非线性方程边值问题的解及其应用
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摘要
非线性泛函分析是现代分析数学的一个重要分支,因其能很好的解释自然界中的各种各样的自然现象受到了越来越多的数学工作者的关注。其中,非线性边值问题来源于应用数学和物理的多个分支,是目前分析数学中研究最为活跃的领域之一。本文利用锥理论,不动点理论,Leggett-williams不动点定理:Krasnoselskii不动点定时等研究了几类微分方程奇异多点边值问题解的情况,得到了一些新成果。根据内容本文分为下列三节:
本文第一节中,研究了一类多点边值问题
正解的存在性,其中:
(H_1)q(t)∈L~1[0,1],q(t)≥0,(?)t∈[0,1];
(H_2)f∈C([0,1]×[0,+∞),[0,+∞));
(H_3)a_i≥0,b_i≥0,sam from i=1 to m-2 a_i<1,sam from i=1 to m-2 b_i<1,p(t)∈C[0,1],且p(t)>0,(?)t∈[0,1];
(H_4)存在非负连续函数 a_i(t),g_i(y)(i=1,2),t∈[0,1],y∈[0,+∞),满足integral from 0 to 1 (q(t)a_i(t)dt)>0,且对(?)t∈[0,1],有r>0,0≤y≤r时a_1(t)g_1(y)≤f(t,y),R>0,y≥R时f(t,y)≤a_2(t)g_2(y);
(H_4~*)存在非负连续函数 a_i(t),g_i(y)(i=1,2),(?)t∈[0,1],y∈[0,+∞),满足integral from 0 to 1 (q(t)a_i(t)dt)>0,且a_1(t)g_1(y)≤f(t,y)≤a_2(t)g_2(y)。
我们得到了如下结果:
定理1 若(H_4)满足,且
Nonlinear functional analysis is an important branch of morderm analysis mathmatics, because it can explain all kinds of natural phenomenal, more and more mathematicans are devoting their time to it.Among them, the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analyse mathematics. The present paper employs the cone theory, fixed point index theory,Leggett-williams fixed point theorem and Krasnoselskii fixed point theorem and so on, to investigate the existence of positive solutions of several classes of differential equations singular m-point boundary value problem. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions. The paper is divided into three sections according to contents.In the first Section, we shall study a class of m-point boundary value problemsThe following conditions hold:(H_1) q(t) ∈ L~1[0,1], q(t) ≥ 0,(?) t∈ [0,1];(H_2)f∈C([0,1]×[0,+∞),[0,+∞));(H_3) a_i ≥ 0, b_i ≥ 0, ∑_(i=1)~(m-2)a_i < 1,∑_(i=1)~(m-2)b_i < 1, p(t) ∈ C[0,1],and p(t)>0,(?)t∈[0,1] ;(H_4) there exist nonnegative functions a_i(t), g_i(y)(i = 1,2), t ∈ [0,1], y ∈ [0, +∞), satisfy ∫_0~1q(t)a_i(t)dt > 0,and (?)t∈ [0,1], (?)r > 0,if 0 ≤ y ≤ r then a_1(t)g_1(y) ≤ f(t,y); if R>0,y≥R then f(t,y) ≤ a2(t)g_2{y);(H_4~*) there exist nonnegative functions a_i(t),g_i(y)(i = 1,2), t ∈ [0,1], y∈ [0, +∞), satisfy ∫_0~1 q(t)a_i(t)dt > 0,and a_1(t)g_1(y) ≤ f(t, y) ≤ a_2(t)g_2(y).
We obtain the following result: Theorem 1 If (H4) hold,andwhere λ_i(i = 1,2) are the first eigenvalue of T_i defined by (2.1.6), then BVP (2.1.2)has at least a positive solution. Theorem 2 If (H_4~(?)) hold,andwhere λ_2 is the first eigenvalue of T_2 defined by (2.1.6).then BVP (2.1.2)has at least a positive solution.Remark 1 If p(t) ≡ 1,then t/(p(t))≥∫_0~t1/(p(s))ds, (?)t ∈ [0,1].Remark 2 When p(t) = 1,q(t) = 1,f(t,y) = a(t)f(y), we obtain the same results with [11] under weaker condition,so in fact this paper of theorem includes and popularizes the relevant results of [11].By making use of Leggett-williams fixed point theorem,we obtain the exitence of two positive solutions of the m-point BVP
The following conditions hold:(#1) di > 0, bt > 0, 0 < Yh=i ai < !? EI^2 &i < 1; (F2)/€C([0,+oo),[0,+oo));(H3) a(t) E La[0,1], a(t) > 0, V t e [0, l],and 0 < /on o(t)rfi < +oo; (tf4) p(t) € C[0,l],p(t) > 0,^ > /o^yds.V t 6 [0,l],and if -72 6i ^ 0 thendx 1 ^ f1 dx \ /?* rfa;We obtain the following result:Theorem 1 If there exist constantO < a < b < c, such that / satisfy:(#5) :/M > f, c < W < ^c;(Jf7) : /(w) > 2, 0p{u'))' + a(t)f(t, u) = 0, 0 < t < 1,?Z? 3.1.4cm(0) - pu'(0) = 0, u(l) = } at1=1where 4>p{s) is p — Laplacian function,^(s) is inverse function to 0p(s),i.e 0P = |s|p-2s, p > 1, J + i = 1, 0 < -ei < ... < U-2 < 1, a > 0, P > 0, a2 + (32 > 0.The following conditions hold:(^2) nonnegative function a(i) 6 C(0,1), 0 < Jo a(s)ds < +00, and there exists Xq G (^m-2,1) such that a(x0) > 0;
(H3) B = a{l- ZZ'i2 ?*&) + P(l- Etf 0.We obtain the following result:Theorem 1 Suppose one of the following hold:(H4) : 3 pi,p2 ￡ (0, +oo), and px < 7p2, such that Z^1 < $p(m),(H5) : 3 pi,pi € (0, +00), and px < p2, such that f￡2 < <&p(m),then BVP (3.1.4)has at least a positive solution. Theorem 2 Suppose one of the following hold: (H6) : 3 pi, p2, P3 e (0, +oo), and pi < 7p2, p2 < p3)such that(H7) : 3 pi, p2, P3 G (0, +oo), and pi < p2 < 7p3,such that f￡2 < $p(m), f%px > $p(^7)>u $" Au, V u G d KP2, f!*3 > 3>P(M7). then BVP (3.1.4)has at least two positive solutions.Remark2 Whenp = 2, m — 3,under conditions hmu_>0+ ^^, limu_+00+ ^ one equal 0 the other equal +00,[20] obtained the existence of a positive solution of BVP (3.1); When a = 1,(3 = 0,we improve [22] to singular conditions.In the third section, by making use of the fixed point theorem of cone expansion or compression type , we set up the existence of positive solutions for third-order three-point boundary value problem with semipositonenonlinearity' u'"(t) - \f(t, u,u') = 0, 0 0 such thatf(t,u,v)>-M(t)
本文第一节中,研究了一类多点边值问题
正解的存在性,其中:
(H_1)q(t)∈L~1[0,1],q(t)≥0,(?)t∈[0,1];
(H_2)f∈C([0,1]×[0,+∞),[0,+∞));
(H_3)a_i≥0,b_i≥0,sam from i=1 to m-2 a_i<1,sam from i=1 to m-2 b_i<1,p(t)∈C[0,1],且p(t)>0,(?)t∈[0,1];
(H_4)存在非负连续函数 a_i(t),g_i(y)(i=1,2),t∈[0,1],y∈[0,+∞),满足integral from 0 to 1 (q(t)a_i(t)dt)>0,且对(?)t∈[0,1],有r>0,0≤y≤r时a_1(t)g_1(y)≤f(t,y),R>0,y≥R时f(t,y)≤a_2(t)g_2(y);
(H_4~*)存在非负连续函数 a_i(t),g_i(y)(i=1,2),(?)t∈[0,1],y∈[0,+∞),满足integral from 0 to 1 (q(t)a_i(t)dt)>0,且a_1(t)g_1(y)≤f(t,y)≤a_2(t)g_2(y)。
我们得到了如下结果:
定理1 若(H_4)满足,且
Nonlinear functional analysis is an important branch of morderm analysis mathmatics, because it can explain all kinds of natural phenomenal, more and more mathematicans are devoting their time to it.Among them, the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analyse mathematics. The present paper employs the cone theory, fixed point index theory,Leggett-williams fixed point theorem and Krasnoselskii fixed point theorem and so on, to investigate the existence of positive solutions of several classes of differential equations singular m-point boundary value problem. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions. The paper is divided into three sections according to contents.In the first Section, we shall study a class of m-point boundary value problemsThe following conditions hold:(H_1) q(t) ∈ L~1[0,1], q(t) ≥ 0,(?) t∈ [0,1];(H_2)f∈C([0,1]×[0,+∞),[0,+∞));(H_3) a_i ≥ 0, b_i ≥ 0, ∑_(i=1)~(m-2)a_i < 1,∑_(i=1)~(m-2)b_i < 1, p(t) ∈ C[0,1],and p(t)>0,(?)t∈[0,1] ;(H_4) there exist nonnegative functions a_i(t), g_i(y)(i = 1,2), t ∈ [0,1], y ∈ [0, +∞), satisfy ∫_0~1q(t)a_i(t)dt > 0,and (?)t∈ [0,1], (?)r > 0,if 0 ≤ y ≤ r then a_1(t)g_1(y) ≤ f(t,y); if R>0,y≥R then f(t,y) ≤ a2(t)g_2{y);(H_4~*) there exist nonnegative functions a_i(t),g_i(y)(i = 1,2), t ∈ [0,1], y∈ [0, +∞), satisfy ∫_0~1 q(t)a_i(t)dt > 0,and a_1(t)g_1(y) ≤ f(t, y) ≤ a_2(t)g_2(y).
We obtain the following result: Theorem 1 If (H4) hold,andwhere λ_i(i = 1,2) are the first eigenvalue of T_i defined by (2.1.6), then BVP (2.1.2)has at least a positive solution. Theorem 2 If (H_4~(?)) hold,andwhere λ_2 is the first eigenvalue of T_2 defined by (2.1.6).then BVP (2.1.2)has at least a positive solution.Remark 1 If p(t) ≡ 1,then t/(p(t))≥∫_0~t1/(p(s))ds, (?)t ∈ [0,1].Remark 2 When p(t) = 1,q(t) = 1,f(t,y) = a(t)f(y), we obtain the same results with [11] under weaker condition,so in fact this paper of theorem includes and popularizes the relevant results of [11].By making use of Leggett-williams fixed point theorem,we obtain the exitence of two positive solutions of the m-point BVP
The following conditions hold:(#1) di > 0, bt > 0, 0 < Yh=i ai < !? EI^2 &i < 1; (F2)/€C([0,+oo),[0,+oo));(H3) a(t) E La[0,1], a(t) > 0, V t e [0, l],and 0 < /on o(t)rfi < +oo; (tf4) p(t) € C[0,l],p(t) > 0,^ > /o^yds.V t 6 [0,l],and if -72 6i ^ 0 thendx 1 ^ f1 dx \ /?* rfa;We obtain the following result:Theorem 1 If there exist constantO < a < b < c, such that / satisfy:(#5) :/M > f, c < W < ^c;(Jf7) : /(w) > 2, 0
(H3) B = a{l- ZZ'i2 ?*&) + P(l- Etf
引文
[1] 郭大钧.非线性泛函分析.山东科技出版社.2001年8月第二版.
[2] Guo.D,Lakshmikantham.V.Nonlinear Problems in Abstract Cone [M].Academic Press,Sandiego. 1988.
[3] Dajun Guo,V.Lakshmikantham,Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers(1996)
[4] Deimling K. Nonlinear Functional Analysis, Berlin: Springer-Verlag. (1980)
[5] 郭大钧,孙经先,刘兆理.非线性常微分方程的泛函方法济南:山东科学技术出版社,1995.
[6] O'Regan D. Theory of Singular Boundary Value Problems.Singapore: World Scientic Press,1994.
[7] 郭大钧.非线性分析中的半序方法.山东科技出版社.2001.3
[8] 郭大钧,孙经先.非线性积分方程.山东科技出版社.1987
[9] Liu Zhaoli, Li Fuyi.Multiple positive solutions of nonlinear two-point boundary value problems. J. Math. Anal. Appl.,1996,203:610-625.
[10] Zhang Guowei, Sun Jingxian.Positive solutions of m-point boundary value problems. J. Math. Anal. Appl.,2004,291:406-418.
[11] Ma Rurun.Exitence of solutions of nonlinear m-point boundary value problems. J. Math. Anal. Appl.,2001,256:556-567.
[12] V.A.II'in and E.I.Moiseev. Nonlocal boundary value problem of the second kind for a Strum-Liouvolle operator. Differential Equations.,1987,23:979-987.
[13] Ma Runyun. Positive solutions for second-order three-point boundary value problems. J. Comput. Appl. Math., 1984,10: 203-217.
[14] Ma Runyun. Positive solutions of a nonlinear three-point boundary value problem. Electron, J. Differential Equations, 1999,34: 1-8.
[15] Ma Runyun. Existence theorems for a second-order three-point boundary value problem. J. Math. Anal. Appl., 1997, 212: 430-442.
[16] Gupta C P. A sharper condition for the a three-poition second order boundary value problem. J. Math. Anal. Appl., 1997, 205: 586-597.
[17] Gupta C P. A generalized multi-point boundary value problem for second order ordinary differential equarions. Appl. Math. comput..,1998,89:133-146.
[18] Yao Qingliu, Ma Ruyun. Multiplicity of positive solutions for secondorder three-point boundary value problems. Compute. Math. Appl., 2000, 40: 193-204.
[19] Avery R. I, Henderson J.Twin solutions of BVP for ordinary differential equations and finite difference equations.[J].Computers Math Applic 2001,42:695-704.
[20] 马宇红,马如云.一类奇异非线性三点边值问题的正解.数学物理学报,2003,23A(5):583-588.
[21] Ma Rurun. Positive solutions of nonlinear m-point boundary value problems. J Math Annl Appl, 2001, 42: 755-765.
[22] Bai Chuanzhi, Fang Jinxuan. Existence of nultiple positive solutions for nonlinear m-point boundary value problems. J Math Annl Appl, 2003, 281: 76-85.
[23] Weizhong Li. Positive solutions of singular sublinear scond order boundary valur problems. Systems Science and Mathematical Science, 1998,10(1):82-88.
[24] 马巧珍.非线性三点边值问题正解的存在性.应用泛函分析学报,2001,3(2):178-182.
[25] 姚庆六.非线性二阶三点边值问题正解的一个存在定理.系统科学与数学,2003,23(4):495-500.
[26] Zhang Yong. Positive Solutions of Singular Enden-Fowler Boundary Value Problems. J. Math. Anal. Appl., 1994, 185(1): 215-222.
[27] 毛安民.正指数超线性Emden-Fowler方程奇异边值问题的正解.数学学报,2000,43(4):623-632.
[28] 韦忠礼.超线性Emden-Fowler方程奇异边值问题的正解.数学物理学报,1998,18(增刊):115-118.
[29] 郝兆才,毛安民.一类奇异二阶边值问题正解存在的充分必要条件.系统科学与数学,2001,21(1):93-100.
[30] Mao A., Xue M., Positive solutions of singular boundary value problems, Acta Math. Sin,, 2001, 44(5): 899-908 (in Chinese).
[31] Ma R., Positive solutions of singular second order boundary value problems, Acta Math. Sin.,1998, 14(Suppl.): 691-698.
[32] Yao Qingliu. Existence Of Positive Solutions For A Third-order Threepoint Boundary Value Problem With Semi-positive Nonlinearity. Journal of Mathematical Research & Exposition.,2003,23(4):591-596.
[33] Yao Qingliu. Several existence result Of some nonlinear third-order ordinary differential equations. Pure and Applied Mathematics.,2003,19(3):248-252.
[34] Agarwal R P., O'Regan D., A note on existence of nonnegative solutions to singular semi-positone problems, Nonlinear Anal., 1999, 36: 615-622.
[35] Ma R., Wang R., Ren L., Existence results for semipositone boundary
[2] Guo.D,Lakshmikantham.V.Nonlinear Problems in Abstract Cone [M].Academic Press,Sandiego. 1988.
[3] Dajun Guo,V.Lakshmikantham,Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers(1996)
[4] Deimling K. Nonlinear Functional Analysis, Berlin: Springer-Verlag. (1980)
[5] 郭大钧,孙经先,刘兆理.非线性常微分方程的泛函方法济南:山东科学技术出版社,1995.
[6] O'Regan D. Theory of Singular Boundary Value Problems.Singapore: World Scientic Press,1994.
[7] 郭大钧.非线性分析中的半序方法.山东科技出版社.2001.3
[8] 郭大钧,孙经先.非线性积分方程.山东科技出版社.1987
[9] Liu Zhaoli, Li Fuyi.Multiple positive solutions of nonlinear two-point boundary value problems. J. Math. Anal. Appl.,1996,203:610-625.
[10] Zhang Guowei, Sun Jingxian.Positive solutions of m-point boundary value problems. J. Math. Anal. Appl.,2004,291:406-418.
[11] Ma Rurun.Exitence of solutions of nonlinear m-point boundary value problems. J. Math. Anal. Appl.,2001,256:556-567.
[12] V.A.II'in and E.I.Moiseev. Nonlocal boundary value problem of the second kind for a Strum-Liouvolle operator. Differential Equations.,1987,23:979-987.
[13] Ma Runyun. Positive solutions for second-order three-point boundary value problems. J. Comput. Appl. Math., 1984,10: 203-217.
[14] Ma Runyun. Positive solutions of a nonlinear three-point boundary value problem. Electron, J. Differential Equations, 1999,34: 1-8.
[15] Ma Runyun. Existence theorems for a second-order three-point boundary value problem. J. Math. Anal. Appl., 1997, 212: 430-442.
[16] Gupta C P. A sharper condition for the a three-poition second order boundary value problem. J. Math. Anal. Appl., 1997, 205: 586-597.
[17] Gupta C P. A generalized multi-point boundary value problem for second order ordinary differential equarions. Appl. Math. comput..,1998,89:133-146.
[18] Yao Qingliu, Ma Ruyun. Multiplicity of positive solutions for secondorder three-point boundary value problems. Compute. Math. Appl., 2000, 40: 193-204.
[19] Avery R. I, Henderson J.Twin solutions of BVP for ordinary differential equations and finite difference equations.[J].Computers Math Applic 2001,42:695-704.
[20] 马宇红,马如云.一类奇异非线性三点边值问题的正解.数学物理学报,2003,23A(5):583-588.
[21] Ma Rurun. Positive solutions of nonlinear m-point boundary value problems. J Math Annl Appl, 2001, 42: 755-765.
[22] Bai Chuanzhi, Fang Jinxuan. Existence of nultiple positive solutions for nonlinear m-point boundary value problems. J Math Annl Appl, 2003, 281: 76-85.
[23] Weizhong Li. Positive solutions of singular sublinear scond order boundary valur problems. Systems Science and Mathematical Science, 1998,10(1):82-88.
[24] 马巧珍.非线性三点边值问题正解的存在性.应用泛函分析学报,2001,3(2):178-182.
[25] 姚庆六.非线性二阶三点边值问题正解的一个存在定理.系统科学与数学,2003,23(4):495-500.
[26] Zhang Yong. Positive Solutions of Singular Enden-Fowler Boundary Value Problems. J. Math. Anal. Appl., 1994, 185(1): 215-222.
[27] 毛安民.正指数超线性Emden-Fowler方程奇异边值问题的正解.数学学报,2000,43(4):623-632.
[28] 韦忠礼.超线性Emden-Fowler方程奇异边值问题的正解.数学物理学报,1998,18(增刊):115-118.
[29] 郝兆才,毛安民.一类奇异二阶边值问题正解存在的充分必要条件.系统科学与数学,2001,21(1):93-100.
[30] Mao A., Xue M., Positive solutions of singular boundary value problems, Acta Math. Sin,, 2001, 44(5): 899-908 (in Chinese).
[31] Ma R., Positive solutions of singular second order boundary value problems, Acta Math. Sin.,1998, 14(Suppl.): 691-698.
[32] Yao Qingliu. Existence Of Positive Solutions For A Third-order Threepoint Boundary Value Problem With Semi-positive Nonlinearity. Journal of Mathematical Research & Exposition.,2003,23(4):591-596.
[33] Yao Qingliu. Several existence result Of some nonlinear third-order ordinary differential equations. Pure and Applied Mathematics.,2003,19(3):248-252.
[34] Agarwal R P., O'Regan D., A note on existence of nonnegative solutions to singular semi-positone problems, Nonlinear Anal., 1999, 36: 615-622.
[35] Ma R., Wang R., Ren L., Existence results for semipositone boundary