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Journal articles on the topic 'Weakly nonlinear analysi'

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Published: 14 March 2023

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1

Zheng, Kelong, Wenqiang Feng, and Chunxiang Guo. "Some New Nonlinear Weakly Singular Inequalities and Applications to Volterra-Type Difference Equation." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/912874.

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Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known weakly singular inequalities and can be used in the analysis of nonlinear Volterra-type difference equations with weakly singular kernel. An application to the upper bound of solutions of a nonlinear difference equation is also presented.

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2

Cheng, Kelong, Chunxiang Guo, and Qingke Zeng. "On Weakly Singular Versions of Discrete Nonlinear Inequalities and Applications." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/795456.

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Some new weakly singular versions of discrete nonlinear inequalities are established, which generalize some existing weakly singular inequalities and can be used in the analysis of nonlinear Volterra type difference equations with weakly singular kernels. A few applications to the upper bound and the uniqueness of solutions of nonlinear difference equations are also involved.

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3

HUNTER, JOHN K. "SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES." Journal of Hyperbolic Differential Equations 03, no. 02 (June 2006): 247–67. http://dx.doi.org/10.1142/s0219891606000781.

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We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

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4

Spagnolo, Sergio, and Giovanni Taglialatela. "Analytic Propagation for Nonlinear Weakly Hyperbolic Systems." Communications in Partial Differential Equations 35, no. 12 (November 4, 2010): 2123–63. http://dx.doi.org/10.1080/03605300903440490.

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5

Christianson, Hans, Jeremy Marzuola, Jason Metcalfe, and Michael Taylor. "Nonlinear Bound States on Weakly Homogeneous Spaces." Communications in Partial Differential Equations 39, no. 1 (December 13, 2013): 34–97. http://dx.doi.org/10.1080/03605302.2013.845044.

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6

Rodriguez, Jesús, and Padraic Taylor. "Weakly nonlinear discrete multipoint boundary value problems." Journal of Mathematical Analysis and Applications 329, no. 1 (May 2007): 77–91. http://dx.doi.org/10.1016/j.jmaa.2006.06.024.

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7

Ibrahim, E. A., and S. P. Lin. "Weakly Nonlinear Instability of a Liquid Jet in a Viscous Gas." Journal of Applied Mechanics 59, no. 2S (June 1, 1992): S291—S296. http://dx.doi.org/10.1115/1.2899503.

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The weakly nonlinear instability of a viscous liquid jet emanated into a viscous gas contained in a coaxial vertical circular pipe is investigated as an initial-value problem. The linear stability theory predicted that the jet may become unstable either due to capillary pinching or due to interfacial stress fluctuation. The results of nonlinear stability analysis shows no tendency of supercritical stability for both of the linearly unstable modes. In fact, the nonlinear growth rate of the disturbance is faster than the exponential growth rate of the linear normal mode disturbance for the same flow parameters. Moreover, the most amplified linear normal mode disturbance evolves nonlinearly into a nonsinusoidal wave of shorter wavelength. No nonlinear instability is found for the linearly stable disturbances. Thus, while the linear theory is adequate for the prediction of the onset of jet breakup, nonlinear theory is required to describe the outcome of the jet breakup.

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8

Demenchuk, A. K. "Weakly irregular quasiperiodic solutions of nonlinear Pfaff systems." Differential Equations 44, no. 2 (February 2008): 186–91. http://dx.doi.org/10.1134/s0012266108020055.

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9

Dieter, Sabine. "Nonlinear degenerate curvature flows for weakly convex hypersurfaces." Calculus of Variations 22, no. 2 (February 2005): 229–51. http://dx.doi.org/10.1007/s00526-004-0279-4.

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10

Xu, Run. "Some new nonlinear weakly singular integral inequalities and their applications." Journal of Mathematical Inequalities, no. 4 (2017): 1007–18. http://dx.doi.org/10.7153/jmi-2017-11-76.

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11

FEDDERSEN, FALK. "Weakly nonlinear shear waves." Journal of Fluid Mechanics 372 (October 10, 1998): 71–91. http://dx.doi.org/10.1017/s0022112098002158.

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Alongshore propagating low-frequency O(0.01 Hz) waves related to the direction and intensity of the alongshore current were first observed in the surf zone by Oltman-Shay, Howd & Birkemeier (1989). Based on a linear stability analysis, Bowen & Holman (1989) demonstrated that a shear instability of the alongshore current gives rise to alongshore propagating shear (vorticity) waves. The fully nonlinear dynamics of finite-amplitude shear waves, investigated numerically by Allen, Newberger & Holman (1996), depend on α, the non-dimensional ratio of frictional to nonlinear terms, essentially an inverse Reynolds number. A wide range of shear wave environments are reported as a function of α, from equilibrated waves at larger α to fully turbulent flow at smaller α. When α is above the critical level αc, the system is stable. In this paper, a weakly nonlinear theory, applicable to α just below αc, is developed. The amplitude of the instability is governed by a complex Ginzburg–Landau equation. For the same beach slope and base-state alongshore current used in Allen et al. (1996), an equilibrated shear wave is found analytically. The finite-amplitude behaviour of the analytic shear wave, including a forced second-harmonic correction to the mean alongshore current, and amplitude dispersion, agree well with the numerical results of Allen et al. (1996). Limitations in their numerical model prevent the development of a side-band instability. The stability of the equilibrated shear wave is demonstrated analytically. The analytical results confirm that the Allen et al. (1996) model correctly reproduces many important features of weakly nonlinear shear waves.

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12

PORNPROMMIN, Adichai, Norihiro IZUMI, and Tetsuro TSUJIMOTO. "WEAKLY NONLINEAR ANALYSIS OF MULTIMODAL FLUVIAL BARS." PROCEEDINGS OF HYDRAULIC ENGINEERING 48 (2004): 1009–14. http://dx.doi.org/10.2208/prohe.48.1009.

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13

Ji, Zhen-Gang, and Cesar Mendoza. "Weakly Nonlinear Stability Analysis for Dune Formation." Journal of Hydraulic Engineering 123, no. 11 (November 1997): 979–85. http://dx.doi.org/10.1061/(asce)0733-9429(1997)123:11(979).

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14

Gross, L. K., and V. A. Volpert. "Weakly Nonlinear Stability Analysis of Frontal Polymerization." Studies in Applied Mathematics 110, no. 4 (May 2003): 351–75. http://dx.doi.org/10.1111/1467-9590.00242.

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15

Gérard-varet, D. "Weakly nonlinear analysis of the α effect." Geophysical & Astrophysical Fluid Dynamics 101, no. 3-4 (June 2007): 171–84. http://dx.doi.org/10.1080/03091920701472535.

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16

Ostadzadeh, S. R., M. Salehi, and B. Jafari. "Efficient Analysis of Infinite Arrays of Nonlinear Antennas in the Frequency Domain." Advanced Electromagnetics 11, no. 2 (May 3, 2022): 28–36. http://dx.doi.org/10.7716/aem.v11i2.1844.

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In this paper, different arrangements of infinite arrays of nonlinearly loaded antennas are analyzed in the frequency domain by an efficient approximate method and compared with the exact one with the aim of validity. The approximate method is based on the nonlinear current approach, while the second one is based on the harmonic balance. In one hand, although the harmonic balance technique is suitable for strongly nonlinear load, it is suffering from gradient operation and initial guess in the iteration process especially under multi tone excitations. On the other hand, although the approximate approach is very efficient, it is limited to weakly nonlinear loads and low-valued incident waves. Finally, validity ranges for the approximate method versus different parameters such as nonlinearity effect of the load are extracted.

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17

Taghipour, M., and H. Aminikhah. "Pell Collocation Method for Solving the Nonlinear Time–Fractional Partial Integro–Differential Equation with a Weakly Singular Kernel." Journal of Function Spaces 2022 (May 23, 2022): 1–15. http://dx.doi.org/10.1155/2022/8063888.

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This article focuses on finding the numerical solution of the nonlinear time–fractional partial integro–differential equation. For this purpose, we use the operational matrices based on Pell polynomials to approximate fractional Caputo derivative, nonlinear, and integro–differential terms; and by collocation points, we transform the problem to a system of nonlinear equations. This nonlinear system can be solved by the fsolve command in Matlab. The method’s stability and convergence have been studied. Also included are five numerical examples to demonstrate the veracity of the suggested strategy.

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18

Rigas, Georgios, Aimee S. Morgans, and Jonathan F. Morrison. "Weakly nonlinear modelling of a forced turbulent axisymmetric wake." Journal of Fluid Mechanics 814 (February 9, 2017): 570–91. http://dx.doi.org/10.1017/jfm.2017.32.

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A theory is presented where the weakly nonlinear analysis of laminar globally unstable flows in the presence of external forcing is extended to the turbulent regime. The analysis is demonstrated and validated using experimental results of an axisymmetric bluff-body wake at high Reynolds numbers, $Re_{D}\sim 1.88\times 10^{5}$, where forcing is applied using a zero-net-mass-flux actuator located at the base of the blunt body. In this study we focus on the response of antisymmetric coherent structures with azimuthal wavenumbers $m=\pm 1$ at a frequency $St_{D}=0.2$, responsible for global vortex shedding. We found experimentally that axisymmetric forcing ($m=0$) couples nonlinearly with the global shedding mode when the flow is forced at twice the shedding frequency, resulting in parametric subharmonic resonance through a triadic interaction between forcing and shedding. We derive simple weakly nonlinear models from the phase-averaged Navier–Stokes equations and show that they capture accurately the observed behaviour for this type of forcing. The unknown model coefficients are obtained experimentally by producing harmonic transients. This approach should be applicable in a variety of turbulent flows to describe the response of global modes to forcing.

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19

Goubet, Olivier, and Marilena Poulou. "Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system." Communications on Pure and Applied Analysis 13, no. 4 (February 2014): 1525–39. http://dx.doi.org/10.3934/cpaa.2014.13.1525.

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20

Boichuk, A., J. Diblík, D. Khusainov, and M. Růžičková. "Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems." Abstract and Applied Analysis 2011 (2011): 1–19. http://dx.doi.org/10.1155/2011/631412.

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Conditions are derived of the existence of solutions of nonlinear boundary-value problems for systems ofnordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of argument in nonlinearity:ż(t)=Az(t-τ)+g(t)+εZ(z(hi(t),t,ε), t∈[a,b], assuming that these solutions satisfy the initial and boundary conditionsz(s):=ψ(s) if s∉[a,b], lz(⋅)=α∈Rm. The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to anexplicitandanalyticalform of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functionall) does not coincide with the number of unknowns in the differential system with a single delay.

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21

Goubet, O., and L. Molinet. "Global attractor for weakly damped nonlinear Schrödinger equations in." Nonlinear Analysis: Theory, Methods & Applications 71, no. 1-2 (July 2009): 317–20. http://dx.doi.org/10.1016/j.na.2008.10.078.

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22

Srinivasan, Gopala Krishna, and V. D. Sharma. "On weakly nonlinear waves in media exhibiting mixed nonlinearity." Journal of Mathematical Analysis and Applications 285, no. 2 (September 2003): 629–41. http://dx.doi.org/10.1016/s0022-247x(03)00452-9.

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23

Carles, Rémi, Eric Dumas, and Christof Sparber. "Multiphase Weakly Nonlinear Geometric Optics for Schrödinger Equations." SIAM Journal on Mathematical Analysis 42, no. 1 (January 2010): 489–518. http://dx.doi.org/10.1137/090750871.

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24

Gonçalves, Patrícia, and Milton Jara. "Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems." Archive for Rational Mechanics and Analysis 212, no. 2 (December 10, 2013): 597–644. http://dx.doi.org/10.1007/s00205-013-0693-x.

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25

Zeng, Biao. "Optimal control for elliptic hemivariational inequalities involving nonlinear weakly continuous operators." Journal of Mathematical Inequalities, no. 2 (2021): 575–90. http://dx.doi.org/10.7153/jmi-2021-15-42.

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26

Moreno-Pulido, Soledad, Francisco Javier García-Pacheco, Alberto Sánchez-Alzola, and Alejandro Rincón-Casado. "Convergence Analysis of the Straightforward Expansion Perturbation Method for Weakly Nonlinear Vibrations." Mathematics 9, no. 9 (May 3, 2021): 1036. http://dx.doi.org/10.3390/math9091036.

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There are typically several perturbation methods for approaching the solution of weakly nonlinear vibrations (where the nonlinear terms are “small” compared to the linear ones): the Method of Strained Parameters, the Naive Singular Perturbation Method, the Method of Multiple Scales, the Method of Harmonic Balance and the Method of Averaging. The Straightforward Expansion Perturbation Method (SEPM) applied to weakly nonlinear vibrations does not usually yield to correct solutions. In this manuscript, we provide mathematical proof of the inaccuracy of the SEPM in general cases. Nevertheless, we also provide a sufficient condition for the SEPM to be successfully applied to weakly nonlinear vibrations. This mathematical formalism is written in the syntax of the first-order formal language of Set Theory under the methodology framework provided by the Category Theory.

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27

Ali, Usman, Haifa A. Alyousef, Umar Ishtiaq, Khalil Ahmed, and Shajib Ali. "Solving Nonlinear Fractional Differential Equations for Contractive and Weakly Compatible Mappings in Neutrosophic Metric Spaces." Journal of Function Spaces 2022 (March 21, 2022): 1–19. http://dx.doi.org/10.1155/2022/1491683.

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In this article, we aim to prove various unique fixed point results for contractive and weakly compatible mappings in the sense of neutrosophic metric spaces. Several nontrivial examples are also imparted. To support main result, uniqueness of solution of nonlinear fractional differential equations is examined.

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28

Pedas, Arvet. "POLYNOMIAL SPLINE COLLOCATION METHOD FOR NONLINEAR TWO‐DIMENSIONAL WEAKLY SINGULAR INTEGRAL EQUATIONS." Mathematical Modelling and Analysis 2, no. 1 (December 15, 1997): 122–29. http://dx.doi.org/10.3846/13926292.1997.9637075.

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29

WATANABE, Yasuharu. "WEAKLY NONLINEAR ANALYSIS OF BAR MODE REDUCTION PROCESS." PROCEEDINGS OF HYDRAULIC ENGINEERING 50 (2006): 967–72. http://dx.doi.org/10.2208/prohe.50.967.

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30

Rodrigues, Savio B., and Jayme De Luca. "Weakly nonlinear analysis of short-wave elliptical instability." Physics of Fluids 21, no. 1 (January 2009): 014108. http://dx.doi.org/10.1063/1.3068188.

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31

Wang, L. F., W. H. Ye, Z. F. Fan, Y. J. Li, X. T. He, and M. Y. Yu. "Weakly nonlinear analysis on the Kelvin-Helmholtz instability." EPL (Europhysics Letters) 86, no. 1 (April 2009): 15002. http://dx.doi.org/10.1209/0295-5075/86/15002.

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32

ALEXAKIS, ALEXANDROS, YUAN-NAN YOUNG, and ROBERT ROSNER. "Weakly nonlinear analysis of wind-driven gravity waves." Journal of Fluid Mechanics 503 (March 25, 2004): 171–200. http://dx.doi.org/10.1017/s0022112003007699.

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33

Wang, L. F., W. H. Ye, Z. F. Fan, Y. J. Li, X. T. He, and M. Y. Yu. "Weakly nonlinear analysis on the Kelvin-Helmholtz instability." EPL (Europhysics Letters) 87, no. 6 (September 1, 2009): 69901. http://dx.doi.org/10.1209/0295-5075/87/69901.

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34

Graf, Tobias, Jerome Moloney, and Shankar Venkataramani. "Asymptotic analysis of weakly nonlinear Bessel–Gauß beams." Physica D: Nonlinear Phenomena 243, no. 1 (January 2013): 32–44. http://dx.doi.org/10.1016/j.physd.2012.09.004.

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35

Gupta, Karan, and Paresh Chokshi. "Weakly nonlinear stability analysis of polymer fibre spinning." Journal of Fluid Mechanics 776 (July 8, 2015): 268–89. http://dx.doi.org/10.1017/jfm.2015.284.

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The extensional flow of a polymeric fluid during the fibre spinning process is studied for finite-amplitude stability behaviour. The spinning flow is assumed to be inertialess and isothermal. The nonlinear extensional rheology of the polymer is described with the help of the eXtended Pom-Pom (XXP) model, which is known to exhibit a significant strain hardening effect necessary for fibre spinning applications. The linear stability analysis predicts an instability known as draw resonance when the draw ratio, $\mathit{DR}$, defined as the ratio of the velocities at the two ends of the fibre in the air gap, exceeds a certain critical value, $\mathit{DR}_{c}$. The critical draw ratio $\mathit{DR}_{c}$ depends on the fluid elasticity represented by the Deborah number, $\mathit{De}={\it\lambda}v_{0}/L$, the ratio of the polymer relaxation time to the flow time scale, thus constructing a stability diagram in the $\mathit{DR}_{c}$–$\mathit{De}$ plane. Here, ${\it\lambda}$ is the characteristic relaxation time of the polymer, $v_{0}$ is the extrudate velocity through the die exit and $L$ is the length of the air gap for the spinning flow. In the present study, we carry out a weakly nonlinear stability analysis to examine the dynamics of the disturbance amplitude in the vicinity of the transition point. The analysis reveals the nature of the bifurcation at the transition point and constructs a finite-amplitude manifold providing insight into the draw resonance phenomena. The effect of the fluid elasticity on the nature of the bifurcation and the finite-amplitude branch is examined, and the findings are correlated to the extensional rheological behaviour of the polymer fluid. For flows at small Deborah number, the Landau constant, which captures the role of nonlinearities, is found to be negative, indicating supercritical Hopf bifurcation at the transition point. In the linearly unstable region, the equilibrium amplitude of the disturbance is estimated and shows a limit cycle behaviour. As the fluid elasticity is increased, initially the equilibrium amplitude is found to decrease below its Newtonian value, reaching the lowest value for $\mathit{De}$ when the strain hardening effect is maximum. With further increase in elasticity, the material undergoes strain softening behaviour which leads to an increase in the equilibrium amplitude of the oscillations in the fibre cross-section area, indicating a destabilizing effect of elasticity in this regime. Interestingly, at a certain high Deborah number, the bifurcation crosses over from supercritical to subcritical nature. In the subcritical regime, a threshold amplitude branch is constructed from the amplitude equation.

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36

Chen, Feng. "Asymptotic analysis of weakly nonlinear and forced oscillations." Journal of Central South University of Technology 3, no. 2 (November 1996): 191–95. http://dx.doi.org/10.1007/bf02652203.

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37

Pedas, Arvet, and Gennadi Vainikko. "The Smoothness of Solutions to Nonlinear Weakly Singular Integral Equations." Zeitschrift für Analysis und ihre Anwendungen 13, no. 3 (1994): 463–76. http://dx.doi.org/10.4171/zaa/501.

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38

vom Scheidt, J., S. Mehlhose, and R. Wunderlich. "Distribution Approximations for Nonlinear Functionals of Weakly Correlated Random Processes." Zeitschrift für Analysis und ihre Anwendungen 16, no. 1 (1997): 201–16. http://dx.doi.org/10.4171/zaa/759.

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39

Goubet, Olivier. "Asymptotic Smoothing Effect for a Weakly Damped Nonlinear Schrodinger Equation in T2." Journal of Differential Equations 165, no. 1 (July 2000): 96–122. http://dx.doi.org/10.1006/jdeq.2000.3763.

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40

Orchini, Alessandro, Georgios Rigas, and Matthew P. Juniper. "Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube." Journal of Fluid Mechanics 805 (September 22, 2016): 523–50. http://dx.doi.org/10.1017/jfm.2016.585.

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In this study we present a theoretical weakly nonlinear framework for the prediction of thermoacoustic oscillations close to Hopf bifurcations. We demonstrate the method for a thermoacoustic network that describes the dynamics of an electrically heated Rijke tube. We solve the weakly nonlinear equations order by order, discuss their contribution on the overall dynamics and show how solvability conditions at odd orders give rise to Stuart–Landau equations. These equations, combined together, describe the nonlinear dynamical evolution of the oscillations’ amplitude and their frequency. Because we retain the contribution of several acoustic modes in the thermoacoustic system, the use of adjoint methods is required to derive the Landau coefficients. The analysis is performed up to fifth order and compared with time domain simulations, showing good agreement. The theoretical framework presented here can be used to reduce the cost of investigating oscillations and subcritical phenomena close to Hopf bifurcations in numerical simulations and experiments and can be readily extended to consider, e.g. the weakly nonlinear interaction of two unstable thermoacoustic modes.

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41

Krichen, Bilel, and Donal O'Regan. "On the class of relatively weakly demicompact nonlinear operators." Fixed Point Theory 19, no. 2 (June 1, 2018): 625–30. http://dx.doi.org/10.24193/fpt-ro.2018.2.49.

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42

Benzoni-Gavage, Sylvie, and Jean-François Coulombel. "On the Amplitude Equations for Weakly Nonlinear Surface Waves." Archive for Rational Mechanics and Analysis 205, no. 3 (May 4, 2012): 871–925. http://dx.doi.org/10.1007/s00205-012-0522-7.

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43

de Suzzoni, Anne-Sophie, and Nikolay Tzvetkov. "On the Propagation of Weakly Nonlinear Random Dispersive Waves." Archive for Rational Mechanics and Analysis 212, no. 3 (March 27, 2014): 849–74. http://dx.doi.org/10.1007/s00205-014-0728-y.

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44

CLAVIN, PAUL. "SELF-SUSTAINED MEAN STREAMING MOTION IN DIAMOND PATTERNS OF A GASEOUS DETONATION." International Journal of Bifurcation and Chaos 12, no. 11 (November 2002): 2535–46. http://dx.doi.org/10.1142/s0218127402006060.

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A weakly nonlinear analysis of an overdriven detonation is carried out in the neighborhood of the instability threshold. The result leads to a nonlinear integral-differential equation describing the dynamics of the cellular front and exhibiting "diamond" patterns similar to those observed in experiments. An unexpected outcome of the analysis is a self-sustained mean streaming motion associated with the nonlinear dynamics of a weakly unstable detonation front.

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45

Shi, Xiulian, Yunxia Wei, and Fenglin Huang. "Spectral collocation methods for nonlinear weakly singular Volterra integro-differential equations." Numerical Methods for Partial Differential Equations 35, no. 2 (September 2, 2018): 576–96. http://dx.doi.org/10.1002/num.22314.

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46

Deng, Guo, Christopher J. Lustri, and Mason A. Porter. "Nanoptera in Weakly Nonlinear Woodpile Chains and Diatomic Granular Chains." SIAM Journal on Applied Dynamical Systems 20, no. 4 (January 2021): 2412–49. http://dx.doi.org/10.1137/21m1398410.

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47

Čiegis, Raimondas, and Violeta Pakenienė. "Numerical Approximation of The Weakly Damped Nonlinear Schrödinger Equation." Computational Methods in Applied Mathematics 1, no. 4 (2001): 319–32. http://dx.doi.org/10.2478/cmam-2001-0021.

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AbstractIn this paper we consider the one-dimensional nonlinear Schrödinger equation. The equation includes an absorption term, and the solution is periodically amplified in order to compensate the lose of the energy. The problem describes propa- gation of a signal in optical fibers. In our previous work we proved that the well-known Crank—Nicolson scheme is unconditionally unstable for this problem. We present in this paper two finite difference approximations. The first one is given by a modified Crank—Nicolson scheme and the second one is obtained by a splitting scheme. The stability and convergence of these schemes are proved. The results of numerical exper- iments are presented and discussed.

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48

Shukla, Rahul, and Andrzej Wiśnicki. "Iterative methods for monotone nonexpansive mappings in uniformly convex spaces." Advances in Nonlinear Analysis 10, no. 1 (January 1, 2021): 1061–70. http://dx.doi.org/10.1515/anona-2020-0170.

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Abstract We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1 n ∑ i = 0 n − 1 T i ( x ) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {T n (x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

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49

Goubet, Olivier. "Regularity of the attractor for a weakly damped nonlinear schrödinger equation." Applicable Analysis 60, no. 1-2 (February 1996): 99–119. http://dx.doi.org/10.1080/00036819608840420.

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50

Zhang, Fengrong, Xiangshuai Zhang, and Yan Hao. "Common Fixed Point Theorems for Contractive Mappings of Integral Type in G -Metric Spaces and Applications." Journal of Function Spaces 2021 (January 31, 2021): 1–15. http://dx.doi.org/10.1155/2021/6619964.

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Two common fixed point theorems for weakly compatible mappings satisfying contractive conditions of integral type in G -metric spaces are demonstrated. The results obtained in this paper generalize and differ from a few results in the literature and are used to prove the existence and uniqueness of common bounded and continuous solutions for certain functional equations and nonlinear Volterra integral equations. A nontrivial example is included.

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