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In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form$ \begin{equation*} \partial_t u + \partial_x(\Lambda^s u + u\Lambda^r u^2) = 0, \end{equation*} $where $ \Lambda^s, \Lambda^r $ are Bessel-type Fourier multipliers. The linear operator may be of low fractional order, $ s>0 $, while the operator on the nonlinear part is assumed to act slightly smoother, $ r

This paper mainly deals with the monotonicity and Liouville type theorems of nonnegative solutions for a class of fully nonlinear fractional equations in half-space. Compared with serial works in [9,14,20], the restrictions on the monotonicity of nonlinear terms and the decay assumption on solutions are not necessary here. Moreover, the monotonicity result of nonnegative bounded solutions for corresponding cooperative systems is also derived. Specifically, we establish some new types of maximum principles for fully nonlinear fractional equations and systems in unbounded domains, which will play crucial roles in carrying on the method of moving planes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 풟 ⁢ ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 풟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation Au = b. We prove that if A is a quasi-uniformly monotone and hemi-continuous operator, then A−1 is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic p-curl systems. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a new concept of Z-Z-B spaces, which is a generalization of Z-C-X spaces. Meanwhile, the new concept of the superior cone is introduced. Secondly, we study some new problems for semi-closed 1-set-contractive operators in the Z-Z-B space and obtain some new results. These new theorems are proven by combining partial order theory with fixed point index theory. Regarding these theorems, in the latter part of the paper, the proofs are omitted since the methods of proving these theorems are similar. Moreover, two important inequality lemmas are proven. [ABSTRACT FROM AUTHOR]

In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source. [ABSTRACT FROM AUTHOR]

In this paper, we prove Liouville type theorem of positive weak solution of nonlinear system for Grushin operator. We give some integral inequalities, which combine the method of moving plane with the integral inequality to get the result for nonlinear system. [ABSTRACT FROM AUTHOR]

In this paper we investigate homogenization results for the principal eigenvalue problem associated to 1-homogeneous, uniformly elliptic, second-order operators. Under rather general assumptions, we prove that the principal eigenpair associated to an oscillatory operator converges to the eigenpair associated to the effective one. This includes the case of fully nonlinear operators. Rates of convergence for the eigenvalues are provided for linear and nonlinear problems, under extra regularity/convexity assumptions. Finally, a linear rate of convergence (in terms of the oscillation parameter) of suitably normalized eigenfunctions is obtained for linear problems. [ABSTRACT FROM AUTHOR]

In this paper we characterize the nonlinear superposition operators mapping the space \mathcal {Q}_s into the Dirichlet-type space \mathcal {D}^p_\alpha and answer basically the remaining open cases provided by Galanopoulos, Girela, and Auxiliadora Márquez [J. Math. Anal. Appl. 463 (2018), pp. 659–680]. Meanwhile, we give some sufficient-necessary conditions for the superposition operators between \mathcal {Q}_s spaces in all cases. [ABSTRACT FROM AUTHOR]

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an α - θ - Geraghty–Pata–Suzuki-type proximal contraction; an α - θ - generalized Geraghty–Pata–Suzuki-type proximal contraction; and an α - θ - modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches. [ABSTRACT FROM AUTHOR]

The prior-based blind image deblurring methods have recently achieved good performance. However, many state-of-art algorithms are time-consuming since some nonlinear operators are involved. Presented in this paper is a fast blind image deblurring algorithm which uses the salience map and gradient cepstrum. The inspiration for this work comes from the fact that the extreme values of the salience map of the clear image are more sparse than those of the blurred one. By enforcing the L 0 norm constraint to the terms involving salience map and incorporating them into the traditional deblurring framework, an effective optimization scheme is explored. Furthermore, gradient cepstrum is used to adjust the number of iterations in each scale and determine the size of the initial kernel. Experimental results illustrate that our algorithm outperforms the state-of-art deblurring algorithms in both benchmark datasets and real blur scenes. Besides, this algorithm greatly shortens the running time since it restrains excessive iterations and does not involve any nonlinear operators. [ABSTRACT FROM AUTHOR]

In this paper, we study an integral boundary value problem involving a Hadamard-type fractional differential equation. Using fixed point theory and upper-lower solutions, we present some sufficient conditions to obtain existence theorems of positive solutions for the problem. Examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper deals with a discrete-time dynamical system generated by a modified susceptible–infected–recovered–dead model (SIRD model; nonlinear operator) in three-dimensional simplex. We introduce a novel approach that incorporates the SIRD model with the quadratic stochastic operator (QSO) that allows for real-time forecasting. The basic reproductive number R 0 is obtained. We describe the set of fixed points of the operator and demonstrate that all fixed points are non-hyperbolic. Further, we study the asymptotical behavior of the trajectories of this system and show that SIRD operators have a regularity property. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new coincidence and common fixed point theorem of hybrid mappings using the C-class function and T-weak commutativity. Finally, we give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

In this paper, we consider the properties of solutions for the following fractional parabolic equation \begin{document}$ \begin{equation*} \frac{\partial u}{\partial t}(x, t)+(-\Delta)^su(x, t) = f(u(x, t)), \ (x, t)\in\mathbb{R}^N\times\mathbb{R}. \end{equation*} $\end{document} Without assuming any decay behavior of $ u $ near infinity, we first establish a narrow region principle in unbounded domains, then employing the direct method of moving planes, we derive the monotonicity, antisymmetry and non-existence of the solutions. In most previous articles, to carry out the direct method of moving planes in the whole space, they often needed to assume that the solution tends to zero near infinity or that the solution is divided by a function to make the new solution satisfy such an asymptotic decay. Here we develop a new approach–estimating the singular integrals defining $ (-\Delta)^s $ and the derivative of the solution with respect to time along a sequence of approximate extreme points.We believe that the new method employed here will be very helpful to study a class of parabolic equations involving nonlinear nonlocal operators. [ABSTRACT FROM AUTHOR]

Recent advances in modeling large-scale, complex physical systems have shifted research focuses towards data-driven techniques. However, generating datasets by simulating complex systems can require significant computational resources. Similarly, acquiring experimental datasets can prove difficult. For these systems, often computationally inexpensive, but in general inaccurate models, known as the low-fidelity models, are available. In this paper, we propose a bi-fidelity modeling approach for complex physical systems, where we model the discrepancy between the true system's response and a low-fidelity response in the presence of a small training dataset from the true system's response using a deep operator network, a neural network architecture suitable for approximating nonlinear operators. We apply the approach to systems that have parametric uncertainty and are partially unknown. Three numerical examples are used to show the efficacy of the proposed approach to model uncertain and partially unknown physical systems. [ABSTRACT FROM AUTHOR]

In a quantum system, equilibrium points are usually defined by the equation of evolution. The analysis of this process is often done by Schrödinger equation, and by linear operators on a Hilbert space. Regardless of the fact that calculations are based on static-point metrics, due to the chaotic behavior, the realization of practical conditions on space and mappings will be relatively difficult. In addition, access to constructive arguments will enable us to provide a computational method. In this paper, at first we define a probabilistic measurement and combine it with probabilistic domains to obtain a probabilistic model suitable for quantum systems. By extending the mappings to nonlinear operators, we examine the conditions under which stable equilibrium points can be reached. In the application section, using the evolution equation, we will determine the stationary points based on their spectral properties. Also there will be possible to generalize this method to simultaneous measurements. [ABSTRACT FROM AUTHOR]

This paper is concerned with the relaxation dynamics of a large population of identical phase oscillators, each of which interacts with all the others through random couplings whose parameters obey the same Gaussian distribution with the average equal to zero and are mutually independent. The results obtained by numerical simulation suggest that for the infinite-size system, the absolute value of Kuramoto's order parameter exhibits superslow relaxation, i.e., 1/ln t as time t increases. Moreover, the statistics on both the transient time T for the system to reach a fixed point and the absolute value of Kuramoto's order parameter at t = T are also presented together with their distribution densities over many realizations of the coupling parameters. [ABSTRACT FROM AUTHOR]

We show two more approaches for generating trajectory-based dynamics in the phase space formulation of quantum mechanics: 'equilibrium continuity dynamics' (ECD) in the spirit of the phase space continuity equation in classical mechanics, and 'equilibrium Hamiltonian dynamics' (EHD) in the spirit of the Hamilton equations of motion in classical mechanics. Both ECD and EHD can recover exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Both ECD and EHD conserve the quasi-probability within the infinitesimal volume dxtdpt around the phase point (xt, pt) along the trajectory. Numerical tests of both approaches in the Wigner phase space have been made for two strongly anharmonic model problems and a double well system, for each potential auto-correlation functions of both linear and nonlinear operators have been calculated. The results suggest EHD and ECD are two additional potential useful approaches for describing quantum effects for complex systems in condense phase. [ABSTRACT FROM AUTHOR]

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase. [ABSTRACT FROM AUTHOR]

Mahale and Nair (2009) considered a iterated Lavrentive regularization method for obtaining stable approximate solution of nonlinear ill-posed operator equation F(x) = y, where F : D (F) ⊂ X → X is a nonlinear monotone operator on Hilbert space X. In this paper, we do convergence analysis of the method under the weaker non-linearity condition than the one considered by Mahale and Nair. We obtain convergence and rate of convergence results for the method merely under Lipschitz's condition. At the end of the paper, we give a numerical example to demonstrate our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation (WSVIE), u(x) = f(x)+∫0x t μ-1/xµ [u(t)]βdt. Al-Jawary and Shehan used the Daftardar-Jafari Method (DJM) and solved the above integral equation for the investigation parameter µ > 1 using specific force functions with µ and β values and obtained unique solutions. We have discovered a force function f(x) = x k1-x γk1/γk1+µ that allows the introduction of noise terms phenomena discovered by Wazwaz that cancel out the terms of the power series in the successive solution terms um, m = 0, 1, 2, ..., n: we thus obtain a maximum finite power series terms for each solution term called truncation point and denoted by x g(n) . Such that the integral solution can be written as u(x) = u0+∑ m n=1 um, where n is finite. Simplifying the solution terms, we get the unique solution u(x) = x k1, irrespective of the n−value in the truncation point. We have discovered a formula relation between the last solution term un and the truncation point as un = anx g(n) . Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter µ > 1. We extend the parameter range to include µ > 1 and 0 < µ ≤ 1 for our solution. In addition, for any chosen rational parameter k1, the solution u(x) = x k1 is extrapolated to be valid for all integer parameter values β ≥ 2 and positive rational parameter values µ > 0 and for any finite value of n ≥ 2. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide sufficient conditions implying that the effective domain $ D(A\phi) $ of an $ m $-accretive operator $ A\phi $ in $ L^1 $ is dense in $ L^1 $. Here, $ A\phi $ refers to the composition $ A\circ \phi $ in $ L^1 $ of the part $ A = (\partial\mathcal{E})_{\vert L^{1\cap \infty}} $ in $ L^{1\cap\infty} $ of the subgradient $ \partial\mathcal{E} $ in $ L^2 $ of a convex, proper, lower semicontinuous functional $ \mathcal{E} $ on $ L^2 $ and a continuous, strictly increasing function $ \phi $ on the real line $ \mathbb{R} $. To illustrate the role of the sufficient conditions, we apply our main result to the class of doubly nonlinear operators $ A\phi $, where $ A $ is a classical Leray-Lions operator. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p -Laplace operator, namely: λ 1 (β , Ω) = min ψ ∈ W 1 , p (Ω) ∖ { 0 } ⁡ ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N − 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, anisotropic mean convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ 1 in the case β > 0 , and a upper bound in the case β < 0. As a consequence, we prove, for β > 0 , a lower bound for λ 1 (β , Ω) in terms of the anisotropic inradius of Ω and, for β < 0 , an upper bound of λ 1 (β , Ω) in terms of β. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear systems involving fully nonlinear nonlocal operators { F α (u (x)) = v p (x) + k 1 (x) u r (x) , x ∈ ℝ N , G β (v (x)) = u q (x) + k 2 (x) v s (x) , x ∈ ℝ N and { F α (u (x)) = v p (x) | x | a + u r (x) | x | b , x ∈ ℝ N \ { 0 } , G β (v (x)) = u q (x) | x | c + v s (x) | x | d , x ∈ ℝ N \ { 0 } , where ki(x) ≥ 0, i = 1, 2, 0 < α, β < 2, p, q, r, s > 1, a, b, c, d > 0. By proving a narrow region principle and other key ingredients for the above systems and extending the direct method of moving planes for the fractional p-Laplacian, we derive the radial symmetry of positive solutions about the origin. During these processes, we estimate the local lower bound of the solutions by constructing sub-solutions. [ABSTRACT FROM AUTHOR]

In this paper we will extend the concept of multi-valued Feng--Liu contraction, by imposing a nonlinear assumption on the operator. Then, fixed point, strict fixed point and stability theorems for the fixed point inclusion with multi-valued nonlinear Feng--Liu contractions are given. An application illustrates the main theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to study the approximate controllability of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the approximate controllability results of the system are obtained using C0$$ {C}_0 $$‐semigroup approach and some other simple conditions on the nonlinear term and operators involved in the model. [ABSTRACT FROM AUTHOR]

This paper proposes a distributed collision-free control scheme for connected and automated vehicles (CAVs) at a non-signalized intersection. We first divide an intersection area into three sections, i.e., the free zone, the platoon zone, and the control zone. In order to enable the following vehicles to track the trajectory of their leading vehicle in the platoon zone and the control zone, as well as to guarantee the desired distance between any two adjacent vehicles, the distributed platoon controllers are designed. In the control zone, each vehicular platoon is taken as a whole to be coordinated via an intersection coordination unit (ICU). To avoid collision between each pair of the conflicting platoons approaching from different directions, a platoon-based coordination strategy is designed by scheduling the arrival time of each leading vehicle of different platoons. Specially, considering traffic efficiency and fuel economy, the optimal control problem of the leading vehicle is formulated subject to the constraint of allowable minimum arrival time, which is derived from coordination with other approaching platoons. The Pontryagin Minimum Principle (PMP) and phase-plane method are applied to find the optimal control sequences. Numerical simulations show the effectiveness of this scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider the symmetry properties of positive solutions for nonlocal parabolic equations in the whole space. We obtain various asymptotic maximum principles for carrying out the asymptotic method of moving planes. With the help of these results, we show that if the equation converges to a symmetric one, then the solutions will converge to radially symmetric functions. The methods and techniques used here can be easily applied to study a variety of nonlocal parabolic equations with more general operators and nonlinear terms. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the regularity in time of solutions for nonlinear fractional reaction–subdiffusion equations (FrRSEs). By constituting regularity estimates of resolvent operator and employing fixed point theorems, we establish some results on the global existence and regularity in time of solutions to FrRSE in two different cases of nonlinear perturbations, namely, sublinear and superlinear cases. As an application of these results, several results on the existence and regularity of solutions in an inverse source problem governed by FrRSE with the additional measurements given at terminal time will be shown. [ABSTRACT FROM AUTHOR]

In this paper, using sunny nonexpansive retractions which are different from the metric projection in Banach spaces, we develop the CR-iteration algorithm in view of two quasi-nonexpansive nonself mappings and also give the convergence analysis for the proposed method in the setting of uniformly convex Banach spaces. Furthermore, our results can be applied for the purpose of finding common zeros of accretive operators, convexly constrained least square problems and convex minimization problems. Regarding application, some numerical experiments involving real-world problems are provided, with focus on differential problems, image restoration problems and signal recovery problems. [ABSTRACT FROM AUTHOR]

This paper focuses on solving a class of equilibrium problems, namely, the fixed point problem of set-valued mappings with second-order cone double constraints. Under certain conditions, the variational inequality form of the fixed point problem of set-valued mappings with second-order cone double constraints is obtained by using the generalized saddle point theory three times. The alternating direction method is used to solve the fixed point problem of set-valued mappings with second-order cone double constraints, and the global convergence of the algorithm is proved. Finally, numerical results of solving five examples with an inexact alternating direction method are given, and the feasibility and effectiveness of the algorithm are demonstrated by comparing with other algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce orthogonal generalized O-α-ψ-Geraghty contractive type mappings and prove some fixed point theorems in O-complete O-b-metric spaces. We also provide an illustrative example to support our theorem. The results proved here will be utilized to show the existence of a solution to an integral equation as an application. [ABSTRACT FROM AUTHOR]

In this paper, some fixed point results for multivalued contractions are established in setting G-complete extended fuzzy b-metric spaces. An example is furnished to demonstrate the validity of results. An application of integral type inclusion is given to authenticate the theorems. Our results extend and generalize many existing results in literature. [ABSTRACT FROM AUTHOR]

In this paper, inertial splitting algorithms for nonlinear operators of pseudocontractive and accretive types are proposed. Weak and strong convergence theorems are established in uniformly convex and q-uniformly smooth Banach spaces. Numerical examples are given to illustrate the effectiveness of our proposed algorithms. [ABSTRACT FROM AUTHOR]

In this study, we suggest a new iterative family of iterative methods for approximating the roots with multiplicity in nonlinear equations. We found a lack in the approximation of multiple roots in the case that the nonlinear operator be non-differentiable. So, we present, in this paper, iterative methods that do not use the derivative of the non-linear operator in their iterative expression. With our new iterative technique, we find better numerical results of Planck's radiation, Van Der Waals, Beam designing, and Isothermal continuous stirred tank reactor problems. Divided difference and weight function approaches are adopted for the construction of our schemes. The convergence order is studied thoroughly in the Theorems 1 and 2, for the case when multiplicity p ≥ 2 . The obtained numerical results illustrate the preferable outcomes as compared to the existing ones in terms of absolute residual errors, number of iterations, computational order of convergence (COC), and absolute error difference between two consecutive iterations. [ABSTRACT FROM AUTHOR]

In this paper we formulate a setvalued fixed point problem by combining four prevalent trends of fixed point theory. We solve the problem by showing that the set of fixed points is nonempty. Further we have a data dependence result pertaining to the problem and also a stability result for the fixed point sets. The main result is extended to metric spaces with a graph. The results are obtained without the use of metric completeness assumption which is replaced by some other conditions suitable for solving the fixed point problem. There are some consequences of the main result. The main result is illustrated with an example. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions. [ABSTRACT FROM AUTHOR]

The main result of the paper provides the existence of a solution to a quasilinear inclusion problem with Dirichlet boundary condition which exhibits a term with full dependence on the solution and its gradient (convection term) and is driven by the degenerated p -Laplacian with weight. The multivalued term in the differential inclusion is in form of the generalized gradient of a locally Lipschitz function expressed through the primitive of a locally essentially bounded function, which makes the problem to be of a hemivariational inequality type. The novelty of our result is that we are able to simultaneously handle three major features: degenerated leading operator, convection term and discontinuous nonlinearity. Results of independent interest regard certain nonlinear operators associated to the differential inclusion. [ABSTRACT FROM AUTHOR]

An important problem in analysis on fractals is the existence of a self‐similar energy on finitely ramified fractals. The self‐similar energies are constructed in terms of eigenforms, that is, eigenvectors of a special nonlinear operator. Previous results by C. Sabot and V. Metz give conditions for the existence of an eigenform. In this paper, I prove this type of result in a different way. The proof given in this paper is based on a general fixed‐point theorem for anti‐attracting maps on a convex set. [ABSTRACT FROM AUTHOR]

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems. [ABSTRACT FROM AUTHOR]

Runaway electron modelling efforts are motivated by the risk these energetic particles pose to large fusion devices. The sophisticated kinetic models can capture most features of the runaway electron generation but have high computational costs, which can be avoided by using computationally cheaper reduced kinetic codes. This paper compares the reduced kinetic and kinetic models to determine when the former solvers, based on analytical calculations assuming quasi-stationarity, can be used. The Dreicer generation rate is calculated by two different solvers in parallel in a workflow developed in the European integrated modelling framework, and this is complemented by calculations of a third code that is not yet integrated into the framework. Runaway Fluid, a reduced kinetic code, NORSE, a kinetic code using non-linear collision operator, and DREAM, a linearized Fokker–Planck solver, are used to investigate the effect of a dynamic change in the electric field for different plasma scenarios spanning across the whole tokamak-relevant range. We find that on time scales shorter than or comparable to the electron–electron collision time at the critical velocity for runaway electron generation, kinetic effects not captured by reduced kinetic models play an important role. This characteristic time scale is easy to calculate and can reliably be used to determine whether there is a need for kinetic modelling or cheaper reduced kinetic codes are expected to deliver sufficiently accurate results. This criterion can be automated, and thus it can be of great benefit for the comprehensive self-consistent modelling frameworks that are attempting to simulate complex events such as tokamak start-up or disruptions. [ABSTRACT FROM AUTHOR]

In this paper, using sunny nonexpansive retractions, which are different from the metric projection in Banach spaces, a new type of study regarding the iterative methods in view of two quasi‐nonexpansive nonself mappings is presented. We also give the convergence analysis for the proposed method in the background of uniformly convex Banach spaces. Moreover, we apply our results to find solutions of common zeros of accretive operators, convexly constrained least square problems, and convex minimization problems. Furthermore, we also discuss novel applications of these methods to differential problems, image deblurring, and signal recovering problems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a new family of iterative methods for approximating the solution of nonlinear systems for non-differentiable operators. The novelty of this family is that it is a m-step generalization of the Steffensen-type method by updating the divided difference operator in the first two steps but not in the following ones. This procedure allows us to increase both the order of convergence and the efficiency index with respect to that obtained in the family that updates divided differences only in the first step. We perform a semilocal convergence study that allows us to fix the convergence domain and uniqueness for real applied problems, where the existence of a solution is not known a priori. After this study, some numerical tests are developed to apply the semilocal convergence theoretical results obtained. Finally, mediating the dynamic planes generated by the different numerical methods that compose the family, we study the symmetry of the basins of attraction generated by each solution, the shape of these basins, and the convergence to each root of a polynomial function. [ABSTRACT FROM AUTHOR]

Multiscale transforms have become a key ingredient in many data processing tasks. With technological development we observe a growing demand for methods to cope with nonlinear data structures such as manifold values. In this paper we propose a multiscale approach for analyzing manifold-valued data using a pyramid transform. The transform uses a unique class of downsampling operators that enable a noninterpolating subdivision schemes as upsampling operators. We describe this construction in detail and present its analytical properties, including stability and coefficient decay. Next, we numerically demonstrate the results and show the application of our method to denoising and anomaly detection. [ABSTRACT FROM AUTHOR]

In this paper, we study the Liouville-type behaviors of the generalized Greiner operators with nonlinear boundary value conditions. We use the technique based upon the method of moving planes. [ABSTRACT FROM AUTHOR]