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

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(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