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**Computation of topological degree of unilaterally asymptotically linear operators and its applications.**
*(English)*
Zbl 1191.47076

Suppose that \(E\) is a Banach space, \(P\) a cone in \(E\), and let \(w\in E\). A nonlinear operator \(A: E\to E\) is said to be unilaterally asymptotically linear along \(w+P\) if there is a bounded linear operator \(L\) in \(E\) such that

\[ \lim_{\|x\|\to\infty,\;x\geq w}\frac{\|Ax-Lx\|}{\|x\|}=0. \]

\(A\) is said to be a cone mapping if \(A(P)\subseteq P\).

The authors present several theorems on the computation of the topological degree and of fixed point indices for unilaterally asymptotically linear operators that are not cone mappings. These theorems are applied to a class of semilinear elliptic boundary value problems with asymptotically linear nonlinearity. Under a variety of conditions, the existence of one or three solutions is proved, including an existence result for a sign changing solution.

\[ \lim_{\|x\|\to\infty,\;x\geq w}\frac{\|Ax-Lx\|}{\|x\|}=0. \]

\(A\) is said to be a cone mapping if \(A(P)\subseteq P\).

The authors present several theorems on the computation of the topological degree and of fixed point indices for unilaterally asymptotically linear operators that are not cone mappings. These theorems are applied to a class of semilinear elliptic boundary value problems with asymptotically linear nonlinearity. Under a variety of conditions, the existence of one or three solutions is proved, including an existence result for a sign changing solution.

Reviewer: Nils Ackermann (México)

### MSC:

47H11 | Degree theory for nonlinear operators |

35J61 | Semilinear elliptic equations |

35J25 | Boundary value problems for second-order elliptic equations |

47N20 | Applications of operator theory to differential and integral equations |

### Keywords:

asymptotically linear operator; topological degree; vector lattice; elliptic boundary value problem### References:

[1] | Amann, H., Multiple positive fixed points of asymptotically linear maps, J. Funct. Anal., 17, 174-213 (1974) · Zbl 0287.47037 |

[2] | Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044 |

[3] | Amann, H., On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal., 11, 346-384 (1972) · Zbl 0244.47046 |

[4] | Cac, N. P.; Gatica, J. A., Fixed point theorems for mappings in ordered Banach spaces, J. Math. Anal. Appl., 71, 547-557 (1979) · Zbl 0448.47035 |

[5] | Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0559.47040 |

[6] | Guo, D., Nonlinear Functional Analysis (2001), Shandong Science and Techonlogy Press: Shandong Science and Techonlogy Press Jinan, (in Chinese) |

[7] | Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045 |

[8] | Krein, M. G.; Rutman, M. A., Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., 10, 199-325 (1962) · Zbl 0030.12902 |

[9] | Lucxemburg, W. A.J.; Zaanen, A. C., Riesz Space, vol. I (1971), London North-Holland Publishing Company · Zbl 0231.46014 |

[10] | Sun, J. X.; Liu, X. Y., Computation of topological degree for nonlinear operators and applications, Nonlinear Anal. TMA, 69, 4121-4130 (2008) · Zbl 1169.47043 |

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