Differential Equations Belonging To The Limit Circle Criteria For Solution Boundedness

The classification of limit point case or the limit circle case for the secondorderdifferential equations was first mentioned and researched by H.Weyl. He pointed out that the second-order linear ordinary differential equation can be divided into two cases: limit circle case and limit point case. If each solution of the linear equations is the square integrable solution, then it is called this equation is of the limit circle case, otherwise, it is called the limit point case. The boundedness of the solution for ordinary differential equation is proposed most early in the researches such as biology,ecology,physiology,physics,neural network question,etc. which is one of the most important in the research of ordinary differential equation.This article use the improved integral inequalities with the deviate variable and some skills of inequalities as well as related knowledge about second-order differential equation,to obtain the following results: the classification of limit point case or limit circle case and boundedness for second order differential or difference equation with deviative variable,discussed the classification for certain n-order differential equations,Criteria for the classification of second order differential equation with deviative variable.This article divides into five chapters according to contents.The first chapter is an introduction, which outline the background of this research.In the second chapter, we consider the second order equation(r(t)x′)′+ a(t)x = 0, (2.1.1)(r(t)x′)′+ [a(t)+b(t)]x = (?)f_i(t,x(t),x(φ(t))), (2.1.2)on 0≤t < +∞where r(t) > 0 is absolutely continuous real function, a(t), b(t) is real continuous function on R⁺ = [0,+∞),φ(t) is a continuously differentiable function, which satisfiesφ(t)≤t,φ′(t) > 0, lim_{tâ†'∞}φ(t) > 0, f_i(t,x,y) are continuous functions defined on [0,+∞)×R₂.The equation (2.1.1) or (2.1.2) is called limit circle case(denoted L·C), if all the solutions of equation (2.1.1) or (2.1.2) belong to L²[0, +∞); the equation (2.1.1). or (2.1.2) is called the Lagrange stably (denoted by L cdotS),if all the solutions of equation (2.1.1) or (2.1.2) are bounded on [0,+00).In this chapter,we use the improved inequality in article [9] to proved that under certain conditions,the equation (2.1.2) belongs to L·S∩L·C can be decided by the equation (2.1.1) belonging to L·S∩L·C.H.Weyl discussed the equationx″+ a(t)x = 0, (2.1.3)If it is of L·C,then when b(t) = 0(1),the equationx″+ [a(t) + b(t)]x = 0, (2.1.4)is L·C.In 1985,Ou Yang Liang [2] research the equation (2.1.1) and the equation(r(t)x′)′+ [a(t)+b(t)]x = O, (2.1.5)and obtained the following results :If the equation (2.1.1) belongs to L·S∩L·C, and |b(t)|∈L^p[0,+∞)(p > 1),thenthe equation (2.1.5)belongs to L·S∩L·C too.In 2001, Xu Run [8] proves that under certain conditions,the problem of the equation(r(t)x′)′+ (a(t) + b(t))x = f(t,x(t),x(φ(t))) (2.1.9)belongs to L·S∩L·C can be decided by the equation (2.1.1)belongs to L·S∩L·C.If r(t) = 1, f_i(t,x(t),x(φ(t))) = 0 then (2.1.2) turns into (2.1.4);If f_i(t,x(t),x(φ(t))) = 0,(i = 1,2,...,m), the equation (2.1.2) turns into the equation (2.1.5); and when m = 1, the equation (2.1.2) turns into the equation (2.1.9).So the results of this article improve the results of the paper cited before. In the third chapter,we consider the n order differential equation:(r(t)y^n-1(t))′+ (?)a_i(t)yⁱ(t) = f(t,y(t),y(φ(t))), (3.1.1) in 0≤t < +∞.where r(t) > 0 is continuous and differentiable on t∈R⁺ = [0,+∞), a_i(t) is continuous on R⁺ (i = 0, ...,n - 2), f(t,x,y) is continuous function which defined on R⁺×R×R, and we assume the equation (3.1.1) satisfies the local existence of Cauchy problem,φ(t)is continuous and differentiable and satisfyφ(t)≤t,φ′(t) > 0, lim_{tâ†'∞}φ(t) > 0.The main purpose of this chapter is discussed conditions for the solutions of equation (3.1.1) belong to L²[0, +∞) and the boundedness condition of the solutions,by the integral inequalities with deviation variable mentioned in article [9].In the fourth chapter,we studies the second-order difference equationx(n + 2) + q(n)x(n + 1) +p(n + 1)x(n) = 0 (4.1.1)x(n + 2) + q(n)x(n + 1) +p(n + 1)x(n) = f(n) (4.1.2)where n G N_n0 = {n₀,n₀ + 1,...}, n₀∈N, q(n), p(n), f(n) are the real sequences defined on N.According to the auxiliary function,we obtain sufficient conditions for the limit circle type of the equation (4.1.1), (4.1.2) and the criteria about the boundedness of the solution of (4.1.1), and (4.1.2).Considering the second difference equation (4.1.1), (4.1.2), The classification of limit circle case or limit point case is useful in the difference operator theory and the expansion theory of the characteristic function for difference equation. This problem had researched early. Ouyang Liang studied the boundedness and the limit cycle question of a kind of second differential operator in article [3],he obtained necessary and sufficient condition of all the solution of the equation having boundedness,as well as the sufficient conditions for all the solutions of the second-order differential equation with perturbation are bounded.Cheng Yuan Ji gave the discipline in paper [4] to judged a kind of second-order differential equation belonging to the limit circle or has boundedness. Meng Fan Wei gave the criteria of the second-order nonlinear differential equation belong to the limit cycle in article [5] . The main purpose of this chapter is discusses the difference equation in a similar nature,which the results of this aspect are not little.In the fifth chapter ,we consider the bounded solution of second-order nonlinear differential equation with deviating argument(a(t)x′(t))′+ f(t,x(t),x(φ(t))) =0. (5.1.1)whereφ(t) is a continuous and differentiable which satisfyingφ(t)≤t,φ′(t) > 0,φ(t) is eventually positive ,using the integral inequality with deviating argument, At last we give an example to show the results which we get is effective.