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The paper [39] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. The results of [39] include regularity assumptions on the perturbation and involve excluding parameters. The paper [39] also constructs response solutions to a quasi-periodically perturbed (ill-posed evolution) problem. In this paper, we use several classical methods (freezing of coefficients, alternative methods for nonlinear elliptic equations) to extend the results of [39]. We weaken the regularity assumptions on the perturbation and we describe the phenomena that happens for all parameters. In the ill-posed problem, we use a recently developed time-dependent center manifold theorem which allows to reduce the problem to a finite-dimensional ODE with quasi-periodic dependence on time. The bounded and sufficiently small solutions of these ODE give solutions of the ill-posed PDE. [ABSTRACT FROM AUTHOR]

In this short communication the solution of a steady state model of a sextuple evaporator system used in a paper industry has been presented. With the help of the residual values of model equations (a set of 12 nonlinear algebraic equations) it has been shown that the presently obtained results are superior to the previously published results. The importance of the number of significant digits in calculation are also briefly discussed. [ABSTRACT FROM AUTHOR]

Abstract The used methods in our papers [1–13] named new modified and extended auxiliary equation methods. The calculations of our article were correct. All solutions satisfied the partial differential equations in our papers. Highlights • The hydrodynamic mathematical methods. • We present nonlinear Schrodinger equation. • Solitary wave to the generalized nonlinear higher order of dynamical equation. • Fractional generalized reaction duffing equations. [ABSTRACT FROM AUTHOR]

In this note, we show by simple calculation that the new auxiliary equation method (Khater method) used in the above two papers is absolutely wrong. Consequently, all the exact solutions of the proposed nonlinear equations obtained in the above two papers are not true. [ABSTRACT FROM AUTHOR]

This paper purports to study the optical solitons solutions regarding the nonlinear schr¨odinger equation with dual power law of nonlinearity along with self steeping and repercussion. The optical solutions mostly comprise of singular solitons with nonlinear chirping. The adopted solitary wave ansatz and the importance of the obtained chirped soliton pulses are also discussed. [ABSTRACT FROM AUTHOR]

Abstract This comment concerns the doubtful new auxiliary equation method (Khater method) included in the above paper. We prove by simple calculation that this method is wrong. The exact solutions of the proposed nonlinear equation obtained in the above paper are all wrong. Highlights • The new auxiliary equation method is called Khater method. • We prove by simple calculation that Khater method is wrong. • We show that the exact solutions obtained in the above paper are all wrong. [ABSTRACT FROM AUTHOR]

In this note we correct and improve the proofs of the main lemmas in a very recent publication (Zeng, 2022), in which the hypotheses of boundedness, coerciveness and relaxed monotonicity for the operators are considered. [ABSTRACT FROM AUTHOR]

In this study, simultaneous brightness contrast was investigated using three different devices: 2D (paper), 3D (space), and D‐up. The paper stimuli were made by printing papers, the space stimuli were produced by illumination in a two‐rooms experimental booth, and the D‐up stimuli were shown to subjects using a D‐up viewer. The brightness of test patches was judged by the amounts of whiteness and blackness. The results were different among the devices; however, if the whiteness of the test patches was plotted for the luminance contrast of the test patch versus the surround, all the data points were expressed by nonlinear equations. Data from the space stimuli were expressed by only one equation; those from the paper stimuli were also expressed by the nonlinear equation but with different coefficients for each test patch. The results from D‐up were between those of the space and paper experiments. All the results were explained in relation to the hypothesized illuminated space over the surrounds. The difference among devices was explained by the whiteness of the surround being transferred to the illumination. [ABSTRACT FROM AUTHOR]

The comment concentrate on the solutions of mapping method that included in the above two papers. We prove by simple calculation and also by using computer software (Maple program V. 16) that this technique for solving nonlinear partial differential equation (PDE) is wrong. [ABSTRACT FROM AUTHOR]

The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB®2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB®2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the validity of the result of Khuangsatung and Kangtunyakarn [Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions, Filomat32(1) (2018) 305–309]. [ABSTRACT FROM AUTHOR]

Summary: This paper investigates the finite‐time robust simultaneous stabilization problem of a set of nonlinear time‐delay systems with general forms and proposes some new simultaneous stabilization results. First, by developing an equivalent form and applying augmented technique, this paper obtains an augmented equivalent form of the original systems. Secondly, based on the equivalent form, we study finite‐time simultaneous stabilization problem and present some new stabilization results by constructing some suitable Lyapunov functionals. Thirdly, using the simultaneous stabilization results obtained, this paper investigates the finite‐time robust simultaneous stabilization problem for the set systems and proposes a delay‐dependent robust simultaneous stabilization result. Finally, the study of an illustrative example shows that the results obtained by this paper work well in the finite‐time robust simultaneous stabilization the set systems. It is shown that, by using the method in this paper, the developed conditions do not contain delay terms, which can avoid solving nonlinear mixed matrix inequalities and reduce effectively computational burden in studying nonlinear time‐delay systems. [ABSTRACT FROM AUTHOR]

This paper focusses on the nonlinear fractional Kadomtsev-Petviashvili (FKP) equation in space-time, employing the conformable fractional derivative (CFD) approach. The main objective of this paper is to examine the application of the (G'/G)-expansion method in order to find analytical solutions to the FKP equation. The (G'/G)-expansion method is a powerful tool for constructing traveling wave solutions of nonlinear evolution equations. However, its application to the FKP equation remains relatively unexplored. By employing traveling wave transformation, the FKP equation was transformed into an ordinary differential equation (ODE) to acquire exact wave solutions. A range of exact analytical solutions for the FKP equation is obtained. Graphical illustrations are included to elucidate the physical characteristics of the acquired solutions. To demonstrate the impact of the fractional operator on results, the acquired solutions are exhibited for different values of the fractional order α, with a comparison to their corresponding exact solutions when taking the conventional scenario where α equals 1. The results indicate that the (G'/G)-expansion method serves as an efficient method and dependable in solving the nonlinear FKP equation. [ABSTRACT FROM AUTHOR]

This paper considers theoretical foundations and mathematical methods of data analysis under the conditions of the Rice statistical distribution. The problem involves joint estimation of the signal and noise parameters. It is shown that this estimation requires the solution of a complex system of essentially nonlinear equations with two unknown variables, which implies significant computational costs. This study is aimed at mathematical optimization of computer algebra methods for numerical solution of the problem of Rician data analysis. As a result of the optimization, the solution of the system of two nonlinear equations is reduced to the solution of one equation with one unknown variable, which significantly simplifies algorithms for the numerical solution of the problem, reduces the amount of necessary computational resources, and enables the use of advanced methods for parameter estimation in information systems with priority of real-time operation. Results of numerical experiments carried out using Wolfram Mathematica confirm the effectiveness of the developed methods for two-parameter analysis of Rician data. The data analysis methods considered in this paper are useful for solving many scientific and applied problems that involve analysis of data described by the Rice statistical model. [ABSTRACT FROM AUTHOR]

Tensile testing was used to measure the mechanical properties of paper at various orientations relative to the fiber alignment of paper. New, artificially aged, and naturally aged paper were shown to behave as orthotropic materials, as shown by the orientation dependence of the Young's modulus, E. The Young's modulus data could be fit to a simplified equation, indicating the shear modulus is relatively insensitive to the specific orientation. The shear modulus as measured by ultrasonic measurements was similar to that measured by fitting the modulus vs orientation data obtained by tensile testing to the orthotropic equation. The inelastic tensile strength could also be described by a simple non-linear equation, and there was more deviation of the data from this prediction for brittle papers than for non-brittle papers. The extent of fiber orientation (E x /E y) could be inferred by either the Young's modulus or the tensile strength, TS, i.e. E x /E y ≈ TS x /TS y. The measured strains were lowest for brittle paper. All brittle papers gave a strain to failure <1% whereas the non-brittle papers gave a strain to failure >1%. • Paper was shown to behave as an orthotropic material for new, artificially aged, and naturally aged papers. • The Young's modulus versus fiber orientation data could be fit to a simplified equation. • The shear modulus is relatively insensitive to the fiber orientation. • The Ramberg-Osgood equation can be used to describe the stress-strain measurements. • All brittle papers gave a strain to failure <1% whereas the non-brittle papers gave a strain to failure >1%. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models. [ABSTRACT FROM AUTHOR]

In this comment, we show by simple calculation that the new auxiliary equation method (Khater method) used in the above paper is absolutely wrong. Consequently, all the exact solutions of the proposed nonlinear equation obtained in the above paper are also wrong. [ABSTRACT FROM AUTHOR]

A prediction of rod wave type with great precision is extremely important in theoretical analysis and practical applications. Besides the well-known periodic motion and resonance, this paper studies the rod wave in a fractal space, and a fractal solitary wave is unlocked by the variational approach, the results reveal that the rod strain non-linearity and fractal dimensions affect greatly the wave travelling properties. This paper offers a new window for identifying a solitary wave from periodic motion easily and accurately. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for the nonlinear Schrodinger system {idtu1(x,t)== ∆u1(x, t)− |u1(x, t)| p−1u1(x, t)− |u2(x, t)| p−1u1(x, t), idtu2(x, t) = ∆u2(x, t) − |u2(x, t)| p−1u2(x, t) − |u1(x, t)| p−1u2(x, t), was investigated in d space dimensions. For 1 < p ≤ 1 + 2/d, the nonexistence of asymptotically free solutions for the nonlinear Schrödinger system was proved based on mathematical analysis and scattering theory methods. The novelty of this paper was to give the proof of pseudo-conformal identity on the nonlinear Schrödinger system. The present results improved and complemented these of Bisognin, Sepúlveda, and Vera(Appl. Numer. Math. 59(9)(2009): 2285–2302), in which they only proved the nonexistence of asymptotically free solutions when d = 1, p = 3. [ABSTRACT FROM AUTHOR]

Given the increasing severity of energy shortages, the exploration of effective strategies to optimize energy structures has become imperative. This requires careful consideration of energy consumption patterns, especially since these data are fundamental inputs for policy formulation. Given the uncertainty in the rate of change in energy consumption, this paper proposes an optimized multivariable grey Bernoulli model that is rooted in the grey Bernoulli model and incorporates background values and genetic algorithms. The grey Bernoulli model effectively linearizes nonlinear problems, thus simplifying computational procedures. In addition, to account for random fluctuations of relevant factors that may affect the model's predictions, this model introduces nonlinear correction terms that allow simulation and prediction values to adhere to the grey index law. The incorporation of background values enhances the model's ability to process information, providing it with a superior grasp of real data. Genetic algorithms can be used to refine the model's parameters, increasing its adaptability and precision. Finally, this paper applies the refined model to examine China's energy consumption patterns, validating its efficacy and versatility. Furthermore, energy consumption patterns for the next four years are forecast, with the analysis revealing that the growth rate of energy consumption from 2021 to 2024 shows a downward trend, particularly notable in 2024, where the growth rate is 1.64%. [ABSTRACT FROM AUTHOR]

Orthogonal maps are two-dimensional mappings that are solutions of the so-called origami problem obtained when folding a paper. These mappings are piecewise linear, and the discontinuities of their gradient form a singular set composed of straight lines representing the folding edges. The proposed algorithm relies on the minimization of a variational principle discussed in Caboussat et al. (2019). A splitting algorithm for the corresponding flow problem derived from the first-order optimality conditions alternates between local nonlinear problems and linear elliptic variational problems at each time step. Anisotropic adaptive techniques allow to obtain refined triangulations near the folding edges while keeping the number of vertices as low as possible. Numerical experiments validate the accuracy and efficiency of the adaptive method in various situations. Appropriate convergence properties are exhibited, and solutions with sharp edges are recovered. [ABSTRACT FROM AUTHOR]

There is no correct form of the term HT ht in the literature. In Physics it is not allowed to add quantities with different units and the term HT ht is wrong. [Extracted from the article]

Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation κ (u) = v , where κ : D (κ) ⊆ X ⟶ X is a nonlinear monotone operator and X is a Hilbert space. The operator in the example used in Tautenhahn's paper was not a monotone operator. So, the following question arises. Can we use LRM for ill-posed nonlinear equations when the involved operator is not monotone? This paper provides a sufficient condition to employ the Lavrentiev regularization technique to such equations whenever the operator involved is non-monotone. Under certain assumptions, the error analysis and adaptive parameter choice strategy for the method are discussed. Moreover, the developed theory is applied to two well-known ill-posed problems—inverse gravimetry and growth law problems. [ABSTRACT FROM AUTHOR]

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems. [ABSTRACT FROM AUTHOR]

The coherence of axial force between steel struts in excavation (axial force coherence) is an important factor affecting the axial force control of servo struts. To understand the law of axial force loss of adjacent struts caused by servo struts loading, this paper proposes a calculation method based on the theory of nonlimiting earth pressure. First, the diaphragm wall is simplified to a simply supported beam with both endpoints free horizontally, the soil on both sides of the diaphragm wall is divided into n small sections (n as big as possible) along the depth direction, and the average nonlimiting earth pressure combined force within each section is calculated. Second, a system of nonlinear force-displacement equations is constructed by applying the "graph multiplication" method to calculate the combined nonlimiting soil pressure and deflection of the wall from the axial force of the servo steel strut at each level. The Newton–Raphson method is applied to obtain a recursive equation for the wall displacements. The accuracy of the method is verified by comparison with field measurements. Then, based on the method in this paper, the effect of prestressing and adjustment on the axial variation of each strut course is investigated. The method can provide a reference for the loading scheme of servo steel struts in deep excavation in soft-soil areas. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence result of solutions for fuzzy nonlinear fractional differential equations involving Caputo differentiability of an arbitrary order 0 < q < 1. As application, an example is included to show the applicability of our result. [ABSTRACT FROM AUTHOR]

The classical linear discriminant analysis (LDA) algorithm has three primary drawbacks, i.e., small sample size problem, sensitivity to noise and outliers, and inability to deal with multi-modal-class data. This paper reviews LDA technology and its variants, covering the taxonomy and characteristics of these technologies and comparing their innovations and developments in addressing these three shortcomings. Additionally, we describe the application areas and emphasize the kernel extensions of these technologies to solve nonlinear problems. Most importantly, this paper presents perspectives on future research directions and potential research areas in this field. [ABSTRACT FROM AUTHOR]

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-W8 flat Lorentzian Kahler space-time manifolds. [ABSTRACT FROM AUTHOR]

This paper is devoted to some inverse problems of recovering the nonlinearity for the Jordan–Moore–Gibson–Thompson equation, which is a third order nonlinear acoustic equation. This equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained with the small initial and boundary data. By the second order linearization to the nonlinear equation, and construction of complex geometric optics solutions for the linearized equation, the uniqueness of recovering the nonlinearity is derived. [ABSTRACT FROM AUTHOR]

In this paper, a new family of iterative methods for the simultane-ous approximation of simple complex polynomial zeros is presented. The proposed family of simultaneous methods is constructed on the basis of the well-known third order Ehrlich iteration, combined with an iterative correction from the sixth order Neta's method for nonlinear equations. It is proved that the use of this iterative correction allows to increase the convergence order of the basic method from three to eight. Numerical examples are given to illustrate the convergence and effectiveness of the proposed combined method. [ABSTRACT FROM AUTHOR]

The development of machine tools in the last twenty years includes, among other things, the application of mechanisms with a non-linear kinematic structure as the mechanical basis of machines. This results in significant improvements in kinematic characteristics and problems related to non-linear dependencies of the accuracy of the drive elements and the realization of movement in the machine's external coordinates. The paper presents an approach to machine tool calibration based on the original O-X glide mechanism based on the ISO 230-4 standard with the mono- and bi-directional compensation of systematic errors and adaptation to the specifics of the mechanism's kinematics. A machine tool prototype was designed and built for the research presented in the paper. The obtained results indicate the possibility of applying the existing recommendations and standards for testing the accuracy of machine tools with the need to correct the methodology by using linear and non-linear kinematic structures in machine tools. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several Liouville-type theorems related to k-Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S_k[D^2u]\ge h(u,|Du|) in \mathbb {R}^n. The results depend on some qualified growth conditions of h at infinity. A Liouville-type result to subsolutions of a prototype equation S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S_k[D^2u]=f(u)+g(u)|Du|^q for 0\le q

In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper. [ABSTRACT FROM AUTHOR]

This is the second paper in a series of papers addressing the characteristic gluing problem for the Einstein vacuum equations. We solve the codimension-10 characteristic gluing problem for characteristic data which are close to the Minkowski data. We derive an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges that are conserved by the linearized null constraint equations and act as obstructions to the gluing problem. The gauge-dependent charges can be matched by applying angular and transversal gauge transformations of the characteristic data. By making use of a special hierarchy of radial weights of the null constraint equations, we construct the null lapse function and the conformal geometry of the characteristic hypersurface, and we show that the aforementioned charges are in fact the only obstructions to the gluing problem. Modulo the gauge-invariant charges, the resulting solution of the null constraint equations is C m + 2 for any specified integer m ≥ 0 in the tangential directions and C 2 in the transversal directions to the characteristic hypersurface. We also show that higher-order (in all directions) gluing is possible along bifurcated characteristic hypersurfaces (modulo the gauge-invariant charges). [ABSTRACT FROM AUTHOR]

The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors' previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems. [ABSTRACT FROM AUTHOR]

Large deployable space mechanisms are widely used in the field of aerospace and have been paid increasingly high attention recently. The satellite antenna expansion system is the classic large deployable space mechanism. However, during the expansion of the satellite antenna deployable mechanism, the expansion accuracy is affected by the existing various uncertain factors which even result in scraping the satellite. For example, the hinge locking error has the significant influence on the deployment accuracy of the satellite antenna panel. In term of this issue, considering the advantage of the back‐propagation (BP) neural network for the high dimensional nonlinear problem, this paper mainly adopts it to analyze the impact of hinge locking error on the expansion accuracy of the antenna panel. The results show that the probability of the actual flatness deviation of the satellite antenna panel falling in the required accuracy area is 99.56%, and the probability of the actual pointing angle deviation falling in the required accuracy area is 99.84%. [ABSTRACT FROM AUTHOR]

To address the challenge of accurately locating unmanned aerial vehicles (UAVs) in situations where radar tracking is not feasible and visual observation is difficult, this paper proposes an innovative acoustic source localization method based on improved Empirical Mode Decomposition (EMD) within an adaptive frequency window. In this study, the collected flight signals of UAVs undergo smoothing filtering. Additionally, Robust Empirical Mode Decomposition (REMD) is applied to decompose the signals into Intrinsic Mode Function (IMF) components for spectrum analysis. We introduce a sliding frequency window with adjustable bandwidth, which is automatically determined using a Grey Wolf Optimizer (GWO) with a sliding index. This window is used to lock and extract specific frequencies from the IMFs. Based on predefined criteria, the extracted IMF components are reconstructed, and trigger signal times are analyzed and recorded from these reconstructed IMFs. The time differences between sensor receptions are then calculated. Furthermore, this study introduces the Chan–Taylor localization algorithm based on weighted least squares. This advanced algorithm takes sensor time delay parameters as input and solves a set of nonlinear equations to determine the target's location. Simulations and real-world signal tests are used to validate the robustness and performance of the proposed method. The results indicate that the localization error remains below 5% within a 15 m × 15 m measurement area. This provides an efficient and real-time method for detecting the location of small UAVs. [ABSTRACT FROM AUTHOR]

This paper investigates the fixed-time adaptive event-triggered control problem for a class of nonlinear multiagent systems with full-state constraints. The problem of full-state constraints is solved for multiagent systems by utilising a one-to-one nonlinear mapping. The design conditions of the controllers are more easily satisfied than the existing barrier Lyapunov functions strategy. A fixed-time adaptive control scheme is proposed for unconstrained systems that are converted from constrained systems. In contract with the existing finite-time control schemes that rely on the initial states, this restriction condition can be removed in this paper. An improved event-triggered mechanism with an adaptable switching threshold is designed to provide flexible switching between fixed threshold and variable threshold policies, resulting in reduced execution and sampling. It is proved via the Lyapunov stability method that all signals of the closed-loop systems are semi-global practical fixed-time stable. Finally, the effectiveness of the proposed control strategy is verified by some simulation results. [ABSTRACT FROM AUTHOR]

This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When β>3, with initial data (uτ,ωτ)∈V1×V2 and external forces (f1,f2)H(f01× H(f02), some uniform estimates of the solution in different function spaces are given. Based on these uniform estimates, the ((V1×V2)×(H(f01)×H(f02)),V1×V2)-continuity of the family of processes {U(f1,f2)(t,τ)}t≥τ is demonstrated. Meanwhile, the (V1×V2,H²(Ω)×H²(Ω))-uniform compactness of {U(f1,f2)(t,τ)}t≥τ is proved. Finally, the existence of a (V1×V2,V1×V2)-uniform attractor and a (V1× V2,H²(Ω)×H²(Ω))-uniform attractor are obtained. Furthermore, the (V1×V2,V1×V2)-uniform attractor and the (V1×V2,H²(Ω)×H²(Ω))-uniform attractor are verified to be the same. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (− Δ) s u + μ u = ( I α * F (u)) F ′ (u) in R N , (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, "Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities," Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, "Nonlinear scalar field equations II: existence of infinitely many solutions," Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C1-regularity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the general dual fractional parabolic problem ∂ t α u (x , t) + L u (x , t) = f (t , u (x , t)) in R n × R. We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons' conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L and the Marchaud time derivative ∂ t α , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons' conjecture. It is worth noting that our results are novel even for a special case of L , the fractional Laplacian (−Δ)s, and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators. [ABSTRACT FROM AUTHOR]

Purpose: This study presents a two-step numerical iteration method specifically designed to solve absolute value equations. The proposed method is valuable and efficient for solving absolute value equations. Several numerical examples were taken to demonstrate the accuracy and efficiency of the proposed method. Design/methodology/approach: We present a two-step numerical iteration method for solving absolute value equations. Our two-step method consists of a predictor-corrector technique. The new method uses the generalized Newton method as the predictor step. The four-point open Newton-Cotes formula is considered the corrector step. The convergence of the proposed method is discussed in detail. This new method is highly effective for solving large systems due to its simplicity and effectiveness. We consider the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solve it using the proposed method. Findings: The paper provides empirical insights into how to solve a system of absolute value equations. Originality/value: This paper fulfills an identified need to study absolute value equations. [ABSTRACT FROM AUTHOR]

Consider an infinite-dimentional Banach space denoted as X, and designate ∂(X) as the algebra of all bounded linear operators on X. Moreover, let s (A) denote the spectrum of ..., and ...(A)) indicate the boundary of s (A). A map ... is termed a ρ-spectrum if ... (A) for all A ... In this paper, we characterize all surjective maps f1 and f2 on ρ(X) satisfying ... = (AB) for all A,B(X). [ABSTRACT FROM AUTHOR]

In this paper, the semi-local convergence of the Cordero's sixth-order iterative method in Banach space was proved by the method of recursion relation. In the process of proving, the auxiliary sequence and three increasing scalar functions can be derived using Lipschitz conditions on the first-order derivatives. By using the properties of auxiliary sequence and scalar function, it was proved that the iterative sequence obtained by the iterative method was a Cauchy sequence, then the convergence radius was obtained and its uniqueness was proven. Compared with Cordero's process of proving convergence, this paper does not need to ensure that G (s) is continuously differentiable in higher order, and only the first-order Fréchet derivative was used to prove semi-local convergence. Finally, the numerical results showed that the recursion relationship is reasonable. [ABSTRACT FROM AUTHOR]

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order p of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order p + 3 (for p ≥ 3 ) and have extended it to a generalized (m + 2) -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to p + 3 m , m ∈ N . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order (for ) and have extended it to a generalized -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to , . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. [ABSTRACT FROM AUTHOR]

This paper presents a new probabilistic coordinated search technique for finding a randomly located target in n-disjoint known regions using n-searchers. Each region contains one searcher. The searchers use advanced technology to communicate with each other. This paper aims to obtain the candidate utility function, namely the expected value of the time for detecting the target, and minimize this expected value considering a restricted amount of time. We present a particular case when the target has a multinomial distribution, which is essential for searching for a valuable target that is missing at sea or lost in a wilderness area. [ABSTRACT FROM AUTHOR]