This research uses the extended exp-
 $( -\varphi(\vartheta ) ) $
 -expansion and the Jacobi elliptical function methods to obtain a fashionable explicit format for solutions to the fragmented biological population and the same width models that depict popular logistics because of deaths or births. In mathematical terminology, the linear, hyperbolic, and trigonometric equation solutions that have been found describe several innovative aspects from the two models. Sketching these solutions in different types is used to give them more details. The accuracy and performance of the method adopted show their ability to be applied to various nonlinear developmental equations.

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Oblique explicit wave solutions of the fractional biological population (BP) and equal width (EW) models

We considered nonlinear singular third order periodic boundary value problem{u″'+ρ3u=f(t,u),t∈I=(0,2π),ρ∈(0,1/3~(1/2))u(i)(0)=u(i)(2π),i=0,1,2 and obtained positive solution under some conditions concern to the eigenvalue of relevant linear operator by fixed index theory on cone.

Positive Solution of Singular Nonlinear Third Order Periodic Boundary Value Problem

In this paper, we investigate and obtain a new discrete $ q $-fractional version of the Gronwall inequality. As applications, we consider the existence and uniqueness of the solution of $ q $-fractional damped difference systems with time delay. Moreover, we formulate the novel sufficient conditions such that the $ q $-fractional damped difference delayed systems is finite time stable. Our result extend the main results of the paper by Abdeljawad et al. [A generalized $ q $-fractional Gronwall inequality and its applications to nonlinear delay $ q $-fractional difference systems, J.Inequal. Appl. 2016,240].

Finite-time stability of $ q $-fractional damped difference systems with time delay

This article discusses some basic properties of solutions to fractional hybrid q-difference equations. First, the existence theorem is presented by applying a fixed point theorem. Then, the stability result is derived by establishing a q-Gronwall inequality. This stability result also implies the uniqueness of the solution. Finally, a simple example is given to illustrate the effectiveness of our main results. Some conclusions in the literature are extended greatly.

The solution theory for the fractional hybrid q-difference equations

In this paper, we investigate and prove a new discrete - fractional version of the coupled Gronwall inequality. By applying this result, the finite-time stability criteria of solutions for a class of nonlinear - fractional difference coupled delay systems are obtained. As an application, an example is provided to demonstrate the effectiveness of our result.

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Finite-Time Stability of Solutions for Nonlinear q -Fractional Difference Coupled Delay Systems

In this paper, we investigate the existence of positive solutions for a class of fractional boundary value problems involving q-difference. By using the fixed point theorem of cone mappings, two existence results are obtained. Examples are given to illustrate the abstract results.

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Existence of positive solutions for the fractional q-difference boundary value problem

This paper deals with the existence of positive ω-periodic solutions for nth-order ordinary differential equation with delays in Banach space E of the form $$L_{n}u(t)=f\bigl(t,u(t-\tau_{1}),\ldots,u(t- \tau_{m})\bigr),\quad t\in\mathbb{R}, $$ where $L_{n}u(t)=u^{(n)}(t)+\sum_{i=0}^{n-1}a_{i} u^{(i)}(t)$ is the nth-order linear differential operator, $a_{i}\in\mathbb {R}$ ($i=0,1,\ldots,n-1$) are constants, $f: \mathbb{R}\times E^{m}\rightarrow E$ is a continuous function which is ω-periodic with respect to t, and $\tau_{i}>0$ ($i=1,2,\ldots,m$) are constants which denote the time delays. We first prove the existence of ω-periodic solutions of the corresponding linear problem. Then the strong positivity estimation is established. Finally, two existence theorems of positive ω-periodic solutions are proved. Our discussion is based on the theory of fixed point index in cones.

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Hong Li

Papers

Existence of positive solutions for the fractional q-difference boundary value problem

Positive periodic solutions for high-order differential equations with multiple delays in Banach spaces