On the Existence of Solutions for Boundary Value Problems in Banach Spaces

Yıl 2015, Cilt: 12 Sayı: 2, - , 01.11.2015

Hero Salahifard S. Mansour Vaezpour Abdolrahman Razani

Öz

In this paper, by applying the theory of condensing multimaps and the topological degree, we
deal with the existence of solutions for boundary value problems with second order differential inclusions
in different cases where the underlying space is a Banach space. Indeed, we investigate the existence of
solutions for the BVP
(
x
′′(t) ∈ F(t, x(t)) t ∈ I = [0,1],
x(0) = x(1) = 0,
where X is a real Banach space and the multifunction F : I ×X ⊸ K(X), in one case, has convex values and
in another case has non-convex values (K(X) denotes compact subsets of X). Moreover, some results on the
existence of solutions for the extended version of BVP
(
u
′′(t) ∈ Q(u) t ∈ I,
u(0) = u(1) = 0,
are presented, where Q :C(I,X) ⊸C(L 2
) is a multimap satisfying some appropriate conditions. Finally, we
show how the results can be used to study periodic feedback control systems

Anahtar Kelimeler

Boundary Value Problems, Condensing map, Feedback control systems, Topological degree

Kaynakça

• [1] R. R. Akhmerov, M. I. Kamenskii, A. S. Patapov, A. E. Rodkina, B. N. Sadovskii, Measures of noncompactness and condensing operators, Operator Theory, Advances and Applications, 55, (1992).
• [2] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, (1990).
• [3] M. Benchohra, S. K. Ntouyas, A. Ouahab, A note on a nonlinear m-point boundary value problem for p-Laplacian differential inclusions, Miskolc Math. Notes, 1, (2005), 19–26.
• [4] A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values, Studia Math., 90, (1988), 69–86.
• [5] Z. Cai, L. Huang, Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays, Commun Nonlinear Sci Numer Simulat. 19, (2014), 1279–1300.
• [6] F. S. De Blasi, L. G´orniewicz, G. Pianigiani, Topological degree and periodic solutions of differential inclusions, Nonlinear Anal. 37, (1999), 217–245.
• [7] L. Erbe, R. Ma and C. C. Tisdell, On two point boundary value problems for second order differential inclusions, Dynamic Systems and Applications, 15(1), (2006), 79–88.
• [8] H. Frankowska, A priori estimates for operational differential inclusions, J. Diff. Eqns. 84, (1990), 100–128.
• [9] M. Frigon, Application de la Th´eorie de la Transversalit´e Topologique `a des Probl´emes non Lin´eaires pour des Equations Diff´erentielles Ordinaires, ´ Dissertationes Mathematicae Warszawa, CCXCVI, (1990).
• [10] L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, second edition, Springer, Dordrecht, (2006).
• [11] C. S. Goodrich, Positive solutions to differential inclusions with nonlocal, nonlinear boundary conditions, Applied Mathematics and Computation. 219, (2013), 11071-11081.
• [12] D. Guo, V. Lakshmikantham, Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces, J. Math. Anal. Appl., 129, (1988), 211–222.
• [13] S. Hu, N. S. Papageorgiou, On the existence of periodic solutions for nonconvex valued differential inclusions in R^n, Proc. Amer. Math. Soc. 123, (1995), 3043–3050.
• [14] S. Hu, N. S. Papageorgiou, Periodic solutions for nonconvex differential inclusions, Proc. Amer. Math. Soc. 127, (1999), 89–94.
• [15] M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications, 7, (2001).
• [16] V. I. Korobov, Reduction of a controllability problem to a boundary value problem, Different. Uranen, 12, (1976), 1310–1312.
• [17] G. Li, X. Xue, On the existence of periodic solutions for differential inclusions, J. Math. Anal. Appl. 276, (2002), 168–183.
• [18] N. V. Loi, V. Obukhovskii, On global bifurcation of periodic solutions for functional differential inclusions, Funct. Diff. Equat, 17(1-2), (2010), 157–168.
• [19] N. V. Loi, Global behaviour of solutions to a class of feedback control systems, Research and Communications in Mathematics and Mathematical Sciences, 2, (2013), 77–93.
• [20] N. V. Loi, V. Obukhovskii, On the existence of solutions for a class of second-order differential inclusions and applications, J. Math. Anal. Appl. 385, (2012), 517-533.
• [21] D. O’Regan, Y. J. Cho, Y. Q. Chen, Topological degree theory and applications, Serries in Mathematical Analysis and Applications, 10, 2006.
• [22] V. Obukhovskii, P. Zecca, On boundary value problems for degenerate differential inclusions in Banach spaces, Abstract and Applied Analysis, 13, (2003), 769–784.
• [23] H. K. Pathak, R. P. Agarwal, Y. J. Chod, Coincidence and fixed points for multi-valued mappings and its application to nonconvex integral inclusions, Journal of Computational and Applied Mathematics, 283, (2015), 201–217.
• [24] C. Ravichandran, D. Baleanu, Existence results for fractional neutral functional integro-differential evolution equations with infinite delay in Banach spaces, Advances in Difference Equations, (2013), 2013–215.
• [25] G. V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics, American Mathematical Society, Providence, (2002).
• [26] S. Qin, X. Xue, Periodic solutions for nonlinear differential inclusions with multivalued perturbations, J. Math. Anal. Appl.424, (2015), 988–1005.
• [27] J.-Z. Xiao, Y.-H. Cang, Q.-F. Liu, Existence of solutions for a class of boundary value problems of semilinear differential inclusions, Mathematical and Computer Modelling, 57, (2013), 671–683.