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It is shown that many systems of nonlinear differential equations of interest in various fields are naturally embedded in a new family of differential equations. In this paper, we improve new and effective methods for nonautonomous systems and they produce new exact solutions to some important biological systems. The exact solution of predator and prey population for different particular cases has been derived. The numerical examples show that new exact solutions can be obtained for many biological systems such as SIR model, Lotka–Volterra model. The methods perform extremely well in terms of efficiency and simplicity to solve this historical biological models. The Lotka–Volterra nonlinear differential equations for two competing species, namely X and Y, contain six independent parameters. Their general analytic solutions, valid for arbitrary values of the parameters, are at present unknown. However, when two or more of these parameters are interrelated, it is possible to obtain the exact solutions in the X, Y phase plane, and six cases of solvability are given in this paper. The dependence of the solutions on the parameters and the initial conditions can thus be readily investigated. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new viscosity approximation method based on the conjugate gradient method and an averaged mapping approach for finding a common element of the set of solutions of a constrained convex minimization problem and the set of solutions of a split variational inclusion problem. Under suitable conditions, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of the split variational inclusion problem and the set of solutions of the constrained convex minimization problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area. Finally, preliminary numerical results indicate the feasibility and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, by using the C -class functions, we present some coupled coincidence point results for a pair of mappings satisfying generalized (ϕ, ψ)-weakly contractive condition in the context of ordered b -metric spaces. Also, an application to study the existence of solutions for the nonlinear integral equations and example are given to support our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, some new theorems for various cyclic contractions are established in G b -metric spaces to discuss the existence and uniqueness of the solutions for a class of the operator equations, which generalize many known results in corresponding literatures. On the other hand, we introduce the notion of the cyclic α - ψ φ-contractive mappings in G b -metric spaces and establish a new fixed point theorem, which is applied to consider the existence of the solutions of the integral equations and ordinary differential equations. Moreover, some examples are given to support our main results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the asymptotic behavior of solutions to the linear spatially homogeneous Boltzmann equation for hard potentials without angular cutoff. We obtain an optimal rate of exponential convergence towards equilibrium in a L 1-space with a polynomial weight. Our strategy is taking advantage of a spectral gap estimate in the Hilbert space L 2 (μ − 1 2 ) and a quantitative spectral mapping theorem developed by Gualdani et al. (2017). [ABSTRACT FROM AUTHOR]

The generalized KdV equation arises in many problems in mathematical physics. In this paper, an effective numerical method is proposed to solve two types of time-fractional generalized fifth-order KdV equations, the time-fractional Sawada-Kotera equation and Ito equation, the idea is to use Bernstein polynomials. Firstly, Bernstein basis polynomials are utilized to approximate unknown function and the error bound is given. Secondly, the representation of Bernstein basis polynomials are proposed to easily and quickly obtain the integer and fractional differential operator of unknown function, by which the studied equations can be displayed as the combination of operator matrices. Finally, comparison with Chebyshev wavelets method and the error data are presented to demonstrate the high accuracy and efficiency of Bernstein polynomials method for this kind of wave equations. [ABSTRACT FROM AUTHOR]

In this paper, the problem of uniform stability for fractional-order complex-valued neural networks with both leakage and discrete delays is considered. Base on the contracting mapping principle, a sufficient condition is proposed for the existence and uniqueness of the equilibrium point of the addressed neural networks. By employing analysis technique, some delay-dependent criteria are established for checking the global uniform stability of the fractional-order complex-valued neural networks. Two numerical examples are presented to demonstrate the validity and feasibility of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we describe an approach to estimate the cyclicity of centers in maps given by f ( x ) = − x − ∑ k = 1 ∞ a k x k + 1 . The main motivation for this problem originates from the study of cyclicity of planar systems of ODEs. We also consider the bifurcation of limit cycles from each component of the center variety of some particular cases of maps f ( x ) = − x − ∑ k = 1 ∞ a k x k + 1 arising from algebraic equations of the form x + y + h . o . t . = 0 where higher order terms up to degree four are present. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of discontinuous periodic solutions for a class of Liénard systems with state impulses, whose solutions are interrupted by abrupt changes of state. By employing Poincaré mapping method, a criterion for the existence of at least one discontinuous periodic solution in the impulsive Liénard systems is established. Finally, we give an example to illustrate the effectiveness of our result, and the corresponding numerical simulation is presented to show that there is a good agreement between our theoretical result with the computer numerical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate several new subclasses of close-to-convex functions and their related mappings. Various interesting properties including sufficiency criteria, coefficient estimates, arc-length problem and radius of convexity are investigated. [ABSTRACT FROM AUTHOR]

In this paper we obtain some equivalences between cyclic contractions and non-cyclic contractions in the framework of b -metric spaces. Our results improve and complement several recent fixed point results for cyclic contractions in b -metric spaces established by George et al. (2015) and Nashine et al. (2014). Moreover, all the results are with much shorter proofs. In addition, an application to integral equations is given to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]

Let φ be a holomorphic self-map of B and g ∈ H ( B ) such that g ( 0 ) = 0 , where H ( B ) is the space of all holomorphic functions on the unit ball B of C n . In this paper we investigate the following integral-type operator D φ g f ( z ) = ∫ 0 1 D f ( φ ( t z ) ) g ( t z ) d t t , f ∈ H ( B ) , where D f is the fractional derivative of f ∈ H ( B ). The boundedness and compactness of the operators D φ g between mixed norm spaces and Bloch spaces in the unit ball are studied. [ABSTRACT FROM AUTHOR]

The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined ‘Mustard’ convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis. The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two finite families of demicontractive mappings. The strong convergence theorem of the proposed algorithm is established under some suitable control conditions in a real Hilbert space. Our result generalizes several recent results in the current literature. [ABSTRACT FROM AUTHOR]

Recently, Cui and Wang (2014) constructed an algorithm for demicontractive operators that converges weakly, under mild assumptions, to some solution of the split common fixed point problem. In this paper, based on Halpern’s type method (1967), we construct an algorithm for demicontractive operators that produces sequences that always converge strongly to a specific solution of the split common fixed point problem. Particular cases of directed operators and quasi-nonexpansive mappings are also considered. [ABSTRACT FROM AUTHOR]

In this paper we unify the system of functional equations defining a multi-additive-quadratic mapping to obtain a single equation. We also prove, using the fixed point method, the generalized Hyers–Ulam stability of this equation thus generalizing some known results. [ABSTRACT FROM AUTHOR]

Let ( X, d ) be a cone metric space over a solid vector space ( Y , ⪯). In this paper, we prove a convergence theorem with error estimates and localization formula for Jungck iteration process for approximating points of coincidence and common fixed points of two selfmappings T and f of X satisfying a quasi-contraction condition of the type d ( T x , T y ) ⪯ λ co { d ( f x , f y ) , d ( f x , T x ) , d ( f y , T y ) , d ( f x , T y ) , d ( f y , T x ) } for all x, y ∈ X , where λ ∈ (0, 1) is a constant. Our result complements the recent result of Ding et al. [9]. The main result is new even in the classical metric space setting. Moreover, this result answers an open question posed by Olaleru [17] concerning common fixed points in metric spaces. [ABSTRACT FROM AUTHOR]

We consider a class of nonlinear matrix equations of the type (1) X = Q + ∑ i = 1 m A i ∗ G ( X ) A i - ∑ j = 1 k B j ∗ K ( X ) B j , where Q is a positive definite matrix, A i , B j are arbitrary n × n matrices and G , K are two order-preserving or order-reversing continuous maps from H ( n ) into P ( n ) . In this paper we first discuss existence and uniqueness of coupled fixed points in a L -space endowed with reflexive relation. Next on the basis of the coupled fixed point theorems, we prove the existence and uniqueness of positive definite solutions to such equations. [ABSTRACT FROM AUTHOR]

Tabulation method is usually adopted to describe the mapping relation of the analytic function of the local rules of Cellular Automaton. Although tabulation method is quite suitable to describe the mapping relation among discrete variables, there are numerous if-else sentences for the state decision of independent variables in computer programming. In this paper, we define an analytic function of the local rules of an Elementary Cellular Automaton by constructing the recognition function of the states of the cell and the neighbors. Therefore, a huge number of calculations by if-else sentences are transferred to the function calculation due to the adoption of the function of the local rules during the evolution of the ECA. Finally, the simulation results show that the programming structure is optimized and the programming efficiency is enhanced. In addition, the proposed method can also be extended to other 2-state cellular automata. [ABSTRACT FROM AUTHOR]

Krasnoselskii noted that many problems in analysis can be formulated as a mapping which is the sum of a contraction and compact map. He proved a theorem covering such cases which is the union of the contraction mapping principle and Schauder’s second fixed point theorem. In putting the two results together he found it necessary to add a condition which has been difficult to fulfill, although a great many problems have been solved using his result and there have been many generalizations and simplifications of his result. In this paper we point out that when the mapping is defined by an integral plus a contraction term, the integral can generate an equicontinuous map which is independent of the smoothness of the functions. Because of that, it is possible to set up that mapping, not as a sum of contraction and compact map, but as a continuous map on a compact convex subset of a normed space. An application of Schauder’s first fixed point theorem will then yield a fixed point without any reference to that difficult condition of Krasnoselskii. Finite and infinite intervals are handled separately. For the class of problems considered, application is parallel to the much simpler Brouwer fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized contraction of Krasnoselskii-type using auxiliary functions, and obtained some sufficient conditions for existence and uniqueness of fixed point for such mappings on bimetric spaces. We also establish a result on coincidence point of two mappings, and derive several corollaries of our main theorems. As application, we establish an existence result for a two-point boundary value problem of second order differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the strong convergence of viscosity iteration and modified viscosity iteration process for strongly continuous semigroup of uniformly Lipschitzian asymptotically pseudocontractive mappings. [ABSTRACT FROM AUTHOR]

In this paper, a class of impulsive stochastic Nicholson’s blowflies model with patch structure and nonlinear harvesting terms is introduced and studied on time scales. By using contraction mapping principal and Gronwall–Bellman inequality technique, some sufficient conditions for the existence and exponential stability of piecewise mean-square almost periodic solutions for the model with infinite delays are established on time scales. Finally, an example is given to demonstrate the validity of the conditions of the main theorem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an iterative scheme for finding a common point of the solution set of a variational inequality and the fixed point set of a Bregman relatively nonexpansive mapping. Moreover, a numerical example which shows the conclusion of our result is presented. Our results improve and generalize many known results in the current literature. [ABSTRACT FROM AUTHOR]

Let H be a real Hilbert spae and F , K : H → H be mappings such that D ( K ) = D ( F ) = H . Suppose that Hammerstein equation of the type u + KFu = 0 has a solution in H , then we studied in this paper methods that contain an auxiliary mapping (defined on an appropriate real Hilbert space in terms of the mappings K and F ) which is pseudocontractive whenever K and F are monotone; and approximation of a fixed point of this pseudocontractive mapping induces approximation of a solution of the equation u + KFu = 0 . Moreover, the mappings K and F need not be defined on compact subset of H or angle bounded on H . Furthermore, our methods which do not involve K - 1 provide an implicit algorithm for approximation of solutions of the equation u + KFu = 0 whenever K and F are assumed to be bounded and continuous; if K and F are assumed to be Lipschitz continuous, then an explicit iterative algorithm for computation of solutions of the equation u + KFu = 0 is provided, still without involving K - 1 . [ABSTRACT FROM AUTHOR]

In this paper we investigate a special kind of almost geodesic mapping of the third type of spaces with non-symmetric affine connection. Also we find some relations for curvature tensors of associated affine connection spaces of almost geodesic mappings of the third type. Finally, we investigate equitorsion almost geodesic mapping having the property of reciprocity and find an invariant geometric object of this mapping. [ABSTRACT FROM AUTHOR]