Existence and uniqueness theorems for fourth-order boundary value problems.

*(English)*Zbl 0634.34009The differential equation (1) \(y^{(IV)}=f(x,y,y'')\) is considered under the following types of boundary conditions:

(2) \(y(0)=y_ 0\), \(y(1)=y_ 1\), \(y''(0)=\bar y_ 0\), \(y''(1)=\bar y_ 1\);

(3) \(y(0)=y_ 0\), \(y(1)=y_ 1\), \(y''(0)=\bar y_ 0\), \(y'''(1)=\bar y_ 1\);

(4) \(y(0)=y(1)=y''(0)=y''(1)=0\) or

(5) \(y(0)=y(1)=0\); \(y'''(0)-hy''(0)=0\), \(y'''(1)+ky''(1)=0\), with \(h, k\geq 0\), \(h+k>0\).

Existence theorems for all of the boundary value problems (1-2)–(1-5) are obtained by application of Schauder’s fixed point theorem under continuity and boundedness hypotheses on \(f\) and its partial derivative \(f_ 3\). A uniqueness theorem is obtained for the problem (1-4) under the additional assumption of a bound on the partial derivative \(f_ 2\). Further uniqueness results for the problem (1-2)–(1-5) are reduced to uniqueness questions for solutions of corresponding second order boundary value problems for the integrodifferential equation \[ u''=f(x,y_ 0+x(y_ 1- y_ 0)+\int^{1}_{0}G(x,t)u(t)\,dt,u) \] obtained by setting \(y''=u\), with \(G(x,t)\) the Green’s function for the problem \[ u''=0,\quad u(0)=u(1)=0. \] Boundary conditions of the form (2)–(5) have been less extensively studied than the familiar conjugate or focal-type problems. R. A. Usmani [Proc. Am. Math. Soc. 77, 329–335 (1979; Zbl 0424.34019)] has studied a problem of the form (1-2) in which the equation is linear and independent of \(y''\).

(2) \(y(0)=y_ 0\), \(y(1)=y_ 1\), \(y''(0)=\bar y_ 0\), \(y''(1)=\bar y_ 1\);

(3) \(y(0)=y_ 0\), \(y(1)=y_ 1\), \(y''(0)=\bar y_ 0\), \(y'''(1)=\bar y_ 1\);

(4) \(y(0)=y(1)=y''(0)=y''(1)=0\) or

(5) \(y(0)=y(1)=0\); \(y'''(0)-hy''(0)=0\), \(y'''(1)+ky''(1)=0\), with \(h, k\geq 0\), \(h+k>0\).

Existence theorems for all of the boundary value problems (1-2)–(1-5) are obtained by application of Schauder’s fixed point theorem under continuity and boundedness hypotheses on \(f\) and its partial derivative \(f_ 3\). A uniqueness theorem is obtained for the problem (1-4) under the additional assumption of a bound on the partial derivative \(f_ 2\). Further uniqueness results for the problem (1-2)–(1-5) are reduced to uniqueness questions for solutions of corresponding second order boundary value problems for the integrodifferential equation \[ u''=f(x,y_ 0+x(y_ 1- y_ 0)+\int^{1}_{0}G(x,t)u(t)\,dt,u) \] obtained by setting \(y''=u\), with \(G(x,t)\) the Green’s function for the problem \[ u''=0,\quad u(0)=u(1)=0. \] Boundary conditions of the form (2)–(5) have been less extensively studied than the familiar conjugate or focal-type problems. R. A. Usmani [Proc. Am. Math. Soc. 77, 329–335 (1979; Zbl 0424.34019)] has studied a problem of the form (1-2) in which the equation is linear and independent of \(y''\).

Reviewer: L.J.Grimm

##### MSC:

34B15 | Nonlinear boundary value problems for ordinary differential equations |

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\textit{A. R. Aftabizadeh}, J. Math. Anal. Appl. 116, 415--426 (1986; Zbl 0634.34009)

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##### References:

[1] | Bebernes, J.W.; Gaines, R., Dependence on boundary data and a generalized boundary-value problem, J. differential equations, 4, 359-368, (1968) · Zbl 0169.10602 |

[2] | Corduneanu, C., Sopra I problemi ai limiti per alcuni sistemi di equazioni differenziali non lineari, Rend. acad. napoli, 4, 98-106, (1958) · Zbl 0091.26201 |

[3] | Reiss, E.L.; Callegari, A.J.; Ahluwalia, D.S., Ordinary differential equations with applications, (1976), Holt, Rinehart & Winston Berlin/New York/Heidelberg · Zbl 0334.34002 |

[4] | Usmani, R.A., A uniqueness theorem for a boundary value problem, (), 329-335, 3 · Zbl 0424.34019 |

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