# An existence theorem for an hyperbolic differential inclusion in Banach spaces

Mouffak Benchohra; Sotiris K. Ntouyas

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2002)

- Volume: 22, Issue: 1, page 5-16
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topMouffak Benchohra, and Sotiris K. Ntouyas. "An existence theorem for an hyperbolic differential inclusion in Banach spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 22.1 (2002): 5-16. <http://eudml.org/doc/271469>.

@article{MouffakBenchohra2002,

abstract = {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.},

author = {Mouffak Benchohra, Sotiris K. Ntouyas},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {hyperbolic differential inclusion; convex multivalued map; existence; condensing map; fixed point; Fréchet space; multivalued map; fixed point arguments},

language = {eng},

number = {1},

pages = {5-16},

title = {An existence theorem for an hyperbolic differential inclusion in Banach spaces},

url = {http://eudml.org/doc/271469},

volume = {22},

year = {2002},

}

TY - JOUR

AU - Mouffak Benchohra

AU - Sotiris K. Ntouyas

TI - An existence theorem for an hyperbolic differential inclusion in Banach spaces

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2002

VL - 22

IS - 1

SP - 5

EP - 16

AB - 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.

LA - eng

KW - hyperbolic differential inclusion; convex multivalued map; existence; condensing map; fixed point; Fréchet space; multivalued map; fixed point arguments

UR - http://eudml.org/doc/271469

ER -

## References

top- [1] L. Byszewski, Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation ${u}_{xt}^{\text{'}\text{'}}=F(x,t,u,{u}_{x})$, J. Appl. Math. Stoch. Anal. 3 (3) (1990), 163-168. Zbl0725.35059
- [2] L. Byszewski, Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation, Appl. Anal. 40 (1991), 173-180. Zbl0725.35060
- [3] L. Byszewski and V. Lakshmikantham, Monotone iterative technique for nonlocal hyperbolic differential problem, J. Math. Phys. Sci. 26 (4) (1992), 345-359. Zbl0811.35083
- [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. Zbl0936.35203
- [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 ${z}_{xy}^{\text{'}\text{'}}\in F(x,y,z)$ in Banach space, Ann. Soc. Math. Pol. Ser. I, Comment. Math. 30 (1) (1990), 41-49. Zbl0759.35029
- [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. Zbl0594.35069
- [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. Zbl0607.34010
- [11] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin-New York 1992. Zbl0760.34002
- [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. Zbl0937.55001
- [14] S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York 1994. Zbl0804.34001
- [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. Zbl0887.47001
- [16] I. Kubiaczyk, Kneser's theorem for hyperbolic equations, Funct. Approx. Comment. Math. 14 (1984), 183-196. Zbl0555.35092
- [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. Zbl0913.35077
- [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. Zbl0151.10703
- [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. Zbl0781.34045
- [21] H. Schaefer, Über die methode der a priori schranken, Math. Ann. 129 (1955), 415-416. Zbl0064.35703
- [22] K. Yosida, Functional Analysis, 6th edn. Springer-Verlag, Berlin 1980.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.