REDUCTION OF AN OPERATOR EQUATION IN TO AN EQUIVALENT BIFURCATION EQUATION THROUGH SCHAUDERS FIXED POINT THEOREM

Volume 3, Issue 3, pp 164-178 http://dx.doi.org/10.22436/jnsa.003.03.02

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Authors

PALLAV KUMAR BARUAH - Department of Mathematics and Computer Science, Sri Sathya Sai University, Prashanthi Nilayam, Puttaparthy, India. B V K BHARADWAJ - Department of Mathematics and Computer Science, Sri Sathya Sai University, Prashanthi Nilayam, Puttaparthy, India. M VENKATESULU - Department of Mathematics and Computer Applications, Kalasalingam University, Krishnankoil, Tamilnadu, India.

Abstract

In this paper we deal with the Nonlinear Coupled Ordinary Differential Equations(Nonlinear CODE). A Multipoint Boundary Value Problem(MBVP) associated with these Nonlinear Equations is defined as an Operator Equation. This equation(infinite dimensional) is reduced to an Equivalent Bifurcation Equation(finite dimensional) using Schauder's Fixed Point Theorem. This Bifurcation Equation being on a finite dimensional space can be easily solved by using standard approximation techniques.

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Keywords

• Coupled Differential Operator
• Nonlinear Operator
• Hilbert Space.

MSC

•  34G20
•  34L30
•  34B05

References

• [1] A. Aliouche, Common Fixed Point Theorems for Hybrid Mappings Satysfying Generalised Contractive Conditions, Journal of Nonlinear Sciences and Applications, 2 (2009), 136-145.


• [2] A. Azam, M. Arshad, I. Beg, Common Fixed Point Theorems in cone Metric Spaces , Journal of Nonlinear Sciences and Applications, 1 (2009), 204-213.


• [3] L. Cesari, Functional analysis and Galerkin's Method, Mich. Math J., 11 (1964), 385-359.


• [4] L. Cesari, R. Kannan , Functional analysis and nonlinear differential equations, Contributions to differential equations, vol 1, No 2, Interscience, Newyork (1956)


• [5] P. C. Das, M. Venkatesulu, An Alternative Method for Boundary Value Problems with Ordinary Differential Equations , Riv. Math. Univ. Parma, 4 (1983), 15-25.


• [6] P. C. Das, M. Venkatesulu, An Existential Analysis for a Nonlinear Differential Equation with Nonlinear Multipoint Boundary Conditions via the alternative mathod, Att. Sem. Mat. Fis.Univ, Modena, XXXII , (1983), 287-307.


• [7] P. C. Das, M. Venkatesulu, An Existential Analysis for a Multipoint Boundary Value Problem via Alternative Method, Riv. Math. Univ. Parma, 4 (1984), 183-193.


• [8] L. Kantarovich, Functional Analysis and Applied Mathematics, Uspehi Mat.Nauk, 3 (1948), 89-185.


• [9] J. Locker , Functional Analysis and Two Point Differential operators, Longman scientific and Technical, Harlow (1986)


• [10] J. Locker, An existential analysis for nonlinear boundary value problems, SIAM J. Appl. Math., 19 (1970), 109-207.


• [11] P. K. Baruah, B. V. K. Bharadwaj, M. Venkatesulu, Characterization of Maximal, Minimal and Inverse Operators and an Existence Theorem for Coupled Ordinary Differential Operator, International Journal of Mathematics and Analysis, 1 (2009), 25-56.


• [12] P. K. Baruah, B. V. K. Bharadwaj, M. Venkatesulu, Green's Matrix for Linear Coupled Ordinary Differential Operator, International Journal of Mathematics and Analysis, 1(1) (2009), 57-82.


• [13] R. Ali Khan, N. Ahmad Asif, Positive Solutions for a Class of Singular Two Point Boundary Value Problems, Journal of Nonlinear Sciences and Applications, 2 (2009), 126- 135.


• [14] S. Sinha, O. P. Mishra, J. Dhar, Study of a prey-predator dynamics under the simultaneous effectof toxicant and disease, The Journal of Nonlinear Sciences and Applications, 1 (2008), 36-44.


• [15] M. Venkatesulu, P. K. Baruah, A. Prabu, A Existence theorem for IVPs of Coupled Ordinary Differential Equations, International Journal of Differential Equations and Applications, 10 (2005), 435-447.