Determination of Fractional Chromatic Numbers in the Operation of Adding Two Different Graphs

The development of graph theory has provided many new pieces of knowledge, one of them is graph color. Where the application is spread in various fields such as the coding index theory. Fractional coloring is multiple coloring at points with different colors where the adjoining point has a different color. The operation in the graph is known as the sum operation. Point coloring can be applied to graphs where the result of operations is from several special graphs. In this case, the graph summation results of the path graph and the cycle graph will produce the same fractional chromatic number as the sum of the fractional chromatic numbers of each graph before it is operated.

Multiplicity of Positive Solutions to Nonlocal Boundary Value Problems with Strong Singularity

In this paper, we consider generalized Laplacian problems with nonlocal boundary conditions and a singular weight, which may not be integrable. The existence of two positive solutions to the given problem for parameter λ belonging to some open interval is shown. Our approach is based on the fixed point index theory.

On a power-type coupled system with mean curvature operator in Minkowski space

We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space $$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{in } B, \\ u|_{\partial B}=v|_{\partial B}=0, \end{cases} $$ { M ( u ) + v α = 0 in B , M ( v ) + u β = 0 in B , u | ∂ B = v | ∂ B = 0 , where $\mathcal{M}(w)=\operatorname{div} ( \frac{\nabla w}{\sqrt{1-|\nabla w|^{2}}} )$ M ( w ) = div ( ∇ w 1 − | ∇ w | 2 ) and B is a unit ball in $\mathbb{R}^{N} (N\geq 2)$ R N ( N ≥ 2 ) . We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β.

The well-posedness and long-time behavior of the nonlocal diffusion porous medium equations with nonlinear term

In this paper, we mainly consider the well-posedness and long-time behavior of solutions for the nonlocal diffusion porous medium equations with nonlinear term. Firstly, we obtain the well-posedness of the solutions in L1(Ω) for the equations. Secondly, we prove the existence of a global attractor by proving there exists a compact absorbing set. Finally, we apply index theory to consider the dimension of the attractor and prove that there exists an infinite dimensional attractor of the equations under proper conditions.

On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs

The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short).

Application of Fixed-Point Index Theory for a Nonlinear Fractional Boundary Value Problem with an Advanced Argument

In this paper, we investigate the existence of positive solutions of a class of fractional three-point boundary value problem with an advanced argument by using fixed-point index theory. Our results improve and extend some known results in the literature. Two examples are given to demonstrate the effectiveness of our results.

Multiple solutions for singular semipositone boundary value problems of fourth-order differential systems with parameters

The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: $$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$ { u ( 4 ) ( t ) + β 1 u ″ ( t ) − α 1 u ( t ) = f 1 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; v ( 4 ) ( t ) + β 2 v ″ ( t ) − α 2 v ( t ) = f 2 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 ; v ( 0 ) = v ( 1 ) = v ″ ( 0 ) = v ″ ( 1 ) = 0 , where $f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]$ f 1 , f 2 ∈ C [ ( 0 , 1 ) × R 0 + × R , R ] , $\mathbb{R}_{0}^{+}=(0,+\infty )$ R 0 + = ( 0 , + ∞ ) . By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.

Research on Health Evaluation and Operation and Maintenance Decision of Intelligent Distribution Network Based on Health Index Theory

To solve the problems of dynamic state evaluation of large-scale distribution equipment and complex and changeable smart grid, and realize efficient planning, operation and maintenance. Based on the existing literature, this paper explores the distribution network health index theory and distribution network operation and maintenance strategy. The distribution network planning method based on health index is studied. Based on the existing power system research indexes, the suitable intelligent grid evaluation indexes are selected from the aspects of safety, economy, reliability and environmental protection, and an index system for comprehensive evaluation of smart distribution network health is selected. The critical characteristics of distribution network are classified, the indexes are established and the distribution network health grades are divided by analytic hierarchy process and entropy weight method. In this paper, the evaluation method is used to calculate the health index, classify the health degree and generate the health degree image for seven lines in a 110kv substation in oil field. According to the optimal operation and maintenance strategy of distribution network based on health index theory, it provides a reliable basis for distribution network planning, construction, operation and maintenance.