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Existence and uniqueness theorems for fourth-order boundary value problems. (English) Zbl 0634.34009
The differential equation (1) $y\sp{(IV)}=f(x,y,y'')$ is considered under the following types of boundary conditions: (2) $y(0)=y\sb 0$, $y(1)=y\sb 1$, $y''(0)=\bar y\sb 0$, $y''(1)=\bar y\sb 1$; (3) $y(0)=y\sb 0$, $y(1)=y\sb 1$, $y''(0)=\bar y\sb 0$, $y'''(1)=\bar y\sb 1$; (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 $h, k\ge 0$, $h+k>0$. 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 $f\sb 3$. A uniqueness theorem is obtained for the problem (1-4) under the additional assumption of a bound on the partial derivative $f\sb 2$. 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 $$ u''=f(x,y\sb 0+x(y\sb 1- y\sb 0)+\int\sp{1}\sb{0}G(x,t)u(t)\,dt,u) $$ 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. {\it 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

34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[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) · Zbl 0334.34002
[4] Usmani, R. A.: A uniqueness theorem for a boundary value problem. Proc. amer. Math. soc. 77, 329-335 (1979) · Zbl 0424.34019