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Although data-driven models, especially deep learning, have achieved astonishing results on many prediction tasks for nonlinear sequences, challenges still remain in finding an appropriate way to embed prior knowledge of physical dynamics in these models. In this work, we introduce a convex relaxation approach for learning robust Koopman operators of nonlinear dynamical systems, which are intended to construct approximate space spanned by eigenfunctions of the Koopman operator. Different from the classical dynamic mode decomposition, we use the layer weights of neural networks as eigenfunctions of the Koopman operator, providing intrinsic coordinates that globally linearize the dynamics. We find that the approximation of space can be regarded as an orthogonal Procrustes problem on the Stiefel manifold, which is highly sensitive to noise. The key contribution of this paper is to demonstrate that strict orthogonal constraint can be replaced by its convex relaxation, and the performance of the model can be improved without increasing the complexity when dealing with both clean and noisy data. After that, the overall model can be optimized via backpropagation in an end-to-end manner. The comparisons of the proposed method against several state-of-the-art competitors are shown on nonlinear oscillators and the lid-driven cavity flow. [ABSTRACT FROM AUTHOR]

There are many risk factors in EPB shield construction. The traditional fuzzy analytic hierarchy process (FAHP) method usually uses a linear analysis method to determine the risk level, but there are often some risk factors with prominent influence, which will reduce the accuracy of the evaluation results. In this paper, a new risk assessment model of Earth pressure balance (EPB) shield construction based on a nonlinear FAHP method is established by introducing nonlinear factors into the comprehensive calculation of the traditional FAHP. First, the new model establishes the framework of EPB shield construction risk analysis based on the work breakdown structure (WBS) and risk breakdown structure (RBS) methods. Then, it constructs an EPB shield construction risk index system by coupling the units of the WBS and RBS. The model constructs a fuzzy consistent judgment matrix, which replaces the 1∼9 scale. Finally, the nonlinear operator is introduced into the FAHP comprehensive calculation, considering the influence of some prominent risk factors, which improves the accuracy of the risk assessment. By applying the new model to the risk analysis of the EPB shield construction section of a tunnel project in Hangzhou, the effectiveness of the model is further verified. [ABSTRACT FROM AUTHOR]

The aim of the paper is to discuss data dependence, existence of fixed points, strict fixed points, and well posedness of some multivalued generalized contractions in the setting of complete metric spaces. Using auxiliary functions, we introduce Wardowski type multivalued nonlinear operators that satisfy a novel class of contractive requirements. Furthermore, the existence and data dependence findings for these multivalued operators are obtained. A nontrivial example is also provided to support the results. The results generalize, improve, and extend existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear fractional-order PI (NL-FO-PI) controller is proposed for primary frequency control (PFC) of a wind farm based on the squirrel cage induction generator. The new structure composites a fractional-order operator and nonlinear function to achieve better control performance for the PFC system. The benchmarking process is demonstrated by investigating the performance of fractional-order PI (FO-PI) and nonlinear PI (NL-PI) controllers. Initially, the controller is applied to a single-area power system for design and stability study and then extended to the two-area interconnected wind farm to validate the applicability in the more realistic power system. The proposed control method ensures the balance of power and keeps the system frequency within a suitable range. The simulation results demonstrate that the proposed NL-FO-PI controller provides less percentage overshoot, settling time, rise time, and peak time than other controllers. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables. [ABSTRACT FROM AUTHOR]

Ćirić and Prešić developed the concept of Prešić contraction to Ćirić-Prešić type contractive mappings in the background of a metric space. On the other hand, Altun and Olgun introduced Perov type F-contractions. In this paper, we extend the concept of Ćirić-Prešić contractions to Perov-Ćirić-Prešić type F-contractions. Our results modify some known ones in the literature. To support our main result, an example and an application to nonlinear operator systems are presented. [ABSTRACT FROM AUTHOR]

Ćirić and Prešić developed the concept of Prešić contraction to Ćirić-Prešić type contractive mappings in the background of a metric space. On the other hand, Altun and Olgun introduced Perov type F-contractions. In this paper, we extend the concept of Ćirić-Prešić contractions to Perov-Ćirić-Prešić type F-contractions. Our results modify some known ones in the literature. To support our main result, an example and an application to nonlinear operator systems are presented. [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]

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]

The main purpose of this paper is to establish and prove some new common fixed point theorems for intuitionistic fuzzy maps in the context of (α,β)-cut sets of intuitionistic fuzzy sets on a complete metric space in association with the Hausdorff metric. Furthermore, the technique of Meir-Keeler (shortly, M-K) contraction is applied to obtain common fixed point of intuitionistic fuzzy compatible maps and fixed points of Kannan type intuitionistic fuzzy set-valued contractive mappings. Our results generalize M-K type fixed point theorem along with its various generalizations. Some nontrivial examples have been furnished in the support of the main results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find out fixed point results for semi-α⁎-dominated multivalued mappings fulfilling a new generalized locally F-dominated multivalued contractive condition on a closed ball in complete dislocated metric space. Example and application both are given to show the novelty of our results. Our results merge, extend, and infer many results. [ABSTRACT FROM AUTHOR]

We introduce the notion of generalized θ-ϕ contraction and establish some new fixed point theorems for this contraction in the setting of complete G-metric spaces. The results presented in the paper improve, extend, and unify some known results. Finally, we give an example to illustrate them. [ABSTRACT FROM AUTHOR]

This paper deals with solvability of the third-order nonlinear partial difference equation with delays Δn{am,n Δ2m{xm,n + bm,nxm-τ0,n-σ0))+ f {m,n,xm-τ1,n-σ1,n ...,xm-τk,m,n-σk,n) = cm,n, m ≥ m0, n ≥ n0. With the help of the Banach fixed-point theorem, the existence results of uncountably many bounded positive solutions for the partial difference equation are given; some Mann iterative schemes with errors are suggested, and the error estimates between the iterative schemes and the bounded positive solutions are discussed. Three nontrivial examples illustrating the results presented in this paper are also provided. [ABSTRACT FROM AUTHOR]

In this paper, two classes of first-order neutral functional differential equations with periodic delays are considered. Some results on the existence of positive periodic solutions for the equations are obtained by using the Krasnoselskii fixed point theorem. Four examples are included to illustrate our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equation ?(a(n, ya1n, . . ., yarn)?(yn + bnyn-t )) + f(n, yf1n, . . ., yfkn) = cn, ∀n = n0. By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature. [ABSTRACT FROM AUTHOR]

It is well known that the variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest and analyze some new proximal point methods for solving the variational inequalities. These new methods include the explicit, the implicit, and the extragradient methods as special cases. The convergence analysis of the new methods is considered under some suitable conditions. Results proved in this paper may stimulate further research in this direction. [ABSTRACT FROM AUTHOR]

This paper presents that the kernel of the fractional Fourier transform (FRFT) satisfies the operator version of Kramer's Lemma (Hong and Pfander, 2010), which gives a new applicability of Kramer's Lemma. Moreover, we give a new sampling formulae for reconstructing the operators which are band limited in the FRFT sense. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to use a new hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family of quasi-φ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present two new forward-backward splitting schemes with relaxations and errors for finding a common element of the set of solutions to the variational inclusion problem with two accretive operators and the set of fixed points of strict pseudocontractions in infinite-dimensional Banach spaces. Under mild conditions, some weak and strong convergence theorems for approximating these common elements are proved. The methods in the paper are novel and different from those in the early and recent literature. Further, we consider the problem of finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a strict pseudocontractions. [ABSTRACT FROM AUTHOR]

Bhaskar and Lakshmikantham (2006) showed the existence of coupled coincidence points of a mapping F from X x X into X and a mapping g from X into X with some applications. The aim of this paper is to extend the results of Bhaskar and Lakshmikantham and improve the recent fixed-point theorems due to Bessem Samet (2010). Indeed, we introduce the definition of generalized g-Meir-Keeler type contractions and prove some coupled fixed point theorems under a generalized g-Meir-Keeler-contractive condition. Also, some applications of the main results in this paper are given. [ABSTRACT FROM AUTHOR]

In this paper, the numerical range for two operators both linear and nonlinear have been studied in semi-inner product spaces. The inclusion relations between numerical range, approximate point spectrum, compression spectrum, eigenspectrum, and spectrum have been established for two linear operators. We also show the inclusion relation between approximate point spectrumand closure of the numerical range for two nonlinear operators. An approximation method for solving the operator equation involving two nonlinear operators is also established. [ABSTRACT FROM AUTHOR]

Lakzian and Samet (2010) studied some fixed-point results in generalized metric spaces in the sense of Branciari. In this paper, we study the existence of fixed-point results of mappings satisfying generalized weak contractive conditions in the framework of a generalized metric space in sense of Branciari. Our results modify and generalize the results of Laksian and Samet, as well as, our results generalize several well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

In the recent paper "common fixed point theorems for commutating mappings in fuzzy metric spaces," the authors proved that a common fixed point theorem for commutating mappings in G-complete fuzzy metric spaces and gave an example to illustrate the main result. In this note, we point out that the above example is incorrect because it does not satisfy the condition of Gcompleteness, and then two appropriate examples are given. In addition, we prove that the theorem proposed by Zheng and Lian actually holds in an M-complete fuzzy metric space. Our results improve and extend some existing results in the relevant literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained. [ABSTRACT FROM AUTHOR]

Cho et al. [Comput. Math. Appl. 61(2011), 1254-1260] studied common fixed point theorems on cone metric spaces by using the concept of c-distance. In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept of c-distance in cone metric spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present some fixed point theorem in dislocated quasimetric space for expansive type mappings. [ABSTRACT FROM AUTHOR]

This paper is about a problem concerning nonlinear Yamabe-type operators of negative admissible metrics. We first give a result on sk Yamabe problem of negative admissiblemetrics by virtue of the degree theory in nonlinear functional analysis and the maximum principle and then establish an existence and uniqueness theorem for the solutions to the problem. [ABSTRACT FROM AUTHOR]

A new concept of the c-distance in conemetric space has been introduced recently in 2011. The aim of this paper is to extend and generalize some coupled fixed-point theorems on c-distance in cone metric space. Some examples are given. [ABSTRACT FROM AUTHOR]

We generalize and extend Brosowski-Meinardus type results on invariant points from the set of best approximation to the set of ε-simultaneous approximation. As a consequence some results on ε-approximation and best approximation are also deduced. The results proved in this paper generalize and extend some of the known results on the subject. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to present some best proximity point theorems for K-cyclic mappings and C-cyclic mappings in the frameworks of metric spaces and uniformly convex Banach spaces, thereby furnishing an optimal approximate solution to the equations of the form Tx = x where T is a non-self mapping. [ABSTRACT FROM AUTHOR]

This paper outlines a detailed study of the coupling of He's polynomials with correction functional of variational iteration method (VIM) for solving various initial and boundary value problems. The elegant coupling gives rise to the modified versions of VIM which is very efficient in solving nonlinear problems of diversified nature. It is observed that the variational iteration method using He's polynomials (VIMHP) is very efficient, easier to implements, and more user friendly. Several examples are given to reconfirm the efficiency of the proposed VIMHP. [ABSTRACT FROM AUTHOR]

Starting from the study of the Shepard nonlinear operator of max-prod type by Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324-326, the Bernstein max-prod-type operator is introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form Cω1(f; 1/√n) (with an unexplicit absolute constant C > 0) and the question of improving the order of approximation ω1(f; 1/√n) is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of ω1(f; 1/√n) and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions f including, for example, that of concave functions, we find the order of approximation ω1(f; 1/√n),which for many functions f is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained. [ABSTRACT FROM AUTHOR]

We introduced n-tupled coincidence point for a pair of maps T:Xn → X and A : X→X in Menger space. Utilizing the properties of the pseudometric and the triangular norm, we will establish n-tupled coincidence point theorems under weak compatibility as well as n-tupled ixed point theorems for hybrid probabilistic ψ-contractions with a gauge function. Our main results do not require the conditions of continuity and monotonicity of ψ. At the end of this paper, an example is given to support our main theorem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the existence of solutions to a general elliptic problem Au+g(x,u,∇u)=f-div⁡F, with f∈L1(Ω) and F∈∏i=1NLp'(Ω,ωi*), where A is a Leray-Lions operator from a weighted Sobolev space into its dual and g(x,s,ξ) is a nonlinear term satisfying gx,s,ξsgn⁡(s)≥ρ∑i=1Nωi|ξi|p, |s|≥h>0, and a growth condition with respect to ξ. Here, ωi, ωi* are weight functions that will be defined in the Preliminaries. [ABSTRACT FROM AUTHOR]

The digital power grid system is used to maintain stability and to provide a continuous supply of electrical energy to consumers in the event of a power outage or other interruption of service. This is due to their ability to respond quickly in the event of an interruption. However, unpredictable failures and cascading accidents occur regularly, resulting in a blackout that can cause significant disruption to modern life. Furthermore, the rotor angle behavior can indicate the overall synchronism and stability of the complete digital power grid system. The digital dynamic nonlinear system operators can profit economically and technically from implementing protection measures that are appropriately designed and implemented. This study has utilized data transitions over digital power grids, connectivity used in digital power grids, power quality, network congestion, and stability issues as independent variables. However, the digital dynamic nonlinear system has been used as the dependent variable. The data was collected using questionnaires, and the data was collected from various digital power grids and solar system users. The collected data was analyzed by using SEM PLS 3. The reliability of the collected data was tested by Cronbach's alpha and the values above 0.7. The results indicated that there are significant values between the variables. [ABSTRACT FROM AUTHOR]

Recently, Azamet al. introduced the notion of complex valued metric spaces and proved fixed point theorems under the contraction condition. Rao et al. introduced the notion of complex valued b-metric spaces. In this paper, we obtain some fixed point results for the mapping satisfying rational expressions in complex valued b-metric spaces. Also, an example is given to illustrate our obtained result. [ABSTRACT FROM AUTHOR]

The aim of this paper is to unify the concept of greatest lower bound (g.l.b) property and establish some generalized common fixed results. We support our results by a nontrivial example. [ABSTRACT FROM AUTHOR]

This paper studies the stability of fixed points for multi-valued mappings in relation to selections. For multi-valued mappings admitting Michael selections, some examples are given to show that the fixed point mapping of these mappings are neither upper semi-continuous nor almost lower semi-continuous. Though the set of fixed points may be not compact for multi-valued mappings admitting Lipschitz selections, by finding sub-mappings of such mappings, the existence of minimal essential sets of fixed points is proved, and we show that there exists at least an essentially stable fixed point for almost all these mappings. As an application, we deduce an essentially stable result for differential inclusion problems. [ABSTRACT FROM AUTHOR]

There are many articles in the literature dealing with differential subordination problems for analytic functions in the unit disk, and only a few articles deal with the above problems in the upper half-plane. In this paper, we aim to derive several differential subordination results for analytic functions in the upper half-plane by investigating certain suitable classes of admissible functions. Some useful consequences of our main results are also pointed out. [ABSTRACT FROM AUTHOR]

We consider a strong mixed vector equilibrium problem in topological vector spaces. Using generalized Fan-Browder fixed point theorem (Takahashi 1976) and generalized pseudomonotonicity for multivalued mappings, we provide some existence results for strong mixed vector equilibrium problem without using KKM-Fan theorem. The results in this paper generalize, improve, extend, and unify some existence results in the literature. Some special cases are discussed and an example is constructed. [ABSTRACT FROM AUTHOR]

This paper focuses on the multidirectional associative memory (MAM) neural networks with m fields which is more advanced to realize associative memory. Based on the Brouwer fixed point theorem and Dini upper right derivative, it is confirmed that the multidirectional associativememory neural network can have 3l equilibria and 2l equilibria of themare stable, where l is a parameter associated with the number of neurons. Furthermore, an example is given to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

In this paper we study semilinear equations of the form Au + λF(u) = f, where A is a linear self-adjoint operator, satisfying a strong positivity condition, and F is a nonlinear Lipschitz operator. As applications we develop Krasnoselskii and Ky Fan type approximation results for certain pair of maps and to illustrate the usability of the obtained results, the existence of solution of an integral equation is provided. [ABSTRACT FROM AUTHOR]

New fixed-point theorems on metric spaces are established, and analogous results on partial metric spaces are deduced. This work can be considered as a continuation of the paper Samet et al. (2013). [ABSTRACT FROM AUTHOR]

Fixed-point theory in complex valuedmetric spaces has greatly developed in recent times. In this paper, we prove certain common fixed-point theorems for two single-valuedmappings in such spaces. The mappings we consider here are assumed to satisfy certain metric inequalities with generalized fixed-point theorems due to Rouzkard and Imdad (2012). This extends and subsumes many results of other authors which were obtained for mappings on complex-valued metric spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show the use of the coupled quasisolutions method as a useful technique when dealing with ordinary differential equations with functional arguments of bounded variation. We will do this by looking for solutions for a first-order ordinary differential equation with an advanced argument of bounded variation. The main trick is to use the Jordan decomposition of this argument in a nondecreasing part and a nonincreasing one. As a necessary step, we will also talk about coupled fixed points of multivalued operators. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define modified weak α-ψ-contractive mappings and to establish fixed point results for such mappings defined on partialmetric spaces using the notion of triangular ψ-admissibility. As an application, we prove newfixed point results for graphic weak ψ-contractivemappings. Moreover, some examples and an application to integral equation are given here to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]

This paper establishes the existence of periodic solution for a kind of second-order singular nonautonomous coupled systems. Our approach is based on fixed point theorem in cones. Examples are given to illustrate the main result. [ABSTRACT FROM AUTHOR]

In this paper we study and prove some new fixed points theorems for pointwise and asymptotic pointwise contraction mappings in modular metric spaces. [ABSTRACT FROM AUTHOR]