Systems of first-order nabla dynamic equations on time scales

Bouharket Bendouma; Amine Benaissa Cherif; Ahmed Hammoudi

doi:10.26637/MJM0604/0009

Abstract

In this article, we study the existence of solutions to systems of first-order $\nabla$-dynamic equations on time scales with periodic boundary or terminal value conditions. where the right member of the system is $\nabla$-Carathéodory. We employ the method of solution-tube and Schauder's fixed-point theorem.

Keywords:

Ñ-dynamic equation, systems of first-order equation on time scales, solution-tube, existence theorems, Schauder’s fixed-point theorem

Mathematics Subject Classification:

Mathematics

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Bouharket Bendouma Ibn Khaldoun, Tiaret University PB78, 14000 zaâroura, Tiaret, Algeria. https://orcid.org/0000-0001-5479-0413
Amine Benaissa Cherif Department of Mathematics, University of Ain Temouchent, BP 284, 46000, Ain Temouchent, Algeria.
Ahmed Hammoudi Department of Mathematics, University of Ain Temouchent, BP 284, 46000, Ain Temouchent, Algeria.

Pages: 757-765
Date Published: 01-10-2018
Vol. 6 No. 04 (2018): Malaya Journal of Matematik (MJM)

R.P. Agarwal, V. Otero-Espinar, K. Perera and DR. Vivero, Basic properties of Sobolev's spaces on time scales, Adv. Differ. Equ. 2006. Article ID 38121 (2006).

M.F. Atici and D.C. Biles, First order dynamic inclusions on time scales, J. Math. Anal. Appl. 292 (2004), no. 1, 222-237.

M. F. Atici and G. Sh. Guseinov, On Green's functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002), no. 1-2, $75-99$.

D. Anderson, J. Bullock, L. Erbe, A. Peterson and H. Tran, Nabla dynamic equations on time scales, Panamer. Math. J. 13 (2003), no. 1, 15-47.

B. Bayour, A. Hammoudi, D.F.M. Torres, Existence of solution to a nonlinear first-order dynamic equation on time scales, J. Math. Anal. 1 (2016), no. 7, 2217-3412.

A. Benaissa cherif, A. Hammoudi, F. Z. Ladrani, Density problems in $L_{Delta}^p(mathbb{T}, mathbb{R})$ space, Electro. J. Math. Anal. Appl. vol. 1(2) July $2013,178-187$.

M. Bohner and A. Peterson, Dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2001.

M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2003.

A. Cabada and D.R. Vivero, Criterions for absolute continuity on time scales, J. Differ. Equ. Appl. 11 (2005), 1013-1028.

A. Cabada and D.R. Vivero, Expression of the Lebesgue $Delta$-integral on time scales as a usual Lebesgue integral: application to the calculus of $Delta$-antiderivatives, Math. Comput. Modelling 43 (2006), 194-207.

Q. Dai and C.C. Tisdell, Existence of solutions to firstorder dynamic boundary value problems, Int. J. Difference. Equ. 1 (2006), no. 1, 1-17.

J.Diblik, M. Ruzickovà and Z.Smarda, Wazewski's method for systems of dynamic equations on time scales, Nonlinear Anal. 71. (2009). 1124-1131.

M. Frigon and $mathrm{H}$. Gilbert, Boundary value problems for systems of second-order dynamic equations on time scales with Carathéodory functions, Abstr. Appl. Anal. 2010 , Art. ID 234015,26 pp.

$mathrm{H}$. Gilbert, Existence theorems for first-Order equations on time scales with $Delta$-Carathédory functions, Adv. Difference. Equ. vol. 2010, Article ID 650827, (2010), 20 pp.

Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285(1), 2003, 107-127.

S. Hilger, Analysis on measure chainsa unified approach to continuous and discrete calculus, Results Math.18 $(1990), 18-56$.

B. Mirandette, Résultats d'existence pour des systèmes d'équations différentielles du premier ordre avec tubesolution, Master Thesis (1996), Université de Montréal, Montréal.

B-R. Satco and C-O. Turcu, Hybrid Systems By Methods of Time Scales Analysis, Adv. Dyn. Syst. Appl. 8 (2013), no. $2,317-326$.

C.C. Tisdell and A.H. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal. 68 (2008), no. 11, 3504-3524.

A. H. Zaidi, Existence of solutions and convergence results for dynamic initial value problems using lower and upper solutions, Electro. J. Differ. Equ. (2009), No. 161, $1-13$.