On some three-point problems for third-order differential equations

Irena Rachůnková

Mathematica Bohemica (1992)

  • Volume: 117, Issue: 1, page 98-110
  • ISSN: 0862-7959

Abstract

top
This paper is concerned with existence and uniqueness of solutions of the three-point problem u ' ' ' = f ( t , u , u ' , u ' ' ) , u ( c ) = 0 , u ' ( a ) = u ' ( b ) . u ' ' ( a ) = u ' ' ( b ) , a ≤ c ≤ b . The problem is at resonance, in the sense that the associated linear problem has non-trivial solutions. We use the method of lower and upper solutions.

How to cite

top

Rachůnková, Irena. "On some three-point problems for third-order differential equations." Mathematica Bohemica 117.1 (1992): 98-110. <http://eudml.org/doc/29386>.

@article{Rachůnková1992,
abstract = {This paper is concerned with existence and uniqueness of solutions of the three-point problem $u^\{\prime \prime \prime \}=f(t,u,u^\{\prime \},u^\{\prime \prime \}), u(c)=0,u^\{\prime \}(a)=u^\{\prime \}(b). u^\{\prime \prime \}(a)=u^\{\prime \prime \}(b), a\le c\le b$. The problem is at resonance, in the sense that the associated linear problem has non-trivial solutions. We use the method of lower and upper solutions.},
author = {Rachůnková, Irena},
journal = {Mathematica Bohemica},
keywords = {existence; uniqueness; three-point mixed problem; method of lower and upper solutions; lower and upper solutions; resonance; Carathéodory conditions; existence; uniqueness; three-point mixed problem; method of lower and upper solutions},
language = {eng},
number = {1},
pages = {98-110},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On some three-point problems for third-order differential equations},
url = {http://eudml.org/doc/29386},
volume = {117},
year = {1992},
}

TY - JOUR
AU - Rachůnková, Irena
TI - On some three-point problems for third-order differential equations
JO - Mathematica Bohemica
PY - 1992
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 117
IS - 1
SP - 98
EP - 110
AB - This paper is concerned with existence and uniqueness of solutions of the three-point problem $u^{\prime \prime \prime }=f(t,u,u^{\prime },u^{\prime \prime }), u(c)=0,u^{\prime }(a)=u^{\prime }(b). u^{\prime \prime }(a)=u^{\prime \prime }(b), a\le c\le b$. The problem is at resonance, in the sense that the associated linear problem has non-trivial solutions. We use the method of lower and upper solutions.
LA - eng
KW - existence; uniqueness; three-point mixed problem; method of lower and upper solutions; lower and upper solutions; resonance; Carathéodory conditions; existence; uniqueness; three-point mixed problem; method of lower and upper solutions
UR - http://eudml.org/doc/29386
ER -

References

top
  1. A. R. Aftabizadeh J. Wiener, Existence and uniqueness theorems for third order boundary value problems, Rend. Sem. Mat. Univ. Padova 75 (1986), 130-141. (1986) MR0847662
  2. A. R. Aftabizadeh J. M. Xu C. P. Gupta, 10.1137/0520049, SIAM J. Math. Anal. 20 (1989), 716-726. (1989) MR0990873DOI10.1137/0520049
  3. R. P. Agarwal, On boundary value problems for y'" = f(x,y,y',y"), Bull. of the Inst. Math. Acad. Sinica 12 (1984), 153-157. (1984) MR0765109
  4. R. P. Agarwal, Existence-uniqueness and iterative methods for third order boundary value problems, J. Comp. Anal. Math., to appear. Zbl0617.34008MR0883170
  5. J. Andres, On a boundary value problem for x'" = f(t,x,x',x"), Acta UPO, ser. mat. 27 (1988), 289-298. (1988) MR1039896
  6. D. Barr T. Sherman, 10.1016/0022-0396(73)90014-4, J. Diff. Eqs. 13 (1973), 197-212. (1973) MR0333326DOI10.1016/0022-0396(73)90014-4
  7. S. A. Bespalova J. A. Klokov, A three-point boundary value problem for a third-order nonlinear ordinary differential equation, Diff. uravn. 12 (1976), 963-970. (In Russian.) (1976) MR0425230
  8. G. Carristi, A three-point boundary value problem for a third order differential equation, Boll. Um. Mat. Ital, C 4 1 (1985), 259-269. (1985) MR0805218
  9. K. M. Das B. S. Lalli, 10.1016/0022-247X(81)90064-0, J. Math. Anal. Appl. 81 (1981), 300-307. (1981) MR0622819DOI10.1016/0022-247X(81)90064-0
  10. A. Granas R. Guenther J. Lee, Nonlinear Boundary Value Problems for Ordinary Differential Equatins, Polish Acad, of Sciences, 1985. (1985) MR0808227
  11. M. Greguš, Third Order Linear Boundary Value Problems, D. Reidel Publishing Co., 1987. (1987) MR0882545
  12. C. P. Gupta, On a third-order three-point boundary value problem at resonance, Diff. Int. Equations 2 (1989), 1-12. (1989) Zbl0722.34014MR0960009
  13. G. H. Hardy J. E. Littlewood G. Polya, Inequalities, IL, Moscow, 1970. (In Russian.) (1970) 
  14. J. Henderson, 10.1137/0518023, SIAM J. Math Anal. 18 (1987), 293-305. (1987) Zbl0668.34017MR0876272DOI10.1137/0518023
  15. S. Hu V. Lakshmikantham, 10.1016/0362-546X(86)90059-3, Nonlinear Anal. 10 (1986), 1203-1208. (1986) Zbl0622.45007MR0866253DOI10.1016/0362-546X(86)90059-3
  16. I. T. Kiguradze, Some Singular Boundary Value Problems for Ordinary Differential Equations, Univ. Press, Tbilisi, 1975. (In Russian.) (1975) MR0499402
  17. E. Lepina A. Lepin, Existence of a solution of the three-point BVP for a nonlinear third-order ordinary differential equation, Latv. M. E. 4 (1986), 247-256. (In Russian.) (1986) 
  18. E. Lepina A. Lepin, Necessary and sufficient conditions for existence of a solution of a three-point BVP for a nonlinear third order differential equation, Latv. M. E. 8 (1970), 149-154. (In Russian.) (1970) 
  19. K. N. Murthy D. R. K. S. Rao, On existence and uniqueness of solutions of two and three point boundary value problems, Bull. Calcuta Math. Soc. 73,3 (1981), 164-172. (1981) MR0669619
  20. K. N. Murthy B. D. C. N. Prasad, Three-point boundary value problems, existence and uniqueness, Yokohama Math. J. 29 (1981), 101-105. (1981) MR0649612
  21. K. N. Murthy B. D. C. N. Prasad, Application of Lyapunov theory to three-point boundary value problems, J. Math. Phys. Sci. 19 (1985), 225-234. (1985) MR0863375
  22. L. I. Pospelov, Necessary and sufficient conditions for existence of a solution for some BVPs for the third order nonlinear ordinary differential equation, Latv. M. E. 8 (1970), 205-213. (In Russian.) (1970) 
  23. D. J. O'Regan, 10.1137/0518048, SIAM J. Math. Anal. 18 (1987), 630-641. (1987) Zbl0628.34017MR0883557DOI10.1137/0518048
  24. J. Rusnák, A three-point boundary value problem for third order differential equations, Math. Slovaca 33 (1983), 307-320. (1983) MR0713954
  25. N. I. Vasiljev J. A. Klokov, Elements of the Theory of Boundary Value Problems for Ordinary Differential Equations, Zinatne, Riga, 1978. (In Russian.) (1978) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.