Homotopy Method On Finding Periodic Solutions For Differential Equation

This paper deals with the problems of finding periodic solutions for vector ordinary differential equations of the formwhere T is a fixed positive number and / satisfies some additional conditions which will be given later.As is well known, the periodicity problem plays a central role in the qualitative theory of ordinary differential equations for its significance in the physical sciences (see [19] and [28]). Hence finding periodic solutions of ordinarydifferential equations is naturally an attracting topic. Some works of this area can be found in [9,17,23,25]. Since Kellogg, Li and Yorke [18], and Smale [29] proposed the notable homotopy method, this method has become a powerful tool in finding solutions for various nonlinear problems, for example, zerosor fixed points of maps; see [1,2,4,5,6,22,23,24,25,27,33,34,35,37] and the references therein. Chow et al.[5] presented the first application of such methods to two-point boundary value problems for ordinary differential equations. Since then, there has been a series of remarkable works in this direction; see, for example, [6,33,37]. An explicit advantage of that method is that the reduced algorithm is with the global convergence (comparing with the classical method, for example, the Newton method). Another is avoiding the discussion of the transversality by utilizing the parametrized Sard theorem ( comparing with the usual continuation method). Recently, there have been some applications of the homotopy method to the periodicity problems [22,23,25].In the present paper, we investigate the following problem with use of the homotopy method.Let be an open set such that for each t (E R, fit C Rn is a bounded open set and satisfies R. If the boundary U is not tangential relative to the equation (6.1), then prove the existence of T-periodic solutions and give a feasible global method of finding that solutions. Often Rn \ dtlo also satisfies some additional continuation conditions.It should be pointed out that during the last decades, there has been a large amount of remarkable works related to nontangential conditions; see [3,8,10,12,14,19,20,26,31]. However these results deal with only the existence of the solutions and say no about finding the solutions, and hence are different from our approach. On the other hand, our existence conditions seem moregeneral than in some results mentioned above. First, our comparison result does not require the convexity of Liapunov functions. Secondly, we remove the restriction of the independence of time variable to guiding functions.This paper is organized as follows.In Section ?3 ?1 we state and prove our main results. Then the more general cases and applications are given in section ?3 - 2.Let us state our main results as followsConsider the first order differential equation(6.1).Set The boundary dfl is said to be nontangential relativeWe require the following assumptions.is continuous and twice continuously differentiablein the second variable.where V denotes the Jacob! matrix of g.We haveTheorem 6.1 Let (A1)-(A3) be fulfilled. Then (6.1) has a T-periodic solution in fi. Moreover, consider a homotopy map: H defined byc is a given small positive number such that c e range(g), and c is not an extreme value. Then the equation (6.1) has a T-periodic solution in fi. Moreover,for almost every there exists a Cl path (p(s), X(s)) of dimension 1 such thatThen (6.1) has a T-periodic solution in fl Theorem 6.3 Let the following conditions be fulfilled, (i) f(t, z) satisfies the Lipschitz condition in x.(ii) There exist a C1 function . which is Lipschitz in u, such thatThen (6.1) has a T-periodic solution x(t] with z(0) e fi. Moreover, if we assume that V(0, z) and f(t,x) are of class (72 with respect to the variable x, then for almost every x0 e Q, there exists a (71 path passing to .TO suqh that following this path, one can find a point x* €. ft for which the solution x(t,x*} of (6.1) with initial value z(0) = z* is the T-preiodic solution of (6.1).Now let us compare our result wit...