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

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.

Anahtar Kelimeler

• Space of continuous function,operator with Volterra property,max-norm,Bielecki norm,contraction,G-contraction,fiber contraction,progressive contraction,step by step contraction,Picard operator,weakly Picard operator,, differential equation,integral equation,conjecture,fixed point

Kaynakça

1. [1] D.D. Bainov, S.G. Hristova, Differential Equations with Maxima, CRC Press, 2011.
2. [2] V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007.
3. [3] O.-M. Bolojan, Fixed Point Methods for Nonlinear Differential Systems with Nonlocal Conditions, Casa Cartii de Ştiinta, Cluj-Napoca, 2013.
4. [4] A. Boucherif, First-order differential inclusions with nonlocal initial conditions, Appl. Math. Letters, 15 (2002), 409-414.
5. [5] A. Boucherif, R. Precup, On the nonlocal initial value problem for first-order differential equations, Fixed Point Theory, 4 (2003), 205-212.
6. [6] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publ., New York, 2008.
7. [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. [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. [9] T.A. Burton, I.K. Purnaras, Progressive contractions, product contractions, quadratic integro-differential equations, AIMS Math., 4 (2019), No. 3, 482-496.
10. [10] C. Corduneanu, Abstract Volterra equations: a survey, Mathematical and Computer Modeling, 32 (2000), 1503-1528.
11. [11] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Acad. Press, New York, 1966.
12. [12] V. Kolmanovskii, A. Myshkis, Applied theory of functional-differential equations, Kluwer, 1992.
13. [13] V. Lakshmikantham, L. Wen, B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer, 1994.
14. [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. [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. [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. [17] I.A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, Cluj-Napoca, 2001.
18. [18] I.A. Rus, Picard operators and applications, Sc. Math. Jpn., 58 (2003), no. 1, 191-219.
19. [19] I.A. Rus, Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian J. Math., 20 (2004), No. 1,125-134.
20. [20] I.A. Rus, Cyclic reprezentations and fixed points, Ann. T. Popoviciu Seminar of Functional Eq. Approximation and Convexity, 3 (2005), 171-178.
21. [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. [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. [23] I.A. Rus, Some problems in the fixed point theory, Adv. Theory of Nonlinear Analysis Appl., 2 (2018), No. 1, 1-10.
24. [24] I.A. Rus, A. Petruşel, G. Petruşel, Fixed Point Theory, Cluj Univ. Press, Cluj-Napoca, 2008.
25. [25] I.A. Rus, M.A. .erban, Operators on infinite dimensional cartesian product, Analele Univ. de Vest Timi³oara, 48 (2010), 253-263.
26. [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. [27] M.A. Şerban, Teoria punctului fix pentru operatori definiti pe produs cartezian, Presa Univ. Clujeana, Cluj-Napoca, 2002.