This article investigates the existence of solutions to second-order boundary value problems (BVPs) for systems of ordinary differential inclusions. The boundary conditions may involve two or more points. Some new inequalities are presented that guarantee a priori bounds on solutions to the differential inclusion under consideration. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow the treatment of systems of BVPs in the absence of maximum principles and growth conditions. The results are also new for differential equations involving Carathéodory or even continuous right-hand sides.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1070,
author = {Christopher C. Tisdell},
title = {Systems of differential inclusions in the absence of maximum principles and growth conditions},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {26},
year = {2006},
pages = {129-141},
zbl = {1139.34008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1070}
}
Christopher C. Tisdell. Systems of differential inclusions in the absence of maximum principles and growth conditions. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 26 (2006) pp. 129-141. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1070/
[000] [1] J. Andres and L. Górniewicz, Topological fixed point principles for boundary value problems. Topological Fixed Point Theory and Its Applications, 1, Kluwer Academic Publishers, Dordrecht 2003. | Zbl 1029.55002
[001] [2] A. Arara, M. Benchohra, S.K. Ntouyas and A. Ouahab, Existence results for boundary value problems for fourth-order differential inclusions with nonconvex valued right hand side, Arch. Math. (Brno) 40 (3) (2004), 219-227. | Zbl 1117.34005
[002] [3] J.P. Aubin and A. Cellina, Differential inclusions. Set-valued maps and viability theory, Fundamental Principles of Mathematical Sciences, 264, Springer-Verlag, Berlin 1984.
[003] [4] E.P. Avgerinos, N.S. Papageorgiou and N. Yannakakis, Periodic solutions for second order differential inclusions with nonconvex and unbounded multifunction, Acta Math. Hungar. 83 (4) (1999), 303-314. | Zbl 0939.34013
[004] [5] R. Bader, B.D. Gelḿan, M. Kamenskii and V. Obukhovskii, On the topological dimension of the solutions sets for some classes of operator and differential inclusions, Discuss. Math. Differ. Incl. Control Optim. 22 (1) (2002), 17-32. | Zbl 1041.47031
[005] [6] M. Benchora and S.K. Ntouyas, Multi point boundary value problems for second order differential inclusions, Mat. Vesnik 53 (1-2) (2001), 51-58. | Zbl 1213.34021
[006] [7] M. Benchohra and S.K. Ntouyas, On three and four point boundary value problems for second order differential inclusions, Math. Notes (Miskolc) 2 (2) (2001), 93-101. | Zbl 1010.34005
[007] [8] M. Benchohra and S.K. Ntouyas, Multi-point boundary value problems for lower semicontinuous differential inclusions, Math. Notes (Miskolc) 3 (2) (2002), 91-99. | Zbl 1017.34012
[008] [9] A. Boucherif and B. Chanane, Boundary value problems for second order differential inclusions, Int. J. Differ. Equ. Appl. 7 (2) (2003), 147-151. | Zbl 1048.34032
[009] [10] A. Boucherif and B. Chanane, Second order multivalued boundary value problems, Comm. Appl. Nonlinear Anal. 11 (1) (2004), 85-91. | Zbl 1055.34017
[010] [11] Y. Daido, M. Ikehata and G. Nakamura, Reconstruction of inclusions for the inverse boundary value problem with mixed type boundary condition, Appl. Anal. 83 (2) (2004), 109-124. | Zbl 1106.35132
[011] [12] B.C. Dhage, On boundary value problems of second order differential inclusions, Discuss. Math. Differ. Incl. Control Optim. 24 (2004), 73-96. | Zbl 1078.34003
[012] [13] T. Donchev and M. Quincampoix, A two point boundary value problem for a class of differential inclusions, J. Nonlinear Convex Anal. 5 (1) (2004), 59-69. | Zbl 1058.34080
[013] [14] L.H. Erbe and W. Krawcewicz, Boundary value problems for second order nonlinear differential inclusions. Qualitative theory of differential equations (Szeged, 1988), 163-171, Colloq. Math. Soc. Janos Bolyai, 53, North-Holland, Amsterdam 1990.
[014] [15] L.H. Erbe and W. Krawcewicz, Boundary value problems for differential inclusions. Differential equations (Colorado Springs, CO, 1989), 115-135, Lecture Notes in Pure and Appl. Math., 127, Dekker, New York 1991.
[015] [16] L.H. Erbe and W. Krawcewicz, Nonlinear boundary value problems for differential inclusions y''∈ F(t,y,y'), Ann. Polon. Math. 54 (3) (1991), 195-226. | Zbl 0731.34078
[016] [17] L. Erbe, R. Ma and C.C. Tisdell, On two point boundary value problems for second order differential inclusions, Dynam. Systems Appl. 15 (1) (2006), 79-88. | Zbl 1112.34008
[017] [18] L. Erbe, C.C. Tisdell and P.J.Y. Wong, On Systems of boundary value problems for differential inclusions, Acta Math. Sinica (in press). | Zbl 1126.34012
[018] [19] M. Filippakis, L. Gasiński and N.S. Papageorgiou, Positive solutions for second order multivalued boundary value problems, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, Vol. 1, 2, 531-547, Kluwer Acad. Publ., Dordrecht 2003. | Zbl 1051.34013
[019] [20] L. Gasiński and N.S. Papageorgiou, Strongly nonlinear multivalued boundary value problems, Nonlinear Anal. 52 (4) (2003), 1219-1238. | Zbl 1034.34017
[020] [21] L. Gasiński and N.S. Papageorgiou, Nonlinear second-order multivalued boundary value problems, Proc. Indian Acad. Sci. Math. Sci. 113 (3) (2003), 293-319. | Zbl 1052.34022
[021] [22] Y.E. Gliklikh and A.V. Obukhovskiĭ, On a two-point boundary value problem for second-order differential inclusions on Riemannian manifolds, Abstr. Appl. Anal. (2003), no. 10, 591-600. | Zbl 1034.58005
[022] [23] A.M. Gomaa, On the solution sets of three-point boundary value problems for nonconvex differential inclusions, J. Egyptian Math. Soc. 12 (2) (2004), 97-107. | Zbl 1089.34011
[023] [24] A. Granas and J. Dugundji, Fixed point theory, Springer Monographs in Mathematics, Springer-Verlag, New York 2003. | Zbl 1025.47002
[024] [25] N. Halidias and N.S. Papageorgiou, Existence and relaxation results for nonlinear second-order multivalued boundary value problems in , J. Differential Equations 147 (1) (1998), 123-154. | Zbl 0912.34020
[025] [26] D.A. Kandilakis and N.S. Papageorgiou, Existence theorems for nonlinear boundary value problems for second order differential inclusions, J. Differential Equations 132 (1) (1996), 107-125. | Zbl 0859.34011
[026] [27] M. Kisielewicz, Differential inclusions and optimal control, Mathematics and its Applications (East European Series), 44. Kluwer Academic Publishers Group, Dordrecht; PWN - Polish Scientific Publishers, Warsaw 1991.
[027] [28] T. Pruszko, Some applications of the topological degree theory to multivalued boundary value problems, Dissertationes Math. (Rozprawy Mat.) 229 (1984). | Zbl 0543.34008
[028] [29] S. Sędziwy, Fredholm alternative and boundary value problems, Proc. Amer. Math. Soc. 132 (6) (2004), 1779-1784. | Zbl 1054.34035
[029] [30] G.V. Smirnov, Introduction to the theory of differential inclusions, Graduate Studies in Mathematics, 41, American Mathematical Society, Providence, RI 2002. | Zbl 0992.34001
[030] [31] A.N. Vityuk, Solvability of a three-point boundary value problem for a second-order differential inclusion, (Russian) UkraŻn. Mat. Zh. 55 (1) (2003), 132-137; translation in Ukrainian Math. J. 55 (1) (2003), 164-170.