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Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle

Yıl 2019, Cilt: 3 Sayı: 3, 111 - 120, 31.08.2019
https://doi.org/10.31197/atnaa.604962

Öz

Following the idea of T.A. Burton, of progressive contractions, presented in some examples (T.A. Burton, \emph{A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions}, Fixed Point Theory, 20 (2019), No. 1, 107-113) and the forward step method (I.A. Rus, \emph{Abstract models of step method which imply the convergence of successive approximations}, Fixed Point Theory, 9 (2008), No. 1, 293-307), in this paper we give some variants of contraction principle in the case of operators with Volterra property. The basic ingredient in the theory of step by step contraction is $G$-contraction (I.A. Rus, \emph{Cyclic representations and fixed points}, Ann. T. Popoviciu Seminar of Functional Eq. Approxim. Convexity, 3 (2005), 171-178). The relevance of step by step contraction principle is illustrated by applications in the theory of differential and integral equations.

Kaynakça

  • [1] D.D. Bainov, S.G. Hristova, Differential Equations with Maxima, CRC Press, 2011.
  • [2] V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007.
  • [3] O.-M. Bolojan, Fixed Point Methods for Nonlinear Differential Systems with Nonlocal Conditions, Casa Cartii de Ştiinta, Cluj-Napoca, 2013.
  • [4] A. Boucherif, First-order differential inclusions with nonlocal initial conditions, Appl. Math. Letters, 15 (2002), 409-414.
  • [5] A. Boucherif, R. Precup, On the nonlocal initial value problem for first-order differential equations, Fixed Point Theory, 4 (2003), 205-212.
  • [6] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publ., New York, 2008.
  • [7] T.A. Burton, A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions,Fixed Point Theory, 20 (2019), No. 1, 107-113.
  • [8] T.A. Burton, I.K. Purnaras, The shrinking fixed point map, Caputo and integral equations: progressive contraction, J.Fractional Calculus Appl., 9 (2018), No. 1, 188-194.
  • [9] T.A. Burton, I.K. Purnaras, Progressive contractions, product contractions, quadratic integro-differential equations, AIMS Math., 4 (2019), No. 3, 482-496.
  • [10] C. Corduneanu, Abstract Volterra equations: a survey, Mathematical and Computer Modeling, 32 (2000), 1503-1528.
  • [11] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Acad. Press, New York, 1966.
  • [12] V. Kolmanovskii, A. Myshkis, Applied theory of functional-differential equations, Kluwer, 1992.
  • [13] V. Lakshmikantham, L. Wen, B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer, 1994.
  • [14] O. Nica, R.Precup, On the nonlocal initial value problem for first order differential systems, Studia Univ. Babeş-Bolyai Math., 56 (2011), No. 3, 125-137.
  • [15] D. Otrocol, Ulam stabilities of differential equation with abstract Volterra operator in a Banach space, Nonlinear Funct.Anal. Appl., 15 (2010), No. 4, 613-619.
  • [16] D. Otrocol, I.A. Rus, Functional-differential equations with "maxima" via weakly Picard operator theory, Bull. Math. Soc. Sci. Math. Roumanie, 51 (2008), No. 3, 253-261.
  • [17] I.A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, Cluj-Napoca, 2001.
  • [18] I.A. Rus, Picard operators and applications, Sc. Math. Jpn., 58 (2003), no. 1, 191-219.
  • [19] I.A. Rus, Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian J. Math., 20 (2004), No. 1,125-134.
  • [20] I.A. Rus, Cyclic reprezentations and fixed points, Ann. T. Popoviciu Seminar of Functional Eq. Approximation and Convexity, 3 (2005), 171-178.
  • [21] I.A. Rus, Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 9 (2008), No. 1, 293-307.
  • [22] I.A. Rus, Some nonlinear functional differential and integral equations, via weakly Picard operator theory: a survey, Carpathian J. Math., 26 (2010), No. 2, 230-258.
  • [23] I.A. Rus, Some problems in the fixed point theory, Adv. Theory of Nonlinear Analysis Appl., 2 (2018), No. 1, 1-10.
  • [24] I.A. Rus, A. Petruşel, G. Petruşel, Fixed Point Theory, Cluj Univ. Press, Cluj-Napoca, 2008.
  • [25] I.A. Rus, M.A. .erban, Operators on infinite dimensional cartesian product, Analele Univ. de Vest Timi³oara, 48 (2010), 253-263.
  • [26] I.A. Rus, M.A. .erban, Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem, Carpathian J. Math., 29 (2013), No. 2, 239-258.
  • [27] M.A. Şerban, Teoria punctului fix pentru operatori definiti pe produs cartezian, Presa Univ. Clujeana, Cluj-Napoca, 2002.
Yıl 2019, Cilt: 3 Sayı: 3, 111 - 120, 31.08.2019
https://doi.org/10.31197/atnaa.604962

Öz

Kaynakça

  • [1] D.D. Bainov, S.G. Hristova, Differential Equations with Maxima, CRC Press, 2011.
  • [2] V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007.
  • [3] O.-M. Bolojan, Fixed Point Methods for Nonlinear Differential Systems with Nonlocal Conditions, Casa Cartii de Ştiinta, Cluj-Napoca, 2013.
  • [4] A. Boucherif, First-order differential inclusions with nonlocal initial conditions, Appl. Math. Letters, 15 (2002), 409-414.
  • [5] A. Boucherif, R. Precup, On the nonlocal initial value problem for first-order differential equations, Fixed Point Theory, 4 (2003), 205-212.
  • [6] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publ., New York, 2008.
  • [7] T.A. Burton, A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions,Fixed Point Theory, 20 (2019), No. 1, 107-113.
  • [8] T.A. Burton, I.K. Purnaras, The shrinking fixed point map, Caputo and integral equations: progressive contraction, J.Fractional Calculus Appl., 9 (2018), No. 1, 188-194.
  • [9] T.A. Burton, I.K. Purnaras, Progressive contractions, product contractions, quadratic integro-differential equations, AIMS Math., 4 (2019), No. 3, 482-496.
  • [10] C. Corduneanu, Abstract Volterra equations: a survey, Mathematical and Computer Modeling, 32 (2000), 1503-1528.
  • [11] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Acad. Press, New York, 1966.
  • [12] V. Kolmanovskii, A. Myshkis, Applied theory of functional-differential equations, Kluwer, 1992.
  • [13] V. Lakshmikantham, L. Wen, B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer, 1994.
  • [14] O. Nica, R.Precup, On the nonlocal initial value problem for first order differential systems, Studia Univ. Babeş-Bolyai Math., 56 (2011), No. 3, 125-137.
  • [15] D. Otrocol, Ulam stabilities of differential equation with abstract Volterra operator in a Banach space, Nonlinear Funct.Anal. Appl., 15 (2010), No. 4, 613-619.
  • [16] D. Otrocol, I.A. Rus, Functional-differential equations with "maxima" via weakly Picard operator theory, Bull. Math. Soc. Sci. Math. Roumanie, 51 (2008), No. 3, 253-261.
  • [17] I.A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, Cluj-Napoca, 2001.
  • [18] I.A. Rus, Picard operators and applications, Sc. Math. Jpn., 58 (2003), no. 1, 191-219.
  • [19] I.A. Rus, Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian J. Math., 20 (2004), No. 1,125-134.
  • [20] I.A. Rus, Cyclic reprezentations and fixed points, Ann. T. Popoviciu Seminar of Functional Eq. Approximation and Convexity, 3 (2005), 171-178.
  • [21] I.A. Rus, Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 9 (2008), No. 1, 293-307.
  • [22] I.A. Rus, Some nonlinear functional differential and integral equations, via weakly Picard operator theory: a survey, Carpathian J. Math., 26 (2010), No. 2, 230-258.
  • [23] I.A. Rus, Some problems in the fixed point theory, Adv. Theory of Nonlinear Analysis Appl., 2 (2018), No. 1, 1-10.
  • [24] I.A. Rus, A. Petruşel, G. Petruşel, Fixed Point Theory, Cluj Univ. Press, Cluj-Napoca, 2008.
  • [25] I.A. Rus, M.A. .erban, Operators on infinite dimensional cartesian product, Analele Univ. de Vest Timi³oara, 48 (2010), 253-263.
  • [26] I.A. Rus, M.A. .erban, Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem, Carpathian J. Math., 29 (2013), No. 2, 239-258.
  • [27] M.A. Şerban, Teoria punctului fix pentru operatori definiti pe produs cartezian, Presa Univ. Clujeana, Cluj-Napoca, 2002.
Toplam 27 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

İoan A. Rus

Yayımlanma Tarihi 31 Ağustos 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 3 Sayı: 3

Kaynak Göster