Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

Petersen Cores and the Oddness of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1‐factors of G. Let be the set of edges contained in precisely i members of the k 1‐factors. Let be the smallest over all lists of k 1‐factors of G. We study lists by three 1‐factors, and call with a ‐core of G. If G is not 3‐edge‐colorable, then . In Steffen (J Graph Theory 78 (2015), 195–206) it is shown that if , then is an upper bound for the girth of G. We show that bounds the oddness of G as well. We prove that . If , then every ‐core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4‐edge‐connected cubic graph G with . On the other hand, the difference between and can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer , there exists a bridgeless cubic graph G such that .     

On the Hamilton–Waterloo Problem with Odd Orders

Given nonnegative integers , the Hamilton–Waterloo problem asks for a factorization of the complete graph into α ‐factors and β ‐factors. Without loss of generality, we may assume that . Clearly, v odd, , , and are necessary conditions. To date results have only been found for specific values of m and n. In this paper, we show that for any integers , these necessary conditions are sufficient when v is a multiple of and , except possibly when or 3. For the case where we show sufficiency when with some possible exceptions. We also show that when are odd integers, the lexicographic product of with the empty graph of order n has a factorization into α ‐factors and β ‐factors for every , , with some possible exceptions.     

Tempered fractional differential equation: variational approach

In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative: (FD) where α∈(1/2,1), λ>0, are the left and right tempered fractional derivatives, is the fractional Sobolev spaces, and . Assuming that f satisfies the Ambrosetti–Rabinowitz condition and another suitable conditions, by using mountain pass theorem and minimization argument over Nehari manifold, we show that (FD) has a ground state solution. Furthermore, we show that this solution is a radially symmetric solution. Copyright © 2017 John Wiley & Sons, Ltd.     

Global stability of a multi‐group model with distributed delay and vaccination

A delayed multi‐group SVEIR epidemic model with vaccination and a general incidence function has been formulated and studied in this paper. Mathematical analysis shows that the basic reproduction number plays a key role in the dynamics of the model: the disease‐free equilibrium is globally asymptotically stable when , while the endemic equilibrium exists uniquely and is globally asymptotically stable when . For the proofs, we exploit a graph‐theoretical approach to the method of Lyapunov functionals. Our results show that distributed delay has no impact on the global stability of equilibria, and the results improve and generalize some known results. Copyright © 2016 John Wiley & Sons, Ltd.     

Attractors for the strongly damped wave equation with p‐Laplacian

This paper is concerned with the initial‐boundary value problem for one‐dimensional strongly damped wave equation involving p‐Laplacian. For p > 2 , we establish the existence of weak local attractors for this problem in . Under restriction 2 < p < 4, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in , and this attractor is a bounded subset of . Copyright © 2017 John Wiley & Sons, Ltd.     

Pseudo asymptotically periodic solutions for Volterra integro‐differential equations

In this paper, we propose a new class of functions called pseudo ‐asymptotically ω‐periodic function in the Stepanov sense and explore its properties in Banach spaces including composition results. Furthermore, the existence and uniqueness of the pseudo ‐asymptotically ω‐periodic mild solutions to Volterra integro‐differential equations is investigated. Applications to integral equations arising in the study of heat conduction in materials with memory are shown. Copyright © 2014 John Wiley & Sons, Ltd.     

On the Minimum Number of Spanning Trees in k‐Edge‐Connected Graphs

We show that a k‐edge‐connected graph on n vertices has at least spanning trees. This bound is tight if k is even and the extremal graph is the n‐cycle with edge multiplicities . For k odd, however, there is a lower bound , where . Specifically, and . Not surprisingly, c₃ is smaller than the corresponding number for 4‐edge‐connected graphs. Examples show that . However, we have no examples of 5‐edge‐connected graphs with fewer spanning trees than the n‐cycle with all edge multiplicities (except one) equal to 3, which is almost 6‐regular. We have no examples of 5‐regular 5‐edge‐connected graphs with fewer than spanning trees, which is more than the corresponding number for 6‐regular 6‐edge‐connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity.     

Liouville theorems for the weighted Lane–Emden equation with finite Morse indices

In this paper, we study the nonexistence result for the weighted Lane–Emden equation: (0.1) and the weighted Lane–Emden equation with nonlinear Neumann boundary condition: (0.2) where f(|x|) and g(|x|) are the radial and continuously differential functions, is an upper half space in , and . Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems 0.1 and 0.2 under appropriate assumptions on f(|x|) and g(|x|). Copyright © 2017 John Wiley & Sons, Ltd.     

Block‐Transitive and Point‐Primitive 2‐ Designs with Sporadic Socle

The purpose of this paper is to classify all pairs , where is a nontrivial 2‐ design, and acts transitively on the set of blocks of and primitively on the set of points of with sporadic socle. We prove that there exists only one such pair : is the unique 2‐(176,8,2) design and , the Higman–Sims simple group.     

The Hamilton–Waterloo Problem for ‐Factors and ‐Factors

The Hamilton–Waterloo problem asks for a 2‐factorization of (for v odd) or minus a 1‐factor (for v even) into ‐factors and ‐factors. We completely solve the Hamilton–Waterloo problem in the case of C₃‐factors and ‐factors for .     

Nonexistence results for higher order pseudo‐parabolic equations in the Heisenberg group

Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear higher order pseudo‐parabolic equation where is the Kohn‐Laplace operator on the (2N + 1)‐dimensional Heisenberg group , m≥1,p > 1. Then, this result is extended to the case of a 2 × 2‐system of the same type. Our technique of proof is based on judicious choices of the test functions in the weak formulation of the sought solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

On Flag‐Transitive Symmetric Designs of Affine Type

It is shown that, if is a nontrivial 2‐ symmetric design, with , admitting a flag‐transitive automorphism group G of affine type, then , p an odd prime, and G is a point‐primitive, block‐primitive subgroup of . Moreover, acts flag‐transitively, point‐primitively on , and is isomorphic to the development of a difference set whose parameters and structure are also provided.     

Multiple sign‐changing solutions for a class of quasilinear elliptic equations

In this paper, we study the following quasilinear Schrödinger equations: where Ω is a bounded smooth domain of , . Under some suitable conditions, we prove that this equation has three solutions of mountain pass type: one positive, one negative, and sign‐changing. Furthermore, if g is odd with respect to its second variable, this problem has infinitely many sign‐changing solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Global solutions to Keller‐Segel‐Navier‐Stokes equations with a class of large initial data in critical Besov spaces

In this article, we consider the Cauchy problem to Keller‐Segel equations coupled to the incompressible Navier‐Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let u_F:=e^tΔu₀; we prove that there exist 2 positive constants σ₀ and C₀ such that if the gravitational potential and the initial data (u₀,n₀,c₀) satisfy for some p,q with and , then the global solutions can be established in critical Besov spaces.     

Wave solutions for variants of the KdV–Burger and the K(n,n)–Burger equations by the generalized G′/G‐expansion method

An application of the ‐expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for variants of the Korteweg–de Vries–Burger and the K(n,n)–Burger equations. The generalized ‐expansion method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. It is shown that the generalized ‐expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems. Copyright © 2017 John Wiley & Sons, Ltd.     

On random k‐out subgraphs of large graphs

We consider random subgraphs of a fixed graph with large minimum degree. We fix a positive integer k and let G_k be the random subgraph where each independently chooses k random neighbors, making kn edges in all. When the minimum degree then G_k is k‐connected w.h.p. for ; Hamiltonian for k sufficiently large. When , then G_k has a cycle of length for . By w.h.p. we mean that the probability of non‐occurrence can be bounded by a function (or ) where . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 143–157, 2017     

Subcubic Edge‐Chromatic Critical Graphs Have Many Edges

We consider graphs G with such that and for every edge e, so‐called critical graphs. Jakobsen noted that the Petersen graph with a vertex deleted, , is such a graph and has average degree only . He showed that every critical graph has average degree at least , and asked if is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if G is a subcubic critical graph other than , then G has average degree at least . This bound is best possible, as shown by the Hajós join of two copies of .     

Asymptotic behavior of solutions to a class of nonlocal non‐autonomous diffusion equations

In this paper, we consider the nonlocal non‐autonomous evolution problems where Ω is a bounded smooth domain in , N≥1, β is a positive constant, the coefficient a is a continuous bounded function on , and K is an integral operator with symmetric kernel , being J a non‐negative function continuously differentiable on and . We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter β is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.