Abstract:
We investigate nonlinear singular fourth-order eigenvalue problems with nonlocal boundary condition , , , , where , , may be singular at and/or . Moreover may also have singularity at and/or . By using fixed point theory in cones, an explicit interval for is derived such that for any in this interval, the existence of at least one symmetric positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and nonsingular cases. The associated Green's function for the above problem is also given. 1. Introduction Boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics and so on, and the existence of positive solutions for such problems has become an important area of investigation in recent years. To identify a few, we refer the reader to [1–7] and references therein. At the same time, a class of boundary value problems with nonlocal boundary conditions appeared in heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. Such problems include two-point, three-point, multipoint boundary value problems as special cases and have attracted the attention of Gallardo [1], Karakostas and Tsamatos [2], and Lomtatidze and Malaguti [3] (and see the references therein). For more information about the general theory of integral equations and their relation to boundary value problems we refer the reader to the book of Corduneanu [8] and Agarwal and O'Regan [9]. Motivated by the works mentioned above, in this paper, we study the existence of symmetric positive solutions of the following fourth-order nonlocal boundary value problem (BVP): where , , may be singular at and/or . Moreover may also have singularity at and/or . The main features of this paper are as follows. Firstly, comparing with [4–7], we discuss the boundary value problem with nonlocal boundary conditions, that is, BVP (1.1) including fourth-order two-point, three-point, multipoint boundary value problems as special cases. Secondly, comparing with [4–7], we discuss the boundary value problem when nonlinearity contains second-derivatives . Thirdly, here we not only allow have singularity at and/or but also allow have singularity at and/or . Finally, in [4–7], authors only studied the existence of positive solutions. However, they did not further provide characters of positive solutions, such as symmetry. It is now natural to consider the existence of symmetric positive solutions. To our knowledge, no paper has considered the existence

Abstract:
We study the following third-order -Laplacian -point boundary value problems on time scales , , , , , where is -Laplacian operator, that is, , , , . We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.

Abstract:
We study the following third-order p-Laplacian m-point boundary value problems on time scales ( p(uΔ )) +a(t)f(t,u(t))=0, t∈[0,T]Tκ, u(0)=∑i=1m 2biu(ξi), uΔ(T)=0, p(uΔ (0))=∑i=1m 2ci p(uΔ (ξi)), where p(s) is p-Laplacian operator, that is, p(s)=|s|p 2s, p>1, p 1= q,1/p+1/q=1, 0<ξ1< <ξm 2<ρ(T). We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term f(t,u) is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.

Abstract:
In this paper, we study the nonlinear second-order m-point boundary value problem $$displaylines{ u''(t)+f(t,u)=0,quad 0leq t leq 1, cr eta u(0)-gamma u'(0)=0,quad u(1)=sum _{i=1}^{m-2}alpha_{i} u(xi_{i}), }$$ where the nonlinear term $f$ is allowed to change sign. We impose growth conditions on $f$ which yield the existence of at least two positive solutions by using a fixed-point theorem in double cones. Moreover, the associated Green's function for the above problem is given.

Abstract:
We study a three-point nonlinear boundary value problem with higher-order p-Laplacian. We show that there exist countable many positive solutions by using the fixed point index theorem for operators in a cone.

Abstract:
The reciprocal complementary Wiener number of a connected graph G is defined as where is the vertex set. is the distance between vertices u and v, and d is the diameter of G. A tree is known as a caterpillar if the removal of all pendant vertices makes it as a path. Otherwise, it is called a non-caterpillar. Among all n-vertex non-cater- pillars with given diameter d, we obtain the unique tree with minimum reciprocal complementary Wiener number, where . We also determine the n-vertex non-caterpillars with the smallest, the second smallest and the third smallest reciprocal complementary Wiener numbers.

Abstract:
For the problems of large water demand and the high cost of raw water in most cities of our country,most of the water plants began to study how to reduce the water consumption in water supply systems.The full-scale experiment study on optimization of water-saving processes in water supply plant was conducted at the Bijiashan Water Plant in Shenzhen.The water consumption of each process unit has achieved a significant decline by optimizing the water plant sludge conditions,adjusting the discharge time and dischanging period exactly.The entire treatment system can save raw water nearly 7.683×105 m3/a.The total water consumption rate can be reduced from 2.08% to 1.49%.Moreover,the economic benefits were considerable as well as that of the environment.It can save nearly 6×105 yuan/a.