The existence and compactness of the set of solutions for a nonlinear integrodifferential equation in N variables in a Banach space
The existence and compactness of the set of solutions for a nonlinear integrodifferential equation in N variables in a Banach space
The paper is devoted to the study of a nonlinear integrodifferential equation in N variables with values ina general Banach space. By applying fixed point theorems in a suitable Banach space under appropriate conditions forsubsets to be relatively compact, we prove the existence and the compactness of the set of solutions. In order to illustratethe results, we give two examples.
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- [1] Abdou MA, Badr AA,El-Kojok MM. On the solution of a mixed nonlinear integral equation. Applied Mathematics
and Computation 2011; 217 (12): 5466-5475. doi: 10.1016/j.amc.2010.12.016
- [2] Aghajani A, Pourhadi E, Rivero M, Trujillo J. Application of Perov’s fixed point theorem to Fredholm type integrodifferential equations in two variables. Mathematica Slovaca 2016; 66 (5): 1207-1216. doi: 10.1515/ms-2016-0216
- [3] Bica AM, Caus VA, Muresan S. Application of a trapezoid inequality to neutral Fredholm integro-differential
equations in Banach spaces. J. Inequal. Pure and Appl. Math. 2006; 7 (5): Art. 173. http://www.kurims.kyotou.ac.jp/EMIS/journals/JIPAM/images/028_06_JIPAM/028_06.pdf
- [4] Corduneanu C. Integral equations and applications. NY, USA: Cambridge University Press, 1991.
- [5] Deimling K. Nonlinear functional analysis. New York, NY, USA: Springer, 1985.
- [6] Danh PH, Dung HTH, Long NT, Ngoc LTP. On nonlinear integrodifferential equations in two variables. Results in
Mathematics 2017; 71 (1): 251-281. doi: 10.1007/s00025-015-0508-5
- [7] Zbyněk Kubáček. On the structure of fixed point sets of some compact maps in the Fréchet space. Mathematica
Bohemica 1993; 118 (4): 343-358. http://eudml.org/doc/29314
- [8] Meehan M ,O’Regan D. Continuums of solutions in a Fréchet space of abstract Volterra equations. Appl. Anal.
2000; 74 (1-2): 95-112. doi: 10.1080/00036810008840805
- [9] Ngoc LTP, Long NT. Applying a fixed point theorem of Krasnosel’skii type to the existence of asymptotically stable
solutions for a Volterra-Hammerstein integral equation. Nonlinear Analysis. TMA. 2011; 74 (11): 3769-3774. doi:
10.1016/j.na.2011.03.021
- [10] Ngoc LTP, Long NT. On a nonlinear Volterra-Hammerstein integral equation in two variables. Acta Mathematica
Scientia 2013; 33B (2): 484-494. doi: 10.1016/S0252-9602(13)60013-2
- [11] Ngoc LTP, Long NT. Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation
in N variables. Mathematische Nachrichten 2015; 288 (5-6): 633-647. doi: 10.1002/mana.201300065
- [12] Ngoc LTP, Long NT. A continuum of solutions in a Fréchet space of a nonlinear functional integral equation in N
variables. Mathematische Nachrichten 2016; 289 (13): 1665-1679. doi: 10.1002/mana.201500008
- [13] Nieto J. Hukuhara-Kneser property for a nonlinear Dirichlet problem. Journal of Mathematical Analysis and
Applications 1987; 128: 57-63. doi: 10.1016/0022-247X(87)90213-7
- [14] Pachpatte BG. On Fredholm type integrodifferential equation. Tamkang Journal of Mathematics 2008; 39 (1):
85-94. doi: 10.5556/j.tkjm.39.2008.48
- [15] Pachpatte BG. On Fredholm type integral equation in two variables. Differential Equations & Applications 2009; 1
(1): 27-39. doi: 10.7153/dea-01-02
- [16] Pachpatte BG. Volterra integral and integrodifferential equations in two variables. Journal inequalities in Pure and Applied Mathematics 2009; 10 (4): Art. 108, 10 pp.
https://www.emis.de/journals/JIPAM/images/160_09_JIPAM/160_09.pdf
- [17] Purnaras IK. A note on the existence of solutions to some nonlinear functional integral equations. Electronic Journal
of Qualitative Theory of Differantial Equations. 2006; 2006 (17): pp. 1-24. doi: 10.14232/ejqtde.2006.1.17