Research article Special Issues

Progressive contractions, product contractions, quadratic integro-differential equations

  • Received: 11 March 2019 Accepted: 19 April 2019 Published: 13 May 2019
  • MSC : 34A08, 34A12, 45D05, 45G05, 47H09, 47H10

  • Fixed point theory has been used very successfully to obtain properties of solutions of integral and integro-differential equations of the form $ x(t) = g(t) +\int^t_0 A(t-s) v(t, s, x(s))ds $ because under general conditions the integral term may map bounded sets of continuous functions into equi-continuous sets. But quadratic integral equations have a coefficient of the integral terms of the form $ f(t, x(t))\int^t_0 A(t-s)v(t, s, x(s))ds $ which destroys the compactness of the map. Investigators have resorted to deep solutions often involving measures of non-compactness and Darbo's fixed point theorem. In an effort to obtain some elementary approaches, in this paper we develop an apparently new technique by showing that by using progressive contractions we can show conditions under which the product of two contractions is a contraction. We focus on integro-differential equations and use direct fixed point mappings which convert Lipschitz conditions into progressive contraction conditions.

    Citation: Theodore A. Burton, Ioannis K. Purnaras. Progressive contractions, product contractions, quadratic integro-differential equations[J]. AIMS Mathematics, 2019, 4(3): 482-496. doi: 10.3934/math.2019.3.482

    Related Papers:

  • Fixed point theory has been used very successfully to obtain properties of solutions of integral and integro-differential equations of the form $ x(t) = g(t) +\int^t_0 A(t-s) v(t, s, x(s))ds $ because under general conditions the integral term may map bounded sets of continuous functions into equi-continuous sets. But quadratic integral equations have a coefficient of the integral terms of the form $ f(t, x(t))\int^t_0 A(t-s)v(t, s, x(s))ds $ which destroys the compactness of the map. Investigators have resorted to deep solutions often involving measures of non-compactness and Darbo's fixed point theorem. In an effort to obtain some elementary approaches, in this paper we develop an apparently new technique by showing that by using progressive contractions we can show conditions under which the product of two contractions is a contraction. We focus on integro-differential equations and use direct fixed point mappings which convert Lipschitz conditions into progressive contraction conditions.


    加载中


    [1] M. A. Darwish and J. Henderson, Existence and asymptotic stability of solutions of a perturbed quadratic fractional integral equation, Fract. Calc. Appl. Anal., 12 (2009), 71-86.
    [2] M. A. Darwish, On a quadratic fractional integral equation with linear modification of the argument, Can. Appl. Math. Q., 16 (2008), 45-58.
    [3] T. A. Burton, Existence and uniqueness results by progressive contractions for integro-differential equations, Nonlinear Dyn. Syst. Theory, 16 (2016), 366-371.
    [4] V. Volterra, Theory of Functionals and of Integral and Integro-Differential Equations, New York: Dover Publications, 1959.
    [5] R. K. Miller, Nonlinear Volterra Integral Equations, Menlo Park, California: W. A. Benjamin, 1971.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3416) PDF downloads(766) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog