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Solvability of multi-point boundary value problem at resonance. IV. (English) Zbl 1071.34014
Summary: In this part, we consider the following second-order ordinary differential equation $x''= f(t, x(t), x'(t))+ e(t),\quad t\in (0,1),\tag{1}$ subject to one of the following boundary value conditions: \begin{alignedat}{2} x(0) &= \sum^{m-2}_{i=1} \alpha_i x(\xi_i),\quad &&x(1)= \sum^{n-2}_{j=1} \beta_j x(\eta_j),\tag{2}\\ x(0) &= \sum^{m-2}_{i=1} \alpha_i x(\xi_i),\quad && x'(1)= \sum^{n-2}_{j=1} \beta_j x'(\eta_j),\tag{3}\\ x'(0) &= \sum^{m-2}_{i=1} \alpha_i x'(\xi_i),\quad && x(1)= \sum^{n-2}_{j=1} \beta_j x(\eta_j),\tag{4}\end{alignedat} with$$\alpha_i\in \mathbb{R}$$, $$1\leq i\leq m-2$$, $$\beta_j\in \mathbb{R}$$, $$1\leq j\leq n-2$$, $$0< \xi_1< \xi_2<\cdots< \xi_{m-2}< 1$$, and $$0< \eta_1< \eta_2<\cdots< \eta_{n-2}< 1$$. When all $$\alpha_i$$ have no the same sign and all $$\beta_j$$ have no the same sign, some existence result are given for (1) with boundary conditions (2)–(4) at resonance case. We give some examples to demonstrate our result, too.
For the parts I, II and III see [the author and J. Yu, Indian J. Pure Appl. Math. 33, 475–494 (2002; Zbl 1021.34013), the author, Appl. Math. Comput. 136, 353–377 (2003; Zbl 1053.34016) and the author and J. Yu, ibid. 129, 119–143 (2002; Zbl 1054.34033)].

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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##### References:
  Feng, W.; Webb, J.R.L., Solvability of m-point boundary value problems with nonlinear growth, J. math. anal. appl., 212, 467-480, (1997) · Zbl 0883.34020  Feng, W.; Webb, J.R.L., Solvability of three point boundary value problems at resonance, Nonlinear anal. theory meth. appl., 30, 3227-3238, (1997) · Zbl 0891.34019  C.P. Gupta, S.K. Ntouyas, P.Ch. Tsamatos, Existence results for multi-point boundary value problems for second order ordinary differential equations, preprint · Zbl 1185.34023  Gupta, C.P., A generalized multi-point boundary value problem for second order ordinary differential equations, Appl. math. comput., 89, 133-146, (1998) · Zbl 0910.34032  Liu, B.; Yu, J.S., Solvability of multi-point boundary value problems at resonance (I), Ind. J. pure appl. math., 34, 475-494, (2002) · Zbl 1021.34013  Liu, B.; Yu, J.S., Solvability of multi-point boundary value problems at resonance (II), Appl. math. comput., 136, 353-377, (2003) · Zbl 1053.34016  Liu, B.; Yu, J.S., Solvability of multi-point boundary value problems at resonance (III), Appl. math. comput., 129, 119-143, (2002) · Zbl 1054.34033  Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, () · Zbl 0798.34025  J. Mawhin, Topological degree methods in nonlinear boundary value problems, in: NSF-CBMS Regional Conference Series in Math, American Mathematics Society, Providence, RI, 1979  Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the first kind for a Sturm-liouvilla operator in its differential and finite difference aspects, Differen. equat., 23, 803-810, (1987) · Zbl 0668.34025  Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problems of the second kind for a Sturm-Liouville operator, Differen. equat., 23, 979-987, (1987) · Zbl 0668.34024  Bitsadze, A.V., On the theory of nonlocal boundary value problems, Soviet math. dokl., 30, 8-10, (1984) · Zbl 0586.30036  Bitsadze, A.V., On a class of conditionally solvable nonlocal boundary value problems for harmonic functions, Soviet math. dokl., 31, 91-94, (1985) · Zbl 0607.30039  Bitsadze, A.V.; Samarskii, A.A., On some simple generalizations of linear elliptic boundary problems, Soviet math. dokl., 10, 398-400, (1969) · Zbl 0187.35501  Ma, R.Y., Existence theorems for second order m-point boundary value problems, J. math. anal. appl., 211, 545-555, (1997) · Zbl 0884.34024  Gupta, C.P., A second order m-point boundary value problem at resonance, Nonlinear anal. theory meth. appl., 24, 1483-1489, (1995) · Zbl 0824.34023  Gupta, C.P., Solvability of a multi-point boundary value problem at resonance, Res. math., 28, 270-276, (1995) · Zbl 0843.34023  Gupta, C.P., Existence theorems for a second order m-point boundary value problem at resonance, Int. J. math. sci., 18, 705-710, (1995) · Zbl 0839.34027  O’Regan, D., Existence theory for nonlinear ordinary differential equations, (1997), Kluwer Academic Dordrecht · Zbl 1077.34505  O’Regan, D., Theory of singular boundary value problems, (1994), World Scientific Singapore · Zbl 0808.34022
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