NEW RESULTS ON K- STABILITY AND BOUNDEDNESS OF SOLUTIONS OF A CERTAIN DIFFERENTIAL EQUATIONS IN BANACH SPACES INTERACTING WITH WEAKLY SUBSYSTEMS

Main Article Content

E. P. EBIENDELE
F. U. NOSAKHARE
A. A. MOMOH

Abstract

The article gives new results of K- Stability and Boundedness of solutions of certain differential equations in Banach Spaces interacting with weakly subsystems. Some results from the theory of semigroups, and the description of the general method of solution of the posed problem were also given.  The application of the matrix – valued Lyapunov function is discussed and the main theorem of the method is formulated for differential equations in a Banach Space. The vector Lyapunov function was the main tool applied for the analysis of K – Stability and Boundedness of a system in Banach Spaces. All the results of this Article for the ‘equation (2.2)’ are essentially new.

Keywords:
New results, k- stability, bounded solutions, weakly systems, differential equations

Article Details

How to Cite
EBIENDELE, E. P., NOSAKHARE, F. U., & MOMOH, A. A. (2022). NEW RESULTS ON K- STABILITY AND BOUNDEDNESS OF SOLUTIONS OF A CERTAIN DIFFERENTIAL EQUATIONS IN BANACH SPACES INTERACTING WITH WEAKLY SUBSYSTEMS. Asian Journal of Mathematics and Computer Research, 29(4), 1-12. https://doi.org/10.56557/ajomcor/2022/v29i47986
Section
Original Research Article

References

Amann H. Compact Embeddings of Vector – Valued Sobolev and Besor Spaces, Glas.. Mat Ser. Lll. 2000;35(55)N0 1:161–177.

Chill R, Tomilov Y. Stability of operator Semigroups. Ideas and results. In perspectives in operator theory of Banch Center publ. Polish Acad. Sci. Warsaw. 2007;75:71-109.

Chol SK, Ryu HS. H -Stability in differential Systems. Bull. Inst.Math.Acad. Since. 1993;21:205-262.

Cichon M. Weak solutions of differential equations in Banach space, Discuss. Math. Differential Incl. 1995;15(1):5-14.

Eisner T, Farkas B, Nagel R, Sereny A. Weakly and almost weakly stable C – Semigroups. Int. J. Dyn. Syst. Differ. Equ. 2007;1:44-57.

Engel KJ, Nagel R. One parameter semigroup for linear Evolution equations. Graduate Texts in Mathematics. Springer – verlag, New York. 2000;94.

Ebiendele EP, Aliu KA. New Conditions for K- like properties of Asymptotically Stable solutions for Weakly Perturbed systems for a certain class of nonlinear differential equations. Journal of Applied Physical Science International.

Ebiendele EP, Agweli AM. Asian Journal of Mathematics and Computer Research. 2019;26(6):251-263.

Ebiendele EP, Asuelinmen O. On the Globally Exponential Stability of solutions for a certain class of nonlinear ordinary differential equations. Asian Journal of Current Research. 2021;6(3):11–17.

Feng SZ. Existence of Generalized solutions for ordinary differential equations in Banach Spaces. J. Math. Anal. Appl. 1987;128(2):405-412.

Gomaa A. Weak and strong solutions for differential equations in Banach Spaces. Chaos Solitons Fractals. 2003;18(4):687-692.

Lakshmikanthan V. Stability and Asymptotic behavior of Solutions of differential equations in a Banach Space. Lecture Notes, Italy. 1974;39-98.

Lakshmikanthan V. Differential equations in Banach Spaces and extension of Lyapunov’s Method. Proc. Camb. Phi. Soc. 1063;59:343–381.

Mass era JL. Contribution to Stability theory. Annals of Math. 1956;64(1):182–206.

Martynyuk AA. The Lyapunov matrix -function and Stability of hybrid Systems. Appl. Mech. 1985;21: 89–96.

Zubov VI. A.M. Lyapunov’s method and their application. Leningrader: Publ. of Leningrad Univ.; 1957.