Existence of solutions of three-point boundary value problems for nonlinear fourth order differential equation.

*(English)*Zbl 0880.34022Summary: 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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\textit{Y. Gao}, Appl. Math. Mech., Engl. Ed. 17, No. 6, 569--576 (1996; Zbl 0880.34022)

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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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