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First order dynamic inclusions on time scales. (English) Zbl 1064.34009
The authors deal with the multi-valued boundary value problem $$y^\nabla(t)\in F(t, y(t))\quad\text{a.e. on }[a, b]_x,\qquad L(y(a), y(b))= 0,\tag1$$ on a time scale $\bbfT$, where $[a,b]_x= \{t\in\bbfT\mid a\le t\le b\}$, $F: [a,b]_x\times \bbfR\to \bbfR\setminus\{0\}$ is a multi-valued map with compact and convex values and $L: \bbfR^2\to \bbfR^2$ is a continuous single-valued map. The proof for the existence of a solution of (1) is based on the method of upper and lower solutions. The authors present some examples to illustrate that if one replaces the $\nabla$-derivative by the $\Delta$-derivative in the dynamic inclusion (1) then such a result holds only under more restrictive assumptions on $F$.

34A60Differential inclusions
34B15Nonlinear boundary value problems for ODE
34A45Theoretical approximation of solutions of ODE
Full Text: DOI
[1] Atici, F. M.; Guseinov, G. Sh.: On Green’s functions and positive solutions for boundary value problems on time scales. J. comput. Appl. math. 141, 75-99 (2002) · Zbl 1007.34025
[2] Biles, D. C.: A necessary and sufficient condition for existence of solutions for differential inclusions. Nonlinear anal. 31, 311-315 (1998) · Zbl 0910.34029
[3] Aubin, J. P.; Cellina, A.: Differential inclusions. (1984) · Zbl 0538.34007
[4] Aubin, J. P.; Frankowska, H.: Set-valued analysis. (1990) · Zbl 0713.49021
[5] Benchohra, M.; Ntouyas, S. K.: The lower and upper solutions method for first order differential inclusions with nonlinear boundary conditions. J. inequal. Pure appl. Math. 3 (2002) · Zbl 1003.34013
[6] Bohner, M.; Peterson, A. C.: Dynamic equations on time scales; an introduction with applications. (2001) · Zbl 0978.39001
[7] Bohner, M.; Peterson, A. C.: Advances in dynamic equations on time scales. (2002) · Zbl 1020.39008
[8] Cabada, A.: The monotone method for first order problems with linear and nonlinear boundary conditions. Appl. math. Comput. 63, 163-186 (1994) · Zbl 0807.34022
[9] Cabada, A.; Otero-Espinar, V.; Pouso, R. L.: Existence and approximation of solutions for first order discontinuous difference equations with nonlinear global conditions in the presense of lower and upper solutions. Comput. math. Appl. 39, 21-33 (2000) · Zbl 0972.39002
[10] Deimling, K.: Multivalued differential equations. (1992) · Zbl 0760.34002
[11] Hu, S.; Papageorgiou, N. S.: Handbook of multivalued analysis, vol. I: theory. (1997) · Zbl 0887.47001
[12] Hu, S.; Papageorgiou, N. S.: Handbook of multivalued analysis, vol. II: applications. (2000) · Zbl 0943.47037
[13] Lasota, A.; Opial, Z.: An application of Kakutani--Ky--Fan theorem in the theory of ordinary differential equations. Bull. acad. Polon. sci. Ser. sci. Math. astronom. Phys. 13, 781-786 (1965) · Zbl 0151.10703
[14] Martelli, M.: A rothe’s type theorem for non compact acyclic-valued maps. Boll. un. Mat. ital. 11, 70-76 (1975) · Zbl 0314.47035
[15] Repovs, D.; Semenov, P. V.: Continuous selections of multivalued mappings. (1998)