Periodic solutions for nonlinear Volterra integrodifferential equations in Banach spaces
Kandilakis, Dimitrios A. ; Papageorgiou, Nikolaos S.
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997), p. 283-296 / Harvested from Czech Digital Mathematics Library
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In this paper we examine periodic integrodifferential equations in Banach spaces. When the cone is regular, we prove two existence theorems for the extremal solutions in the order interval determined by an upper and a lower solution. Both theorems use only the order structure of the problem and no compactness condition is assumed. In the last section we ask the cone to be only normal but we impose a compactness condition using the ball measure of noncompactness. We obtain the extremal solutions for both the Cauchy and periodic problems in a constructive way, using a monotone iterative technique.

Publié le : 1997-01-01
Classification:  34K30,  45G10,  45J05,  45L05,  45N05,  47H17,  47N20 
@article{118927,
     author = {Dimitrios A. Kandilakis and Nikolaos S. Papageorgiou},
     title = {Periodic solutions for nonlinear Volterra integrodifferential equations in Banach spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {38},
     year = {1997},
     pages = {283-296},
     zbl = {0891.45010},
     mrnumber = {1455496},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118927}
}

Kandilakis, Dimitrios A.; Papageorgiou, Nikolaos S. Periodic solutions for nonlinear Volterra integrodifferential equations in Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 283-296. http://gdmltest.u-ga.fr/item/118927/

Amann H. Order structures and fixed points, in Atti 2$^{nd}$ Seminario Anal. Funz. Appl., Univ. Cosenza, Italy, 1977, pp.1-50.

Avgerinos E.; Papageorgiou N.S. Topological properties of the solution set of integrodifferential inclusions, Comment. Math. Univ. Carolinae 36 (1995), 429-442. (1995) | MR 1364483 | Zbl 0836.34019

Banas J.; Goebel K. Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980. | MR 0591679 | Zbl 0441.47056

Barbu V. Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands, 1976. | MR 0390843 | Zbl 0328.47035

Cheng Y.-B.; Zhuang W. The existence of maximal and minimal solutions of the nonlinear integrodifferential equation in Banach space, Applic. Anal. 22 (1986), 139-147. (1986) | MR 0854546

Coppel W. Stability and Asymptotic Behavior of Differential Equations, D.C. Heath and Co., Boston, 1965. | MR 0190463 | Zbl 0154.09301

Diestel J.; Uhl J. Vector Measures, Math. Surveys, Vol. 15, AMS, Providence, Rhode Island, 1977. | MR 0453964 | Zbl 0521.46035

Guo D.; Lakshmikantham V. Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988. | MR 0959889 | Zbl 0661.47045

Hale J. Ordinary Differential Equations, Wiley Interscience, New York, 1969. | MR 0419901 | Zbl 0433.34003

Heikkila S.; Lakshmikantham V.; Sun Y. Fixed point results in ordered normed spaces with applications to abstract and differential equations, J. Math. Anal. Appl. 163 (1962), 422-437. (1962) | MR 1145839

Kisielewisz M. Multivalued differential equations in separable Banach spaces, J. Optim. Theory Appl. 37 (1982), 231-249. (1982) | MR 0663523

Lakshmikantham V. Some problems in integrodifferential equations of Volterra type, J. Integral Equations 10 (1985), 137-146. (1985) | MR 0831240 | Zbl 0598.45015

Lakshmikantham V.; Leela S. An Introduction to Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Oxford, 1980. | MR 0616449

Lakshmikantham V.; Leela S. On the method of upper and lower solutions in abstract cones, Annales Polonici Math. XLII (1983), 159-164. (1983) | MR 0728078 | Zbl 0544.34056

Pachpatte B.G. A note on Gronwall-Bellman inequality, J. Math. Anal. Appl. 44 (1973), 758-762. (1973) | MR 0335721 | Zbl 0274.45011

Papageorgiou N.S. Existence of solutions for integrodifferential inclusions in Banach spaces, Comment. Math. Univ. Carolinae 32 (1992), 687-696. (1992) | MR 1159815

Redheffer R.; Walter W. Remarks on ordinary differential equations in ordered Banach spaces, Monats. Math. 102 (1986), 237-249. (1986) | MR 0863220 | Zbl 0597.34063