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In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper we study the strong maximum principle for equations of the form F [ u ] = H (u , | D u |) where F is either a fully nonlinear elliptic operator or is the p-Laplace operator. We give sufficient conditions on H to ensure that the strong maximum principle (SMP) holds. The condition is also necessary for SMP to hold for the the equation F [ u ] = g (| D u |). [ABSTRACT FROM AUTHOR]

Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point of λ -strictly pseudocontractive mappings, solution of α -inverse-strongly monotone mappings, and solution of integral equations of Fredholm type. [ABSTRACT FROM AUTHOR]

This paper presents a state‐dependent self‐tuning fractional control strategy for a DC‐DC buck converter in order to enhance its output voltage regulation and disturbance attenuation capability. The proposed control scheme primarily employs a ubiquitous proportional‐integral‐derivative (PID) controller, where gains are optimally selected using a linear‐quadratic state‐space tuning approach. The optimal PID controller is then augmented with fractional‐order integral and derivative operators in order to improve the controller's degrees‐of‐freedom as well as the system's overall time‐domain performance. The fractional controller's robustness against bounded exogenous disturbances, contributed by the input fluctuations and load‐step transients, is further enhanced by adaptively modulating the fractional‐orders of the integro‐differential operators as a smooth nonlinear function of controlled‐variable's error‐dynamics. An online dynamic adjustment law, comprising of a zero‐mean Gaussian function of error and its derivative, is used to individually update the two fractional orders after every sampling interval. The error derivative is evaluated by measuring the output capacitor's current in order to compensate the noise injected by parasitic impedance. The other controller parameters are tuned via particle‐swarm‐optimization algorithm. The proposed self‐adaptive control strategy renders rapid transits, minimum transient recovery time, and minimal fluctuations around steady state in the response. Its efficacy is validated through hardware in‐the‐loop experiments conducted on a buck converter prototype. [ABSTRACT FROM AUTHOR]

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ (ℝ n) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1

In this paper we revisit the discrepancy principle for Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces and prove a new convergence result which requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we expand the applied range of the discrepancy principle for Landweber iteration to cover non-smooth ill-posed inverse problems and to handle the situation that the data is contaminated by various types of noise. [ABSTRACT FROM AUTHOR]

In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent (P) (− Δ) s u = λ u γ + | u | 2 α ∗ − 2 u | x | α in Ω , u > --> 0 in Ω , u = 0 in R N ∖ Ω , $$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$ where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < 2 s ∗ $\begin{array}{} \displaystyle 2_s^* \end{array}$ , where 2 s ∗ = 2 N N − 2 s and 2 α ∗ = 2 (N − α) N − 2 s $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by (− Δ) s u (x) = − 1 2 ∫ R N u (x + y) + u (x − y) − 2 u (x) | y | N + 2 s d y , for all x ∈ R N. $$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$ By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P). [ABSTRACT FROM AUTHOR]

This paper presents the potential of nonlinear and linear versions of an observation operator for simulating polarimetric variables observed by weather radars. These variables, deduced from the horizontally and vertically polarized backscattered radiations, give information about the shape, the phase and the distributions of hydrometeors. Different studies in observation space are presented as a first step toward their inclusion in a variational data assimilation context, which is not treated here. Input variables are prognostic variables forecasted by the AROME-France numerical weather prediction (NWP) model at convective scale, including liquid and solid hydrometeor contents. A nonlinear observation operator, based on the T-matrix method, allows us to simulate the horizontal and the vertical reflectivities (ZHH and ZVV), the differential reflectivity ZDR , the specific differential phase KDP and the co-polar correlation coefficient ρHV. To assess the uncertainty of such simulations, perturbations have been applied to input parameters of the operator, such as dielectric constant, shape and orientation of the scatterers. Statistics of innovations, defined by the difference between simulated and observed values, are then performed. After some specific filtering procedures, shapes close to a Gaussian distribution have been found for both reflectivities and for ZDR , contrary to KDP and ρHV. A linearized version of this observation operator has been obtained by its Jacobian matrix estimated with the finite difference method. This step allows us to study the sensitivity of polarimetric variables to hydrometeor content perturbations, in the model geometry as well as in the radar one. The polarimetric variables ZHH and ZDR appear to be good candidates for hydrometeor initialization, while KDP seems to be useful only for rain contents. Due to the weak sensitivity of ρHV , its use in data assimilation is expected to be very challenging. [ABSTRACT FROM AUTHOR]

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [12], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [8], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray-Schauder topological degree theory.

Summary: In this paper, we propose for the first time to extend the application field of the high‐order mesh‐free approach to the stationary incompressible Navier‐Stokes equations. This approach is based on a high‐order algorithm, which combines a Taylor series expansion, a continuation technique, and a moving least squares (MLS) method. The Taylor series expansion permits to transform the nonlinear problem into a succession of continuous linear ones with the same tangent operator. The MLS method is used to transform the succession of continuous linear problems into discrete ones. The continuation technique allows to compute step‐by‐step the whole solution of the discrete problems. This mesh‐free approach is tested on three examples: a flow around a cylindrical obstacle, a flow in a sudden expansion, and the standard benchmark lid‐driven cavity flow. A comparison of the obtained results with those computed by the Newton‐Raphson method with MLS, the high‐order continuation with finite element method, and those of literature is presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method. [ABSTRACT FROM AUTHOR]

Contraction coefficients are distribution dependent constants that are used to sharpen standard data processing inequalities for f-divergences (or relative f-entropies) and produce so-called "strong" data processing inequalities. For any bivariate joint distribution, i.e., any probability vector and stochastic matrix pair, it is known that contraction coefficients for f-divergences are upper bounded by unity and lower bounded by the contraction coefficient for χ2-divergence. In this paper, we elucidate that the upper bound is achieved when the joint distribution is decomposable, and the lower bound can be achieved by driving the input f-divergences of the contraction coefficients to zero. Then, we establish a linear upper bound on the contraction coefficients of joint distributions for a certain class of f-divergences using the contraction coefficient for χ2-divergence, and refine this upper bound for the salient special case of Kullback-Leibler (KL) divergence. Furthermore, we present an alternative proof of the fact that the contraction coefficients for KL and χ2-divergences are equal for bivariate Gaussian distributions (where the former coefficient may impose a bounded second moment constraint). Finally, we generalize the well-known result that contraction coefficients of stochastic matrices (after extremizing over all possible probability vectors) for all nonlinear operator convex f-divergences are equal. In particular, we prove that the so-called "less noisy" preorder over stochastic matrices can be equivalently characterized by any nonlinear operator convex f-divergence. As an application of this characterization, we also derive a generalization of Samorodnitsky's strong data processing inequality. [ABSTRACT FROM AUTHOR]

The object of this paper was to study two majorization results for the subclasses S s ∗ and S c ∗ of starlike functions, which are, respectively, associated with the sine and cosine functions, without acting upon any linear or nonlinear operators to the above function classes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of square-mean piecewise almost automorphic function. By using the theory of semigroups of operators and the contraction mapping principle, the existence of square-mean piecewise almost automorphic mild solutions for linear and nonlinear impulsive stochastic evolution equations is investigated. In addition, the exponential stability of square-mean piecewise almost automorphic mild solutions for nonlinear impulsive stochastic evolution equations is obtained by the generalized Gronwall–Bellman inequality. Finally, we provide an illustrative example to justify the results. [ABSTRACT FROM AUTHOR]

In this paper, we consider equations involving the fully nonlinear fractional order operator with homogeneous Dirichlet condition:{Fα(u)(x)=f(x,u,∇u) in Ω,u>0, in Ω; u≡0, in Rn∖Ω,{Fα(u)(x)=f(x,u,∇u) in Ω,u>0, in Ω; u≡0, in Rn∖Ω,where ΩΩ is a domain(bounded or unbounded) in RnRn which is convex in x1−x1−direction. By using some ideas of maximum principle, we prove that the solution is strictly increasing in x1−x1−direction in the left half of ΩΩ. Symmetry of solution is also proved. Meanwhile we obtain a Liouville type theorem on the half space Rn+R+n. [ABSTRACT FROM AUTHOR]

Existing viscosity approximation schemes have been extensively investigated to solve equilibrium problems, variational inequalities, and fixed-point problems, and most of which contain that contraction is a self-mapping defined on certain bounded closed convex subset C of Hilbert spaces H for standard viscosity approximation. In particular, if the zero point does not belong to C, the standard viscosity approximation cannot be applied to solve the minimum norm fixed point of some nonlinear operators. In this paper, we introduce three generalized viscosity approximation algorithms with boundary point method for quasi-nonexpansive mappings, which overcome this deficiency above. These three proposed algorithms have simple expressions; moreover, they are easy to implement in the actual computation process. Especially, we can find the minimum norm fixed point of quasi-nonexpansive mappings under contraction is a zero operator. [ABSTRACT FROM AUTHOR]

The paper provides results of the study of correctness of the mathematical model that includes the influence of stresses on irradiation-induced swelling of metal in the problems of elastic-plastic deformation mechanics. The present-day approaches to modeling irradiation-induced swelling, which take into account a damaging dose, irradiation temperature, and the effect of the stress state on the swelling deformation, are discussed. The constitutive equations that describe the elasticplastic deformation processes allowing for the influence of a stress mode on the irradiation-induced swelling in metal are put forward. Analysis of these equations has made it possible to find the conditions that ensure correctness of the plasticity equations considered and to make a lower-bound estimate of the maximum permissible value of free swelling and irradiation dose. A priori estimates of the maximum permissible value of free swelling and damaging dose are given for 08Kh18N10T steel under various irradiation temperatures. In the practice of strength design such estimates are needed at the stage of problem formulation in order to analyze adequacy of input data, for they enable one to assess a prior the possibility of solving the problem for a given temperature and irradiation dose. The boundary-value problem that describes non-isothermal processes of elasticplastic deformation including swelling strains has been defined in the form of a nonlinear operator equation. Based on the findings regarding the correctness of the constitutive equations, we have established the existence and uniqueness of the generalized solution and its continuous dependence on applied loads, thermal strains and swelling strains. The convergence of the method of elasticity solutions and the method of variable elasticity parameters has been studied as applied to a thermoplasticity problem including irradiation-induced swelling strains. [ABSTRACT FROM AUTHOR]

In this paper, a normal S-iterative algorithm is studied and analyzed for solving a general class of variational inequalities involving a set of fixed points of nonexpansive mappings and two nonlinear operators. It is shown that the proposed algorithm converges strongly under mild conditions. The rate of convergence of the proposed iterative algorithm is also studied. An equivalence of convergence between the normal S-iterative algorithm and Algorithm 2.6 of Noor (J. Math. Anal. Appl. 331, 810–822, 2007) is established and a comparison between the two is also discussed. As an application, a modified algorithm is employed to solve convex minimization problems. Numerical examples are given to validate the theoretical findings. The results obtained herein improve and complement the corresponding results in Noor (J. Math. Anal. Appl. 331, 810–822, 2007). [ABSTRACT FROM AUTHOR]

In the present paper, we establish some fixed-point theorems for a 2 × 2 block operator matrix involving multivalued maps acting on Banach spaces. These results are formulated in terms of weak sequential continuity and the technique of measures of weak noncompactness. The results obtained are then applied to a coupled system of nonlinear equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem. [ABSTRACT FROM AUTHOR]

One of the tools and techniques concerned with the stability of nonlinear waves is the Evans function which is an analytic function whose zeros give the eigenvalues of the linearized operator. Here, in this paper, we propose a direct approach, which is based essentially upon constructing the eigenfunction solution of the perturbed equation based upon the topological invariance in conjunction with usage of the Legendre polynomials, which have presumably not considered in the literature thus far. The associated Legendre eigenvalue problem arising from the stability analysis of traveling waves solutions is systematically studied here. The present work is of considerable interest in the engineering sciences as well as the mathematical and physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product. This can be controlled by depicting the initial concentration of the reactant, which is determined by its value at the bifurcation point. This analysis leads to the point separating stable and unstable solutions. As far as chemical reactions are described by reaction-diffusion equations, this specific concentration can be found mathematically. On the other hand, the study of stability analysis of solutions may depict whether or not a soliton pulse is well-propagated in fiber optics. This can, and should, be carried out by finding the solutions of the coupled nonlinear Schrödinger equations and by analyzing the stability of these solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of a non‐autonomous porous elastic systems with nonlinear damping and sources terms. By employing nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions. We also prove the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the sources terms. Finally, we prove the upper‐semicontinuity of pullback attractors with respect to non‐autonomous perturbations. [ABSTRACT FROM AUTHOR]

Increasing attention is paid to modeling flexibility of individual components in the multibody simulation of large-scale mechanical systems. Nevertheless, the high model order of common methods such as FEA restricts efficient explorations, especially in dynamic design and iterative optimization. In this paper, a rigid multibody modeling strategy (RMMS) with low DOFs and explicit physical meaning is proposed, which directly discretizes a continuous structure into a number of rigid finite elements (RFEs) connected by spring-damping elements (SDEs). In the RMMS, a new identification method from the perspective of the inverse vibration problem is particularly put forward to resolve the parameters of SDEs, which is crucial to the implementation of RMMS in complex flexible structures. With decoupling and linearization, this nonlinear problem is transformed into solving the incompatible linear equations in R n 2 vector space based on vectorization operator and Kronecker product, and optimal parameters are obtained by calculating the Moore–Penrose generalized inverse. Finally, the comparison of the experimental results with the simulated ones by the RMMS strongly validates the feasibility and correctness of the RMMS in predicting the dynamic behaviors while with few DOFs and explicit physical meaning; the application in a lithography system exhibits the applicability of the RMMS for dynamic modeling of large-scale mechanical systems. [ABSTRACT FROM AUTHOR]

Many iterative algorithms like Picard, Mann, Ishikawa are very useful to solve fixed point problems of nonlinear operators in real Hilbert spaces. The recent trend is to enhance their convergence rate abruptly by using inertial terms. The purpose of this paper is to investigate a new inertial iterative algorithm for finding the fixed points of nonexpansive operators in the framework of Hilbert spaces. We study the weak convergence of the proposed algorithm under mild assumptions. We apply our algorithm to design a new accelerated proximal gradient method. This new proximal gradient technique is applied to regression problems. Numerical experiments have been conducted for regression problems with several publicly available high-dimensional datasets and compare the proposed algorithm with already existing algorithms on the basis of their performance for accuracy and objective function values. Results show that the performance of our proposed algorithm overreaches the other algorithms, while keeping the iteration parameters unchanged. [ABSTRACT FROM AUTHOR]

In this paper we design a family of cell-average nonlinear prediction operators that make use of the generalized harmonic means and we apply the resulting schemes to image processing. The new family of nonlinear schemes conserve the numerical properties of the linear schemes, such as the L 1 -stability, the order of accuracy or compression rate but avoiding Gibbs phenomenon close to the discontinuities. The generalized harmonic mean was introduced in the framework of point-values in [A. Guessab, M. Moncayo, and G. Schmeisser, A class of nonlinear four-point subdivision schemes. Properties in terms of conditions, Adv. Comput. Math. 37 (2012), pp. 151–190] in order to improve the results of the harmonic mean. However, in the cell-average setting our conclusion is that, from a numerical point of view, the advantage of using the new mean is not clear. [ABSTRACT FROM AUTHOR]

The paper investigates the existence of global attractors for a few classes of multi-valued operators. We establish some criteria and give their applications to a strongly damped wave equation with fully supercritical nonlinearities and without the uniqueness of solutions. Moreover, the geometrical structure of the global attractors of the corresponding multi-valued operators is shown. [ABSTRACT FROM AUTHOR]

It is our purpose in this paper to prove some fixed point results and Fejer monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued Banach spaces. We prove that results in complex valued Banach spaces are valid in cone metric spaces with Banach algebras. Furthermore, we apply our results in solving certain mixed type Volterra-Fredholm functional nonlinear integral equation in complex valued Banach spaces. [ABSTRACT FROM AUTHOR]

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise. [ABSTRACT FROM AUTHOR]

In this note, we concern with a class of doubly nonlinear operators whose prototype is ut − div(∣u∣m−1∣Du∣p−2Du) = 0, p > 1, m + p = 2. In the last few years many progresses were made in understanding the right form of the Harnack inequalities for singular parabolic equations. For doubly nonlinear equations the singular case corresponds to the range m+p < 3. For 3−p/N < m+p < 3, where N denotes the space dimension, intrinsic Harnack estimates hold. In the range 2 < m + p ≤ 3 − p/N only a weaker Harnack form survives. In the limiting case m+p = 2, only the case p = 2 was studied. In this paper we fill this gap and we study the behaviour of the solutions in the full range p > 1 and m = 2 − p. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationship between the class of weakly demicompact linear operators, introduced in [KRICHEN, B.—O'REGAN, D.: On the class of relatively weakly demicompact nonlinear operators, Fixed Point Theory 19 (2018), 625–630], and measures of weak noncompactness of linear operators with respect to an axiomatic one. Moreover, some Fredholm and perturbation results involving the class of weakly demicompact linear operators are investigated. Our results are then used to investigate the relationship between the relative essential spectrum of the sum of two linear operators and the relative essential spectrum of each of these operators. [ABSTRACT FROM AUTHOR]

In this paper, we establish the generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces. The strong convergence theorems of the implicit rules proposed are proved under certain assumptions imposed on the control parameters. The results presented in this paper improve and extend some recent corresponding results announced. [ABSTRACT FROM AUTHOR]

In this paper, we consider a family of nonlinear integral operators of Urysohn‐type and study the pointwise convergence of the family at characteristic points of L1−function. The kernel Kλ(x,t,u(t)) depends on the positive parameter λ changing on the set of numbers with the accumulation point at infinity and Kλ(x,t,u(t)) is an entire analytic function of variable u, which is a bounded function belonging to L1(R). [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of local stable manifolds for a semi-linear differential equation, where the linear part is a Hille–Yosida operator on a Banach space and the nonlinear forcing term f satisfies the φ -Lipschitz conditions, where φ belongs to certain classes of admissible function spaces. The approach being used is the fixed point arguments and the characterization of the exponential dichotomy of evolution equations in admissible spaces of functions defined on the positive half-line. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to analyze in a comparative way the convergence of some additive and additive Schwarz–Richardson methods for inequalities with nonlinear monotone operators. We first consider inequalities perturbed by a Lipschitz operator in the framework of a finite dimensional Hilbert space and prove that they have a unique solution if a certain condition is satisfied. For these inequalities, we introduce additive and restricted additive Schwarz methods as subspace correction algorithms and prove their convergence, under a certain convergence condition, and estimate the error. The convergence of the restricted additive methods does not depend on the number of the used subspaces and we prove that the convergence rate of the additive methods depends only on a reduced number of subspaces which corresponds to the minimum number of colors required to color the subdomains such that the subdomains having the same color do not intersect with each other, but not on the actual number of subdomains. The convergence condition of the algorithms is more restrictive than the existence and uniqueness condition of the solution. We then introduce new additive and restricted additive Schwarz algorithms that have a better convergence and whose convergence condition is identical to the condition of existence and uniqueness of the solution. The additive and restricted additive Schwarz–Richardson algorithms for inequalities with nonlinear monotone operators are obtained by taking the Lipschitz operator of a particular form and the convergence results are deducted from the previous ones. In the finite element space, the introduced algorithms are additive and restricted additive Schwarz–Richardson methods in the usual sense. Numerical experiments carried out for three problems confirm the theoretical predictions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop analytic techniques to deal with Hankel-total positivity of sequences. We show two nonlinear operators preserving Stieltjes moment property of sequences. They actually both extend a result of Wang and Zhu that if (an)n≥0 is a Stieltjes moment sequence, then so is (an+2an - a²n+1)n≥0. Using complete monotonicity of functions, we also prove Stieltjes moment properties of the sequences ...and .... Particularly in a new unified manner our results imply the Stieltjes moment properties of binomial coefficients (pn+r-1 n) and Fuss- Catalan numbers r/pn+r (pn+r n) proved by Mlotkowski, Penson, and Zyczkowski, and Liu and Pego, respectively, and also extend some results for log-convexity of sequences proved by Chen-Guo-Wang, Su-Wang, Yu, and Wang-Zhu, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we establish global Sobolev a priori estimates for L p -viscosity solutions of fully nonlinear elliptic equations as follows: F (D 2 u , D u , u , x) = f (x) in  Ω u (x) = φ (x) on  ∂ Ω by considering minimal integrability condition on the data, i.e. f ∈ L p (Ω) , φ ∈ W 2 , p (Ω) for n < p < ∞ and a regular domain Ω ⊂ ℝ n , and relaxed structural assumptions (weaker than convexity) on the governing operator. Our approach makes use of techniques from geometric tangential analysis, which consists in transporting "fine" regularity estimates from a limiting operator, the Recession profile, associated to F to the original operator via compactness methods. We devote special attention to the borderline case, i.e. when f ∈ p − BMO ⊋ L ∞ . In such a scenery, we show that solutions admit BMO type estimates for their second derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space. [ABSTRACT FROM AUTHOR]

Investigation of trajectories of evolution operator has essential roles in the theory of multi-dimensional nonlinear dynamical systems. In this paper we study on the asymptotical behavior of trajectories generated by concrete normalized gonosomal evolution operator of sex linked inheritance on the three dimensional simplex. We get sharp estimation for the trajectories of such evolution operator. Moreover, we obtain an explicit value of the order of convergence. [ABSTRACT FROM AUTHOR]

In this paper we first introduce N̅qΔ − summable difference sequence spaces and prove some properties of these spaces. We then obtain the necessary and sufficient conditions for infinite matrix A to map these sequence spaces on the spaces c, c0 and l∞. Finally, the Hausdorff measure of noncompactness is then used to obtain the necessary and sufficient conditions for the compactness of the linear operators defined on these spaces. [ABSTRACT FROM AUTHOR]

Let U be a Hilbert A-module and L(U) the set of all adjointable A-linear maps on U. Let K = {Λx ∈ L(U, Vx) : x ∈ X } and L = {Γx ∈ L(U, Vx) : x ∈ X } be two continuous g-frames for U, K is said to be similar with L if there exists an invertible operator J ∈ L(U) such that Γx = ΛxJ, for all x ∈ X . In this paper, we define the concepts of closeness and nearness between two continuous g-frames. In particular, we show that K and L are near, if and only if they are similar. [ABSTRACT FROM AUTHOR]

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu-Lσ,μ[φ(u)] = f in RN×(0,T), where Lσ,μ is a general symmetric diffusion operator of Lévy type and φ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators Lσ,μ are the (fractional) Laplacians Δ and -(-Δ)α/2 for α ∈ (0,2), discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal Lévy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions, including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [F. del Teso, J. Endal, and E. R. Jakobsen, C. R. Math. Acad. Sci. Paris, 355 (2017), pp. 1154--1160]. We also present some numerical tests, but extensive testing is deferred to the companion paper [F. del Teso, J. Endal, and E. R. Jakobsen, SIAM J. Numer. Anal., 56 (2018), pp. 3611-3647] along with a more detailed discussion of the numerical methods included in our theory. [ABSTRACT FROM AUTHOR]

In this paper we consider optimal feedback control problems for a general class of nonlinear partially observed stochastic evolution equations on Banach spaces. The system is governed by a pair of (coupled) stochastic evolution equations, one representing the main system and the other representing the observer. Both are governed by stochastic evolution equations on unconditional Martingale difference Banach spaces. The state of the second system, which is observable, is used to provide the input to control the main system. We present the existence of mild solutions of the system equations and then introduce a class of admissible nonlinear feedback operators. The space of feedback operators is endowed with the topology of pointwise convergence on the domain space with respect to the weak topology in the range space giving a compact Hausdorff space. This is then used to prove the existence of an optimal output feedback control law for the Bolza problem. Also we prove the weak compactness of the attainable set of induced measures and prove the existence of optimal feedback control laws for several nontypical control problems involving measures. [ABSTRACT FROM AUTHOR]

This paper is concerned with a host-macroparasite difference model. By applying the contraction mapping fixed point theorem, we prove the existence of unique almost periodic positive solution. Moreover, we investigate the exponential stability of almost periodic solution by means of Lyapunov functional. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new approach for discussing the solvability of a class of operator equations by establishing fixed point theorems in locally convex spaces. Our results are obtained extend some Krasnosel'skii type fixed point theorems. As an application, we investigate the existence and global attractivity of solutions for a general nonlinear integral equation of mixed type of Urysohn and Volterra. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of concave and monotone positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the form $x^{(4)}(t)=f(t,x(t),x'(t),x''(t))$, $t\in[0,1]$, $x(0)=x'(1)=x'''(1)=0$, $x''(0)=\int_{0}^{1}g(s)x''(s) \, \mathrm{d}s$, where $f\in C([0,1]\times[0,+\infty)^{2}\times(-\infty,0],[0,+\infty))$, $g\in C([0,1],[0,+\infty))$. By using a fixed point theorem of cone expansion and compression of norm type, the existence and nonexistence of concave and monotone positive solutions for the above boundary value problems is obtained. Meanwhile, as applications of our results, some examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of positive mild solutions of fractional evolution equations with nonlocal conditions of order $1<\alpha<2$ by using Schauder's fixed point theorem and Krasnoselskii's fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are established for the existence and uniqueness of global solutions to stochastic impulsive systems with expectations in the nonlinear terms. The maximal interval and the estimate of mild solutions are also discussed. These results are obtained by using the fixed point theorem, interval partition, and Lyapunov-like technique. Finally, examples are given to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid three-term conjugate gradient (CG) method is proposed to solve constrained nonlinear monotone operator equations. The search direction is computed such that it is close to the direction obtained by the memoryless Broyden–Fletcher–Goldferb–Shanno (BFGS) method. Without any condition, the search direction is sufficiently descent and bounded. Moreover, based on some conditions, the search direction satisfy the conjugacy condition without using any line search. The global convergence of the method is established under mild assumptions. Comparison with existing methods is done to test the efficiency of the proposed method through some numerical experiments. Lastly, the applicability of the proposed method is shown. [ABSTRACT FROM AUTHOR]

The interest in incorporating environmentally friendly and renewable sources of energy, like photovoltaic (PV) technology, into electricity grids has grown significantly. These sources offer benefits, such as reduced power losses and improved voltage stability. To optimize these advantages, it is essential to determine optimal placement and management of these energy resources. This paper proposes an Improved RUNge–Kutta optimizer (IRUN) for allocating PV-based distributed generations (DGs) and Battery Energy Storage (BES) in distribution networks. IRUN utilizes three strategies to avoid local optima and enhance exploration and exploitation phases: a non-linear operator for smoother transitions, a Chaotic Local Search for thorough exploration, and diverse solution updates for refinement. The efficacy of IRUN is evaluated using 10 benchmark functions from the CEC’20 test suite, followed by statistical analysis. Next, IRUN is used to optimize the allocation of PVDG and BES to minimize energy losses in two standard IEEE distribution networks. The optimization problem is divided into two stages. In the first stage, the optimal size and the location of PV systems are calculated to meet peak load demand. In the second stage, considering time-varying load demand and intermittent PV generation, effective energy management of BES is employed. The effectiveness of IRUN is compared against the original RUN and other well-known optimization algorithms through simulation results. The comprehensive analysis demonstrates that IRUN outperforms the compared algorithms, making it a leading solution for optimizing PV distributed generation and BES allocation in distribution networks and the results show that the energy loss reduction reaches 63.54% and 68.19% when using PVand BES in IEEE 33-bus and IEEE 69 bus respectively. [ABSTRACT FROM AUTHOR]