Existence and uniqueness theorems for fourth-order boundary value problems
The 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. \textit{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 \(y''\).
scientific article; zbMATH DE number 4032116
scientific article