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Uniqueness of solutions for fourth-order nonlocal boundary value problems. (English) Zbl 1151.34017

From the introduction: We are concerned with uniqueness of solutions of certain nonlocal boundary value problems for the fourth-order ordinary differential equation,
\[ y^{(4)}=f(x,y,y',y'',y'''),\quad a<x<b,\tag{*} \]
where
(A) \(f: (a,b)\times \mathbb R^4\to\mathbb R\) is continuous,
(B) solutions of initial value problems for (1.1) are unique and exist on all of \((a,b)\).
In particular, we deal with “uniqueness implies uniqueness” relationships among solutions of (*) satisfying nonlocal 5-point, 4-point, and 3-point nonlocal boundary conditions.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

[1] Agarwal RP: Compactness condition for boundary value problems.Equadiff (Brno) 1997, 9: 1-23.
[2] Clark S, Henderson J: Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations. to appear in Proceedings of the American Mathematical Society · Zbl 1120.34010
[3] Conti M, Terracini S, Verzini G: Infinitely many solutions to fourth order superlinear periodic problems.Transactions of the American Mathematical Society 2004,356(8):3283-3300. 10.1090/S0002-9947-03-03514-1 · Zbl 1074.34047 · doi:10.1090/S0002-9947-03-03514-1
[4] Ehme J, Hankerson D: Existence of solutions for right focal boundary value problems.Nonlinear Analysis 1992,18(2):191-197. 10.1016/0362-546X(92)90093-T · Zbl 0755.34016 · doi:10.1016/0362-546X(92)90093-T
[5] Franco D, O’Regan D, Perán J: Fourth-order problems with nonlinear boundary conditions.Journal of Computational and Applied Mathematics 2005,174(2):315-327. 10.1016/j.cam.2004.04.013 · Zbl 1068.34013 · doi:10.1016/j.cam.2004.04.013
[6] Graef JR, Qian C, Yang B: A three point boundary value problem for nonlinear fourth order differential equations.Journal of Mathematical Analysis and Applications 2003,287(1):217-233. 10.1016/S0022-247X(03)00545-6 · Zbl 1054.34038 · doi:10.1016/S0022-247X(03)00545-6
[7] Gupta CP: A Dirichlet type multi-point boundary value problem for second order ordinary differential equations.Nonlinear Analysis 1996,26(5):925-931. 10.1016/0362-546X(94)00338-X · Zbl 0847.34018 · doi:10.1016/0362-546X(94)00338-X
[8] Gupta CP, Ntouyas SK, Tsamatos PCh:Solvability of an[InlineEquation not available: see fulltext.]-point boundary value problem for second order ordinary differential equations.Journal of Mathematical Analysis and Applications 1995,189(2):575-584. 10.1006/jmaa.1995.1036 · Zbl 0819.34012 · doi:10.1006/jmaa.1995.1036
[9] Hartman P:Unrestricted[InlineEquation not available: see fulltext.]-parameter families.Rendiconti del Circolo Matematico di Palermo. Serie II 1958, 7: 123-142. 10.1007/BF02854523 · Zbl 0085.04505 · doi:10.1007/BF02854523
[10] Hartman P:On[InlineEquation not available: see fulltext.]-parameter families and interpolation problems for nonlinear ordinary differential equations.Transactions of the American Mathematical Society 1971, 154: 201-226. · Zbl 0222.34017
[11] Henderson J: Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations.Journal of Differential Equations 1981,41(2):218-227. 10.1016/0022-0396(81)90058-9 · Zbl 0438.34015 · doi:10.1016/0022-0396(81)90058-9
[12] Henderson J: Existence of solutions of right focal point boundary value problems for ordinary differential equations.Nonlinear Analysis 1981,5(9):989-1002. 10.1016/0362-546X(81)90058-4 · Zbl 0468.34010 · doi:10.1016/0362-546X(81)90058-4
[13] Henderson J:Right[InlineEquation not available: see fulltext.]boundary value problems for third order differential equations.Journal of Mathematical and Physical Sciences 1984,18(4):405-413. · Zbl 0654.34015
[14] Henderson J:Existence theorems for boundary value problems for[InlineEquation not available: see fulltext.]th-order nonlinear difference equations.SIAM Journal on Mathematical Analysis 1989,20(2):468-478. 10.1137/0520032 · Zbl 0671.34017 · doi:10.1137/0520032
[15] Henderson J: Uniqueness implies existence for three-point boundary value problems for second order differential equations.Applied Mathematics Letters 2005,18(8):905-909. 10.1016/j.aml.2004.07.032 · Zbl 1092.34507 · doi:10.1016/j.aml.2004.07.032
[16] Henderson J, Karna B, Tisdell CC: Existence of solutions for three-point boundary value problems for second order equations.Proceedings of the American Mathematical Society 2005,133(5):1365-1369. 10.1090/S0002-9939-04-07647-6 · Zbl 1061.34009 · doi:10.1090/S0002-9939-04-07647-6
[17] Henderson J, McGwier RW Jr.: Uniqueness, existence, and optimality for fourth-order Lipschitz equations.Journal of Differential Equations 1987,67(3):414-440. 10.1016/0022-0396(87)90135-5 · Zbl 0642.34005 · doi:10.1016/0022-0396(87)90135-5
[18] Henderson J, Yin WKC: Existence of solutions for fourth order boundary value problems on a time scale.Journal of Difference Equations and Applications 2003,9(1):15-28. · Zbl 1030.39023 · doi:10.1080/10236100309487532
[19] Ill’in VA, Moiseev EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator.Differential Equations 1987,23(8):979-987. · Zbl 0668.34024
[20] Jackson LK: Uniqueness of solutions of boundary value problems for ordinary differential equations.SIAM Journal on Applied Mathematics 1973,24(4):535-538. 10.1137/0124054 · Zbl 0237.34030 · doi:10.1137/0124054
[21] Jackson LK: Existence and uniqueness of solutions of boundary value problems for third order differential equations.Journal of Differential Equations 1973,13(3):432-437. 10.1016/0022-0396(73)90002-8 · Zbl 0256.34018 · doi:10.1016/0022-0396(73)90002-8
[22] Jackson, LK; Ahmed, S. (ed.); Keener, M. (ed.); Lazer, A. (ed.), Boundary value problems for Lipschitz equations, 31-50 (1980), New York
[23] Karna B: Extremal points for the fourth order boundary value problems.Mathematical Sciences Research Journal 2003,7(10):382-393. · Zbl 1050.34008
[24] Klaasen GA:Existence theorems for boundary value problems for[InlineEquation not available: see fulltext.]th order ordinary differential equations.The Rocky Mountain Journal of Mathematics 1973, 3: 457-472. 10.1216/RMJ-1973-3-3-457 · Zbl 0268.34025 · doi:10.1216/RMJ-1973-3-3-457
[25] Kong L, Kong Q:Positive solutions of nonlinear[InlineEquation not available: see fulltext.]-point boundary value problems on a measure chain.Journal of Difference Equations and Applications 2003,9(6):615-627. 10.1080/1023619031000060963 · Zbl 1037.34015 · doi:10.1080/1023619031000060963
[26] Lasota A, Łuczyński M: A note on the uniqueness of two point boundary value problems I.Zeszyty Naukowe Uniwersytetu Jagiellonskiego, Prace Matematyczne 1968, 12: 27-29. · Zbl 0275.34022
[27] Lasota A, Opial Z: On the existence and uniqueness of solutions of a boundary value problem for an ordinary second-order differential equation.Colloquium Mathematicum 1967, 18: 1-5. · Zbl 0155.41401
[28] Liu Y, Ge W: Existence theorems of positive solutions for fourth-order four point boundary value problems.Analysis and Applications (Singapore) 2004,2(1):71-85. · Zbl 1050.34009 · doi:10.1142/S0219530504000254
[29] Lü H, Yu H, Liu Y: Positive solutions for singular boundary value problems of a coupled system of differential equations.Journal of Mathematical Analysis and Applications 2005,302(1):14-29. 10.1016/j.jmaa.2004.08.003 · Zbl 1076.34022 · doi:10.1016/j.jmaa.2004.08.003
[30] Ma R: Positive solutions for second-order three-point boundary value problems.Applied Mathematics Letters 2001,14(1):1-5. 10.1016/S0893-9659(00)00102-6 · Zbl 0989.34009 · doi:10.1016/S0893-9659(00)00102-6
[31] Ma R:Existence of positive solutions for superlinear semipositone[InlineEquation not available: see fulltext.]-point boundary-value problems.Proceedings of the Edinburgh Mathematical Society. Series II 2003,46(2):279-292. 10.1017/S0013091502000391 · Zbl 1069.34036 · doi:10.1017/S0013091502000391
[32] Ma R, Castaneda N:Existence of solutions of nonlinear[InlineEquation not available: see fulltext.]-point boundary-value problems.Journal of Mathematical Analysis and Applications 2001,256(2):556-567. 10.1006/jmaa.2000.7320 · Zbl 0988.34009 · doi:10.1006/jmaa.2000.7320
[33] Ma R: Multiple positive solutions for a semipositone fourth-order boundary value problem.Hiroshima Mathematical Journal 2003,33(2):217-227. · Zbl 1048.34048
[34] Ma R: Some multiplicity results for an elastic beam equation at resonance.Applied Mathematics and Mechanics 1993,14(2):193-200. 10.1007/BF02453362 · Zbl 0776.73037 · doi:10.1007/BF02453362
[35] Ma JJ, Zhang FR: Solvability of a class of fourth-order two-point boundary value problems.Heilongjiang Daxue Ziran Kexue Xuebao 2001,18(2):19-22. · Zbl 1076.34505
[36] Marchant TR: Higher-order interaction of solitary waves on shallow water.Studies in Applied Mathematics 2002,109(1):1-17. 10.1111/1467-9590.00001 · Zbl 1152.76343 · doi:10.1111/1467-9590.00001
[37] Palamides PK:Multi point boundary-value problems at resonance for[InlineEquation not available: see fulltext.]-order differential equations: positive and monotone solutions.Electronic Journal of Differential Equations 2004,2004(25):1-14. · Zbl 1066.34013
[38] Pei MH, Chang SK: Existence and uniqueness theorems of solutions for a class fourth-order boundary value problems.Kyungpook Mathematical Journal 2001,41(2):299-309. · Zbl 1006.34020
[39] Peterson AC: Focal Green’s functions for fourth-order differential equations.Journal of Mathematical Analysis and Applications 1980,75(2):602-610. 10.1016/0022-247X(80)90104-3 · Zbl 0439.34026 · doi:10.1016/0022-247X(80)90104-3
[40] Peterson DE: Uniqueness, existence, and comparison theorems for ordinary differential equations, Doctoral dissertation. , Nebraska; 1973.
[41] Spanier EH: Algebraic Topology. McGraw-Hill, New York; 1966:xiv+528. · Zbl 0810.55001
[42] Vidossich G: On the continuous dependence of solutions of boundary value problems for ordinary differential equations.Journal of Differential Equations 1989,82(1):1-14. 10.1016/0022-0396(89)90164-2 · Zbl 0725.34007 · doi:10.1016/0022-0396(89)90164-2
[43] Webb JRL: Positive solutions of some three point boundary value problems via fixed point index theory.Nonlinear Analysis 2001,47(7):4319-4332. 10.1016/S0362-546X(01)00547-8 · Zbl 1042.34527 · doi:10.1016/S0362-546X(01)00547-8
[44] Zhang Z, Wang J: Positive solutions to a second order three-point boundary value problem.Journal of Mathematical Analysis and Applications 2003,285(1):237-249. 10.1016/S0022-247X(03)00396-2 · Zbl 1035.34011 · doi:10.1016/S0022-247X(03)00396-2
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