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Monotone‐Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”

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  • S. Hristova
  • A. Golev

Abstract

The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone‐iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

Suggested Citation

  • S. Hristova & A. Golev, 2012. "Monotone‐Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:493271
    DOI: 10.1155/2012/493271
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    References listed on IDEAS

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    1. Leonid Berezansky & Elena Braverman, 2011. "New Stability Conditions for Linear Differential Equations with Several Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    2. J. Diblík & M. Kúdelčíková, 2011. "Existence and Asymptotic Behavior of Positive Solutions of Functional Differential Equations of Delayed Type," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-16, December.
    3. Jaromír Baštinec & Leonid Berezansky & Josef Diblík & Zdeněk Šmarda, 2010. "On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    4. Leonid Berezansky & Elena Braverman, 2011. "New Stability Conditions for Linear Differential Equations with Several Delays," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-19, June.
    5. Jaromír Baštinec & Leonid Berezansky & Josef Diblík & Zdeněk Šmarda, 2010. "On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-20, September.
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    Cited by:

    1. Angel Golev & Snezhana Hristova & Svetoslav Nenov, 2013. "Monotone‐Iterative Method for Solving Antiperiodic Nonlinear Boundary Value Problems for Generalized Delay Difference Equations with Maxima," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).

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