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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]

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]

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]

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]

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]

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators. [ABSTRACT FROM AUTHOR]

Predicting complex dynamics in physical applications governed by partial differential equations in real-time is nearly impossible with traditional numerical simulations due to high computational cost. Neural operators offer a solution by approximating mappings between infinite-dimensional Banach spaces, yet their performance degrades with system size and complexity. We propose an approach for learning neural operators in latent spaces, facilitating real-time predictions for highly nonlinear and multiscale systems on high-dimensional domains. Our method utilizes the deep operator network architecture on a low-dimensional latent space to efficiently approximate underlying operators. Demonstrations on material fracture, fluid flow prediction, and climate modeling highlight superior prediction accuracy and computational efficiency compared to existing methods. Notably, our approach enables approximating large-scale atmospheric flows with millions of degrees, enhancing weather and climate forecasts. Here we show that the proposed approach enables real-time predictions that can facilitate decision-making for a wide range of applications in science and engineering. Real-time prediction of dynamics for complex physical systems governed by partial differential equations is challenging and computationally expensive. The authors propose a framework for learning neural operators in latent spaces that allows real-time predictions of high-dimensional nonlinear systems. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a dichotomy of iterated means of nonlinear operators acting on a compact convex subset of a finite-dimensional real Banach space. As an application, we study the mean ergodicity of nonhomogeneous Markov chains. [ABSTRACT FROM AUTHOR]

The existence of some class of multi-order nonlinear fractional systems is investigated in this paper. Some sufficient conditions of solutions for multi-order nonlinear systems are obtained based on fixed point theorems. Our results in this paper improve some known results. [ABSTRACT FROM AUTHOR]

In 1968, H. Brezis introduced a notion of operator pseudomonotonicity which provides a unified approach to monotone and nonmonotone variational inequalities. A closely related notion is that of Ky-Fan hemicontinuity, a continuity property which arises if the famous Ky-Fan minimax inequality is applied to the variational inequality framework. It is clear from the corresponding definitions that Ky-Fan hemicontinuity implies Brezis pseudomonotonicity, but quite surprisingly, a recent publication by Sadeqi and Paydar (J Optim Theory Appl 165(2):344-358, 2015) claims the equivalence of the two properties. The purpose of the present note is to show that this equivalence is false; this is achieved by providing a concrete example of a nonlinear operator which is Brezis pseudomonotone but not Ky-Fan hemicontinuous. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial value problem(IVP henceforth) associated with the higher order nonlinear dispersive equation given in Jones et al. (Int J Math Math Sci 24:371–377, 2000): ∂ t u + α ∂ x 7 u + β ∂ x 5 u + γ ∂ x 3 u + μ ∂ x u + λ u ∂ x u = 0 , x ∈ R , t ∈ R , u (x , 0) = u 0 (x) , x ∈ R with the initial data in the Sobolev space H s (R). Benefited from the ideas of Huo and Jia (Z Angew Math Phys 59:634–646, 2008), Zhang et al. (Acta Math Sci 37B(2):385–394, 2017) and Zhang and Huang (Math Methods Appl Sci 39(10):2488–2513, 2016) that is, using Fourier restriction norm method, Tao's [k, Z]-multiplier method and the contraction mapping principle, we prove that IVP is locally well-posed for the initial data u 0 ∈ H s (R) with s ≥ - 5 8 . Moreover, based on the local well-posedness and conservation law, we establish the global well-posedness for the initial data u 0 ∈ H s (R) with s = 0 . [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial-value problem for nonlinear first-order impulsive integro-differential equations on time scales T . We establish certain existence criteria by using a fixed-point theorem for operator on cones, under which such problems have aminimal and amaximal solution lying in a corresponding region bounded by their lower and upper solutions. [ABSTRACT FROM AUTHOR]

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce a viscosity iterative forward-backward splitting method with errors to find zeros of the sum of two accretive operators in Banach spaces. We shall prove the strong convergence of the method under mild conditions. We also discuss applications of these methods to monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new idea to investigate the important basic problem: how to extend Darbo' s fixed point theorem? We establish some new generalizations of Darbo's fixed point theorem by using 'integral conditions'. Our fixed point theorems extend the existing results on the problem above. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a proximal point iterative algorithm with general errors for monotone mappings in Banach spaces. We prove that the proposed algorithm converges strongly to a proximal point for monotone mappings. Our theorems in this paper improve and unify most of the results that have been proposed for this important class of nonlinear mappings. [ABSTRACT FROM AUTHOR]

In this paper, we consider two families Ψ1 and Ψ2 of mappings defined on [0,+∞) satisfy some certain properties. Using the mentioned properties for Ψ1 and Ψ2, we prove the analogous PPF dependent fixed point theorems for mappings as in Drici et al. (Nonlinear Anal. 67:641-647, 2007) in partially ordered Banach spaces where mappings satisfy the weaker contractive conditions without assuming the topological closedness with respect to the norm topology for the Razumikhin class Rc. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of nontrivial solution to a quasi-linear problem where (-Δ)psu(x)=2limϵ→0∫RN\Bε(X)|u(x)-u(y)|p-2(u(x)-u(y))|x-y|N+spdy,x∈RN is a nonlocal and nonlinear operator and p∈(1,∞), s∈(0,1), λ∈R, Ω⊂RN(N≥2) is a bounded domain which smooth boundary ∂Ω. Using the variational methods based on the critical points theory, together with truncation and comparison techniques, we show that there exists a critical value λ∗>0 of the parameter, such that if λ>λ∗, the problem (P)λ has at least two positive solutions, if λ=λ∗, the problem (P)λ has at least one positive solution and it has no positive solution if λ∈(0,λ∗). Finally, we show that for all λ≥λ∗, the problem (P)λ has a smallest positive solution. [ABSTRACT FROM AUTHOR]

In this paper we consider the following fractional system F ( x , u ( x ) , v ( x ) , F α ( u ( x ) ) ) = 0 , G ( x , v ( x ) , u ( x ) , G β ( v ( x ) ) ) = 0 , $$\begin{array}{} \displaystyle \left\{ \begin{gathered} F(x,u(x),v(x),{\mathcal{F}_\alpha }(u(x))) = 0,\\ G(x,v(x),u(x),{\mathcal{G}_\beta }(v(x))) = 0, \\ \end{gathered} \right. \end{array}$$ where 0 < α, β < 2, 𝓕α and 𝓖β are the fully nonlinear fractional operators: F α ( u ( x ) ) = C n , α P V ∫ R n f ( u ( x ) − u ( y ) ) x − y n + α d y , G β ( v ( x ) ) = C n , β P V ∫ R n g ( v ( x ) − v ( y ) ) x − y n + β d y. $$\begin{array}{} \displaystyle {\mathcal{F}_\alpha }(u(x)) = {C_{n,\alpha }}PV\int_{{\mathbb{R}^n}} {\frac{{f(u(x) - u(y))}} {{{{\left| {x - y} \right|}^{n + \alpha }}}}dy} ,\\ \displaystyle{\mathcal{G}_\beta }(v(x)) = {C_{n,\beta }}PV\int_{{\mathbb{R}^n}} {\frac{{g(v(x) - v(y))}} {{{{\left| {x - y} \right|}^{n + \beta }}}}dy}. \end{array}$$ A decay at infinity principle and a narrow region principle for solutions to the system are established. Based on these principles, we prove the radial symmetry and monotonicity of positive solutions to the system in the whole space and a unit ball respectively, and the nonexistence in a half space by generalizing the direct method of moving planes to the nonlinear system. [ABSTRACT FROM AUTHOR]

In this paper, a new iterative scheme by hybrid projection method is proposed for a finite family of Bregman totally quasi-D-asymptotically nonexpansive mappings. Conditions ensuring strong convergence are imposed to common elements of set of common fixed points of the mappings and set of common solutions to a system of generalized mixed equilibrium problems in a reflexive Banach space. These results extend many important recent ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new fixed point theorems for a 2×2 block operator matrix defined on nonempty, bounded, closed, and convex subsets of Banach algebras, where the inputs are multi-valued mappings. The obtained results are then applied to a coupled system of perturbed functional differential inclusions of initial and boundary value problems in order to prove the existence of solutions under a Carathéodory condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the time periodic traveling wave solutions for a periodic SIR epidemic model with diffusion and standard incidence. We establish the existence of periodic traveling waves by investigating the fixed points of a nonlinear operator defined on an appropriate set of periodic functions. Then we prove the nonexistence of periodic traveling via the comparison arguments combined with the properties of the spreading speed of an associated subsystem. [ABSTRACT FROM AUTHOR]

This paper deals with long-time behaviors of nonoscillatory solutions of a system of first-order dynamic equations on time scales. Some well-known fixed point theorems and double improper integrals are used to prove the main results. [ABSTRACT FROM AUTHOR]

We simplify some technical steps from Savin (Ann Math. (2) 169(1):41-78, 2009) in which a conjecture of De Giorgi was addressed. For completeness we make the paper self-contained and reprove the classification of certain global bounded solutions for semilinear equations of the type where W is a double well potential. [ABSTRACT FROM AUTHOR]

The first purpose of this paper is to define a homotopy for fuzzy spaces. We continue our work by showing that the property of having a fixed point is invariant by this homotopy. These theorems generalize and improve well-known results. [ABSTRACT FROM AUTHOR]

To reflect both fuzziness and hesitation in the evaluation of interactivity between attributes in the identification process of 2-order additive fuzzy measure, this work uses the hesitant fuzzy linguistic term set (HFLTS) to describe and depict the interactivity between attributes. Firstly, the interactivity between attributes is defined by the supermodular game theory. According to this definition, a linguistic term set is established to characterize the interactivity between attributes. Under the linguistic term set, the experts employ linguistic expressions generated by context-free grammar to qualitatively describe the interactivity between attributes. Secondly, through the conversion function, the linguistic expressions are transformed into the hesitant fuzzy linguistic term sets (HFLTSs). The individual evaluation results of all experts were further aggregated with the defined hesitant fuzzy linguistic weighted power average operator (HFLWPA). Thirdly, based on the standard Euclidean distance formula of the hesitant fuzzy linguistic elements (HFLEs), the hesitant fuzzy linguistic interaction degree (HFLID) between attributes is defined and calculated by constructing a piecewise function. As a result, a 2-order additive fuzzy measure identification method based on HFLID is proposed. Based on the proposed method, using the Choquet fuzzy integral as nonlinear integration operator, a multi-attribute decision making (MADM) process is then presented. Taking the credit assessment of the big data listed companies in China as an application example, the analysis results of application example prove the feasibility and effectiveness of the proposed method. This work successfully reflects both the fuzziness and hesitation in evaluating the interactivity between attributes in the identification process of 2-order additive fuzzy measure, enriches the theoretical framework of 2-order additive fuzzy measure, and expands the applicability and methodology of 2-order additive fuzzy measure in multi-attribute decision making. [ABSTRACT FROM AUTHOR]

The present research is aimed to analyze the existence of strict fixed points (SFPs) and fixed points of multivalued generalized contractions on the platform of controlled metric spaces (CMSs). Wardowski-type multivalued nonlinear operators have been introduced employing auxiliary functions, modifying a new contractive requirement form. Well-posedness of obtained fixed point results is also established. Moreover, data dependence result for fixed points is provided. Some supporting examples are also available for better perception. Many existing results in the literature are particular cases of the results established. [ABSTRACT FROM AUTHOR]

In this paper, by using some fixed point theorems and the measure of noncompactness, we discuss the existence of solutions for a boundary value problem of impulsive integrodifferential equations of fractional order $\alpha\in(1,2]$ . Our results improve and generalize some known results in (Zhou and Chu in Commun. Nonlinear Sci. Numer. Simul. 17:1142-1148, 2012; Bai et al. in Bound. Value Probl. 2016:63, 2016). Finally, an example is given to illustrate that our result is valuable. [ABSTRACT FROM AUTHOR]

Meir and Keeler formulated their fixed point theorem for contractive mappings with purely metric condition. This idea was extended by numerous mathematicians. In this paper, we present a simple method of proving such theorems and give new results. [ABSTRACT FROM AUTHOR]

In this paper, an Nicholson's blowflies model with time-varying delays and a harvesting term is investigated. By applying the fixed point theorem, the properties of pseudo almost periodic function, inequality analysis technique and constructing appropriate Lyapunov functionals, we establish some new criteria for the existence and convergence dynamics of pseudo almost periodic solutions for the model. An illustrative example with its numerical simulation is presented to demonstrate the effectiveness of the derived results. Our results complement some previous studies. [ABSTRACT FROM AUTHOR]

In this paper, we present a convergence rate analysis for the inexact Krasnosel'skiĭ-Mann iteration built from non-expansive operators. The presented results include two main parts: we first establish the global pointwise and ergodic iteration-complexity bounds; then, under a metric sub-regularity assumption, we establish a local linear convergence for the distance of the iterates to the set of fixed points. The obtained results can be applied to analyze the convergence rate of various monotone operator splitting methods in the literature, including the Forward-Backward splitting, the Generalized Forward-Backward, the Douglas-Rachford splitting, alternating direction method of multipliers and Primal-Dual splitting methods. For these methods, we also develop easily verifiable termination criteria for finding an approximate solution, which can be seen as a generalization of the termination criterion for the classical gradient descent method. We finally develop a parallel analysis for the non-stationary Krasnosel'skiĭ-Mann iteration. [ABSTRACT FROM AUTHOR]

The first part of this paper is a general approach towards chaotic dynamics for a continuous map $$f:X\supset M\rightarrow X$$ which employs the fixed point index and continuation. The second part deals with the differential equation with state-dependent delay. For a suitable parameter $$\alpha $$ close to $$5\pi /2$$ we construct a delay functional $$d_{{\varDelta }}$$ , constant near the origin, so that the previous equation has a homoclinic solution, $$h(t)\rightarrow 0$$ as $$t\rightarrow \pm \infty $$ , with certain regularity properties of the linearization of the semiflow along the flowline $$t\mapsto h_t$$ . The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence and uniqueness of couple fixed point theorems for three mappings satisfying some new rational contractive conditions. We prove our results in the frame work of G -metric space which is recently introduced by Aghajani et al. (Filomat 28(6):1087-1101, 2014). Illustrative example is also given to support our result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of anti-periodic (or anti-periodic differentiable) mild solutions to the semilinear differential equation $$u'(t) = Au(t) + f (t, u(t))$$ with nondense domain. Furthermore, an example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

One key point in the multiple attribute decision making is to determine the associated weights. In this paper, we first briefly review some main methods for determining the weights by using distribution functions. Then, motivated by the idea of data distribution, we develop some novel methods for obtaining the weights associated with the weighted arithmetic aggregation operators. The methods can relieve the influence of biased data on the decision results by weighting these data with small values based on the corresponding probability of data. Furthermore, some commonly used probability distribution methods are used to solve the problems in different conditions. Finally, four practical examples are provided to illustrate the weighting method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn-Hilliard equation. [ABSTRACT FROM AUTHOR]

This paper analyzes the computational complexity of fixed point search in a nonincreasing additive operator. A power algorithm determining a fixed point is proposed. A constructive proof of fixed point existence is given in a special case of a nonincreasing additive bounded-variation operator. Possible applications include voluntary financing of public good, the Cournot oligopoly and others. [ABSTRACT FROM AUTHOR]

We study the existence and nonexistence of traveling waves of a general diffusive Kermack-McKendrick SIR model with standard incidence where the total population is not constant. The three classes, susceptible S, infected I and removed R, are all involved in the traveling wave solutions. We show that the minimum wave speed of traveling waves for the three-dimensional non-monotonic system can be derived from its linearizaion at the initial disease-free equilibrium. The proof in this paper is based on Schauder fixed point theorem and Laplace transform. Our study provides a promising method to deal with high dimensional epidemic models. [ABSTRACT FROM AUTHOR]

This paper concerns with the stabilization of a Timoshenko beam with bounded constraints on boundary feedback controls. Since the resulting controlled system is nonlinear, the weak well-posedness is proven by theories of the nonlinear monotone operators and the optimization. Then, the asymptotical stability of the controlled beam is analyzed by the weak topology, and its exponential stability is also proven by the Lyapunov's second method. In the end, the numerical experiment indicates that the control design is feasible. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of fixed points for nonlinear and semilinear operators on order intervals. The abstract results unified some methods in studying the existence of positive solutions for boundary and initial value problems of nonlinear difference and differential equations. Applications are shown by examples. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharpened versions of some classical order-theoretic metrical fixed point theorems due to Nieto and Rodríguez-López (Order 22(3):223-239, 2005) using order-theoretic variants of completeness and continuity besides some another notions such as: the ICC property, the DCC property, and the MCC property. In this continuation, we further extend our results for Boyd-Wong type nonlinear contractions. Finally, as an application of our certain newly proved results, we establish the existence and uniqueness of solution of a first order periodic boundary value problem. [ABSTRACT FROM AUTHOR]

Let $(X,\|\cdot\|)$ be a Banach space. Let C be a nonempty, bounded, closed, and convex subset of X and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, it is shown that a technique of Mann which is defined by is fruitful in finding a fixed point of monotone nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In this paper, we present a new proof of the Mazur-Orlicz theorem, which uses the Markov-Kakutani common fixed point theorem, and a new proof of the Markov-Kakutani common fixed point theorem, which uses the Mazur-Orlicz theorem. [ABSTRACT FROM AUTHOR]

In this paper, several kinds of generalized vector quasi-equilibrium problems are introduced and studied in abstract convex spaces. Using the properties of Γ-convex and $\mathfrak{KC}$-maps, some sufficient conditions are given to guarantee the existence of solutions in connection with these generalized vector quasi-equilibrium problems. As applications, some existence theorems of solutions for the generalized semi-infinite programs with vector quasi-equilibrium constraints are also given. [ABSTRACT FROM AUTHOR]

In the paper, we introduce a new hybrid algorithm which is not based on the modification to weak convergence algorithms. The strong convergence theorem of the proposed algorithm is presented. Finally, the numerical experiments suggest that the new algorithm could be faster than Nakajo and Takahashi's algorithm in J. Math. Anal. Appl. 279:372-379, 2003. [ABSTRACT FROM AUTHOR]