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Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for \(n\)th order differential equations. (English) Zbl 1396.34011

Summary: For the \(n\)th order differential equation, \(y^{(n)}= f(x,y,y',\ldots,y^{(n-1)})\), we consider uniqueness implies existence results for solutions satisfying certain nonlocal \((k+2)\)-point boundary conditions, \(1\leq k\leq n-1\). Uniqueness of solutions when \(k=n-1\) is intimately related to uniqueness of solutions when \(1\leq k\leq n-2\). These relationships are investigated as well.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] S. Clark, J. Henderson, Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations, Proc. Amer. Math. Soc., in press; S. Clark, J. Henderson, Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations, Proc. Amer. Math. Soc., in press · Zbl 1120.34010
[2] Harris, G. A.; Henderson, J.; Lanz, A.; Yin, W. K.C., Third order right focal boundary value problems on a time scale, J. Difference Equ. Appl., 12, 525-533 (2006) · Zbl 1101.34010
[3] Hartman, P., Unrestricted \(n\)-parameter families, Rend. Circ. Mat. Palermo (2), 7, 123-142 (1958) · Zbl 0085.04505
[4] Hartman, P., On \(n\)-parameter families and interpolation problems for nonlinear differential equations, Trans. Amer. Math. Soc., 154, 201-226 (1971) · Zbl 0222.34017
[5] Henderson, J., Uniqueness of solutions of right focal point boundary value problems, J. Differential Equations, 41, 218-227 (1981) · Zbl 0438.34015
[6] Henderson, J., Existence of solutions of right focal point boundary value problems, Nonlinear Anal., 5, 989-1002 (1981) · Zbl 0468.34010
[7] Henderson, J., Existence theorems for boundary value problems for \(n\) th order nonlinear difference equations, SIAM J. Math. Anal., 20, 468-478 (1989) · Zbl 0671.34017
[8] Henderson, J., Focal boundary value problems for nonlinear difference equations, I, J. Math. Anal. Appl., 141, 559-567 (1989) · Zbl 0706.39001
[9] Henderson, J., Focal boundary value problems for nonlinear difference equations, II, J. Math. Anal. Appl., 141, 568-579 (1989) · Zbl 0706.39002
[10] Henderson, J., Uniqueness implies existence for three-point boundary value problems for second order differential equations, Appl. Math. Lett., 18, 905-909 (2005) · Zbl 1092.34507
[11] Henderson, J.; Karna, B.; Tisdell, C. C., Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc., 133, 1365-1369 (2005) · Zbl 1061.34009
[12] J. Henderson, D. Ma, Uniqueness of solutions for fourth order nonlocal boundary value problems, Bound. Value Probl., in press; J. Henderson, D. Ma, Uniqueness of solutions for fourth order nonlocal boundary value problems, Bound. Value Probl., in press · Zbl 1151.34017
[13] Henderson, J.; Yin, W. K.C., Existence of solutions for fourth order boundary value problems on a time scale, J. Difference Equ. Appl., 9, 15-28 (2003) · Zbl 1057.39011
[14] Jackson, L. K., Uniqueness of solutions of boundary value problems for ordinary differential equations, SIAM J. Appl. Math., 24, 525-538 (1973) · Zbl 0237.34030
[15] Jackson, L. K., Existence and uniqueness of solutions for third order differential equations, J. Differential Equations, 13, 432-437 (1973) · Zbl 0256.34018
[16] Klaasen, G., Existence theorems for boundary value problems for \(n\) th order ordinary differential equations, Rocky Mountain J. Math., 3, 457-472 (1973) · Zbl 0268.34025
[17] Lasota, A.; Opial, Z., On the existence and uniqueness of solutions of a boundary value problem for an ordinary second order differential equation, Colloq. Math., 18, 1-5 (1967) · Zbl 0155.41401
[18] Peterson, A. C., Existence-uniqueness for focal-point boundary value problems, SIAM J. Math. Anal., 12, 173-185 (1982) · Zbl 0473.34009
[19] Schrader, K., Uniqueness implies existence for solutions of nonlinear boundary value problems, Abstracts Amer. Math. Soc., 6, 235 (1985)
[20] Spanier, E. H., Algebraic Topology (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0145.43303
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