## Extremal solutions for third-order nonlinear problems with upper and lower solutions in reversed order.(English)Zbl 1084.34013

The authors study the scalar nonlinear third-order boundary value problem in Laplacian form $-\biggl[\varphi\bigl(u''(t)\bigr)\biggr]'= f\bigl(t,u(t)\bigr),\;t\in[a,b];\quad u(a)=A,\;u''(a)=B,\;u''(b)=C.$ Here, $$\varphi:\mathbb{R}\to\mathbb{R}$$ is an increasing homeomorphism, and $$f$$ is a Carathéodory function. Employing the monotone iterative technique, sufficient conditions are obtained for the existence of minimal and maximal solutions. These extremal solutions are constructed as limits of some monotone sequences of functions. One of the imposed assumptions is the presence of a pair of lower and upper solutions given in the reversed order, that is, the lower solution is over the upper one.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations

### Keywords:

third-order nonlinear boundary value problems
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### References:

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