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Existence and uniqueness of solutions for fourth-order boundary-value problems in Banach spaces. (English) Zbl 1191.34079

Using Mönch’s fixed point theorem, an existence result is proved for the solutions of the following boundary value problem
\[ x^{(4)}= f(t,x,x'') \quad (t \in (0,1)), \quad x(0) = x(1) = x''(0) = x''(1) = 0, \]
where \(f : [0,1] \times E \times E \to E\) is continuous and \(E\) is a Banach space. The function \(f\) has to satisfy some linear growth conditions with sufficiently small coefficients, as well as some restrictions with respect to a measure of compactness. The uniqueness is proved under stronger conditions.

MSC:

34G20 Nonlinear differential equations in abstract spaces
47H10 Fixed-point theorems
34B15 Nonlinear boundary value problems for ordinary differential equations
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