Weak solutions of first-order differential inclusions in Banach space

Year 2016, Volume: 1 Issue: 1, 1 - 11, 17.12.2016

Khouni Yassine

https://doi.org/10.30931/jetas.281375

Cited By: 2

Abstract

The aim of this paper is to investigate the existence of pseudo-solutions for a First- order

multivalued differential equation with nonlocal integral boundary condition in a Banach space.

Our approach is based on the use of the technique of measures of weak noncompactness and a fixed

point theorem of Mönch type.

Keywords

Pseudo-solutions, differential inclusions, integral boundary condition, measure of weak noncompactness, pettis integral

References

• [1] G. Adomian and G. E. Adomian, Cellular systems and aging models, Comput. Math. App. 11 (1985) 283-291.
• [2] A. Arara and M. Benchohra, Fuzzy solutions for boundary value problems with integralboundary conditions, Acta Math. Univ. Comenianae LXXV (2006) 119-126.
• [3] O. Arino, S. Gautier, J. P. Penot, A Fixed Point Theorem For Sequentially Continuous Mappings With Application To Ordinary Differential Equations, Funkcialaj Ekvcioj, 27 (1984) 273-279.
• [4] J. P. Aubin, A. Cellina, Differential inclusions, Springer, Berlin, 1984.
• [5] M. Benchohra, S. Hamani, J. Henderson, Functional differential inclusions with integral boundary conditions, Electron. J. Qua. Theory Di er. Equ. 15 (2007) 13 pages.
• [6] M. Benchohra, J. R. Graef , F. Z. Mostefai, Weak solutions for boundary value problems with nonlinear fractional differential inclusions, Nonlinear Dynamics and Systems Theory. 11, 3 (2011) 227-237.
• [7] M. Benchohra, F. Z. Mostefai, Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach space, Opuscula Mathematica, 32, 1 (2012) 31-40.
• [8] K. W. Blayneh, Analysis of age structured host-parasitoid model, FAR; East. J. Dyn. Syst. 4 (2002) 125-145.
• [9] K. Chichon, Differential inclusions and multivalued integrals, Differential Inclusions, Control and Optimization 33 (2013) 171-191.
• [10] F. S. De Blasi, On the property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. Roumanie (N. S.) 21 (1977) 259-262.
• [11] L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, 2nd Edition, Springer, Netherlands, 2006.
• [12] Hind H. G. Hashem, Weak solutions of differential equations in Banach space, Journal of fractional calculus and applications, 1, 3 (2012) 1-9.
• [13] R. W. Ibrahim, The existence of weak solutions for fractional integral inclusions involving Pettis integral, Journal of Scientific and Mathematical Research, (2008)1-8.
• [14] G. Infante, Eigenvalues and positive solutions of ODEs involving integral boundary conditions, Discrete Contin. Dyn. Syst. (2005) 436-442.
• [15] Wu Jianrong Xue Xiaoping Wu Congxin, Existence theorem for weak solutions of Random differential inclusions in Banach spaces, Advences in Mathematics, 30, 4 (2001) 359-366.
• [16] S. K. Pandey, D. K. Singh, P. Kumar and M. Kumar, Existence of measurable selectors in Pettis integrable multi function, 5, 1 (2014) 79-83.
• [17] B. J. Pettis, On integration in vector spaces, Trans. Amer. Maths. Soc. 44 (1938) 277- 304.
• [18] I. I. Vrabie, Compactness methods for nonlinear evolutions, Longman, Harlow, 1987.
• [19] K. Yosida, Functional Analysis, Springer-Verlag, Berlin Heidelberg, New York 1980.