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In this paper, the complete invariant subspace classification of the generalized nonlinear diffusion–convection (D–C) equation is presented firstly. Then the explicit solutions to the nonlinear equations are constructed based on the invariant subspace method. [ABSTRACT FROM AUTHOR]

In this paper, a system of neutral functional differential equations in critical case is investigated. Under some suitable assumptions, we prove that every bounded solution of the considered system tends to zero vector as t → + ∞ . An example and its numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The operator-variation-of-constants formula was derived by Wu et al. (2015) for the general multidimensional nonlinear wave equations, and the authors proved that the formula is adapted to different boundary conditions. Furthermore, an energy-preserving scheme for one-dimensional (1D) nonlinear Hamiltonian wave equations with periodic boundary conditions was proposed by Liu et al. (2016). It is known that the formula is associated with the operator-valued functions which depend on Laplacian. Hence, it is crucial to show the boundness of the operator-valued functions. This motivates the new study of the boundness of the operator-valued functions. As an application, we extend the energy-preserving scheme from 1D to multidimensional nonlinear Hamiltonian wave equations with three different boundary conditions. [ABSTRACT FROM AUTHOR]

This paper focuses on a stochastic SIR model with Beddington–DeAngelis functional response, perturbed by a complex type of noises. By introducing a threshold, sufficient and almost necessary conditions for permanence and extinction of the disease are given. [ABSTRACT FROM AUTHOR]

We establish some new criteria for the oscillation and asymptotic behaviour of solutions of the third-order nonlinear neutral differential equation ( 1 p ( t ) ( 1 r ( t ) [ x ( t ) + a ( t ) x ( γ ( t ) ) ] ′ ) ′ ) ′ + q ( t ) f ( x ( δ ( t ) ) ) = 0 . Applications illustrating the results are given. [ABSTRACT FROM AUTHOR]

In this paper, we study a rank constrained matrix approximation problem in the Frobenius norm: min r ( X ) = k ‖ A X B − C ‖ F 2 where k is a nonnegative integer. First, we derive the feasible interval of the parameter K for the existence of solutions to the problem. Second, on condition that such a solution exists, we give a general expression for the solution to the corresponding rank constrained matrix approximation problem. Last, we provide the feasible interval of the parameter K for the existence of the minimal norm of X . [ABSTRACT FROM AUTHOR]

This paper studies a reaction–diffusion system of a predator–prey model with Holling type II functional response and prey-taxis, proposed by Ainseba et al. (2008), where the prey-taxis means a direct movement of the predator in response to a variation of the prey (which results in the aggregation of the predator). The global existence of classical solutions was established by Tao (2010). In this paper we prove furthermore that the global classical solutions are globally bounded, by means of the Gagliardo–Nirenberg inequality, the L p − L q estimates for the Neumann heat semigroup, and the L p estimates with Moser’s iteration of parabolic equations. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study asymptotic properties of the fourth-order linear trinomial delay differential equation ( E ) y ( 4 ) ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( τ ( t ) ) = 0 . Applying suitable comparison principles, we present new criteria for all solutions of ( E ) to be Kneser ones. Obtained results essentially simplify the examination of the studied equations. [ABSTRACT FROM AUTHOR]

The exact solutions to a class of differential equation are studied. Some special cases are discussed for the central potentials, single ring-shaped potential and the angular Teukolsky equation. A new expression to the associated Legendre polynomials is found. Some new properties of the universal associated-Legendre polynomials (UALPs) including the generating function, Rodrigues’ formula, parity, some special values and the recurrence relations are presented. A new different Rodrigues’ formula of associated Legendre polynomials is also obtained. [ABSTRACT FROM AUTHOR]

The paper deals with one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the form ut=[G (u) ux]x + F(u, ū), where u = u (x, t) and u #772; = u(x, t-τ), with τ denoting the delay time. Generalized and functional separable solutions for this class of equations have been obtained and presented for the first time; these equations have not been known to have such solutions so far. To construct these solutions and solutions of some other delay PDEs, we developed a few exact methods that rely on using invariant subspaces for corresponding nonlinear differential operators. Many of the results are extendable to more complex nonlinear reaction-diffusion equations with several delay times, τ1,...,τm, and equations with time-varying delay, τ = τ (t). All of the equations considered involve several free parameters (or an arbitrary function) and so their solutions can be suitable for testing approximate analytical and numerical methods for nonlinear delay reaction-diffusion equations. The exact methods described may also be applied to other classes of nonlinear delay PDEs. [ABSTRACT FROM AUTHOR]

Differential equations models for functional data play a strong role in economics, medicine, and so on. It is challenging to identify a linear differential operator from functional data. The purpose of this letter is to find a new identification approach for differential equations models based on functional data. The effectiveness of our approaches are verified through numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: Following an idea similar to that given by Dennis and Schnabel (1996) in [2], we prove a local convergence result for Newton’s method under generalized conditions of Kantorovich type. [Copyright &y& Elsevier]

Abstract: We present low-order functional observer existence conditions; the proposed existence conditions always lead to a design of functional observers of order less than or equal to that of a Luenberger observer. The proposed low-order observer always exists if a Luenberger observer exists. [Copyright &y& Elsevier]

Abstract: We study a variable exponent model for image restoration in the case that the exponent attains the critical value one. We prove existence and -convergence. The results answer an open question by Li, Li and Pi [F. Li, Z. Li, L. Pi, Ling, Variable exponent functionals in image restoration, Appl. Math. Comput. 216 (3) (2010) 870–882] [Copyright &y& Elsevier]

Abstract: In this paper, we prove that any approximate linear derivation on a semisimple Banach algebra is continuous. We deal with the functional inequalities associated with additive mappings and some stability theorems are proved. Based on these facts, we obtain some results for the functional inequalities corresponding to the additive mappings and the equation . [Copyright &y& Elsevier]

Abstract: In this work, perturbed systems of sines and cosines are considered. A method for establishing basicity of these systems in the Sobolev spaces is suggested. Therein, specific conditions are not imposed on the perturbation; the basicities of the systems under consideration in the Lebesgue space are used. [Copyright &y& Elsevier]

Abstract: Using the fixed point method, we prove the Hyers–Ulam stability of the Cauchy–Jensen functional inequality in fuzzy Banach algebras. [Copyright &y& Elsevier]

Abstract: The purpose of the present paper is to introduce two subclasses of -valent Bazilevic functions by using the generalized operator and to investigate various properties for these subclasses. [Copyright &y& Elsevier]

Abstract: Fixed point theorems for operators of a certain type on partial metric spaces are given. Orbitally continuous operators on partial metric spaces and orbitally complete partial metric spaces are defined, and fixed point theorems for these operators are given. [Copyright &y& Elsevier]

Abstract: In this paper, we prove the Hyers–Ulam stability of some set-valued functional equations. [Copyright &y& Elsevier]

Abstract: I discuss the problem of time-dependent harmonic oscillators on the basis of a periodic functional approach to the calculus of variations. Both the Lagrangian and Hamiltonian formulations are explored and discussed in some detail. Some interesting consequences are revealed. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a strongly damped wave equation with fractional damping on part of its boundary and also with an internal source. Under some appropriate assumptions on the parameters, and with certain initial data, a blow-up result with positive initial energy is established. [Copyright &y& Elsevier]

Abstract: In this paper the unbounded increasing solution of the scalar nonlinear partial equation of the parabolic type for finite time is investigated. The sufficient condition for nonlinearity is established. Under this condition every solution of the investigated problem is exploded by the absence of restriction to smallness on the initial function. The existence of the solution is proved by smallness of the initial function. [Copyright &y& Elsevier]

Abstract: In this paper, a class of even-order half-linear functional differential equations with damping are studied. By using the generalized Riccati transformation and the integral averaging technique, new oscillation criteria are obtained for all solutions of the equations, which generalize and improve some known results. [Copyright &y& Elsevier]

Abstract: For a positive integer , a -rainbow dominating function of a graph is a function from the vertex set to the set of all subsets of the set such that for any vertex with the condition is fulfilled, where is the neighborhood of . The -rainbow domination is the same as the ordinary domination. A set of -rainbow dominating functions on with the property that for each is called a -rainbow dominating family (of functions) on . The maximum number of functions in a -rainbow dominating family on is the -rainbow domatic number of , denoted by . Note that is the classical domatic number . If is a graph of order and is the complement of , then we prove in this note for the Nordhaus–Gaddum inequality This improves the Nordhaus–Gaddum bound given by Sheikholeslami and Volkmann recently. [Copyright &y& Elsevier]

Abstract: Let be a graph, and a positive integer. Let be a function. If holds for each , then we call a fractional -factor of with indicator function where . A graph is called a fractional -deleted graph if there exists a fractional -factor of with indicator function such that for any , where is any subgraph of with edges. In this paper, we use a binding number to obtain a sufficient condition for a graph to be a fractional -deleted graph. This result is best possible in some sense, and it is an extension of Zhou’s previous results. [Copyright &y& Elsevier]

Abstract: Recently, Wang et al. discussed the properties of fuzzy information systems under homomorphisms in the work [C. Wang, D. Chen, L. Zhu, Homomorphisms between fuzzy information systems, Appl. Math. Lett. 22 (2009) 1045–1050], where homomorphisms are based upon the concepts of consistent functions and fuzzy relation mappings. In this work, we classify consistent functions as predecessor-consistent and successor-consistent, and then proceed to present more properties of consistent functions. In addition, we improve some characterizations of fuzzy relation mappings provided in the above cited work. [Copyright &y& Elsevier]

Abstract: We prove that Euler’s equation , characterising homogeneous functions, is stable in Hyers–Ulam sense if and only if . [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a -analogue of the Szász–Mirakyan–Kantorovich operators and we propose two different modifications of the -Szász–Mirakyan–Kantorovich operators. These modifications preserve some test functions. We also examine the rate of convergence for the constructed operators by means of modulus of continuity. [Copyright &y& Elsevier]

Abstract: An -labeling of a graph is defined as a function from the vertex set into the nonnegative integers such that for any two vertices , , if and if , where is the distance between and in . The -labeling number of is the smallest number such that has an -labeling with . Griggs and Yeh conjectured that for any simple graph with maximum degree . In this paper, we consider the total graph of a graph and derive its upper bound of . Shao, Yeh and Zhang had proved that . We improve the bound to , which shows that the conjecture of Griggs and Yeh is true for the total graph. In addition, we obtain the exact value of for the total graph of a complete bipartite graph with . [Copyright &y& Elsevier]

Abstract: In this work, we establish a strong duality theorem for Mond–Weir type multiobjective higher-order nondifferentiable symmetric dual programs. This fills some gaps in the work of Chen [X. Chen, Higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 290 (2004) 423–435]. [Copyright &y& Elsevier]

Abstract: In this article we have discussed the E-convex functions and studied the theorems of Jensen, Bernstein–Doetsch, Blumberg–Sierpinski and Ostrowski on E-midconvex functions. [Copyright &y& Elsevier]

Abstract: In this paper, a new Picone-type identity for the quasilinear differential operator of the second order is used to establish an integral inequality involving functions and their derivatives which generalizes the classical Wirtinger inequality. [Copyright &y& Elsevier]

Abstract: In this note, we prove that if is a suitable iteration group on an open interval and a function is continuous at least at one point and commutes with two mappings such that is irrational, then . If, moreover, , then the commutativity of with and can be replaced by the inequalities and . As an application of the obtained results, we give a contribution to Schwaiger’s problem concerning iteration groups which are commuting in pairs. [Copyright &y& Elsevier]

Abstract: We construct a complete metric space of random measure-valued image functions. This formalism is an extension of previous work on measure-valued image functions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a Cauchy problem for a nonlinear viscoelastic equation with nonlinear damping and source terms. Under suitable assumptions on the initial data and the relaxation function, we establish a finite-time blow-up result and a global existence result. [Copyright &y& Elsevier]

Abstract: Some methods are considered for obtaining power series of products and powers of elementary and special functions [Copyright &y& Elsevier]

Abstract: For analytic functions in the open unit disk , there are many sufficient conditions for to be starlike and convex in . In view of the paper by P.T. Mocanu [P.T. Mocanu, Some starlikeness conditions for analytic functions, Rev. Roumaine Math. Pures Appl. 33 (1988) 117–124], some extensions of sufficient conditions for to be starlike of order and to be convex of order are discussed. [Copyright &y& Elsevier]

Abstract: A class of virus dynamics model with intracellular delay and nonlinear infection rate of Beddington–DeAngelis functional response is analysed in this paper. By constructing suitable Lyapunov functionals and using LaSalle-type theorem for delay differential equations, we show that the global stability of the infection-free equilibrium and the infected equilibrium depends on the basic reproductive ratio , that is, the former is globally stable if and so is the latter if . Our results extend the known results on delay virus dynamics considered in the other papers and suggest useful methods to control virus infection. [Copyright &y& Elsevier]

Abstract: We deal with a predator–prey interaction model with Ivlev-type functional response which is subject to the homogeneous Dirichlet boundary condition. On the basis of the fixed point index theory, we give a sufficient and necessary condition for the existence of positive solutions of the model. [Copyright &y& Elsevier]

Abstract: The present work aims to study the stability of the following three functional equations: (i) , (ii) , and (iii) for all . The first functional equation arises in the characterization of symmetrically compositive sum form distance measures. [Copyright &y& Elsevier]

Abstract: In this paper we make a new approach to the notions of -summability and -statistical convergence by using ideals and introduce new notions, namely, -summability and -statistical convergence. We mainly examine the relation between these two new methods as also the relation between -statistical convergence and -statistical convergence introduced by the authors recently. We carry out the whole investigation in normed linear spaces. [Copyright &y& Elsevier]

Abstract: By using the bilinear form and the Leray-Schauder principle, we prove the new anti-periodic existence results for second order differential equations. [Copyright &y& Elsevier]

Abstract: We denote by , the class of all analytic functions in the unit disc with the normalization . For a positive number , we denote by the class of all , such that , and satisfying the condition A function is said to be in if . In this paper, we find conditions on , and such that is included in the class of all starlike functions of order , or the class of all strongly starlike functions of order , or , respectively. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) micropolar fluid flows. It is proved that if the pressure satisfies then the weak solution becomes a regular solution on . The methods are based on the innovative function decomposition technique. [ABSTRACT FROM AUTHOR]

Abstract: We propose a general method for obtaining the representation of solutions for linear fractional order differential equations based on the theory of -regularized families of operators. We illustrate the method for the case of the fractional order differential equation where is an unbounded closed operator defined on a Banach space and is an -valued function. [ABSTRACT FROM AUTHOR]

Abstract: We prove the Euler–Lagrange delta-differential equations for problems of the calculus of variations on arbitrary time scales with delta-integral functionals depending on higher-order delta derivatives. [ABSTRACT FROM AUTHOR]

Abstract: This letter is devoted to results on intermediate asymptotics for the heat equation. We study the convergence towards a stationary solution in self-similar variables. By assuming the equality of some moments of the initial data and of the stationary solution, we get improved convergence rates using entropy/entropy-production methods. We establish the equivalence of the exponential decay of the entropies with new, improved functional inequalities in restricted classes of functions. This letter is the counterpart in a linear framework of a recent work on fast diffusion equations; see Bonforte et al. (2009) . The results extend to the case of a Fokker–Planck equation with a general confining potential. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we apply a fixed point theorem in a cone to obtain sufficient conditions for the existence of multiple positive periodic solutions to a class of higher order functional difference equations. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the modeling of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. Various mathematical structures of the hiding–learning dynamics are derived at the mesoscopic scale. Subsequently, these structures are further detailed referring to modeling issues and, in particular, to the learning-hiding competition among tumors and the immune system cells. [Copyright &y& Elsevier]