are investigated. Criterions of nilpotence for differentiable and polynomial mappings are obtained.

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Noor, Muhammad Aslam, Themistocles M. Rassias, and Zhenyu Huang. "Three-step iterations for nonlinear accretive operator equations." Journal of Mathematical Analysis and Applications 274, no. 1 (October 2002): 59–68. http://dx.doi.org/10.1016/s0022-247x(02)00224-x.

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Carl, B., and C. Schiebold. "Nonlinear equations in soliton physics and operator ideals." Nonlinearity 12, no. 2 (January 1, 1999): 333–64. http://dx.doi.org/10.1088/0951-7715/12/2/012.

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Tarafdar, E. U., H. B. Thompson, and R. O. Vyborny. "On the solvability of nonlinear noncompact operator equations." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 43, no. 1 (August 1987): 103–26. http://dx.doi.org/10.1017/s1446788700029025.

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Shtaras, A. L. "THE AVERAGING METHOD FOR WEAKLY NONLINEAR OPERATOR EQUATIONS." Mathematics of the USSR-Sbornik 62, no. 1 (February 28, 1989): 223–42. http://dx.doi.org/10.1070/sm1989v062n01abeh003237.

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Ran, A. C. M., M. C. B. Reurings, and L. Rodman. "A Perturbation Analysis for Nonlinear Selfadjoint Operator Equations." SIAM Journal on Matrix Analysis and Applications 28, no. 1 (January 2006): 89–104. http://dx.doi.org/10.1137/05062873.

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Li, Fuyi, Zhanping Liang, Qi Zhang, and Yuhua Li. "On sign-changing solutions for nonlinear operator equations." Journal of Mathematical Analysis and Applications 327, no. 2 (March 2007): 1010–28. http://dx.doi.org/10.1016/j.jmaa.2006.04.064.

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