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In this paper, a new modified Ishikawa iterative algorithm with errors by a shrinking projection method for generalized mixed equilibrium problems and a countable family of uniformly Bregman totally quasi- D-asymptotically nonexpansive mappings is introduced and investigated in the framework of a real Banach space. Strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions. These results are new and develop some recent results in this field. [ABSTRACT FROM AUTHOR]

In this paper, we prove some strong and △-convergence theorems for a finite family of multivalued quasi-nonexpansive mappings satisfying condition $(E)$ in $\operatorname{CAT}(\kappa)$ spaces. Our results extend the corresponding results of Abkar and Eslamian (Nonlinear Anal. 75:1895-1903, 2012), Panyanak (Fixed Point Theory Appl. 2014:1, 2014), Shahzad and Zegeye (Nonlinear Anal. 71:838-844, 2009) and many others. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to obtain some new common fixed point theorems for Geraghty's type contraction mappings using the monotone property with two metrics and to give some examples to illustrate the main results. Further, by using our main results, we prove some results about multidimensional common fixed points. Our results generalize and extend some recent results given by Kadelburg et al. (Fixed Point Theory Appl. 2015:27, 2015) and Choudhurya and Kundu (J. Nonlinear Sci. Appl. 5:259-270, 2012). [ABSTRACT FROM AUTHOR]

The present paper deals with the common solution method for finding a fixed point of a nonexpansive mapping and a solution of split hierarchical Minty variational inequality problems. We discuss the weak convergence of the sequences generated by the proposed method to a common solution of a fixed point problem and a split hierarchical Minty variational inequality problem. An example is presented to illustrate the proposed algorithm and result. [ABSTRACT FROM AUTHOR]

In this paper, we study strong and △-convergence of a newly defined three-step iteration process for nearly asymptotically nonexpansive mappings in the setting of $\operatorname{CAT}(k)$ spaces. Our results generalize, unify and extend many known results from the existing literature. [ABSTRACT FROM AUTHOR]

In the present paper, measure differential equations involving the distributional Henstock-Kurzweil integral are investigated. Theorems on the existence and structure of the set of solutions are established by using Schauder's fixed point theorem and Vidossich theorem. Two examples of the main results paper are presented. The new results are generalizations of some previous results in the literatures. [ABSTRACT FROM AUTHOR]

Motivated by (Kyoung and Yong-Hoon in Sci. China Math. 53:967-984, 2010) and (Chen and Zhang in Sci. China Math. 54:959-972, 2011), this paper is concerned with a boundary value problem of singular second-order differential systems with quasi-Laplacian operators on the whole line. By constructing a completely continuous nonlinear operator and using a fixed point theorem, sufficient conditions guaranteeing the existence of at least one unbounded solution are established. The methods used are standard, however, their exposition in the framework of such a kind of problems is new and skillful. Three concrete examples are given to illustrate the main theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of integral equations and apply the coincidence and common fixed point theorems for four mappings satisfying a (ψ, α, β)-weakly contractive condition in ordered metric spaces to prove the existence of a common solution to integral equations. Also we furnish suitable examples to demonstrate the validity of the hypotheses of our results. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce the notion of weakly upper semicontinuous set-valued maps and to establish a new fixed-point theorem. The set-valued maps with an approximating upper semicontinuous selection property are also defined. Next, we use our fixed-point result to obtain equilibrium existence in abstract economies with two constraints, which provide a natural scenario for potential applications of our approach to general equilibrium theory. In this regard, we set models of economies with asymmetric informed agents, who are able to improve their initial information through market signals. These economies offer examples in which the informational feasibility requirement represents an additional constraint. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of an α3-proximal admissible mappings and establish the existence of best proximity point theorems for implicit relation type modified α3-proximal contractions. As applications of our theorems, we derive some new best proximity point results for implicit relation type contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some best proximity point results in the literature. Several interesting consequences of our theorems are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we study on q-Stancu-Szasz-Beta type operators. We give these operators convergence properties and obtain a weighted approximation theorem in the interval [0,∞). [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish certain new fixed point results for multi-valued as well as single-valued maps satisfying an α-ψ-contractive conditions in complete metric space. As an application, we derive some new fixed point theorems for ψ-graphic contractions defined on a metric space endowed with a graph as well as an ordered metric space. The presented results complement and extend some very recent results proved by Asl et al. (Fixed Point Theory Appl. 2012:212, 2012) and Samet et al. (Nonlinear Anal. 75:2154-2165, 2012) as well as other theorems given by Hussain et al. (Fixed Point Theory Appl. 2013:212, 2013). Some comparative examples are constructed which illustrate the superiority of our results to the existing ones in the literature. [ABSTRACT FROM AUTHOR]

This paper is devoted to the algorithm of the linear Cauchy problem solution for large systems of first-order ordinary differential equations using parallel calculations. The proof of the convergence of the iteration process using the solution as expansion over orthogonal polynomials for the interval [0,1] is presented. The features of this algorithm are its simplicity, the opportunity to get a solution by parallel calculations, and also the possibility to get a solution for nonlinear problems by changing the operator using the solution from the iteration process. [ABSTRACT FROM AUTHOR]

Seepage prediction is a vital part of the dam safety monitoring system. Traditional statistical models ignore the nonlinear characteristics of the measured variables, resulting in poor accuracy and stability. In this study, an optimized neural network model is constructed to predict the seepage extent of hydropower station dams. An improved gradient-based optimizer (IGBO) is proposed to increase the accuracy and reliability of extreme learning machine (ELM) model predictions. The IGBO introduces an initialization method with elite opposition-based learning to improve population diversity. A crossover operator and a nonlinear parameter are used in the IGBO to enhance the ability of local search and the probability of avoiding local optima. The performance of the IGBO-optimized ELM network (IGBO-ELM) was evaluated on 12 datasets. In addition, the comparison experimental results with actual monitoring data of concrete dams show that IGBO-ELM has strong generalization performance and accuracy among the other four optimization models. [ABSTRACT FROM AUTHOR]

We study the spectral stability of the nonlinear Dirac operator in dimension 1 + 1 , restricting our attention to nonlinearities of the form f (ψ , β ψ C 2) β . We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form e - i ω t ϕ 0 . For the case of power nonlinearities f (s) = s | s | p - 1 , p > 0 , we obtain a range of frequencies ω such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition ϕ 0 , β ϕ 0 C 2 > 0 characterizes groundstates analogously to the Schrödinger case. [ABSTRACT FROM AUTHOR]

The first mixed boundary value problem for a nonlinear functional-differential equation of parabolic type with shifts in the spatial variables is considered. Sufficient conditions are proved under which a nonlinear differential-difference operator is demicontinuous, coercive, and pseudomonotone on the domain of the operator . Based on these properties, existence theorems for a generalized solution are proved. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex real Banach space E with dual space E∗. In this paper, a Krasnoselskii-type subgradient extragradient iterative algorithm is constructed and used to approximate a common element of solutions of variational inequality problems and fixed points of a countable family of relatively nonexpansive maps. The theorems proved are improvement of the results of Censor et al. (J. Optim. Theory Appl. 148:318-335, 2011). [ABSTRACT FROM AUTHOR]

In this paper, we consider a one-dimensional mean curvature equation in Minkowski space and the corresponding one-parameter problem. By using a fixed point theorem of cone expansion and compression of norm type, the existence and multiplicity of positive solutions for the above problems are obtained. Meanwhile, as applications of our results, some examples are given. [ABSTRACT FROM AUTHOR]

In this paper we study the category of discrete G-spectra for a profinite group G. We consider an embedding of module objects in spectra into a category of module objects in discrete G-spectra, and study the relationship between the embedding and the homotopy fixed points functor. We also consider an embedding of module objects in terms of quasi-categories, and show that the two formulations of embeddings are equivalent in some circumstances. [ABSTRACT FROM AUTHOR]

We consider the evolution problem associated to the infinity fractional Laplacian introduced by Bjorland et al. (Adv Math 230(4–6):1859–1894, 2012) as the infinitesimal generator of a non-Brownian tug-of-war game. We first construct a class of viscosity solutions of the initial-value problem for bounded and uniformly continuous data. An important result is the equivalence of the nonlinear operator in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially symmetric and monotone functions. Thanks to this and a comparison theorem between classical and viscosity solutions, we are able to establish a global Harnack inequality that, in particular, explains the long-time behavior of the solutions. [ABSTRACT FROM AUTHOR]

An abstract Cauchy problem for a second-order differential equation with nonlinear operator coefficients is considered. The local solvability of the problem in suitable spaces of abstract continuous and differentiable functions is proved. Sufficient conditions for finite-time blow-up of its solutions are obtained. [ABSTRACT FROM AUTHOR]

We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in a Banach space X of the form ∂ ∂ t F u t = A t F u t + f t , u t , t ≥ s , t , s ∈ R , u s = ϕ ∈ C : = C - r , 0 , X under the conditions that the family of linear partial differential operators (A(t))t∈ℝ generates the evolution family (U(t, s))t≥s with exponential dichotomy on the whole line ℝ; the difference operator F : C → X is bounded and linear, and the nonlinear delay operator f satisfies the φ-Lipschitz condition, i.e., ‖ f t , ϕ - f t , ψ ‖ ≤ φ t ‖ ϕ - ψ ‖ C for ϕ, ψ ∈ 풞, where φ(·) belongs to an admissible function space defined on ℝ. We also prove that an unstable manifold of the admissible class attracts all other solutions with exponential rates. Our main method is based on the Lyapunov – Perron equation combined with the admissibility of function spaces. We apply our results to the finite-delay heat equation for a material with memory. [ABSTRACT FROM AUTHOR]

We describe fixed points of an infinite-dimensional nonlinear operator related to a hard-core (HC) model with a countable set of spin values on a Cayley tree. This operator is defined by a countable set of parameters , , . We find a sufficient condition on these parameters under which the operator has a unique fixed point. When this condition is not satisfied, we show that the operator may have up to five fixed points. We also prove that every fixed point generates a normalizable boundary law and therefore defines a Gibbs measure for the given HC model. [ABSTRACT FROM AUTHOR]

Minimizing finite sums of smooth and strongly convex functions is an important task in machine learning. Recent work has developed stochastic gradient methods that optimize these sums with less computation than methods that do not exploit the finite sum structure. This speedup results from using efficiently constructed stochastic gradient estimators, which have variance that diminishes as the algorithm progresses. In this work, we ask whether the benefits of variance reduction extend to fixed point and root-finding problems involving sums of nonlinear operators. Our main result shows that variance reduction offers a similar speedup when applied to a broad class of root-finding problems. We illustrate the result on three tasks involving sums of n nonlinear operators: averaged fixed point, monotone inclusions, and nonsmooth common minimizer problems. In certain "poorly conditioned regimes," the proposed method offers an n-fold speedup over standard methods. [ABSTRACT FROM AUTHOR]

The article focuses on the introduction of neutrosophic continuity and neutrosophic boundedness, which is a fair extension of intuitionistic fuzzy continuity and intuitionistic fuzzy boundedness, respectively. The article further advances to illustrate the Fréchet derivative of nonlinear operators between neutrosophic normed spaces (NNS). Examples have been provided in compliance with the theory with the aid of some standard sequence spaces. [ABSTRACT FROM AUTHOR]

In this work we present a unified error analysis for abstract space discretizations of wave-type equations with nonlinear quasi-monotone operators. This yields an error bound in terms of discretization and interpolation errors that can be applied to various equations and space discretizations fitting in the abstract setting. We use the unified error analysis to prove novel convergence rates for a non-conforming finite element space discretization of wave equations with nonlinear acoustic boundary conditions and illustrate the error bound by a numerical experiment. [ABSTRACT FROM AUTHOR]

We introduce the notions of semi-uniform input-to-state stability and its subclass, polynomial input-to-state stability, for infinite-dimensional systems. We establish a characterization of semi-uniform input-to-state stability based on attractivity properties as in the uniform case. Sufficient conditions for linear systems to be polynomially input-to-state stable are provided, which restrict the range of the input operator depending on the rate of polynomial decay of the product of the semigroup and the resolvent of its generator. We also show that a class of bilinear systems are polynomially integral input-to-state stable under a certain smoothness assumption on nonlinear operators. [ABSTRACT FROM AUTHOR]

In the research of fully nonlinear elliptic partial differential equations, the concavities of the nonlinear operators is always essential. The concavity is the key to establish the curvature estimates of these equations. However, very few concave operators are known. In the present paper, some new examples are provided, which are the sum Hessian operators and the linear combination of k Hessian operators. We obtain the concavities and the quotient concavities of these new operators. As an application of our concavities, we establish the curvature estimates of convex solutions for the equations with general right-hand side defined by these new operators. [ABSTRACT FROM AUTHOR]

The paper deals with the existence and non-existence of solutions of the following nonlinear non-autonomous boundary value problem governed by the p-Laplacian operator: (P) { (h (t , x (t)) | x ′ (t) | p − 2 x ′ (t)) ′ = g (t , x (t) , x ′ (t)) a.e. t ∈ R , x (− ∞) = a , x (+ ∞) = b with a < b , where a is a positive, continuous function and g is a Caratheódory nonlinear function. We prove an existence result, underlying the relationship between the behavior of g (t , x , ⋅) as y → 0 related to that of g (⋅ , x , y) and h (⋅ , x) as | t | → + ∞ . [ABSTRACT FROM AUTHOR]

This paper is concerned with a boundary value problem for a nonlinear fractional differential equation involving a general form of Caputo fractional derivative operator with respect to new function ψ . The existence and uniqueness results of solutions are obtained. Our analysis relies on a variety of tools of fractional calculus together with fixed point theorems of Banach and Schaefer. The investigation of the results will be illustrated by providing a suitable example. [ABSTRACT FROM AUTHOR]

In this paper, we prove that linear and nonlinear equations with the Caputo–Fabrizio operators can be represented as systems of differential equations with derivatives of integer orders. The order of these equations is not more than one with respect to the integer part of the highest order of the Caputo–Fabrizio operators. We state that the Caputo–Fabrizio operators with exponential kernel cannot describe nonlocality and memory (temporal nonlocality) in processes and systems. Using the principle of nonlocality for fractional derivatives of noninteger orders ("No nonlocality. No fractional derivative"), we can state that the Caputo–Fabrizio operators cannot be considered as a fractional derivative. A general physical and economic interpretation (meaning) of the Caputo–Fabrizio operator is proposed. We state that physical and economic meaning of the Caputo–Fabrizio operators is continuously (exponentially) distributed lags. [ABSTRACT FROM AUTHOR]

This study presents a thorough investigation of the performance comparison of three ensemble data assimilation (DA) methods, including the maximum likelihood ensemble filter (MLEF), the ensemble Kalman filter (EnKF), and the iterative EnKF (IEnKF), with respect to solution accuracy and computational efficiency for nonlinear problems. The convection–diffusion–reaction (CDR) problem is first tested, and then, the chaotic Lorenz 96 model is solved. Both linear and nonlinear observation operators are considered. The study demonstrates that MLEF consistently produces more accurate and efficient solution than the other two methods and provides more information on both states and their uncertainties. The IEnKF and MLEF are used to estimate model parameters and uncertainty in initial conditions using a nonlinear observation operator. The assimilation performance is assessed based on the quality metrics, such as the squared true error, the trace of the error covariance matrix, and the root-mean-square (RMS) error. Based on these DA performance assessments, MLEF demonstrates better convergence and higher accuracy. Results of the CDR problem show significant improvements in the estimate of model parameters and the solution accuracy by MLEF compared to the EnKF family. This study provides evidence supporting the choice of MLEF when solving large nonlinear problems. [ABSTRACT FROM AUTHOR]

The mathematical foundation of quantum mechanics is built on linear algebra, while the application of nonlinear operators can lead to outstanding discoveries under some circumstances, such as the prediction of positron, a direct outcome of the Dirac equation which stems from the square-root of the Klein-Gordon equation. In this article, we propose a model of square-root higher-order Weyl semimetal (SHOWS) by inheriting features from its parent Hamiltonians. It is found that the SHOWS hosts both "Fermi-arc" surface and hinge states that respectively connect the projection of the Weyl points on the side surface and arris. We theoretically construct and experimentally observe the exotic SHOWS state in three-dimensional (3D) stacked electric circuits with honeycomb-kagome hybridizations and double-helix interlayer couplings. Our results open the door for realizing the square-root topology in 3D solid-state platforms. The topological properties of square-root Weyl semimetals are derived from the square of the Hamiltonian. Here, the authors propose a tight-binding model for a square-root higher-order Weyl semimetal hosting both Fermi-arc surface and hinge states. [ABSTRACT FROM AUTHOR]

We present a circle criterion which is necessary and sufficient for absolute stability with respect to a natural class of sector-bounded nonlinear causal operators. This generalized circle criterion contains the classical result as a special case. Furthermore, we develop a version of the generalized criterion which guarantees input-to-state stability. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is threefold. First, to give definition of I-nonexpansive mappings in a Kohlenbach hyperbolic space. Second, to define a new one-step implicit iterative process. Finally, to establish strong and $$\Delta $$ -convergence theorems for this iterative process in a Kohlenbach hyperbolic space. In addition, an example is provided to validate our result. Our results extend some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems of Hadamard-type fractional functional differential equations and inclusions involving both retarded and advanced arguments. We make use of the standard tools of fixed point theory to obtain the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions for boundary value problems of high-order conformable differential equations involving the p-Laplacian operator. By applying properties of the Green's function and the Guo–Krasnosel'skii fixed point theorem, some sufficient conditions for the existence of at least one positive solution are established. In addition, we demonstrate the effectiveness of the main result by using an example. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a generalized singular fractional order Kelvin–Voigt model with a nonlinear operator. By using analytic techniques, the uniqueness of solution and an iterative scheme converging to the unique solution are established, which are very helpful to govern the process of the Kelvin–Voigt model. At the same time, the corresponding eigenvalue problem is studied and the property of solution for the eigenvalue problem is established. Some examples are given to illuminate the main results. [ABSTRACT FROM AUTHOR]

In this paper, by studying the solutions of the abstract operator equation A(x,x)+B(x,x)+e=x on ordered Banach spaces, where A, B are two mixed monotone operators and e∈P with θ≤e≤h, we prove a class of boundary value problems on elastic beam equation to have a unique solution. Furthermore, we also apply our abstract result to establish the existence and uniqueness theorem of nontrivial solutions for nonlinear fractional boundary value problems. The iterative sequences to approximate unique solutions for the above two classes of problems are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of positive solutions of fractional differential equations on the infinite interval (0,+∞). The positive solution of fractional differential equations is gained by using the properties of the Green’s function, Leray-Schauder’s fixed point theorems, and Guo-Krasnosel’skii’s fixed point theorem. As an application, two examples are given to prove our conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fixed point for a class of nonlinear sum-type operators ‘A+B+C’ on an ordered Banach space, where A, B are two mixed monotone operators, C is an increasing operator. Without assuming the existence of upper-lower solutions or compactness or continuity conditions, we prove the unique existence of a positive fixed point and also construct two iterative schemes to approximate it. As applications, we research a nonlinear fractional differential equation with multi-point fractional boundary conditions. By using the obtained fixed point theorems of sum-type operator, we get the sufficient conditions which guarantee the existence and uniqueness of positive solutions. At last, a specific example is provided to illustrate our result. [ABSTRACT FROM AUTHOR]

The Ambartsumian equation, along with its generalizations, is one of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. The Ambartsumian equation plays an important role in the study of light scattering in media of infinite optical thickness. Nowadays the analytical solution of this equation is not known; therefore, the development of approximate methods for its solution is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed, the substantiation of which has been carried out under rather severe conditions. In the previous work of the authors, a spline-collocation method for solving the Ambartsumian equation with zero-order splines was constructed and substantiated. The accuracy of this method is , where N is a number of collocation nodes. It is of considerable interest to construct an iterative method adapted to the smoothness of the coefficients and kernels of the equation. Light scattering in media of finite optical thickness is described by Ambartsumian equation systems, for an approximate solution of which it is necessary to construct and substantiate effective numerical methods. This study is devoted to the construction of such methods. The construction and substantiation of iterative methods for solving systems of Ambartsumian equations is based on a generalization of the continuous method for solving nonlinear operator equations. The method and its generalization are based on the Lyapunov stability theory and are stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. In this work, spline-collocation methods with first-order splines are constructed for solving the Ambartsumian equation and systems of equations, and their justification is given. Model examples that illustrate the effectiveness of the methods are solved. Equations generalizing the classical Ambartsumian equations are considered. To solve them, the computational schemes of spline-collocation methods are constructed and substantiated. The results obtained can be used to solve a number of astrophysical issues. [ABSTRACT FROM AUTHOR]

In this paper we consider the second order nonlinear neutral delay partial difference equation ΔnΔm (xm,n + am,nxm-k,n-l) + f (m, n, xm-τ,n-σ) = bm,n, m ≥ m0, n ≥ n0. Under suitable conditions, by making use of the Banach fixed point theorem, we show the existence of uncountably many bounded positive solutions for the above partial difference equation. Three nontrivial examples are given to illustrate the advantages of our results. [ABSTRACT FROM AUTHOR]

We recall the concepts of {α, ξ}-contractive and α-admissible mappings on complete metric spaces to state related fixed point theorems. In this paper, we obtain some fixed point results for {α, ξ}-expansive locally contractive mappings in complete metric spaces. The contractiveness of the mapping is only on a closed ball instead of the whole space. Our results unify, generalize, and complement various well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain D ⊂ ℝd and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by P . We associate to Problem P an optimal control problem, denoted by Q . Then, using appropriate Tykhonov triples, governed by a nonlinear operator G and a convex K ˜ , we provide results concerning the well-posedness of problems P and Q . Our main results are Theorems 4.2 and 5.2, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem P , constructed with particular choices of G and K ˜ . We prove that Theorems 4.2 and 5.2 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for autonomous nonlinear differential operators defined in the space of functions bounded and continuous on the axis to be C1-diffeomorphisms. [ABSTRACT FROM AUTHOR]

We propose and analyze a new mixed finite element method for the coupling of the Stokes equations with a transport problem modelled by a scalar nonlinear convection–diffusion problem. Our approach is based on the introduction of the Cauchy fluid stress and two vector unknowns involving the gradient and the total flux of the concentration. The introduction of these further unknowns lead to a mixed formulation in a Banach space framework in both Stokes and transport equations, where the aforementioned stress tensor and vector unknowns, together with the velocity and the concentration, are the main unknowns of the system. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with Babuška–Brezzi's theory in Banach spaces, classical results on nonlinear monotone operators and certain regularity assumptions, are applied to prove the unique solvability of the continuous system. As for the associated Galerkin scheme, whose solvability is established similarly to the continuous case by using the Brouwer fixed-point theorem, we employ Raviart–Thomas approximations of order k ≥ 0 for the stress and total flux, and discontinuous piecewise polynomials of degree k for the velocity, concentration, and concentration gradient. With this choice of spaces, momentum is conserved in both Stokes and transport equations if the external forces belong to the piecewise constants and concentration discrete space, respectively, which constitutes one of the main features of our approach. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of periodic solutions of one kind of planar Hamiltonian systems with a nonlinear term satisfying semilinear conditions. Using a generalized Poincaré-Birkhoff fixed point theorem, we prove that the system has infinitely many periodic solutions, provided that the time map tends to zero. [ABSTRACT FROM AUTHOR]

In this paper, we present an existence criterion for multiple positive solutions of nonlinear neutral delay partial difference equations. Such equations can be regarded as a discrete analog of neutral delay partial differential equations. Our main result relies on fixed point index theory. An example is constructed to show the applicability of the obtained result. [ABSTRACT FROM AUTHOR]