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Existence and uniqueness for nonlinear third-order two-point boundary value problems. (English) Zbl 1188.34027

Summary: The upper and lower solutions method, Leray-Schauder degree theory and differential inequality technique are employed to establish existence and uniqueness results for the class of nonlinear third-order two-point boundary value problems with one-sided Nagumo condition.
\[ x''' =f(t,x, x', x"),\quad a<t<b, \tag{1} \]
\[ x(a) = A,\tag{2} \]
\[ g(x'(a))-[x''(a)]^p=B,\tag{3} \]
\[ h(x(b),x'(b))+[x''(b)]^q=C,\tag{4} \]
where \(A, B,C\in\mathbb R\), \(f(t,x,y,z) : [a, b]\times \mathbb R^3\mathbb R\) is continuous, \(g(y) : \mathbb R\to\mathbb R\) is continuous, \(h(x,y):\mathbb R^2\to\mathbb R\) is continuous and decreasing on \(x\), \(p\), \(q\) are odd numbers.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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