Existence of solutions for three-point boundary value problems for second order equations

Authors:
Johnny Henderson, Basant Karna and Christopher C. Tisdell

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1365-1369

MSC (2000):
Primary 34B15; Secondary 34B10

Published electronically:
October 18, 2004

MathSciNet review:
2111960

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Shooting methods are employed to obtain solutions of the three-point boundary value problem for the second order equation, where is continuous, and and conditions are imposed implying that solutions of such problems are unique, when they exist.

**1.**Chuanzhi Bai and Jinxuan Fang,*Existence of multiple positive solutions for nonlinear 𝑚-point boundary value problems*, J. Math. Anal. Appl.**281**(2003), no. 1, 76–85. MR**1980075****2.**Martin Bohner and Allan Peterson,*Dynamic equations on time scales*, Birkhäuser Boston, Inc., Boston, MA, 2001. An introduction with applications. MR**1843232****3.**Chuan Jen Chyan,*Uniqueness implies existence on time scales*, J. Math. Anal. Appl.**258**(2001), no. 1, 359–365. MR**1828110**, 10.1006/jmaa.2001.7512**4.**John M. Davis and Johnny Henderson,*Uniqueness implies existence for fourth-order Lidstone boundary value problems*, Panamer. Math. J.**8**(1998), no. 4, 23–35. MR**1657450****5.**Chaitan P. Gupta and Sergej I. Trofimchuk,*Solvability of a multi-point boundary value problem and related a priori estimates*, Canad. Appl. Math. Quart.**6**(1998), no. 1, 45–60. Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1996). MR**1638415****6.**Philip Hartman,*On 𝑁-parameter families and interpolation problems for nonlinear ordinary differential equations*, Trans. Amer. Math. Soc.**154**(1971), 201–226. MR**0301277**, 10.1090/S0002-9947-1971-0301277-X**7.**Johnny Henderson,*Existence of solutions of right focal point boundary value problems for ordinary differential equations*, Nonlinear Anal.**5**(1981), no. 9, 989–1002. MR**633013**, 10.1016/0362-546X(81)90058-4**8.**Johnny Henderson,*Existence theorems for boundary value problems for 𝑛th-order nonlinear difference equations*, SIAM J. Math. Anal.**20**(1989), no. 2, 468–478. MR**982673**, 10.1137/0520032**9.**Johnny Henderson,*Focal boundary value problems for nonlinear difference equations. I, II*, J. Math. Anal. Appl.**141**(1989), no. 2, 559–567, 568–579. MR**1009063**, 10.1016/0022-247X(89)90197-2**10.**Johnny Henderson and William Yin,*Existence of solutions for fourth order boundary value problems on a time scale*, J. Difference Equ. Appl.**9**(2003), no. 1, 15–28. In honour of Professor Allan Peterson on the occasion of his 60th birthday, Part II. MR**1958300**, 10.1080/1023619031000060954**11.**Johnny Henderson and William K. C. Yin,*Two-point and three-point problems for fourth order dynamic equations*, Dynam. Systems Appl.**12**(2003), no. 1-2, 159–169. Special issue: dynamic equations on time scales. MR**1989029****12.**Lloyd Jackson and Keith Schrader,*Existence and uniqueness of solutions of boundary value problems for third order differential equations*, J. Differential Equations**9**(1971), 46–54. MR**0269920****13.**Gene A. Klaasen,*Existence theorems for boundary value problems of 𝑛th order ordinary differential equations*, Rocky Mountain J. Math.**3**(1973), 457–472. MR**0357944****14.**Andrzej Lasota and Marian Łuczyński,*A note on the uniqueness of two point boundary value problems. I*, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. No.**12**(1968), 27–29. MR**0224900****15.**Ruyun Ma,*Existence theorems for a second order three-point boundary value problem*, J. Math. Anal. Appl.**212**(1997), no. 2, 430–442. MR**1464888**, 10.1006/jmaa.1997.5515**16.**Ruyun Ma,*Existence and uniqueness of solutions to first-order three-point boundary value problems*, Appl. Math. Lett.**15**(2002), no. 2, 211–216. MR**1880760**, 10.1016/S0893-9659(01)00120-3**17.**Allan C. Peterson,*Existence-uniqueness for focal-point boundary value problems*, SIAM J. Math. Anal.**12**(1981), no. 2, 173–185. MR**605428**, 10.1137/0512018**18.**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****19.**Giovanni Vidossich,*On the continuous dependence of solutions of boundary value problems for ordinary differential equations*, J. Differential Equations**82**(1989), no. 1, 1–14. MR**1023298**, 10.1016/0022-0396(89)90164-2**20.**B. Yang,*Boundary Value Problems for Ordinary Differential Equations*, Ph.D. Dissertation, Mississippi State University, Mississippi State, MS, 2002.

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Additional Information

**Johnny Henderson**

Affiliation:
Department of Mathematics, Baylor University, Waco, Texas 76798-7328

Email:
Johnny_Henderson@baylor.edu

**Basant Karna**

Affiliation:
Department of Mathematics, Baylor University, Waco, Texas 76798-7328

Address at time of publication:
Department of Mathematics, Marshall University, Huntington, West Virginia 25755-2560

Email:
Basant_Karna@baylor.edu, karna@marshall.edu

**Christopher C. Tisdell**

Affiliation:
School of Mathematics, The University of New South Wales, Sydney 2052, Australia

Email:
cct@maths.unsw.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-04-07647-6

Keywords:
Boundary value problem,
three-point,
shooting method

Received by editor(s):
October 30, 2003

Received by editor(s) in revised form:
December 30, 2003

Published electronically:
October 18, 2004

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2004
American Mathematical Society