### Global attractivity for a differential-difference population model.

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A class of nonlinear boundary value problems for p-Laplacian differential equations is studied. Sufficient conditions for the existence of solutions are established. The nonlinearities are allowed to be superlinear. We do not apply the Green's functions of the relevant problem and the methods of obtaining a priori bounds for solutions are different from known ones. Examples that cannot be covered by known results are given to illustrate our theorems.

In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory....

This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation $$\left\{\begin{array}{c}{x}^{\text{'}}\left(t\right)=f(t,x\left(t\right),x\left({\alpha}_{1}\left(t\right)\right),\cdots ,x\left({\alpha}_{n}\left(t\right)\right))\text{for}\phantom{\rule{4.0pt}{0ex}}\text{a.e.}\phantom{\rule{4pt}{0ex}}t\in [0,T],\Delta x\left({t}_{k}\right)={I}_{k}\left(x\left({t}_{k}\right)\right),\phantom{\rule{4pt}{0ex}}k=1,\cdots ,m,x\left(0\right)=x\left(T\right).\hfill \end{array}\right.$$ We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin’s continuation theorem. Examples are presented to illustrate the main results.

This paper is concerned with the existence of positive solutions of a multi-point boundary value problem for higher-order differential equation with one-dimensional $p$-Laplacian. Examples are presented to illustrate the main results. The result in this paper generalizes those in existing papers.

Sufficient conditions for the existence of at least one $T-$periodic solution of second order nonlinear functional difference equations are established. We allow $f$ to be at most linear, superlinear or sublinear in obtained results.

A class of impulsive boundary value problems of fractional differential systems is studied. Banach spaces are constructed and nonlinear operators defined on these Banach spaces. Sufficient conditions are given for the existence of solutions of this class of impulsive boundary value problems for singular fractional differential systems in which odd homeomorphism operators (Definition 2.6) are involved. Main results are Theorem 4.1 and Corollary 4.2. The analysis relies on a well known fixed point...

The purpose of this paper is to study global existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. By constructing a special Banach space and employing fixed-point theorems, some sufficient conditions are obtained for the global existence and uniqueness of solutions of this kind of equations involving Caputo fractional derivatives and multiple base points. We apply the results to solve the forced logistic model with multi-term fractional...

In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equation $${x}^{\left(n\right)}\left(t\right)=f(t,x\left(t\right),{x}^{\text{'}}\left(t\right),\cdots ,{x}^{(n-1)}\left(t\right))+e\left(t\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}0<t<1\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}(*)$$ and the following multi-point boundary value conditions $$\begin{array}{cc}\hfill 1*-1{x}^{\left(i\right)}\left(0\right)& =0\phantom{\rule{1.0em}{0ex}}for\phantom{\rule{1.0em}{0ex}}i=0,1,\cdots ,n-3\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {x}^{(n-1)}\left(0\right)& =\alpha {x}^{(n-1)}\left(\xi \right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}{x}^{(n-2)}\left(1\right)=\sum _{i=1}^{m}{\beta}_{i}{x}^{(n-2)}\left({\eta}_{i}\right)\phantom{\rule{0.166667em}{0ex}}.**\hfill \end{array}$$ Sufficient conditions for the existence of at least one solution of the BVP $(*)$ and $(**)$ at resonance are established. The results obtained generalize and complement those in [13, 14]. This paper is directly motivated by Liu and Yu [J. Pure Appl. Math. 33 (4)(2002), 475–494...

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