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Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations. (English) Zbl 1120.34010

The authors study nonlinear boundary value problems for a third-order ordinary differential equation \[ y'''= f(x,y,y',y''),\quad a< x< b \] with continuous right-hand side \(f\). It is assumed that solutions to initial value problems for this equation are unique and extend to \((a,b)\). Some four-point and three-point boundary conditions are considered, in particular, of the form \[ y(x_1)= y_1,\quad y(x_2)= y_2,\quad y(x_3)- y(x_4)= y_3. \] Uniqueness implies uniqueness relationships for these boundary value problems, and uniqueness implies existence results.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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