On multivalued impulsive differential equations

M.Shahin

Consider the following boundary value problem for multivalued differential equations with the interface (impulsive) conditions: $$ { {dx(t)\over dt}-A(t)x(t)\in F(t,x(t),} \quad {Lx(t)=r,}$$ $$ x(t_i^+)-B_ix(t_i^-)=C_i,\quad i\in\{1,2,\cdots, \ell\}. $$ Under suitable hypotheses and the use of a fixed point theorem for multivalued mappings, the existence of solutions for the above boundary value problem is proved. An application to the obtained result to a periodic problem is given.

1. Aubin J.P., Cellina A. Differential Inclusions, Set-valued maps and viability theory, Springer Verlag, New York, 1984.

2. Cellina A. Multivalued differential equations and ordinary differential equations, SIAM J. Appl. Math.(2), 18 (1970), 533--538.

3. Deimling K. Multivalued Differential Equations, deGruyter Series in Nonlinear Analysis and Applications, Vol I, Walter de Gruyter, Berlin, 1991.

4. Gonnelli A. Un teorema di esistenza per un problema di tipo interface, Matematiche, 22 (1967), 201--211.

5. Grandolfi M. Problemi ai limiti per le equazioni differenziali multivoche, Atti Accad. Naz, Lincei Rend. Cl. Sci. Fis. Mat. Natur., 42 (1967), 355--360.

6. Hu Sh., Papageogiou N.S. Handbook of Multivalued Analysis, Vol I. Theory, Mathematics and Its Applications, Vol 419, Kluwer Academic Publishers, Dordrecht, 1997.

7. Ky Fan Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci., U.S.A., 38 (1952), 121--126.

8. Lasota A., Opial Z. An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. S\'er. Sci. Math. Astronom. Phys., 13 (1965), 781--786.

9. Shahin M. On interface problems for systems of non-linear differential equations in Banach spaces, Afrika Mat., 3 (1981), 51--67.

10. Tolstonogov A. Differential Inclusions in Banach Space, Mathematics and Its Applications, Vol. 524, Kluwer Academic Publishers, Dordrecht, 2000.

11. Weiss D., Pham D. Sur quelques problèmes aux limites dans les systèmes d’équations différentielles linéaires et quasilinéaires, Bull. Math. Soc. Sci. Math. R. S. Roumaine, 8 (56), nr. 3-4 (1964), 289--306.

	On multivalued impulsive differential equations
Author	M.Shahin mshahin@desu.edu Delaware State University,Department of Mathematics and,Applied Mathematics Research Center
Abstract	Consider the following boundary value problem for multivalued differential equations with the interface (impulsive) conditions: $$ { {dx(t)\over dt}-A(t)x(t)\in F(t,x(t),} \quad {Lx(t)=r,}$$ $$ x(t_i^+)-B_ix(t_i^-)=C_i,\quad i\in\{1,2,\cdots, \ell\}. $$ Under suitable hypotheses and the use of a fixed point theorem for multivalued mappings, the existence of solutions for the above boundary value problem is proved. An application to the obtained result to a periodic problem is given.
Keywords	boundary value problem, multivalued differential equation, interface condition
DOI	doi:10.30970/ms.26.1.91-96
Reference	1. Aubin J.P., Cellina A. Differential Inclusions, Set-valued maps and viability theory, Springer Verlag, New York, 1984. 2. Cellina A. Multivalued differential equations and ordinary differential equations, SIAM J. Appl. Math.(2), 18 (1970), 533--538. 3. Deimling K. Multivalued Differential Equations, deGruyter Series in Nonlinear Analysis and Applications, Vol I, Walter de Gruyter, Berlin, 1991. 4. Gonnelli A. Un teorema di esistenza per un problema di tipo interface, Matematiche, 22 (1967), 201--211. 5. Grandolfi M. Problemi ai limiti per le equazioni differenziali multivoche, Atti Accad. Naz, Lincei Rend. Cl. Sci. Fis. Mat. Natur., 42 (1967), 355--360. 6. Hu Sh., Papageogiou N.S. Handbook of Multivalued Analysis, Vol I. Theory, Mathematics and Its Applications, Vol 419, Kluwer Academic Publishers, Dordrecht, 1997. 7. Ky Fan Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci., U.S.A., 38 (1952), 121--126. 8. Lasota A., Opial Z. An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. S\'er. Sci. Math. Astronom. Phys., 13 (1965), 781--786. 9. Shahin M. On interface problems for systems of non-linear differential equations in Banach spaces, Afrika Mat., 3 (1981), 51--67. 10. Tolstonogov A. Differential Inclusions in Banach Space, Mathematics and Its Applications, Vol. 524, Kluwer Academic Publishers, Dordrecht, 2000. 11. Weiss D., Pham D. Sur quelques problèmes aux limites dans les systèmes d’équations différentielles linéaires et quasilinéaires, Bull. Math. Soc. Sci. Math. R. S. Roumaine, 8 (56), nr. 3-4 (1964), 289--306.
Pages	91-96
Volume	26
Issue	1
Year	2006
Journal	Matematychni Studii
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