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Extremal Solutions of a Class of Nonlinear Integro-Differential Equations in Banach spaces
Author(s):
Hong-Kun
Xu;
Juan
J.
Nieto
Journal:
Proc. Amer. Math. Soc.
125
(1997),
2605-2614.
MSC (1991):
Primary 45J05;
Secondary 34G20
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Abstract:
The monotone iterative technique is applied to a class of nonlinear first order integro-differential equations in Banach spaces. First a linear system with a ``small'' nonlinear perturbation is solved using Banach's Contraction Principle and the technique of Green's function. Then based upon a comparison result, the existence of minimal and maximal solutions is proved.
References:
- 1.
- C. Corduneanu, Integral Equations and Applications, Cambridge University Press, 1991. MR 92h:45001
- 2.
- K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin/Heidelberg, 1985. MR 86j:47001
- 3.
- S. Hu and V. Lakshmikantham, Periodic boundary value problems for integro-differential equations of Volterra type, Nonlinear Analysis: TMA 10 (1986), 1203-1208. MR 88c:45009
- 4.
- G.S. Ladde, V. Lakshmikantham and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman Advanced Publishing Program, 1985. MR 88g:35006
- 5.
- V. Lakshmikantham and S. Leela, Differential and integral inequalities, vols I, II, Academic Press, New York, 1969. MR 52:837; MR 52:838
- 6.
- E. Liz and J.J. Nieto, Periodic boundary value problem for integro-differential equations with general kernel, Dynamic Systems and Applications 3 (1994), 297-304. MR 95f:45005
- 7.
- H.K. Xu, M. Su and X.W. Lu, The monotone iterative technique for first order differential equations in Banach spaces, Math. Japonica 38 (1993), 667-673. MR 94d:34072
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Additional Information:
Hong-Kun
Xu
Affiliation:
Department of Mathematics, University of Durban-Westville, Private Bag X54001, Durban 4000, South Africa
Email:
hkxu@pixie.udw.ac.za
Juan
J.
Nieto
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15706 Santiago de Compostela, Spain
Email:
amnieto@usc.es
DOI:
10.1090/S0002-9939-97-04149-X
PII:
S 0002-9939(97)04149-X
Keywords:
Periodic boundary value problem,
first order nonlinear integro-differential equation,
Green's function,
monotone iterative technique,
minimal and maximal solutions,
normal cone,
Banach space
Received by editor(s):
March 13, 1996
Additional Notes:
Part of the paper was announced by the first author at the International Symposium on Methods and Applications of Analysis, City University of Hong Kong, Hong Kong, December 16--19, 1994.
The first author was partially supported by the South African Foundation for Research Development.
Communicated by:
Palle E. T. Jorgensen
Copyright of article:
Copyright
1997,
American Mathematical Society
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