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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]

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]

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]

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]

The aim of this paper is to study the existence of extreme solutions and their properties for a general σ$$ \sigma $$‐Hessian equation involving a nonlinear operator. By introducing a suitable growth condition and developing a iterative technique, some new results on existence and asymptotic estimates of minimum and maximum solutions are derived. Moreover, we also establish the iterative sequences that converge uniformly to the extreme solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions to a nonlinear singular transport equation (ie, transport equation with unbounded collision frequency and unbounded collision operator) with vacuum boundary conditions in bounded spatial domain on Lp‐spaces with 1 ≤ p<+∞. This problem was already considered in6,8,9 under the hypothesis that the collision frequency σ(·) and the collision operator are bounded. In this work, we show that these hypotheses are not necessary; it suffices to assume that σ(·) is locally bounded, and the collision operator is bounded between Xpσ (a weighted space) and Xp (cf Section 2). Although the analysis for p∈(1,+∞) is standard in the sense that it uses the Schauder fixed point theorem, the compactness of the involved operator is not easy to derive. However, the analysis in the case p=1 uses the concept of Dunford‐Pettis operators and a new version of the Darbo fixed point theorem for a measure of weak noncompactness introduced in the paper. [ABSTRACT FROM AUTHOR]

The concern of this paper is to continue the investigation of convergence properties of nonlinear approximation operators, which are defined by Karsli. In details, the paper centers around Urysohn‐type nonlinear counterpart of the Bernstein operators. As a continuation of the study of Karsli, the present paper is devoted to obtain Voronovskaya‐type theorems for the Urysohn‐type nonlinear Bernstein operators. [ABSTRACT FROM AUTHOR]

This paper proposes a novel guidance law based on model predictive control for protecting an aircraft from a homing missile. Considering that in the target‐missile‐defender engagement the attacking missile is pursuing the target aircraft by employing a known linear guidance law, a non‐linear predictive model for the engagement is established. This predictive model allows converting the three‐body engagement to a one‐on‐one engagement. The time‐to‐go for the defender towards the future position of the incoming missile is analytically derived, indicating that the optimal time for the defender to intercept the incoming missile is a fixed‐point of a real‐valued function. As a result, the optimal interception duration can be obtained by a standard fixed‐point iteration method, and thus the interception point can be readily predicted via integrating the predictive model. The guidance command is generated by directly leading the defender to the predicted intercept point. Numerical simulations in different scenarios validate the algorithm to be computationally efficient, and demonstrate that the proposed strategy outperforms previous techniques in terms of miss distance, engagement duration, and the control effort. [ABSTRACT FROM AUTHOR]

The scope and purpose of the subject paper by the present authors Int J Quantum Chem 2004, 100, 1086 are clarified. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [ABSTRACT FROM AUTHOR]

In this paper, impulsive Lasota-Wazewska model with infinite delay is studied. By using fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence, which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by Liapunov functional. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we, consider an n‐th order nonlinear difference equation with parameter dependence. An exhaustive study of the related Green's function is done. The exact expression of the function is given. The range of parameter for which either it has constant sign or it changes sign is obtained. Some existence results for the nonlinear problem are deduced by using the classical Krasnosel'skii's fixed point theorem on cones and fixed point index theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized predator‐prey system with prey‐taxis under the Neumann boundary condition. We investigate the local and global asymptotical stability of constant steady states (including trivial, semitrivial, and interior constant steady states). On the basis of a priori estimate and the fixed‐point index theory, several sufficient conditions for the nonexistence/existence of nonconstant positive solutions are given. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with a problem of an equitable allocation. We consider the concept of ψ‐equity, which is a general concept of equity. We provide a series of examples of equity concepts that are captured by ψ‐equity. We show the existence of an efficient and ψ‐equitable allocation by employing Kakutani's fixed‐point theorem. [ABSTRACT FROM AUTHOR]

In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin-Bona-Mahony equation, Burger's equation, Fornberg-Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper proposes a first-order algorithm for unconstrained nonlinear optimal control problems based on the fixed point theorem. The method has the following features: (a) the algorithm is simple; (b) unlike second-order algorithms, the choice of the first approximation is less critical; and (c) unlike gradient methods, experimental selections of parameters during iterative calculations are not required. This paper also describes the conditions of convergence of the algorithm in the space of continuous functions, and the optimality of a solution derived. [ABSTRACT FROM AUTHOR]

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite‐dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators. [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]

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 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]

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, 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 classification is given of blowup solutions at the intersection of the free and fixed boundary corresponding to obstacle problems generated by fully nonlinear, uniformly elliptic operators. As a consequence, nontransversal intersection is shown to hold in any dimension. Several regularity results are also obtained. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of a solution of a C0 Interior Penalty Discontinuous Galerkin (C0 IPDG) method for the numerical solution of a fourth‐order total variation flow problem that has been developed in part I of the paper. The proof relies on a nonlinear version of the Lax‐Milgram Lemma. It requires to establish that the nonlinear operator associated with the C0 IPDG approximation is Lipschitz continuous and strongly monotone on bounded sets of the underlying finite element space. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical method is proposed and analyzed for the Allen–Cahn (AC) equation. We divide the AC equation into linear section and nonlinear section based on the idea of operator splitting. For the linear part, it is discretized by using the Crank–Nicolson scheme and solved by finite element method. The nonlinear part is solved accurately. In addition, a posteriori error estimator of AC equation is constructed in adaptive computation based on superconvergent cluster recovery. According to the proposed a posteriori error estimator, we design an adaptive algorithm for the AC equation. Numerical examples are also presented to illustrate the effectiveness of our adaptive procedure. [ABSTRACT FROM AUTHOR]

In this paper, we are dealing with an analytical study of a singular fractional order nonlinear differential equation with fractional integral and differential boundary conditions and ϕp‐operator, for existence and stability results. Our problem is based on two types of fractional order derivatives, that is, Caputo factional derivative of order σ and Riemann‐Liouville derivative of order β, where m − 1  <  σ,β  ≤  m, and m  ∈  {3,4,5,...}. The suggested problem will be converted into an equivalent integral form by the help of Green function. After the proofs for these properties, some classical fixed point theorems are employed for the existence of positive solution (EPS). For application of the results, an expressive example is included. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of positive solution for four-point nonlocal boundary value problems of fractional order. By means of some fixed point theorems, some results on the existence and multiplicity of positive solutions are obtained. Furthermore, we provide a representative example to illustrate a possible application of the established results. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we construct the pullback exponential attractors for evolution processes in which the difference of 2 solutions lacks the smoothing property. To do this, by the uniform squeezing property of the corresponding discrete process, we add the points to the pullback attractor such that every new set of it has the finite fractal dimension and pullback exponentially attracts every bounded subset of the phase space. As the applications, we establish the existence of pullback exponential attractors for non‐autonomous reaction‐diffusion equation without any restriction on the growing order of nonlinear term and non‐autonomous strongly damped wave equation in R n with critical nonlinearity. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to investigate the decay of Bernstein numbers of the embedding from [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach for the estimation of spectroscopic data by combining the predictions of an ensemble of estimators using the induced ordered weighted averaging (IOWA) fusion operators. For ensemble generation, we use Gaussian process regression (GPR) and extreme learning machine (ELM) estimators associated with different kernels. To render the model selection issue of ELM as efficiently as in the GPR Bayesian estimation method, we develop an automatic solution based on the powerful differential evolution (DE) algorithm. During the fusion process, the IOWA operator needs two things: (1) an order-inducing value; and (2) a way to determine its weights. For the order-inducing value, we propose to use the residual of each estimated output value. Because we cannot compute the true residual, we explore the idea of estimating the residuals themselves by associating to each estimator of the ensemble a second estimator of the same kind called a residual estimator. To learn the weights associated with these nonlinear operators, the proposed method relies on the concept of prioritized aggregation, where we generate the weights directly from the estimated residuals. Experimental results obtained on three real spectroscopic datasets confirm the interesting capabilities of the proposed IOWA fusion method. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras, where the entries are nonlinear operators. Furthermore, we apply the obtained results to a coupled system of nonlinear equations. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we state and prove two convergence results for the solution of a history‐dependent variational‐hemivariational inequality. The first one concerns the pointwise convergence of the solution with respect to the data, including the set of constraints, the nonlinear operator and the two functionals which govern the inequality. The second result, obtained under additional assumptions, concerns the uniform convergence of the solution with respect to the set of constraints. These convergence results allow us to consider two general optimization problems for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to history‐dependent variational‐hemivariational inequalities. To provide an example, we illustrate our results in the study of a nonlinear problem with unilateral constraints and subdifferential boundary conditions. [ABSTRACT FROM AUTHOR]

Some chaotic systems can be realized with different nonlinear functions. These systems consist of a fixed main system block and a changeable nonlinear function block. Studying with these generalized systems is very useful from the educational point of view including general modeling and design issues of chaotic systems. SIMULINK, a graphical programming tool, offers a very good environment for dynamic modeling of such generalized chaotic systems. This paper presents generalized two chaotic systems, which can be utilized in chaotic system modeling for engineering applications and the use of SIMULINK in dynamical modeling and simulation of these generalized systems which can be realized with multiple nonlinear functions. The proposed models have been integrated into undergraduate nonlinear circuits and systems course at Erciyes University, Kayseri, Turkey. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 18: 684-693, 2010; View this article online at ; DOI 10.1002/cae.20273 [ABSTRACT FROM AUTHOR]

It is known that many kinds of problems, including as typical examples the shortest path problem studied in the Operations Research and the data flow analysis problem treated in the compiler technique, can be formulated uniformly as a fixed-point problem on graphs. This paper deals with elimination-type algorithms for solving such problems. First, a unified formulation of the problem is presented and then two categories of elimination-type solution methods are identified and classified by the elimination order and its meaning on graphs. One class is characterized as maintaining a spanning tree and the other as maintaining a maximal acyclic graph in the upper triangular matrix. The former is efficient specifically when applied to reducible graphs. This approach is reexamined from a new viewpoint. The latter is a method newly proposed by the authors and is applicable to general irreducible graphs and has practical features. [ABSTRACT FROM AUTHOR]

This paper considers the characterization of surface skimming waves (SSBW) by two newly defined parameters, i.e., effective electromechanical coupling factor and critical length. The parameters are estimating easily by the effective permittivity and are directly applicable to the delta function model analysis. Experiments on a TVIF SAW filter agreed well with the analysis. In addition, these parameters for SO2/36° YX-LiTaO3 structure were estimated. It is shown both theoretically and experimentally that the SSBW responses can be suppressed by depositing SiO4 with appropriate thickness. [ABSTRACT FROM AUTHOR]

Let V$$ V $$ be a finite set, E⊂2V$$ E\subset {2}^V $$ be a set of hyperedges, and w:E→(0,∞)$$ w:E\to \left(0,\infty \right) $$ be an edge weight. On the (wighted) hypergraph G=(V,E,w)$$ G=\left(V,E,w\right) $$, we can define a multivalued nonlinear operator LG,p$$ {L}_{G,p} $$ (p∈[1,∞)$$ p\in \left[1,\infty \right) $$) as the subdifferential of a convex function on ℝV$$ {\mathbb{R}}^V $$, which is called "hypergraph p$$ p $$‐Laplacian." In this article, we first introduce an inequality for this operator LG,p$$ {L}_{G,p} $$, which resembles the Poincaré–Wirtinger inequality in PDEs. Next, we consider an ordinary differential equation on ℝV$$ {\mathbb{R}}^V $$ governed by LG,p$$ {L}_{G,p} $$, which is referred as "heat" equation on the graph and used to study the geometric structure of the hypergraph in recent researches. With the aid of the Poincaré–Wirtinger type inequality, we can discuss the existence and the large time behavior of solutions to the ODE by procedures similar to those for the standard heat equation in PDEs with the zero Neumann boundary condition. [ABSTRACT FROM AUTHOR]

Lp$$ {L}^p $$ estimates for the fractional Φ$$ \Phi $$‐Laplacian operator defined in bounded domains are established, where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with Moser's iteration, we prove that any weak solution for fractional Φ$$ \Phi $$‐Laplacian operator defined in bounded domains belongs to L∞(Ω)$$ {L}^{\infty}\left(\Omega \right) $$, under appropriate hypotheses on the N$$ N $$‐function Φ$$ \Phi $$. Using the Orlicz space and taking into account the fractional setting for our problem, the main results are stated for a huge class of nonlinear operators and nonlinearities. [ABSTRACT FROM AUTHOR]

In this article, we consider the nonlinear problem ∂u∂t=Au+V(x)um+p−2+λuqinΩ×(0,T),u(x,0)=u0(x)≥0inΩ,|∇u|p−2∂u∂ν=β(x)up−1on∂Ω×(0,T).$$ \left\{\begin{array}{cc}\frac{\partial u}{\partial t}&amp;#x0003D;\mathbf{A}u&amp;#x0002B;V(x){u}&amp;#x0005E;{m&amp;#x0002B;p-2}&amp;#x0002B;\lambda {u}&amp;#x0005E;q\kern0.30em &amp; \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)&amp;#x0003D;{u}_0(x)\ge 0\kern0.30em &amp; \mathrm{in}\kern0.30em \Omega, \\ {}{\left&amp;#x0007C;\nabla u\right&amp;#x0007C;}&amp;#x0005E;{p-2}\frac{\partial u}{\partial \nu }&amp;#x0003D;\beta (x){u}&amp;#x0005E;{p-1}\kern0.30em &amp; \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right).\end{array}\right. $$where Au=div(mum−1|∇u|p−2∇u)$$ \mathbf{A}u&amp;#x0003D;\operatorname{div}\left(m{u}&amp;#x0005E;{m-1}{\left&amp;#x0007C;\nabla u\right&amp;#x0007C;}&amp;#x0005E;{p-2}\nabla u\right) $$ is the doubly nonlinear operator. Here, Ω⊂ℝN$$ \Omega \subset {\mathbb{R}}&amp;#x0005E;N $$ is a bounded domain with smooth boundary, m>0,10,λ∈ℝ,q>0$$ m&gt;0,1&lt;p&lt;N,V\in {L}_{loc}&amp;#x0005E;1\left(\Omega \right),m&amp;#x0002B;p-2&gt;0,\lambda \in \mathbb{R},q&gt;0 $$ and β∈Lloc1(∂Ω)$$ \beta \in {L}_{loc}&amp;#x0005E;1\left(\mathrm{\partial \Omega}\right) $$. We establish some sufficient conditions on the functions V(x),β(x)$$ V(x),\beta (x) $$, and the exponents m+p$$ m&amp;#x0002B;p $$ and q$$ q $$, so that the above problem has no positive solutions. Furthermore, various concrete potentials V(x)$$ V(x) $$ are taken into account to demonstrate applications of our main result. [ABSTRACT FROM AUTHOR]

Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic N‐function, we prove the uniqueness for a broad class of measures. In order to provide it, the framework of capacities in fully anisotropic Orlicz–Sobolev spaces is developed and the capacitary characterization of a bounded measure is given. Moreover, we give an example of an anisotropic Young function Φ, such that |ξ|p≲Φ(ξ)≲|ξ|plogα(1+|ξ|), with arbitrary p ≥ 1, α > 0, but so irregularly growing that the Orlicz–Sobolev‐type space generated by Φ indispensably requires fully anisotropic tools to be handled. [ABSTRACT FROM AUTHOR]

Iterative ensemble filters and smoothers are now commonly used for geophysical models. Some of these methods rely on a factorization of the observation likelihood function to sample from a posterior density through a set of "tempered" transitions to ensemble members. For Gaussian‐based data assimilation methods, tangent linear versions of nonlinear operators can be relinearized between iterations, thus leading to a solution that is less biased than a single‐step approach. This study adopts similar iterative strategies for a localized particle filter (PF) that relies on the estimation of moments to adjust unobserved variables based on importance weights. This approach builds off a "regularization" of the local PF, which forces weights to be more uniform through heuristic means. The regularization then leads to an adaptive tempering, which can also be combined with filter updates from parametric methods, such as ensemble Kalman filters. The role of iterations is analyzed by deriving the localized posterior probability density assumed by current local PF formulations and then examining how single‐step and tempered PFs sample from this density. From experiments performed with a low‐dimensional nonlinear system, the iterative and hybrid strategies show the largest benefits in observation‐sparse regimes, where only a few particles contain high likelihoods and prior errors are non‐Gaussian. This regime mimics specific applications in numerical weather prediction, where small ensemble sizes, unresolved model error, and highly nonlinear dynamics lead to prior uncertainty that is larger than measurement uncertainty. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive periodic solutions to second-order singular differential equations. The proof relies on Schauders fixed point theorem. Our results show that in some situations weak singularities can help create periodic solutions, as pointed out by Torres [J. Differential Equations 232 (2007) 277–284]. [ABSTRACT FROM AUTHOR]

The omega limit sets of a nonlinear operator $T$ which is defined on a positive cone and satisfies certain ray-contractive type conditions are discussed. Under the assumption that the restriction of $T$ to a compact subset is surjective, the following alternatives are proved: the omega limit set of a point in the cone either consists of a fixed point or forms a 2-cycle. In addition, new proofs and extensions to relevant results are given. [ABSTRACT FROM AUTHOR]

We study the problem of the existence and uniqueness of solutions to the Bellman equation in the presence of unbounded returns. We introduce a new approach based both on consideration of a metric on the space of all continuous functions over the state space, and on the application of some metric fixed point theorems. With appropriate conditions we prove uniqueness of solutions with respect to the whole space of continuous functions. Furthermore, the paper provides new sufficient conditions for the existence of solutions that can be applied to fairly general models. It is also proven that the fixed point coincides with the value function and that it can be approached by successive iterations of the Bellman operator. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of μ‐pseudo almost automorphic solution for a delayed non‐autonomous partial functional differential equation in the exponential dichotomic case, where the nonlinear operator F satisfies the φ‐Lipschitz condition and φ belongs to some admissible spaces. We further prove the existence of an invariant stable manifold around the μ‐pseudo almost automorphic solution in that case. An application is given to illustrate our theory. [ABSTRACT FROM AUTHOR]

In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators (which can be either degenerate or singular when "the gradient is small") under strong absorption conditions of the general form: G(x,Du,D2u)=f(u)χ{u>0}inΩ,where the mapping u↦f(u) fails to decrease fast enough at the origin, so allowing that nonnegative solutions may create plateau regions, that is, a priori unknown subsets where a given solution vanishes identically. We establish improved geometric Clocκ regularity along the set F0=∂{u>0}∩Ω (the free boundary of the model), for a sharp value of κ≫1 (obtained explicitly) depending only on structural parameters. Non‐degeneracy among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings. [ABSTRACT FROM AUTHOR]

This work is concerned with the homogenization of initial boundary value parabolic equations with hysteresis, containing nonlinear monotone operators in the diffusion term. We make use of the sigma‐convergence concept together with the properties of hysteresis operator to derive the homogenized problem. The homogenized operator is obtained in terms of a solution of a nonlinear corrector problem posed and solved in the usual sense of distributions. We also provide an approximation scheme for the homogenized coefficient. [ABSTRACT FROM AUTHOR]

Elliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator and the corresponding isomorphism of the linearized operator; second, the definition of an auxiliary problem in deriving the relation between the L2 norm and H1 norm of the Ritz projection error; third, the bilinearity/nonbilinearity of the linearized variational forms. The symmetric homotopy for the discretized equations preserves not only D4 symmetry, but also structural symmetry. With the symmetric homotopy, a filter strategy and a finite element Newton refinement, multiple solutions to a system of semilinear elliptic equations arising from Bose–Einstein condensate are found. [ABSTRACT FROM AUTHOR]

We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy‐type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile of global solutions. [ABSTRACT FROM AUTHOR]

By studying a weakly singular integral whose kernel involves Mittag‐Leffler functions, we obtain some new Gronwall‐type integral inequalities. Applying these inequalities and fixed point theorems, existence and uniqueness of positive solution of initial value problem to nonlinear fractional differential equation with Caputo‐like counterpart hyper‐Bessel operators are established. [ABSTRACT FROM AUTHOR]

We discuss the existence of solutions for a class of nonlinear equations driven by the fractional p‐Laplacian operator of the form (−Δ)psu+λsV(x,λ)|u|p−2u=f(x,u)inRN,where 0ps and λ is a positive parameter. By combining variational techniques with a version of the mountain pass theorem without Palais‐Small (PS) condition, we establish the existence of nontrivial solutions for the above equation under certain appropriate assumptions on nonlinearity and weight functions. [ABSTRACT FROM AUTHOR]

We obtain Calderón–Zygmund‐type estimates in generalized Morrey spaces Lq,φ(Ω) for nonlinear equations of p‐Laplacian type. Our result is obtained under minimal regularity assumptions on both the operator and the domain Ω. This result allows us to study more general class of nonlinear equations which are asymptotically regular, that is, 'near' enough to some nonlinear operator of p‐Laplacian type. As a by‐product, we obtain also generalized Hölder regularity C0,φ(Ω) of the solution under some restrictions on the generalizing function φ. [ABSTRACT FROM AUTHOR]