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A bound sets technique for Dirichlet problem with an upper-Carathéodory right-hand side. (English) Zbl 1237.34024

Consider the vector Dirichlet problem \[ \ddot{x}(t)\in F(t,x(t),\dot{x}(t))\quad \text{a.e.}\,\, t\in [0,T], \tag{1} \]
\[ x(0) = x(T)=0, \tag{2} \] where \(F(t,x(t),\dot{x}(t))\) is a subset of \(\mathbb{R}^{n}.\) By a solution of problem (1), (2) is meant a function \(x: [0,T]\to\mathbb{R}^{n}\) with absolutely continuous first derivative satisfying (1), (2). It is proved that the problem (1), (2) has a solution if some conditions are fulfilled.

MSC:

34A60 Ordinary differential inclusions
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

[1] Andres, J., Górniewicz, L.: Topological Fixed Point Principles for Boundary Value Problems. Topological Fixed Point Theory and Its Applications, vol. 1 Kluwer, Dordrecht, 2003. · Zbl 1029.55002
[2] Andres, J., Pavlačková, M.: Asymptotic boundary value problems for second-order differential systems. Nonlin. Anal. 71, 5-6 (2009), 1462-1473. · Zbl 1182.34038
[3] Appell, J., De Pascale, E., Thái, N. H., Zabreiko, P. P.: Multi-Valued Superpositions. Diss. Math., Vol. 345, PWN, Warsaw, 1995. · Zbl 0855.47037
[4] De Blasi, F. S., Pianigiani, G.: Solution sets of boundary value problems for nonconvex differential inclusions. Topol. Methods Nonlinear Anal. 1 (1993), 303-314. · Zbl 0785.34018
[5] Deimling, K.: Multivalued Differential Equations. de Gruyter, Berlin, 1992. · Zbl 0820.34009
[6] Erbe, L., Krawcewicz, W.: Nonlinear boundary value problems for differential inclusions \(y^{\prime \prime } \in F(t, y, y^{\prime })\). Ann. Pol. Math. 54 (1991), 195-226. · Zbl 0731.34078
[7] Gaines, R., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin, 1977. · Zbl 0339.47031
[8] Halidias, N., Papageorgiou, N. S.: Existence and relaxation results for nonlinear second order multivalued boundary value problems in \(R^n\). J. Diff. Equations 147 (1998), 123-154. · Zbl 0912.34020
[9] Halidias, N., Papageorgiou, N. S.: Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions. J. Comput. Appl. Math. 113 (2000), 51-64. · Zbl 0941.34008
[10] Kožušníková, M.: A bounding functions approach to multivalued Dirichlet problem. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 55 (2007), 1-19. · Zbl 1202.34036
[11] Kyritsi, S., Matzakos, N., Papageorgiou, N. S.: Nonlinear boundary value problems for second order differential inclusions. Czechoslovak Math. J. 55 (2005), 545-579. · Zbl 1081.34020
[12] Miklaszewski, D.: The two-point problem for nonlinear ordinary differential equations and differential inclusions. Univ. Iagell Acta Math. 36 (1998), 127-132. · Zbl 1002.34011
[13] Palmucci, M., Papalini, F.: Periodic and boundary value problems for second order differential inclusions. J. of Applied Math. and Stoch. Anal. 14 (2001), 161-182. · Zbl 1014.34009
[14] Zuev, A. V.: On the Dirichlet problem for a second-order ordinary differential equation with discontinuous right-hand side. Diff. Urav. 42 (2006), 320-326. · Zbl 1133.34309
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