# Article

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Keywords:
existence; uniqueness; three-point mixed problem; method of lower and upper solutions; lower and upper solutions; resonance; Carathéodory conditions
Summary:
This paper is concerned with existence and uniqueness of solutions of the three-point problem $u'''=f(t,u,u',u''), u(c)=0,u'(a)=u'(b). u''(a)=u''(b), a\leq c\leq b$. 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.
References:
[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. MR 0847662
[2] A. R. Aftabizadeh J. M. Xu C. P. Gupta: Existence and uniqueness theorems for three-point boundary value problems. SIAM J. Math. Anal. 20 (1989), 716-726. DOI 10.1137/0520049 | MR 0990873
[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. MR 0765109 | Zbl 0542.34015
[4] R. P. Agarwal: Existence-uniqueness and iterative methods for third order boundary value problems. J. Comp. Anal. Math., to appear. MR 0883170 | Zbl 0617.34008
[5] J. Andres: On a boundary value problem for x'" = f(t,x,x',x"). Acta UPO, ser. mat. 27 (1988), 289-298. MR 1039896 | Zbl 0712.34032
[6] D. Barr T. Sherman: Existence and uniqueness of solutions of three-point boundary value problems. J. Diff. Eqs. 13 (1973), 197-212. DOI 10.1016/0022-0396(73)90014-4 | MR 0333326
[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.) MR 0425230
[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. MR 0805218
[9] K. M. Das B. S. Lalli: Boundary value problems for y'" = f(x,y,y',y"). J. Math. Anal. Appl. 81 (1981), 300-307. DOI 10.1016/0022-247X(81)90064-0 | MR 0622819
[10] A. Granas R. Guenther J. Lee: Nonlinear Boundary Value Problems for Ordinary Differential Equatins. Polish Acad, of Sciences, 1985. MR 0808227
[11] M. Greguš: Third Order Linear Boundary Value Problems. D. Reidel Publishing Co., 1987. MR 0882545
[12] C. P. Gupta: On a third-order three-point boundary value problem at resonance. Diff. Int. Equations 2 (1989), 1-12. MR 0960009 | Zbl 0722.34014
[13] G. H. Hardy J. E. Littlewood G. Polya: Inequalities. IL, Moscow, 1970. (In Russian.)
[14] J. Henderson: Best interval lengths for boundary value problems for third order Lipschitz equations. SIAM J. Math Anal. 18 (1987), 293-305. DOI 10.1137/0518023 | MR 0876272 | Zbl 0668.34017
[15] S. Hu V. Lakshmikantham: Periodic boundary value problems for integro-differential equations of Volterra type. Nonlinear Anal. 10 (1986), 1203-1208. DOI 10.1016/0362-546X(86)90059-3 | MR 0866253 | Zbl 0622.45007
[16] I. T. Kiguradze: Some Singular Boundary Value Problems for Ordinary Differential Equations. Univ. Press, Tbilisi, 1975. (In Russian.) MR 0499402
[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.)
[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.)
[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. MR 0669619
[20] K. N. Murthy B. D. C. N. Prasad: Three-point boundary value problems, existence and uniqueness. Yokohama Math. J. 29 (1981), 101-105. MR 0649612
[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. MR 0863375
[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.)
[23] D. J. O'Regan: Topological transversality: Applications to third order boundary value problems. SIAM J. Math. Anal. 18 (1987), 630-641. DOI 10.1137/0518048 | MR 0883557 | Zbl 0628.34017
[24] J. Rusnák: A three-point boundary value problem for third order differential equations. Math. Slovaca 33 (1983), 307-320. MR 0713954
[25] N. I. Vasiljev J. A. Klokov: Elements of the Theory of Boundary Value Problems for Ordinary Differential Equations. Zinatne, Riga, 1978. (In Russian.)

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