# ON THE BOUNDED AND UNIQUE SOLVABILITY OF THE BOUNDARY VALUE PROBLEM OF THE EQUATION IN THE SPACE OF SCALAR FUNCTIONS WITH ABSOLUTE CONTINUOUS DERIVATIVE OF THE (n -1) ORDER AND ITS ISOTONIC GREEN OPERATOR FOR A CERTAIN CLASS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

## Main Article Content

## Abstract

The objectives of this paper is to investigate the boundary value problem of the equation in the space of scalar functions with absolute continuous derivative of the nth order, and to establishes the effective and sufficient conditions for its bounded and it unique Solvability. Theorems were stated and prove under the Preliminaries note, about four Theorems with applications to prove the main results. The necessary and sufficient conditions that guarantee the studied boundary value problem to satisfy the Isotonic Property of the Green Operator was also established. My approach in this study improved on the literatures, to the case where more than two arguments of the studying equations were established, as in the case of one argument in the authors in [3,4].

## Article Details

*Asian Journal of Mathematics and Computer Research*,

*26*(4), 251–263. Retrieved from http://ikprress.org/index.php/AJOMCOR/article/view/4878

## References

Bravyi E. A note on the Fredholm property of boundary value problems for linear functional differential equations. Mem. Differential Equations Math. Phys. 2000;20:133–135.

Bravyi EI. Solvability of boundary value problems for linear functional differential equations. (Russian) R and C Dynamics, Ishevsk, Russia; 2011.

Bravyi E, Hakel R, Lomtatidse A. Optimal conditions for unique solvability of the Cauchy problem for first order linear functional equations. Csechoslovak Math. J. 2002;52(3):513–530.

Bravyi EI. On the solvability of the Cauchy problem for higher – order linear functional differential equations. Diff. Equations. 2012;48(4):465–476.

Domoshnitsky A, Hakl R. Mutli – point boundary value problem for linear functional differential equations. Georgian Math. J. 2017;24(2):193–206.

Ebiendele PE, Nosakhare FU. On the uniquely solvability of Cauchy problem and dependences of parameters for a certain class of linear functional differential equations. Asian Research Journal of Mathematics. 2018;10(3):1-11.

Ebiendele PE, Asuelinmen O. On uniform boundedness and stability of solutions for predator –prey model for certain class of delay nonlinear differential equations. Asian Journal of Mathematics and Computer Research. 2018;25(4):238–248.

Ebiendele PE. On the stable and unstable state of a certain class of delay differential equations. Archives of Applied Science Research. 2017;9(3):35–40.

Eugene Bravyi. Boundary value problem for families of functional differential equations. Memoirs on Differential Equations and Mathematical Physics. 2017;72:27–35.

Kiguradse I, Sokhade S. On nonlinear boundary value problems for higher order functional differential equations. Georgian Math. J. 2016;23:537–550.

Maksimov AV. The property of being Noetherian of the general boundary value problem for a linear functional differential equation. Differentsial’nye Uravneniya. 1974;10(12):2288-2291,2312. (Russian)

Sremer J. On the initial value problem for two – dimensional linear functional differential systems. Mem. Differ. Equ. Math. Phy. 2010;50:1-12.

Skabachevskil AL. Boundary value problems for elliptic functional differential equations and their applications. Uspekhi Math. Nauk. 2016;71(5):3–112. (Russian)