# Article

Full entry | PDF   (0.2 MB)
Keywords:
fixed point; Banach algebra; integral equation; integro-differential system; epidemic model; blowing-up solution
Summary:
In this paper we prove some fixed point theorems of the Banach and Krasnosel’skii type for mappings on the \$m\$-tuple Cartesian product of a Banach algebra \$X\$ over \$\mathbb {R}\$. Using these theorems existence results for a system of integral equations of the Gripenberg’s type are proved. A sufficient condition for the nonexistence of blowing-up solutions of this system of integral equations is also proved.
References:
[1] Abdeldaim, A.: On some new Gronwall-Bellman-Ou-Iang type integral inequalities to study certain epidemic model. J. Integral Equations Appl. (2011), (to appear). MR 2945799
[2] Banas, J., Sadarangani, K.: Solutions of some functional-integral equations in Banach Algebra. Math. Comput. Modelling 38 (2003), 245–250. DOI 10.1016/S0895-7177(03)90084-7 | MR 2004993 | Zbl 1053.45007
[3] Banas, J., Lacko, M.: Fixed points of the product of operators in Banach algebra. Panamer. Math. J. 12 (2002), 101–109. MR 1895774
[4] Bihari, I.: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Mathematica Hungarica 7, 1 (1956), 81–94. DOI 10.1007/BF02022967 | MR 0079154 | Zbl 0070.08201
[5] Brestovanská, E.: Qualitative behaviour of an integral equation related to some epidemic model. Demonstratio Math. 36, 3 (2003), 604–609. MR 2004545 | Zbl 1044.45001
[6] Djebali, S., Hammache, K.: Furi-Pera fixed point theorems in Banach algebra with applications. Acta Univ. Palacki. Olomouc, Fac. Rer. Nat., Math. 47 (2008), 55–75. MR 2482717
[7] Gripenberg, G.: On some epidmic models. Quart. Appl. Math. 39 (1981), 317–327. MR 0636238
[8] Krasnosel’skii, M. A.: Two remarks on the method of succesive approximations. Uspechi Mat. Nauk 10 (1955), 123–127. MR 0068119
[9] Olaru, I. M.: An integral equation via weakly Picard operators. Fixed Point Theory 11, 1 (2010), 97–106. MR 2656009 | Zbl 1196.45009
[10] Olaru, I. M.: Generalization of an integral equation related to some epidemic models. Carpatian J. Math. 26 (2010), 0–4. MR 2676722 | Zbl 1224.34205
[11] Zeidler, E.: Applied Functional Analysis: Applications to Mathematical Physics. Applied Mathematical Sciences, Vol. 108, Springer-Verlag, New York, 1995. DOI 10.1007/978-1-4612-0815-0 | MR 1347691 | Zbl 0834.46002

Partner of