Asymptotic results on the existence of 4‐RGDDs and uniform 5‐GDDs

In this paper, we continue to investigate the existence of 4‐RGDDs and uniform 5‐GDDs. It is proved that the necessary conditions for the existence of such designs are also sufficient with a finite number of possible exceptions. As an application, the known results on the existence of uniform 4‐frames are also improved. © 2004 Wiley Periodicals, Inc.     

Weighted pseudo‐almost periodic functions on time scales with applications to cellular neural networks with discrete delays

In this paper, we first propose a concept of weighted pseudo‐almost periodic functions on time scales and study some basic properties of weighted pseudo‐almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo‐almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo‐almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of solutions for some non‐Fredholm integro‐differential equations with the bi‐Laplacian

We prove the existence of solutions for some semilinear elliptic equations in the appropriate H⁴ spaces using the fixed‐point technique where the elliptic equation contains fourth‐order differential operators with and without Fredholm property, generalizing the previous results.     

Some existence results for an elliptic equation of Kirchhoff‐type with changing sign data and a logarithmic nonlinearity

The paper deals with some existence results for an elliptic equation of Kirchhoff‐type with changing sign data and a logarithmic nonlinearity by direct variational method, Galerkin approach, and sub‐super solutions method. Our results are natural extension of Boulaaras' work in (Math Methods Appl Sci; 41(13):5203‐5210).     

Topologically‐based characterizations of the existence of solutions of optimization‐related problems

Necessary and sufficient conditions for the existence of solutions of optimization‐related problems, defined on general sets, are established by using topologically‐based structures, without the usual linear structure. First, we prove that the existence of solutions to a variational relation problem is equivalent to the existence of either a KKM‐structure or a connectedness structure, satisfying additionally some verifiable conditions. Then, applying these results, we obtain such full (two‐way) characterizations for the existence of invariant points, solutions of quasiequilibrium problems of the Stampacchia‐Minty type, saddle points, and Nash equilibria for noncooperative games. The main advantages of our scheme with the mentioned two structures are that full characterizations for existence are obtained in a unified way, with no convexity assumptions, and also that one has flexibility to choose a suitable structure in investigations. (This flexibility is decisive when the natural underlying structure of the given problem is scarcely employed.)     

Eigenvalue problems for nonlinear third‐order m‐point p‐Laplacian dynamic equations on time scales

This work deals with the existence and uniqueness of a nontrivial solution for the third‐order p‐Laplacian m‐point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when λ is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley & Sons, Ltd.     

Positive solutions for fourth‐order singular nonlinear differential equation with variable‐coefficient

By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation with variable‐coefficient, which extend and improve significantly existing results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Positive solution for four‐point nonlocal boundary value problems of fractional order

In this paper, we consider the existence of positive solution for four‐point nonlocal boundary value problems of fractional order. By means of some fixed point theorems, some results on the existence and multiplicity of positive solutions are obtained. Furthermore, we provide a representative example to illustrate a possible application of the established results. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and global exponential stability of anti‐periodic solutions for delayed quaternion‐valued cellular neural networks with impulsive effects

In this paper, we consider a class of delayed quaternion‐valued cellular neural networks (DQVCNNs) with impulsive effects. By using a novel continuation theorem of coincidence degree theory, the existence of anti‐periodic solutions for DQVCNNs is obtained with or without assuming that the activation functions are bounded. Furthermore, by constructing a suitable Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of anti‐periodic solutions for DQVCNNs. Our results are new and complementary to the known results even when DQVCNNs degenerate into real‐valued or complex‐valued neural networks. Finally, an example is given to illustrate the effectiveness of the obtained results.     

Existence and non‐existence of steady states to a cross‐diffusion system arising in a Leslie predator–prey model

This paper is concerned with a cross‐diffusion system arising in a Leslie predator–prey population model in a bounded domain with no flux boundary condition. We investigate sufficient condition for the existence and the non‐existence of non‐constant positive solution. We obtain that if natural diffusion coefficient of predator is large enough and cross‐diffusion coefficients are fixed, then under some conditions there exists non‐constant positive solution. Furthermore, we show that if natural diffusion coefficients of predator and prey are both large enough, and cross‐diffusion coefficients are small enough, then there exists no non‐constant positive solution. Copyright © 2012 John Wiley & Sons, Ltd.     

Stationary solutions to the Falk system on shape memory alloys

We study the Falk model system describing martensitic phase transitions in shape memory alloys. Its physically closed stationary state is formulated as a nonlinear eigenvalue problem with a non‐local term. Then, some results on existence, stability, and bifurcation of the solution are proven. In particular, we prove the existence of dynamically stable nontrivial stationary solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamics of a generalized predator‐prey model with delay and impulse via the basic reproduction number

In this paper, we study a generalized predator‐prey model with delay and impulse. The existence of the predator‐free periodic solution is investigated. We employ the approach and techniques coming from epidemiology and calculate the basic reproduction number for the predator. Using the basic reproduction number, we consider the global attraction of the predator‐free periodic solution and permanence of the model. As for application, an example is discussed. Furthermore, some numerical simulations are given to illustrate our results.     

Exponential stability of pseudo almost periodic delayed Nicholson‐type system with patch structure

In this paper, we consider the existence and global exponential stability of positive pseudo almost periodic solutions for a delayed Nicholson‐type system with patch structure. With the aid of ergodic theory and differential inequality technique, we prove the existence and global exponential stability of positive pseudo almost periodic solutions for the addressed system, which generalize and improve some known results. Specially, an example and its numerical simulations are provided to manifest the effectiveness of theoretical results.     

One‐factorizations of complete graphs with vertex‐regular automorphism groups

We consider one‐factorizations of K_2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one‐factors which are fixed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non‐existence statements in case the number of fixed one‐factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 1–16, 2002     

Clique‐coloring some classes of odd‐hole‐free graphs

We consider the problem of clique‐coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, the existence of a constant C such that any perfect graph is C‐clique‐colorable is an open problem. In this paper we solve this problem for some subclasses of odd‐hole‐free graphs: those that are diamond‐free and those that are bull‐free. We also prove the NP‐completeness of 2‐clique‐coloring K₄‐free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006     

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.     

Multiple positive fixed points for the sum of expansive mappings and k‐set contractions

In this work, we are concerned with the existence of multiple positive fixed points for the sum of an expansive mapping with constant h > 1 and a k‐set contraction when 0 ≤ k < h ? 1. In particular, the case of the sum of an expansive mapping with constant h > 1 and an e‐concave operator and an e‐convex operator is considered. Two examples of application illustrate some of the theoretical results.     

Dynamics and bifurcations of a modified Leslie‐Gower–type model considering a Beddington‐DeAngelis functional response

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov‐Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented.     

Analysis of a thermo‐viscoplastic model with Lipschitz continuous constitutive equations

We study a thermo‐mechanical model, where the mechanical inelastic model with a Lipschitz–continuous constitutive relation for the plastic strain is coupled with a heat equation. The main results are the local‐in‐time existence and uniqueness of the solution to the considered model and the existence of the solution for an arbitrarily long time interval for the sufficiently small given data. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence results of periodic solutions for third‐order nonlinear singular differential equation

Using Green's function for third‐order differential equation and some fixed‐point theorems, i.e., Leray‐Schauder alternative principle and Schauder's fixed point theorem, we establish three new existence results of periodic solutions for nonlinear third‐order singular differential equation, which extend and improve significantly existing results in the literature.