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Existence-uniqueness and iterative methods for third-order boundary value problems. (English) Zbl 0617.34008

Various types of existence conditions are given for the solution of the 3-point boundary value problem \[ x\prime''=f(t,x,x',x'');\quad x'(a)=A,\quad x(b)=B,\quad x'(c)=C, \] utilizing the Schauder fixed point principle. The case when f(t,x,y,z) is linear in x,y,z is separately considered. An iterative scheme for approximating the solution is justified. Finally, illustrative examples are considered.
Reviewer: M.M.Konstantinov

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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[1] Agarwal, R. P., The numerical solution of multipoint boundary value problems, J. Comp. Appl. Math., 5, 17-24 (1979) · Zbl 0394.65025
[2] Agarwal, R. P., On the method of complementary functions for nonlinear boundary value problems, J. Optimization Theory Appl., 36, 139-144 (1982) · Zbl 0448.34016
[3] Agarwal, R. P., (Boundary value problems for higher order integrodifferential equations, Nonlinear Analysis:, 7 (1983), Pergamon Press: Pergamon Press New York), 259-270 · Zbl 0505.45002
[4] Agarwal, R. P.; Loi, S. L., (On approximate Picard’s iterates for multipoint boundary value problems, Nonlinear Analysis: TMA, 8 (1984), Pergamon Press: Pergamon Press New York), 381-391 · Zbl 0567.65054
[5] Agarwal, R. P., Quasilinearization and approximate quasilinearization for multipoint boundary value problems, J. Math. Anal. Appl., 107, 317-330 (1985) · Zbl 0602.34014
[6] Agarwal, R. P.; Akrivis, G., Boundary value problems occuring in plate deflection theory, J. Comp. Appl. Math., 8, 145-154 (1982) · Zbl 0503.73061
[7] Agarwal, R. P., On boundary value problems for \(y\)‴ = \(f(x, y, y\)′, \(y\)″), Bull. Inst. Math. Acad. Scinica, 12, 153-157 (1984) · Zbl 0542.34015
[8] Agarwal, R. P.; Chow, Y. M., Iterative methods for a fourth order boundary value problem, J. Comp. Appl. Math., 10, 203-217 (1984) · Zbl 0541.65055
[9] Agarwal, R. P.; Gupta, R. C., On the solution of Holt’s problem, BIT, 24, 342-346 (1984) · Zbl 0551.65058
[10] Agarwal, R. P.; Gupta, R. C., method of Chasing for multipoint boundary value problems, Appl. Math. Comp., 17, 37-43 (1985) · Zbl 0599.34021
[11] Agarwal, R. P.; Usmani, R. A., Iterative methods for solving right focal point boundary value problems, J. Comp. Appl. Math., 14, 371-390 (1986) · Zbl 0593.65056
[12] Barr, D.; Sherman, T., Existence and uniqueness of solutions of three-point boundary value problems, J. Diff. Eqs., 13, 197-212 (1973) · Zbl 0261.34014
[13] Das, K. M.; Lalli, B. S., Boundary value problems for \(y\)‴ = \(f(x, y, y\)′, \(y\)″), J. Math. Anal. Appl., 81, 300-307 (1981) · Zbl 0465.34012
[14] Henderson, J., (Three-point boundary value problems for ordinary differential equations by matching solutions, Nonlinear analysis, TMA 7 (1983), Pergamon Press: Pergamon Press New York), 411-417 · Zbl 0508.34015
[15] Krajcinvic, D., Sandwich beam analysis, J. Appl. Mech., 39, 773-778 (1972) · Zbl 0241.73043
[16] Meyer, G. H., Initial Value Methods for Boundary Value Problems (1973), Academic Press: Academic Press New York · Zbl 0304.34018
[17] Moorti, V. R.G.; Garner, J. B., Existence-uniqueness theorems for three-point boundary value problems for \(n\) th-order nonlinear differential equations, J. Diff. Eqs., 29, 205-213 (1978) · Zbl 0355.34006
[18] Na, T. Y., Computational Methods in Engineering Boundary Value Problems (1979), Academic Press: Academic Press New York · Zbl 0456.76002
[19] Ojika, T.; Kasue, Y., Initial-value adjusting method for the nonlinear multipoint boundary-value problems, J. Math. Anal. Appl., 69, 359-371 (1979) · Zbl 0412.65041
[20] Ojika, T.; Welsh, W., A numerical method for multipoint boundary value problems with applications to a restricted three body problem, Internat. J. Comp. Math., 8, 329-344 (1980) · Zbl 0439.65066
[21] Rao, D. R.K. S.; Murthy, K. N.; Rao, A. S., On three-point boundary value problems associated with third order differential equations, Nonlinear Analysis: TMA, 5, 669-673 (1981) · Zbl 0485.34011
[22] Welsh, W.; Ojika, T., Multipoint boundary value problems with discontinuities. I. Algorithms and applications, J. Comp. Appl. Math., 6, 133-143 (1980) · Zbl 0439.65064
[23] Agarwal, R. P., Boundary Value Problems for Higher Order Differential Equations (1986), World Scientific: World Scientific Singapore/Philadelphia · Zbl 0598.65062
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