SOLVABILITY OF A FOURTH ORDER BOUNDARY VALUE PROBLEM WITH PERIODIC BOUNDARY CONDITIONS

Fourth order boundary value problems arise in the study of the equilibrium of an elastaic beam under an external load. The author earlier investigated the existence and uniqueness of the solutions of the nonlinear analogues of fourth order boundary value problems that arise in the equilibrium ofanelastic beam depending on how the ends of the beam are supported. This paper concerns the existence and uniqueness of solutions of the fourth order boundary value problems with periodic boundary conditions.

However, the methods of [6], [7] do not apply to the more general boundary value problem under consideration in this paper. One needs to show that the set of solutions of the family of homotopy equations for the boundary value problem ( Define a linear operator L" D (L) c X --+ Y by setting Lu --.
Now, for u D (L) we see, using integration by parts, that d4u (Lu, u") J 0 dx--u "(xldx I(u'"(x))2dx "o If the function gr(x) in condition (iii) is required to be in L z(0,1), we say that the function f satisfies L z-Caratheodory conditions.
Next, let f" [0,1]xR 4 ---> R be a given function satisfying Caratheodory's conditions. We define a (nonlinear) mapping N" X --> Y by setting for u X. We see that KPN" X --4 X is a well-defined compact mapping and QN" X ---> X2 is a bounded mapping.
Remark 3. Suppose that a(x)=-a, b(x)=m, c(x)=c, d(x)=-d, where c and d are some constants in Theorem 3, so that n p q 0. Then the conclusion of Theorem 3 remains valid if 4:2a + m < 16rt4.
Remark 4. We refer the reader to [15] for Wirtinger inequalities used in this paper.
Finally, we remark that the theorems of this paper clearly apply to a wider class of boundary value problems than the theorems studied by the author in [6], [7]. But it is easy to find situations where the results of [6] and [7] apply and the results of this paper do not apply. Accordingly, the results of this paper complement the results of [6] and [7].