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A spanning subgraph of a graph G is called a path-factor of G if its each component is a path. A path-factor is called a 𝒫≥k-factor of G if its each component admits at least k vertices, where k ≥ 2. (Zhang and Zhou, Discrete Math. 309 (2009) 2067–2076) defined the concept of 𝒫≥k-factor covered graphs, i.e., G is called a 𝒫≥k-factor covered graph if it has a 𝒫≥k-factor covering e for any e∈ E(G). In this paper, we firstly obtain a minimum degree condition for a planar graph being a 𝒫≥2-factor and 𝒫≥3-factor covered graph, respectively. Secondly, we investigate the relationship between the maximum degree of any pairs of non-adjacent vertices and 𝒫≥k-factor covered graphs, and obtain a sufficient condition for the existence of 𝒫≥2-factor and 𝒫≥3-factor covered graphs, respectively.

With suitable assumptions, uniqueness of nontrivial nonnegative solution is established for the following nonlinear integral equation φ(x)=∫DE(x,y)G(y,φ(y))dy. This result is applied to branching d...

A spanning subgraph H of a graph G is a $$P_{3}$$ -factor of G if every component of H is a path of order three. The square of a graph G, denoted by $$G^{2}$$ , is the graph with vertex set V(G) such that two vertices are adjacent in $$G^{2}$$ if and only if their distance in G is at most 2. A graph G is subcubic if it has maximum degree at most three. In this paper, we give a sharp necessary condition for the existence of $$P_{3}$$ -factors in the square of a tree, which improves a result of Li and Zhang (Graphs Combin 24:107–111, 2008). In addition, we will also present a sufficient condition for the existence of $$P_{3}$$ -factors in the square of a tree, which has the following interesting application to subcubic trees: if T is a subcubic tree of order 3n such that $$|L(T-L(T))|\le 7$$ , then $$T^{2}$$ has a $$P_{3}$$ -factor, where L(T) denotes the set of leaves in T. Examples show that the upper bound 7 on $$|L(T-L(T))|$$ is sharp.

In this paper, we shall study unilateral global bifurcation phenomenon for the following homogeneous Kirchhoff type problem \begin{equation*} \begin{cases} -\left(\int_0^1 \left\vert u'\right\vert^2\,dx\right)u''=\lambda u^3+h(x,u,\lambda)&\text{in}\,\, (0,1),\\ u(0)=u(1)=0. \end{cases} \end{equation*} As application of bifurcation result, we shall determine the interval of $\lambda$ in which there exist nodal solutions for the following homogeneous Kirchhoff type problem \begin{equation*} \begin{cases} -\left(\int_0^1 \left\vert u'\right\vert^2\,dx\right) u''=\lambda f(x,u)&\text{in}\,\, (0,1),\\ u(0)=u(1)=0, \end{cases} \end{equation*} where $f$ is asymptotically cubic at zero and infinity. To do this, we also establish a complete characterization of the spectrum of a homogeneous nonlocal eigenvalue problem.

We establish the unilateral global bifurcation result for the following nonlinear operator equation $$u = L\left( \lambda \right)u + H\left( {\lambda,u} \right),\,\left( {\lambda,u} \right) \in \mathbb{R}{^m} \times X$$ where m is a positive integer, X is a Banach space, L(·) is a positively homogeneous completely continuous operator and H: ℝm·X → X is completely continuous with H = o (||u||) near u = 0 uniformly on bounded λ sets.

We study the following displacement problem in nonlinear elastostatics − I + ∇ u ∇ 2 u = λ f ( u ) in Ω , c u = 0 on ∂ Ω . We obtain some global and local bifurcation results, which are related to an open problem.

Belief Propagation (BP), a distributed, message-passing algorithm, has been widely used in different disciplines including information theory, artificial intelligence, statistics and optimization problems in graphical models such as Bayesian networks and Markov random fields. Despite BP algorithm has a great success in many application fields and many progress about BP algorithm has been made, the rigorous analysis about the correctness and convergence of BP algorithm are known in only a few cases for arbitrary graph. In this paper, we will investigate the correctness and convergence of BP algorithm for determining the optimal solutions of the Chinese postman problem in both undirected and directed graphs. As a main result, we prove that BP algorithm converges to the optimal solution of the undirected Chinese postman problem within O(n) iterations where n represents the number of vertices, provided that the optimal solution is unique. For the directed case, we consider the directed Chinese postman problem with capacity and show that BP algorithm also converges to its optimal solution after O(n) iterations, as long as its corresponding linear programming relaxation has the unique optimal solution.

By variational method, we obtain various existence and multiplicity results for the following double phase problem { − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω . In particular, we find a sign-changing ground state solution. Some properties of double phase operator are also obtained.

We establish a unilateral global bifurcation result for the following problem where Ω is a bounded domain in . Based on this global bifurcation result, we also studied the global structure of one-sign solutions according to different asymptotic behaviors (sublinear/linear/superlinear/jumping) nonlinearity near zero.

We study the global structure of admissible solutions for the following k -Hessian equation S k D 2 u = λ k f ( − u ) . By bifurcation and topological methods, we determine the interval of λ for the existence of admissible solution for this problem.

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.

In this paper, we investigate the global regularity of Leray–Hopf weak solutions to the 3D Navier–Stokes equations in terms of pressure. In particular, we establish three Serrin-type regularity criteria in framework of the anisotropic Lebesgue spaces. Our result can be understood as a generalization of the notable works of Y. Zhou (2006) [23] , C. Cao and E. Titi (2008) [5] , and Y. Zhou and M. Pokorný (2010) [24] .

The uniqueness obtained in [1] may not right if f ( s ) / s k is only decreasing for s > 0 . We correct this error by requiring f ( s ) / s k is “strictly” decreasing for s > 0 .

This paper is concerned with the following semilinear elliptic problem { − Δ u = λ m ( x ) f ( u ) in R N , u → 0 as | x | → + ∞ , where λ is a real parameter and m is a weight function which may be sign-changing. For the linear case, i.e., f ( u ) = u , we investigate the spectral structure. For the semilinear case, we study the existence and asymptotic behavior of one-sign and nodal solutions by bifurcation method.

We study the following eigenvalue problem of k-Hessian equation { S k ( D 2 u ) = λ k f ( − u ) in B , u = 0 on ∂ B . Global bifurcation result is established for this problem. As applications of the bifurcation result, we determine the intervals of λ for the existence, nonexistence, uniqueness and multiplicity of radially symmetric k-admissible solutions for this problem.

Using bifurcation analysis we investigate the structure of the set of positive solutions for the coupled nonlinear Schrodinger system \begin{equation*} \begin{cases} -\Delta u_1+ u_1= u_1^3+\beta u_1u_2^2 & \text{in } \mathbb{R}^N,\\ -\Delta u_2+\lambda u_2=\mu u_2^3+\beta u_2u_1^2 &\text{in } \mathbb{R}^N,\\ u_1(x),u_2(x)\rightarrow 0 &\text{as } \vert x\vert\rightarrow+\infty, \end{cases} \end{equation*} where $N=1,2,3$, $\mu$ is a positive constant, $\lambda$ and $\beta$ are positive real parameters. We prove the existence of two two-dimensional continua $\mathcal{S}_1$ and $\mathcal{S}_2$ emanating from the two sets of semi-positive solutions which cover some regions in term of $(\beta,\lambda)\in \mathbb{R}_+^2$. To do this, we establish a multi-parameter unilateral global bifurcation theorem.

We study the existence/nonexistence and multiplicity of radial solutions of the following mean curvature problem in Minkowski spacetime $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla u}{\sqrt{1-\vert \nabla u\vert ^2}}\right) = \lambda f(x,u)\,\, &{}\quad \text {in}\,\, \mathbb {R}^N,\\ u\rightarrow 0&{}\quad \text {as}\,\, \vert x\vert \rightarrow +\infty . \end{array} \right. \end{aligned}$$ Moreover, we also obtain the rate of decay of solutions at \(\infty \).

In this paper we develop a non-autonomous reaction-diffusion model with the Robin boundary conditions to describe insect dispersal on an isotropically varying domain. We investigate the stability of the reaction-diffusion model. The stability results of the model describe either insect survival or vanishing.

In this paper, we establish a unilateral global bifurcation result from intervals for a class of semilinear problems on the unit ball. By applying the above result, we obtain the spectrum of a class of half-linear problems on the unit ball. Our results partially give a positive answer to the open problem proposed by Berestycki.

We study the following Monge-Ampere equation { ( det ⁡ ( D 2 u ) ) 1 N = λ f ( − u ) in Ω , u = 0 on ∂ Ω by bifurcation technique. We establish some results about the existence, nonexistence, uniqueness and multiplicity of convex solutions for this problem. Our results generalize and improve many important known results from previous literature.

Consider the boundary value problem u ′ ′ + u = α u − + p ( t ) , u ( 0 ) = 0 = u ( π ) , α ≤ 0 . We show that a necessary and sufficient condition for the problem to be solvable is that ∫ 0 π p ( t ) sin t d t ≥ 0 . We thus answer positively a counter part of a long-standing open problem posed by S. Fucik. A more general problem is also considered and the sufficient condition for the existence of solution is obtained.

We consider a double phase problem in RN with a q−1-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Using variational methods based on the topological degree and critical point theory together with the Nehari manifold method and deformation lemma, we find three nontrivial solutions: among them, the first one is the positive ground state solution, the second one is the negative ground state solution, and the third one is the least energy sign-changing solution, which changes sign only once. We also discuss the existence of infinitely many solutions for even energy functional and that of radial solutions.

We study the global structure of positive solutions of the following mean curvature equation in the Minkowski space \[ -\div \bigg (\frac {\nabla u}{\sqrt {1-\vert \nabla u\vert ^2}}\bigg )= \lambda f(x,u), \] on an annular domain with the Robin boundary condition. According to the behavior of $f$ near $0$, we obtain the existence and multiplicity of positive solutions for this problem.

In this paper, we investigate the structure of the nontrivial solution set for the following nonlinear operator equation u = L ( λ ) u + H ( λ , u ) , ( λ , u ) ∈ R m × X , where m is a positive integer, X is a Banach space, L ( ⋅ ) : X → X is a (positively) homogeneous compact operator and H : R m × X → X is compact with H = o ( ‖ u ‖ ) near u = 0 uniformly on bounded λ sets. We obtain some results involving (unilateral) global bifurcation. Two examples of p-Laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied.

In this paper, we introduce the notion of a strict lower/upper solution to nonlinear Sturm-Liouville boundary value problems. Based on the maximum principles, we establish a result of Leray-Schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions. Applying the result and the fixed point index theory in cones, we obtain the global existence results of positive solutions for sub-superlinear Sturm-Liouville problems.

In this paper, we study global bifurcation phenomena for the following Kirchhoff type problem { - M ( ? ? | ? u ( x ) | 2 d x ) Δ u = λ f ( x , u ) in ? , u = 0 on ? ? , where M is a continuous function. Under some natural hypotheses, we show that ( λ 1 ( a ) M ( 0 ) , 0 ) is a bifurcation point and there is a global continuum C emanating from ( λ 1 ( a ) M ( 0 ) , 0 ) , where λ 1 ( a ) denotes the first eigenvalue of the above problem with f ( x , s ) = a ( x ) s . As an application of the above result, we study the existence of positive solution for this problem with asymptotically linear nonlinearity.

In this paper, we establish a unilateral global bifurcation result from interval for a class of $p$-Laplacian problems. By applying above result, we study the spectrum of a class of half-quasilinear problems. Moreover, we also investigate the existence of nodal solutions for a class of half-quasilinear eigenvalue problems.

In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, Digg.com. The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.

We establish the existence of nontrivial nonnegative solution for the following 0-Dirichlet problem with mean curvature operator in the Minkowski space $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla u}{\sqrt{1-\vert \nabla u\vert ^2}}\right) = \lambda f(x,u)\,\, &{}\text {in}\,\, \Omega ,\\ u=0&{}\text {on}\,\, \partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a general bounded domain of $$\mathbb {R}^N$$ . By bifurcation and topological methods, we determine the interval of parameter $$\lambda $$ in which the above problem has nontrivial nonnegative solution according to sublinear or linear nonlinearity at zero.

In this paper, we establish the existence and non-existence of positive solutions for p-Kirchhoff type problems with a parameter on without assuming the usual compactness conditions. We show that the p-Kirchhoff type problems have at least one positive solution when the parameter is small, while the p-Kirchhoff type problems have no positive solutions when the parameter is large. Our argument is based on variational methods, monotonicity methods, cut-off functional techniques, and a priori estimates techniques. Copyright © 2014 John Wiley & Sons, Ltd.

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.

In this paper, we shall establish the unilateral global bifurcation result from an interval for a class of high-dimensional p -Laplacian problems. As an application of the above result, we shall prove the existence of the principal half-eigenvalues for a class of half-quasilinear eigenvalue problems. Our result partially answers the open problem proposed by Berestycki. Moreover, we shall investigate the existence of one-sign solutions for a class of high-dimensional p -Laplacian problems with non- p − 1-homogeneous nonlinearities.

In this paper, we shall establish the unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity of a class of nonlinear Sturm–Liouville problems with nondifferentiable nonlinearity, respectively. As applications of the above results, we shall determine the interval of r, in which there exist nodal solutions for the following problem { u ″ ( t ) + r a ( t ) f ( u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where a ( t ) is a positive continuous function on [ 0 , 1 ] , f ∈ C ( R , R ) but is not necessarily differentiable at the origin and infinity. Moreover, as a special case of the above problem, we shall prove more details about the existence of nodal solutions for a class of half-linear eigenvalue problems.

We are concerned with determining values of γ, for which there exist radial nodal solutions of the boundary-value problem where B is the unit open ball in ℝ N with N ≥ 1. The proofs of our main results are based upon bifurcation techniques. In particular, we establish the global bifurcation theory for a second-order weighted p-Laplacian eigenvalue problem in B.