An existence theorem for an hyperbolic differential inclusion in Banach spaces
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 22 (2002) no. 1, pp. 5-16.

Voir la notice de l'article provenant de la source Library of Science

In this paper, we investigate the existence of solutions on unbounded domain to a hyperbolic differential inclusion in Banach spaces. We shall rely on a fixed point theorem due to Ma which is an extension to multivalued between locally convex topological spaces of Schaefer's theorem.
Keywords: hyperbolic differential inclusion, convex multivalued map, existence, condensing map, fixed point, Fréchet space
@article{DMDICO_2002_22_1_a0,
     author = {Benchohra, Mouffak and Ntouyas, Sotiris},
     title = {An existence theorem for an hyperbolic differential inclusion in {Banach} spaces},
     journal = {Discussiones Mathematicae. Differential Inclusions, Control and Optimization},
     pages = {5--16},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2002},
     zbl = {1039.35148},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMDICO_2002_22_1_a0/}
}
TY  - JOUR
AU  - Benchohra, Mouffak
AU  - Ntouyas, Sotiris
TI  - An existence theorem for an hyperbolic differential inclusion in Banach spaces
JO  - Discussiones Mathematicae. Differential Inclusions, Control and Optimization
PY  - 2002
SP  - 5
EP  - 16
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMDICO_2002_22_1_a0/
LA  - en
ID  - DMDICO_2002_22_1_a0
ER  - 
%0 Journal Article
%A Benchohra, Mouffak
%A Ntouyas, Sotiris
%T An existence theorem for an hyperbolic differential inclusion in Banach spaces
%J Discussiones Mathematicae. Differential Inclusions, Control and Optimization
%D 2002
%P 5-16
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMDICO_2002_22_1_a0/
%G en
%F DMDICO_2002_22_1_a0
Benchohra, Mouffak; Ntouyas, Sotiris. An existence theorem for an hyperbolic differential inclusion in Banach spaces. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 22 (2002) no. 1, pp. 5-16. http://geodesic.mathdoc.fr/item/DMDICO_2002_22_1_a0/

[1] L. Byszewski, Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation $u''_{xt} = F(x,t,u,u_x)$, J. Appl. Math. Stoch. Anal. 3 (3) (1990), 163-168.

[2] L. Byszewski, Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation, Appl. Anal. 40 (1991), 173-180.

[3] L. Byszewski and V. Lakshmikantham, Monotone iterative technique for nonlocal hyperbolic differential problem, J. Math. Phys. Sci. 26 (4) (1992), 345-359.

[4] L. Byszewski and N.S. Papageorgiou, An application of a noncompactness technique to an investigation of the existence of solutions to nonlocal multivalued Darboux problem, J. Appl. Math. Stoch. Anal. 12 (2) (1999), 179-190.

[5] C. Corduneanu, Integral Equations and Applications, Cambridge Univ. Press, New York 1990.

[6] M. Dawidowski and I. Kubiaczyk, An existence theorem for the generalized hyperbolic equation $z''_{xy} ∈ F(x,y,z)$ in Banach space, Ann. Soc. Math. Pol. Ser. I, Comment. Math. 30 (1) (1990), 41-49.

[7] M. Dawidowski and I. Kubiaczyk, Existence theorem for hyperbolic differential inclusion with Carathéodory right hand side, Discuss. Math. Differ. Incl. 10 (1990), 69-75.

[8] M. Dawidowski and I. Kubiaczyk, On bounded solutions of hyperbolic differential inclusion in Banach spaces, Demonstr. Math. 25 (1-2) (1992), 153-159.

[9] F. De Blasi and J. Myjak, On the structure of the set of solutions of the Darboux problem for hyperbolic equations, Proc. Edinburgh Math. Soc. 29 (1986), 7-14.

[10] F. De Blasi and J. Myjak, On the set of solutions of a differential inclusion, Bull. Inst. Math., Acad. Sin. 14 (1986), 271-275.

[11] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin-New York 1992.

[12] J. Dugundji and A. Granas, Fixed Point Theory, Monografie Mat. PWN, Warsaw 1982.

[13] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht 1999.

[14] S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York 1994.

[15] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, Boston, London, 1997, Volume II: Applications, Kluwer, Dordrecht, Boston, London 2000.

[16] I. Kubiaczyk, Kneser's theorem for hyperbolic equations, Funct. Approx. Comment. Math. 14 (1984), 183-196.

[17] I. Kubiaczyk and A.N. Mostafa, On the existence of weak solutions of the Darboux problem for the hyperbolic partial differential equations in Banach spaces, Fasc. Math. 28 (1998), 93-99.

[18] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786.

[19] T.W. Ma, Topological degrees for set-valued compact vector fields in locally convex spaces, Dissertationess Math. 92 (1972), 1-43.

[20] N.S. Papageorgiou, Existence of solutions for hyperbolic differential inclusions in Banach spaces, Arch. Math. (Brno) 28 (1992), 205-213.

[21] H. Schaefer, Über die methode der a priori schranken, Math. Ann. 129 (1955), 415-416.

[22] K. Yosida, Functional Analysis, 6th edn. Springer-Verlag, Berlin 1980.