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Existence of solutions of three-point boundary value problems for nonlinear fourth order differential equation. (English) Zbl 0880.34022
Summary: The author uses the methods in [G. A. Klaasen, J. Differ. Equations 10, 529-537 (1971; Zbl 0218.34024); J. Wang, Northeast. Math. J. 7, No. 2, 181-189 (1991; Zbl 0755.34021)] to study the existence of solutions of three-point boundary value problems of nonlinear fourth-order differential equations $y^{(4)}= f(t,y,y',y'';y''')\tag{$$*$$}$ with the boundary conditions $\begin{cases} g(y(a),y'(a), y''(a), y'''(a))= 0,\;h(y(b), y''(b))= 0\\ y'(b)= b_1,\;k(y(c),y'(c),y''(c),y'''(c))= 0\end{cases}\tag{$$**$$}$ For the boundary value problems of nonlinear fourth-order differential equations $y^{(4)}= f(t,y,y',y'',y''')$ many results have been given at the present time. But the existence of solutions of the boundary value problem $$(*)$$, $$(**)$$ studied in this paper has not been involved by the above researches. Moreover, the corollary of the important theorem, i.e. existence of solutions of the boundary value problem of equation $$(*)$$ with the following boundary conditions $a_0 y(a)+ a_1y'(a)+ a_2y''(a)+ a_3 y'''(a)= y_0,\;b_0y(b)+ b_1y''(b)= y_1,$ $y'(b)= y_2,\;c_0y(c)+ c_1y'(c)+ c_2y''(c)+ c_3y'''(c)= y_3$ has not been dealt with in previous works.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
 [1] G. A. Klasen, Differential inequalities and existence theorems for second and third order boundary value problems, J. Differential Equation, 10 (1971), 529-537. · Zbl 0218.34024 · doi:10.1016/0022-0396(71)90010-6 [2] Wang Zinzi, Existence of solutions of nonlinear two-point boundary value problems for third order nonlinear differential equations, Northeast Math. J., 7, 2 (1991), 181-189. · Zbl 0755.34021 [3] L. K. Jackson and K. Schrader, Subfunction and third order differential inequalities, J. Differential Equation, 8 (1970), 180-194. · Zbl 0194.40902 · doi:10.1016/0022-0396(70)90044-6 [4] P. Hartman, Ordinary Differential Equations, Wiley, New York (1964). · Zbl 0125.32102 [5] Zhao Weili, Existence of solutions of boundary value problems for third order nonlinear differential equations, Acta Scientiarum Naturalium Universitatis Jilinensis, 2 (1984), 10-19. · Zbl 0537.34016
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