Abstract
References
- [1] C.D. Aliprantis and R. Tourky, Cones and Duality, Graduate Studies in Mathematics, Vol. 84, Amer. Math. Soc., Providence, Rhode Island, 2007.
- [2] J.M. Borwein and D.T. Yost, Absolute norms on vector lattices, Proc. Edinb. Math. Soc. 27, 215–222, 1984.
- [3] S. Carl and S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications, Springer, New York, 2011.
- [4] L.H. Erbe, W. Krawcewicz, and D. Guo, Positive solutions of two-point boudary value problems for nonlinear integro-differential equations in Banach spaces, Differ. Equ. Dyn. Syst. 2, 161–171, 1994.
- [5] K.H. Förster and B. Nagy, On the local spectral radius of a nonnegative element with respect to an irreducible operator, Acta Sci. Math. (Szeged), 55, 155–166, 1991.
- [6] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, 1988.
- [7] D. Guo, Multiple positive solutions of impulsive Fredholm integral equations and ap- plications, J. Math. Anal. Appl. 173, 318–324, 1993.
- [8] D. Guo, Y.J. Cho and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Science Publishers Inc., Hauppauge, 2004.
- [9] R.D. Holmes and A.T. Lau, Nonexpansive actions of topological semigroups and fixed points, J. Lond. Math. Soc. (2), 5, 330–336, 1972.
- [10] G.S. Ladde, V. Lakshmikantham and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985.
- [11] Z. Liang, Some properties of nonlinear operators and positive solutions of a class of integral equations, Acta Math. Sinica (Chin. Ser.), 40, 345–350, 1997.
- [12] J.J. Nieto and R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 223–239, 2005.
- [13] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132, 1435–1443, 2004.
- [14] C.A. Stuart, Positive solutions of a nonlinear integral equation, Math. Ann. 192, 119–124, 1971.
- [15] M. Zima, A certain fixed point theorem and its applications to integral-functional equations, Bull. Aust. Math. Soc. 46, 179–186, 1992.
- [16] M. Zima, Positive Operators in Banach Spaces and Their Applications, Wydawnictwo Uniwersytetu Rzeszowskiego, Rzeszow, 2005.
A fixed point result for semigroups of monotone operators and a solution of discontinuous nonlinear functional-differential equations
Abstract
We improve some fixed point theorems by stating a fixed point result for semigroups of monotone operators in the setting of ordered Banach spaces with a normal cone. We illustrate the usefulness of our results by proving the existence and conditional unicity of a solution of an initial value problem for discontinuous nonlinear functional-differential equations under natural hypotheses involving the order structure of the underlying space.
Keywords
Fixed point semitopological semigroup ordered Banach space normal cone functional-differential equation lower and upper solutions
References
- [1] C.D. Aliprantis and R. Tourky, Cones and Duality, Graduate Studies in Mathematics, Vol. 84, Amer. Math. Soc., Providence, Rhode Island, 2007.
- [2] J.M. Borwein and D.T. Yost, Absolute norms on vector lattices, Proc. Edinb. Math. Soc. 27, 215–222, 1984.
- [3] S. Carl and S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications, Springer, New York, 2011.
- [4] L.H. Erbe, W. Krawcewicz, and D. Guo, Positive solutions of two-point boudary value problems for nonlinear integro-differential equations in Banach spaces, Differ. Equ. Dyn. Syst. 2, 161–171, 1994.
- [5] K.H. Förster and B. Nagy, On the local spectral radius of a nonnegative element with respect to an irreducible operator, Acta Sci. Math. (Szeged), 55, 155–166, 1991.
- [6] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, 1988.
- [7] D. Guo, Multiple positive solutions of impulsive Fredholm integral equations and ap- plications, J. Math. Anal. Appl. 173, 318–324, 1993.
- [8] D. Guo, Y.J. Cho and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Science Publishers Inc., Hauppauge, 2004.
- [9] R.D. Holmes and A.T. Lau, Nonexpansive actions of topological semigroups and fixed points, J. Lond. Math. Soc. (2), 5, 330–336, 1972.
- [10] G.S. Ladde, V. Lakshmikantham and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985.
- [11] Z. Liang, Some properties of nonlinear operators and positive solutions of a class of integral equations, Acta Math. Sinica (Chin. Ser.), 40, 345–350, 1997.
- [12] J.J. Nieto and R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 223–239, 2005.
- [13] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132, 1435–1443, 2004.
- [14] C.A. Stuart, Positive solutions of a nonlinear integral equation, Math. Ann. 192, 119–124, 1971.
- [15] M. Zima, A certain fixed point theorem and its applications to integral-functional equations, Bull. Aust. Math. Soc. 46, 179–186, 1992.
- [16] M. Zima, Positive Operators in Banach Spaces and Their Applications, Wydawnictwo Uniwersytetu Rzeszowskiego, Rzeszow, 2005.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | February 4, 2021 |
| Published in Issue | Year 2021 Volume: 50 Issue: 1 |