Approach to the qualitative theory of functional equations by phase plane analysis

(1)考虑一类非线性时滞差分方程，给出了其所有非平凡解振动的充分条件和非振动解存在的充分条件。用相平面分析方法证明了这一结果。（二）利纳德体系的原型是由法国物理学家利纳德于1928年首次提出的。该系统对微分方程定性理论的发展起了重要作用。本文给出了非自治Lienard系统零解全局渐近稳定的充分条件。研究了广义Lienard系统中同斜轨道的存在性，得到了Lienard系统具有同斜轨道的若干条件。作为一个实际应用，我们给出了受Allee效应影响较大的gaes型捕食-食饵模型中同斜轨道存在的充分必要条件。(3)众所周知，线性微分方程的解空间具有齐次性和可加性；也就是说，一个解的任何倍数也是一个解两个解的和是另一个解。近二十年来，半线性微分方程的研究引起了相当大的关注。“半线性微分方程”一词源于这样一个事实，即解空间只具有上述性质的一半，即齐次性（但不具有可加性）。半线性微分方程有时被称为一维p-拉普拉斯微分方程。本文讨论了非自治半线性微分系统的解的渐近行为和零解的全局渐近稳定性。（iv）我们处理了各种微分方程的振动问题，如自伴随微分方程、半线性微分方程、带p-拉普拉斯或带扰动项的非线性微分方程和椭圆方程。我们给出了所有非平凡解是振荡的充分条件和所有非平凡解是非振荡的充分条件。所得到的定理推广了先前关于这个问题的许多结果。我们的主要方法是对每个方程等效的系统进行相平面分析。(v)将Lienard系统和微分方程与一维p-拉普拉斯算子相结合，考虑了带p-拉普拉斯算子的Lienard型系统。给出了该系统至少有一个稳定极限环的充分条件。用庞加莱-本迪克森定理对相平面分析证明了主要结果。少

(i) We considered a nonlinear delay difference equation, and gave sufficient conditions for all nontrivial solutions to be oscillatory and sufficient conditions for the existence of a nonoscillatory solution. Our results were proved by use of phase plane analysis for a system equivalent to this equation.(ii) The prototype of Lienard system was first formulated by the French physicist A. Lienard in 1928. The system played an important role in the development of the qualitative theory of differential equations. In this research, we gave sufficient conditions for the zero solution of nonautonomous Lienard systems to be globally asymptotically stable. We also investigated the existence of homoclinic orbits in generalized Lienard systems and obtained some conditions under which the Lienard system has homoclinic orbits. As a practical application, we gave a necessary and sufficient condition for the the existence of homoclinic orbits in Gause-type predator-prey models which are under the inf … More luence of an Allee effect.(iii) It is well-known that the solution space of linear differential equations has homogeneity and additivity; that is, any multiple of a solution is also a solution and the sum of two solutions is another solution. The investigation of half-linear differential equations has attracted considerable attention in the last two decades. The term half-linear differential equations is derived from the fact that the solution space has just one half of the above properties, namely, homogeneity (but not additivity). Half-linear differential equations are sometimes called differential equations with the one-dimensional p-Laplacian. In this research, we discussed the asymptotic behavior of solutions and the global asymptotical stability of the zero solution of nonautonomous half-linear differential systems.(iv) We dealt with the oscillation problem for various differential equations such as self-adjoint differential equations, half-linear differential equations, nonlinear differential equations with p-Laplacian or with perturbed terms, and elliptic equations. We presented sufficient conditions for all nontrivial solutions to be oscillatory and sufficient conditions for all nontrivial solutions to be nonoscillatory. The obtained theorems extended many previous results on this problem. Our main method is phase plane analysis for a system equivalent to each equation.(v) Combining Lienard system and differential equations with the one-dimensional p-Laplacian, we considered Lienard-type system with p-Laplacian. We gave sufficient conditions under which this new system has at least one stable limit cycle. The main results were proved by means of phase plane analysis with the Poincare-Bendixson theorem. Less