Existence of multiple positive solutions of higher order multi-point nonhomogeneous boundary value problem

In this paper, by using the Avery and Peterson fixed point theorem, we establish the existence of multiple positive solutions for the following higher order multi-point nonhomogeneous boundary value problem $u^{(n)}(t) + f(t,u(t),u'(t),\ldots,u^{(n-2)}(t)) = 0, t\in (0,1)$, $u(0)= u'(0)=\cdots=u^{(n-3)}(0)=u^{(n-2)}(0)=0, u^{(n-2)}(1)-\sum_{i=1}^{m} a_i u^{(n-2)}(\xi_i)=\lambda$, where $n\ge3$ and $m\ge1$ are integers, $00$ for $1\le i\le m$ and $\sum_{i=1}^{m} a_i\xi_i<1$, $f(t,u,u',\cdots,u^{(n-2)})\in C([0,1]\times[0,\infty)^{n-1}, [0,\infty))$. We give an example to illustrate our result

Triple positive solutions for second-order four-point boundary value problem with sign changing nonlinearities

In this paper, we study the existence of triple positive solutions for second-order four-point boundary value problem with sign changing nonlinearities. We first study the associated Green's function and obtain some useful properties. Our main tool is the fixed point theorem due to Avery and Peterson. The results of this paper are new and extent previously known results

Green's function and positive solutions of a singular nth-order three-point boundary value problem on time scales

In this paper, we investigate the existence of positive solutions for a class of singular $n$th-order three-point boundary value problem. The associated Green's function for the boundary value problem is given at first, and some useful properties of the Green's function are obtained. The main tool is fixed-point index theory. The results obtained in this paper essentially improve and generalize some well-known results

Existence of positive solutions for nth-order boundary value problem with sign changing nonlinearity

In this paper, we investigate the existence of positive solutions for singular $n$th-order boundary value problem $u^{(n)}(t)+a(t)f(t,u(t))=0,\quad 0\le t\le1,$ $u^{(i)}(0)=u^{(n-2)}(1)=0,\quad 0\le i\le n-2,$ where $n\ge2$, $a\in C((0,1),[0,+\infty))$ may be singular at $t=0$ and (or) $t=1$ and the nonlinear term $f$ is continuous and is allowed to change sign. Our proofs are based on the method of lower solution and topology degree theorem

Positive solutions for nonlinear semipositone nth-order boundary value problems

In this paper, we investigate the existence of positive solutions for a class of nonlinear semipositone $n$th-order boundary value problems. Our approach relies on the Krasnosel'skii fixed point theorem. The result of this paper complement and extend previously known result

Cluster Analysis Based on Bipartite Network

Clustering data has a wide range of applications and has attracted considerable attention in data mining and artificial intelligence. However it is difficult to find a set of clusters that best fits natural partitions without any class information. In this paper, a method for detecting the optimal cluster number is proposed. The optimal cluster number can be obtained by the proposal, while partitioning the data into clusters by FCM (Fuzzy c-means) algorithm. It overcomes the drawback of FCM algorithm which needs to define the cluster number c in advance. The method works by converting the fuzzy cluster result into a weighted bipartite network and then the optimal cluster number can be detected by the improved bipartite modularity. The experimental results on artificial and real data sets show the validity of the proposed method