Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

Qingliu Yao

Applications of Mathematics (2013)

  • Volume: 58, Issue: 1, page 93-110
  • ISSN: 0862-7940

Abstract

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We consider the classical nonlinear fourth-order two-point boundary value problem u ( 4 ) ( t ) = λ h ( t ) f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) , 0 < t < 1 , u ( 0 ) = u ' ( 1 ) = u ' ' ( 0 ) = u ' ' ' ( 1 ) = 0 . In this problem, the nonlinear term h ( t ) f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) contains the first and second derivatives of the unknown function, and the function h ( t ) f ( t , x , y , z ) may be singular at t = 0 , t = 1 and at x = 0 , y = 0 , z = 0 . By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

How to cite

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Yao, Qingliu. "Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem." Applications of Mathematics 58.1 (2013): 93-110. <http://eudml.org/doc/251427>.

@article{Yao2013,
abstract = {We consider the classical nonlinear fourth-order two-point boundary value problem \[ \{\left\lbrace \begin\{array\}\{ll\} u^\{(4)\}(t)=\lambda h(t)f(t,u(t),u^\{\prime \}(t),u^\{\prime \prime \}(t)),\quad 0<t<1,\\ u(0)=u^\{\prime \}(1)=u^\{\prime \prime \}(0)=u^\{\prime \prime \prime \}(1)=0. \end\{array\}\right.\} \] In this problem, the nonlinear term $h(t)f(t,u(t),u^\{\prime \}(t),u^\{\prime \prime \}(t))$ contains the first and second derivatives of the unknown function, and the function $h(t)f(t,x,y,z)$ may be singular at $t=0$, $t=1$ and at $x=0$, $y=0$, $z=0$. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.},
author = {Yao, Qingliu},
journal = {Applications of Mathematics},
keywords = {nonlinear ordinary differential equation; singular nonlinearity; positive solution; eigenvalue interval; nonlinear ordinary differential equation; singularity; positive solution; eigenvalue interval},
language = {eng},
number = {1},
pages = {93-110},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem},
url = {http://eudml.org/doc/251427},
volume = {58},
year = {2013},
}

TY - JOUR
AU - Yao, Qingliu
TI - Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem
JO - Applications of Mathematics
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 1
SP - 93
EP - 110
AB - We consider the classical nonlinear fourth-order two-point boundary value problem \[ {\left\lbrace \begin{array}{ll} u^{(4)}(t)=\lambda h(t)f(t,u(t),u^{\prime }(t),u^{\prime \prime }(t)),\quad 0<t<1,\\ u(0)=u^{\prime }(1)=u^{\prime \prime }(0)=u^{\prime \prime \prime }(1)=0. \end{array}\right.} \] In this problem, the nonlinear term $h(t)f(t,u(t),u^{\prime }(t),u^{\prime \prime }(t))$ contains the first and second derivatives of the unknown function, and the function $h(t)f(t,x,y,z)$ may be singular at $t=0$, $t=1$ and at $x=0$, $y=0$, $z=0$. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.
LA - eng
KW - nonlinear ordinary differential equation; singular nonlinearity; positive solution; eigenvalue interval; nonlinear ordinary differential equation; singularity; positive solution; eigenvalue interval
UR - http://eudml.org/doc/251427
ER -

References

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  1. Agarwal, R. P., O'Regan, D., 10.1016/S0377-0427(97)00205-7, J. Comput. Appl. Math. 88 (1998), 129-147. (1998) Zbl0902.34017MR1609070DOI10.1016/S0377-0427(97)00205-7
  2. Agarwal, R. P., O'Regan, D., 10.1006/jdeq.2000.3808, J. Differ. Equations 170 (2001), 142-156. (2001) Zbl0978.34018MR1813103DOI10.1006/jdeq.2000.3808
  3. Atanackovic, T. M., Stability Theory of Elastic Rods. Series on Stability, Vibration and Control of Systems, Series A, World Scientific Singapore (1997). (1997) 
  4. Bai, Z., Wang, H., 10.1016/S0022-247X(02)00071-9, J. Math. Anal. Appl. 270 (2002), 357-368. (2002) Zbl1006.34023MR1915704DOI10.1016/S0022-247X(02)00071-9
  5. Bai, Z., 10.1016/j.na.2006.08.009, Nonlinear Anal., Theory Methods Appl. 67 (2007), 1704-1709. (2007) Zbl1122.34010MR2326022DOI10.1016/j.na.2006.08.009
  6. Dunninger, D. R., 10.4064/ap77-2-3, Ann. Pol. Math. 77 (2001), 161-168. (2001) Zbl0989.34014MR1869312DOI10.4064/ap77-2-3
  7. Elgindi, M. B. M., Guan, Z., 10.1155/S0161171297000343, Int. J. Math. Math. Sci. 20 (1997), 257-261. (1997) Zbl0913.34020MR1444725DOI10.1155/S0161171297000343
  8. Feng, H., Ji, D., Ge, W., 10.1016/j.na.2008.07.013, Nonlinear Anal., Theory Methods Appl. 70 (2009), 3561-3566. (2009) Zbl1169.34308MR2502764DOI10.1016/j.na.2008.07.013
  9. Graef, J. R., Yang, B., 10.1080/00036810008840810, Appl. Anal. 74 (2000), 201-214. (2000) Zbl1031.34025MR1742276DOI10.1080/00036810008840810
  10. Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering, 5, Academic Press Boston (1988). (1988) MR0959889
  11. Gupta, C. P., 10.1080/00036818808839715, Appl. Anal. 26 (1988), 289-304. (1988) Zbl0611.34015MR0922976DOI10.1080/00036818808839715
  12. Henderson, J., Wang, H., 10.1006/jmaa.1997.5334, J. Math. Anal. Appl. 208 (1997), 252-259. (1997) Zbl0876.34023MR1440355DOI10.1006/jmaa.1997.5334
  13. Hewitt, E., Stromberg, K., Real and Abstract Analysis. A Modern Treatment of The Theory of Functions of a Real Variable. 3rd printing, Springer Berlin, Heidelberg, New York (1975). (1975) Zbl0307.28001MR0367121
  14. Liu, B., 10.1016/S0096-3003(02)00857-3, Appl. Math. Comput. 148 (2004), 407-420. (2004) Zbl1039.34018MR2015382DOI10.1016/S0096-3003(02)00857-3
  15. Ma, R., Ma, Q., 10.1007/BF02538837, Appl. Math. Mech., Engl. Ed. 16 (1995), 961-969. (1995) Zbl0836.73032MR1365596DOI10.1007/BF02538837
  16. Marcos, J., Lorca, S., Ubilla, P., 10.1016/j.aml.2007.02.025, Appl. Math. Lett. 21 (2008), 279-286. (2008) Zbl1169.34317MR2433742DOI10.1016/j.aml.2007.02.025
  17. Meehan, M., O'Regan, D., 10.1216/jiea/1020282208, J. Integral Equations Appl. 12 (2000), 271-280. (2000) Zbl0988.45004MR1810743DOI10.1216/jiea/1020282208
  18. Staněk, S., 10.1016/j.na.2009.03.043, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 4893-4905. (2009) Zbl1192.34027MR2548721DOI10.1016/j.na.2009.03.043
  19. Wei, Z., 10.1016/S0096-3003(03)00683-0, Appl. Math. Comput. 153 (2004), 865-884. (2004) Zbl1057.34006MR2066157DOI10.1016/S0096-3003(03)00683-0
  20. Yao, Q., 10.1016/j.na.2007.07.002, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69 (2008), 1570-1580. (2008) MR2424530DOI10.1016/j.na.2007.07.002
  21. Yao, Q., 10.1007/s10440-008-9317-0, Acta Appl. Math. 108 (2009), 385-394. (2009) Zbl1188.34013MR2551480DOI10.1007/s10440-008-9317-0
  22. Yao, Q., 10.1016/j.jmaa.2009.07.043, J. Math. Anal. Appl. 363 (2010), 138-154. (2010) Zbl1191.34031MR2559048DOI10.1016/j.jmaa.2009.07.043

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