Quasilinearization method for nonlinear differential equations with causal operators

Volume 9, Issue 3, pp 1356--1364 http://dx.doi.org/10.22436/jnsa.009.03.59

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Authors

Ali Yakar - Department of Mathematics, Gaziosmanpasa University, 60250 Tokat, Turkey. Mehmet Emir Koksal - Department of Mathematics, Ondokuz Mayis University, 55139 Samsun, Turkey.

Abstract

Employing quasilinearization technique coupled with the method of upper and lower solutions, we construct monotone sequences whose iterates are solutions to corresponding linear problems and show that the sequences converge uniformly and monotonically to the unique solution of the nonlinear problem with causal operator. Especially, instead of assuming convexity or concavity assumption on the nonlinear term that is demanded by the method of quasilinearization, we impose weaker conditions to be more useful in applications. The results obtained include several special cases and extend previous results.

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Keywords

• Quasilinearization
• differential equation
• causal operator
• nonlinearity
• initial value problem
• quadratic convergence.

MSC

•  34A12
•  34A34
•  34A99
•  34AXX
•  34LXX

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