##
**Fixed points and approximate solutions for nonlinear operator equations.**
*(English)*
Zbl 0954.47038

The equation
\[
Lu= Nu,\tag{1}
\]
where \(L\) and \(N\) are linear and nonlinear operators defined on a Hilbert space, respectively, is considered. If \(L\) is invertible, then the equation (1) is equivalent to
\[
u= L^{-1}(Nu).\tag{2}
\]
Thus the problem of existence of solutions of (1) leads to the problem of finding fixed points for the operator \(L^{-1}\circ N\).

It may happen that \(L\) is not invertible but for some \(s\in R\), there exists \((L+ sI)^{-1}\). Then the equation (1) is equivalent to \[ u= (L+ sI)^{-1} (Nu+ su).\tag{3} \] To solve such equation one may use the monotone iterative technique for increasing operators on the right-hand side of (3) to find fixed points.

In this paper the authors present some new technique for operators which can be written as difference of monotone operators.

They present the method of finding lower and upper solutions of (1) as well as the criteria for the existence of the unique solution of the equation (1).

The last part of the paper contains applications of the concept of lower and upper solutions to the second-order boundary value problems and to the \(n\)th-order periodic bundary value problems.

It may happen that \(L\) is not invertible but for some \(s\in R\), there exists \((L+ sI)^{-1}\). Then the equation (1) is equivalent to \[ u= (L+ sI)^{-1} (Nu+ su).\tag{3} \] To solve such equation one may use the monotone iterative technique for increasing operators on the right-hand side of (3) to find fixed points.

In this paper the authors present some new technique for operators which can be written as difference of monotone operators.

They present the method of finding lower and upper solutions of (1) as well as the criteria for the existence of the unique solution of the equation (1).

The last part of the paper contains applications of the concept of lower and upper solutions to the second-order boundary value problems and to the \(n\)th-order periodic bundary value problems.

Reviewer: Stefan Czerwik (Gliwice)

### MSC:

47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |

47H10 | Fixed-point theorems |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

47J25 | Iterative procedures involving nonlinear operators |

### Keywords:

monotone iterative technique; increasing operators; lower and upper solutions; second-order boundary value problems; \(n\)th-order periodic bundary value problems
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\textit{A. Cabada} and \textit{J. J. Nieto}, J. Comput. Appl. Math. 113, No. 1--2, 17--25 (2000; Zbl 0954.47038)

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### References:

[1] | H. BrĂ©zis, Analyse Fonctionnelle, Masson, Paris, 1983. |

[2] | Cabada, A., The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. math. anal. appl., 185, 302, (1994) · Zbl 0807.34023 |

[3] | Cabada, A., The method of lower and upper solutions for nth order periodic boundary value problems, J. appl. math. stoch. anal., 7, 33, (1994) · Zbl 0801.34026 |

[4] | Cabada, A., The method of lower and upper solutions for third order periodic boundary value problems, J. math. anal. appl., 195, 568, (1995) · Zbl 0846.34019 |

[5] | Cabada, A.; Lois, S., Maximum principles for fourth and sixth order periodic boundary value problems, Nonlinear anal. T.M.A., 29, 1161, (1997) · Zbl 0886.34018 |

[6] | Cabada, A.; Lois, S., Existence results for nonlinear problems with separated boundary conditions, Nonlinear anal. T.M.A., 35, 449, (1999) · Zbl 0918.34021 |

[7] | G.S. Ladde, V. Lakshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985. · Zbl 0658.35003 |

[8] | Nieto, J.J., An abstract monotone iterative technique, Nonlinear anal. T.M.A., 28, 1923, (1997) · Zbl 0883.47058 |

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