Optimal Solution for the System of Differential Inclusion in Hilbert Space

Document Type : Original Scientific Paper

Authors

1 Department of Pure Mathematics, University of Kashan, Kashan, 87317-53153, I. R. Iran

2 Basic Science Group, Golpayegan College of Engineering, Isfahan University of Technology, Golpayegan, 87717-67498, Iran

Abstract

In this paper, we study the existence of the following optimal solution for the system of differential inclusion
y′ ∈ Φ(t,y(t))  a.e.  t ∈ I = [t0,b]  and  y(t0) = u2,
y′ ∈ Ψ(t,y(t))  a.e.  t ∈ I = [t0,b]  and  y(t0) = u1.
in a Hilbert space, where Φ and Ψ are multivalued maps. Our existence result is obtained via selection technique and the best proximity point methods reducing the problem to a differential inclusion.

Keywords

References

[1] J. P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer, Berlin Heidelberg, New York, Tokyo, 1984.
[2] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
[3] S. Djebali, L. Górniewicz and A. Ouahab, Solution Sets for Differential Equations and Inclusions, De Gruyter, Berlin, 2013.
[4] M. Fakhar, Z. Soltani and J. Zafarani, Existence of best proximity points for set-valued cyclic Mier–Keeler contraction, Fixed Point Theory 19 (1) (2018) 211 − 218.
[5] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Academic Publishers, Dordrecht, 1997.
[6] 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, Walter de Gruyter Co., Berlin, 2001.
[7] E. .N. Mahmudov, Approximation and Optimization of Discrete and Differ ential Inclusions, Elsevier, Boston, USA, 2011.
[8] M. Srebrny, Measurable Selectors of PCA Multifunctions with Applications, Mem. Amer. Math. Soc. 52 (311) (1984) pp. 50.
[9] T. Suzuki, M. Kikkawa and C. Vetro, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal 71 (2009) 2918–2926.
[10] E. Zeidler, Nonlinear Functional Analysis and Its Applications,Vol. 3 : Variational methods and optimization, Springer, New York, 1985.
[11] P. Veeramani and S. Rajesh, Best Proximity Points,In: Ansari Q. (eds) Nonlinear Analysis. Trends in Mathematics. Birkhüser, New Delhi, 2014.
[12] D. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim 15 (1977) 859–903.
Mathematics Interdisciplinary Research
Volume 6, Issue 4
December 2021
Pages 319-327

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