1 [ 1 + a ( t ) b ( t ) ] G ( t , s ) ρ 1 [ 1 + a ( t ) b ( t ) ] ) ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s | γ 1 | b ( t ) b ( t ) | + γ 2 | a ( t ) a ( t ) | 1 + β 1 β 2 + 0 | G ( t , s ) G ( t , s ) | ρ 1 ( 1 + β 1 β 2 ) ϕ ( s ) ( q ( s , y 1 n ( s ) , y 2 n ( s ) ) + q ( s , y 1 ( s ) , y 2 ( s ) ) ) d s γ 1 | b ( t ) b ( t ) | + γ 2 | a ( t ) a ( t ) | 1 + β 1 β 2 + 2 S γ 0 ρ 1 ( 1 + β 1 β 2 ) 0 | G ( t , s ) G ( t , s ) | ϕ ( s ) d s 0 , t t ,

| ( T m x ) ( t ) ( T m x ) ( t ) | | α 1 γ 2 α 2 γ 1 ρ | | p ( t ) p ( t ) | p ( t ) p ( t ) + sup t , t [ 0 , + ) { 1 p ( t ) , 1 p ( t ) } ( t t α 1 b ( s ) + α 2 a ( s ) ρ ( f ( s , x n ( s ) , x n ( s ) ) k 2 x n ( s ) ) d s ) | α 1 γ 2 α 2 γ 1 ρ | | | p ( t ) p ( t ) | | p ( t ) p ( t ) + sup t , t [ 0 , + ) { 1 p ( t ) , 1 p ( t ) } ( S γ 0 t t α 1 b ( s ) + α 2 a ( s ) ρ ϕ ( s ) d s ) 0 , t t .

T m M [ 0 , + ) 上等度连续。由于

| ( T m x ) ( t ) ρ 1 [ 1 + a ( t ) b ( t ) ] ( T m x ) ( ) ρ 1 [ 1 + a ( ) b ( ) ] | γ 1 | b ( t ) b ( ) | + γ 2 | a ( t ) a ( ) | 1 + β 1 β 2 + 0 | G ( t , s ) G ¯ ( s ) | ρ 1 ( 1 + β 1 β 2 ) ϕ ( s ) ( q ( s , y 1 n ( s ) , y 2 n ( s ) ) + q ( s , y 1 ( s ) , y 2 ( s ) ) ) d s γ 1 | b ( t ) b ( ) | + γ 2 | a ( t ) a ( ) | 1 + β 1 β 2 + 2 S r 0 ρ 1 ( 1 + β 1 β 2 ) 0 | G ( t , s ) G ¯ ( s ) | ϕ ( s ) d s 0 , t .

并且可以证明 | ( T m x ) ( t ) ( T m x ) ( ) | 0 , t ,所以 T m M 一致收敛。由引理2.2,对每一个自然数 m T m : K K 是全连续算子。

最后证明, T : K K 是全连续算子。对任意的 t [ 0 , + ) , x K , x 1 ,即

y 1 ( t ) = x ( t ) ρ 1 [ 1 + a ( t ) b ( t ) ] 1 y 2 ( t ) = x ( t ) | x ( t ) | 1 t [ 0 , + )

| ( T x ) ( t ) ( T m x ) ( t ) | ρ 1 [ 1 + a ( t ) b ( t ) ] = 0 1 m G ( t , s ) ρ 1 [ 1 + a ( t ) b ( t ) ] ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s 0 1 m G ( t , s ) ρ 1 [ 1 + a ( t ) b ( t ) ] g ( s , y 1 ( s ) , y 2 ( s ) ) d s 0 1 m G ( t , s ) ρ 1 [ 1 + a ( t ) b ( t ) ] ϕ ( s ) q ( s , y 1 ( s ) , y 2 ( s ) ) d s S 1 0 1 m ϕ ( s ) d s 0 , m + .

其中 S 1 : = sup { q ( t , v 1 , v 2 ) : 0 v 1 , | v 2 | 1 } < + .

| ( T x ) ( t ) ( T m x ) ( t ) | = | α 2 ρ p ( t ) 0 1 m a ( s ) ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s | sup t [ 0 , + ) 1 p ( t ) 0 1 m g ( s , y 1 ( s ) , y 2 ( s ) ) d s sup t [ 0 , + ) 1 p ( t ) 0 1 m ϕ ( s ) q ( s , y 1 ( s ) , y 2 ( s ) ) d s S 1 sup t [ 0 , + ) 1 p ( t ) 0 1 m ϕ ( s ) d s 0 , m + .

由此可知 T x T m 0 , m + , T : K K 是全连续算子。

定理3.1:假设条件(H1) (H2)成立,并且 q , g 满足下列条件

(H3) 0 lim v 1 , , v 2 0 + min t [ 0 , + ) q ( t , v 1 , v 2 ) max { v 1 , | v 2 | } < L , k 2 < l < lim v 1 + | v 2 | + min t [ a * , b * ] g ( t , v 1 , v 2 ) v 1 + | v 2 | + ,

L = max { ( 1 γ 1 b ( 0 ) + γ 2 a ( 0 ) r 1 ( 1 + β 1 β 2 ) ) ( 0 ϕ ( s ) d s ) 1 , ( 1 sup t [ 0 , + ) 1 p ( t ) α 1 γ 2 + α 2 γ 1 ρ r 1 ) ( 0 ϕ ( s ) d s ) 1 } ,

l = max { k 2 + 1 , ( ( a * b * G ( t , s ) d s ) 1 + k 2 ) 1 1 + a ( ) b ( 0 ) ρ } , r 1 > 0.

则BVP(1.1)至少有一个解。

证明:由条件(H3)的第一个不等式,存在 ε 0 > 0 , r > 0 满足

q ( t , v 1 , v 2 ) ( L ε 0 ) max { v 1 , | v 2 | } , 0 r , t [ 0 , + ) . (3.12)

r > r 1 时,对任意的 0 v 1 , | v 2 | r 1 , t [ 0 , + ) ,(3.12)同样成立。记 K r 1 = { x K : x < r 1 } ,对任意的 x K r 1 ,有 y 1 ( t ) = x ( t ) ρ 1 [ 1 + a ( t ) b ( t ) ] r 1 y 2 ( t ) = | x ( t ) | r 1 ,所以

q ( t , y 1 ( t ) , y 2 ( t ) ) ( L + ε ) { y 1 ( t ) , | y 2 ( t ) | } .

对任意的 t [ 0 , + )

| ( T x ) ( t ) | ρ 1 [ 1 + a ( t ) b ( t ) ] = γ 1 b ( t ) ρ ρ 1 [ 1 + a ( t ) b ( t ) ] + γ 2 a ( t ) ρ ρ 1 [ 1 + a ( t ) b ( t ) ] + 0 G ( t , s ) ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s γ 1 b ( t ) + γ 2 a ( t ) [ 1 + a ( t ) b ( t ) ] + 0 G ( t , s ) ρ 1 [ 1 + a ( t ) b ( t ) ] g ( s , y 1 ( s ) , y 2 ( s ) ) d s γ 1 b ( 0 ) + γ 2 a ( ) 1 + β 1 β 2 + 0 ϕ ( s ) q ( s , y 1 ( s ) , y 2 ( s ) ) d s γ 1 b ( 0 ) + γ 2 a ( ) 1 + β 1 β 2 + ( L ε 0 ) max { y 1 ( s ) , | y 2 ( s ) | } 0 ϕ ( s ) d s x = r 1 .

| ( T x ) ( t ) | = | ( γ 1 b ( t ) ρ + γ 2 a ( t ) ρ + 0 G ( t , s ) ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s ) | = | α 1 γ 2 ρ p ( t ) α 2 γ 1 ρ p ( t ) α 2 ρ p ( t ) 0 t a ( s ) ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s + α 1 ρ p ( t ) t b ( s ) ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s | sup t [ 0 , + ) 1 p ( t ) ( α 1 γ 2 + α 2 γ 1 ρ + 0 t α 2 a ( s ) ρ ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s + t α 1 b ( s ) ρ ( f ( s , x ( s ) , x ( s ) ) k 2 x ( s ) ) d s )

sup t [ 0 , + ) 1 p ( t ) ( α 1 γ 2 + α 2 γ 1 ρ + 0 f ( s , x ( s ) , x ( s ) ) d s ) sup t [ 0 , + ) 1 p ( t ) ( α 1 γ 2 + α 2 γ 1 ρ + 0 ϕ ( s ) q ( s , y 1 ( s ) , y 2 ( s ) ) d s ) sup t [ 0 , + ) 1 p ( t ) ( α 1 γ 2 + α 2 γ 1 ρ + ( L ε 0 ) max { y 1 ( s ) , | y 2 ( s ) | } 0 ϕ ( s ) d s ) x = r 1 .

另一方面,由条件(H3)的第二个不等式,存在 r 0 > c * r 1 > 0 满足

g ( t , v 1 , v 2 ) ( l + ε 0 ) ( v 1 + | v 2 | ) , v 1 , | v 2 | r 0 , t [ a * , b * ] . (3.13)

r 2 = r 1 + c * r 1 > r 1 K r 2 = { x K , x < r 2 } x 0 = 1 K 1 下面证明

x T x + μ x 0 , x K r 2 , μ > 0. (3.14)

否则,存在 x K r 2 μ * > 0 ,有 x * = T x * + μ * ,由(3.13)和

y 1 * ( t ) + y 2 * = x * ( t ) ρ 1 ( 1 + a ( t ) b ( t ) ) + | x * ( t ) | c * x * ( u ) ρ 1 ( 1 + a ( u ) b ( u ) ) + | x * ( t ) | c * ( x 0 + x ) > c * r 2 > r 0 , t [ a * , b * ] , u [ 0 , + )

可知

g ( t , y 1 * ( t ) , y 2 * ( t ) ) ( l + ε 0 ) ( y 1 * ( t ) + | y 2 * ( t ) | ) . (3.15)

假设 ξ = min { x * ( t ) : t [ a * , b * ] } ,则

x * ( t ) = γ 1 b ( t ) ρ + γ 2 a ( t ) ρ + 0 G ( t , s ) ( f ( s , x * ( s ) , x * ( s ) ) k 2 x * ( s ) ) d s a * b * G ( t , s ) ( g ( s , y 1 * ( s ) , y 2 * ( s ) ) k 2 x * ( s ) ) d s + μ 2 a * b * G ( t , s ) ( l + ε 0 ) ( y 1 * ( s ) + | y 2 * ( s ) | k 2 x * ( s ) ) d s + μ 2 = a * b * G ( t , s ) ( l + ε 0 ) ( x * ( s ) ρ 1 ( 1 + a ( s ) b ( s ) ) + | x * ( s ) | k 2 x * ( s ) ) d s + μ 2

> min s [ a * , b * ] x * ( s ) a * b * G ( t , s ) ( ( l + ε 0 ) ρ 1 ( 1 + a ( s ) b ( s ) ) k 2 ) d s + μ 2 > min s [ a * , b * ] x * ( s ) a * b * G ( t , s ) ( ( l + ε 0 ) ρ 1 ( 1 + a ( ) b ( 0 ) ) k 2 ) d s + μ 2 > ξ + μ 2 > ξ .

ξ < ξ ,所以(3.14)成立。根据以上的讨论,引理3.1和2.3, T 有不动点 x 满足 0 < r 1 < x r 2 。易知 x 是BVP(1.1)的正解。

类似于定理3.1的证明可得下面的定理3.2成立。

定理3.2:假设条件(H1) (H2)成立,并且 q , g 满足下列条件

(H4) 0 lim v 1 , | v 2 | max t [ 0 , + ) q ( t , v 1 , v 2 ) max { v 1 , | v 2 | } < L , k 2 < l < lim v 1 , | v 2 | 0 + max t [ a * , b * ] g ( t , v 1 , v 2 ) max { v 1 , | v 2 | } + .

L , l 的定义同定理3.1。则BVP(1.1)至少有一个正解。

4. 结论

本文,我们主要讨论的是无穷区间上一类微分方程边值问题正解的存在性,其中BVP(1.1)中的非线性项 f t = 0 点是奇异的,文 [5] [6] 中的非线性项 f 限制了连续的条件下,而且我们所研究的方程中, f 增加了导数项,这是在文 [5] [6] [7] [8] 中都没有涉及的,就要求在更为复杂的空间中,构造特殊的函数来讨论BVP(1.1),同时,相较于文 [5] [6] [7] [8] ,我们研究的边值条件更具有一般性,所以说,本文的结果,在一定程度上,改进和推广了许多已知结果。

基金项目

本文受到临沂大学大学生创新创业训练计划项目(201610452168)部分资助。

文章引用

王 克,王 颖. 无穷区间上奇异边值问题正解的存在性
Existence of Positive Solutions for Singular Boundary Value Problems on the Infinite Interval[J]. 应用数学进展, 2017, 06(09): 1151-1162. http://dx.doi.org/10.12677/AAM.2017.69140

参考文献 (References)

  1. 1. Liu, B.G., Liu, L.S. and Wu, Y.H. (2010) Unbounded Solutions for Three-Point Boundary Value Problems with Non-linear Boundary Conditions on . Nonlinear Analysis, 73, 2923-2932.
    https://doi.org/10.1016/j.na.2010.06.052

  2. 2. Smail, D., Quiza, S. and Yan, B.Q. (2012) Positive Solutions for Singular BVPs on the Positive Half-Line Arising from Epidemiology and Combustion Theory. Acta Mathematica Sci-entia, 32, 672-694.
    https://doi.org/10.1016/S0252-9602(12)60048-4

  3. 3. Liu, L.S., Wang, Z.G. and Wu, Y.H. (2009) Multiple Pos-itive Solutions of the Singular Boundary Value Problems for Secong-Order Differential Equations on the Haly-Line. Nonlinear Analysis, 71, 2564-2575.
    https://doi.org/10.1016/j.na.2009.01.092

  4. 4. Ma, R.Y. and Zhu, B. (2009) Existence of Positive Solutions for a Semipositone Boundary Value Problem on the Half-Line. Computers & Mathematics with Applications, 58, 1672-1686.
    https://doi.org/10.1016/j.camwa.2009.07.005

  5. 5. Zimbabwe, M. (2001) On Solutions of Boundary Value Problems on the Half-Line. Journal of Mathematical Analysis and Applications, 259, 127-136.
    https://doi.org/10.1006/jmaa.2000.7399

  6. 6. Hao, Z.C., Laing, J. and Xiamen, T.J. (2006) Positive Solutions of Operator Equations on Halt-Line. Journal of Mathematical Analysis and Applications, 314, 423-435.
    https://doi.org/10.1016/j.jmaa.2005.04.004

  7. 7. Wang, Y., Liu, L.S. and Wu, Y.H. (2008) Positive Solutions of Singular Boundary Value Problems on the Half-Line. Applied Mathematics and Computation, 197, 789-796.

  8. 8. Liam, H.R. and Ge, W.G. (2006) Existence of Positive for Sturm-Liouville Boundary Value Problems on the Half-Line. Journal of Mathematical Analysis and Applications, 321, 781-792.
    https://doi.org/10.1016/j.jmaa.2005.09.001

  9. 9. Xing, M.H., Zhang, K.M. and Gao, H.L. (2009) Existence of Positive Solutions for General Storm-Liouville Boundary Value Problems. Acta Mathematica Scientia, 29A, 929-939.

  10. 10. Yan, B.Q., Q’Regan, D. and Agartala, R.P. (2006) Unbounded Solutions for Singular Boundary Value Problems on the Semi-Infinite Interval: Upper and Lower Solutions and Multiplicity. Journal of Computational and Applied Mathematics, 1997, 365-386.

  11. 11. Guo, D.J. and Lakshmikantham, V. (1998) Nonlinear Problems in Abstract Cones. Academic Press, New York.

  12. 12. Amani, H. (1976) Fixed Point Equations and Nonlinear Eigenvalue Problems in Erdered Banach Space. SIAM Review, 18, 620-709.
    https://doi.org/10.1137/1018114

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