Abstract
Let \(a<c<b)\) real numbers, \((\mathbb{B},|\cdot|)\) a (real or complex) Banach space, \(H\in C([a,b]\times [a,c]\times\mathbb{B},\mathbb{B})\), \(K\in C([a,b]^{2}\times\mathbb{B},\mathbb{B})\), \(g\in C([a,b]),\mathbb{B}
,A:C([a,c],\mathbb{B})\rightarrow C([a,c],\mathbb{B})\) and \(B:C([a,b],\mathbb{B})\rightarrow C([a,b],\mathbb{B})\).
In this paper we study the following functional integral equation,
\[
x (t) =\int_a^c H( t,s,A) ( x) ( s)) ds \int_a^t K (t,s,B) (x) (s))ds g(t), \quad t\in [a,b]
\]
By a new variant of fibre contraction principle (A. Petrusel, I.A. Rus, M.A. Serban, Some variants of fibre contraction principle and applications: from existence to the convergence of successive approximations, Fixed Point Theory, 22 (2021), no. 2, 795-808) we give existence, uniqueness and convergence of successive approximations results for this equation.
In the case of ordered Banach space \(\mathbb{B}\), Gronwall-type and comparison-type results are also given.
Authors
Veronica Ilea
Babes-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania
Diana Otrocol
Technical University of Cluj-Napoca, Cluj-Napoca, Romania,
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Ioan A. Rus
Babes-Bolyai University, Faculty of Mathematics and Computer Science,Cluj-Napoca, Romania
Marcel-Adrian Serban
Babes-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania
Keywords
Functional integral equation, Volterra operator, Picard operator, fibre contraction principle, Gronwall lemma, comparison lemma.
Paper coordinates
About this paper
Journal
Fixed Point Theory
Publisher Name
Casa Cărţii de Ştiinţă Cluj-Napoca
(House of the Book of Science Cluj-Napoca)
Print ISSN
1583-5022
Online ISSN
2066-9208
google scholar link
[1] S. Andras, Fibre ϕ−contraction on generalized metric spaces and applications, Mathematica, 45(68)(2003), no. 1, 3-8.
[2] C. Avramescu, C. Vladimirescu, Fixed points for some non-obviously contractive operators defined in a space of continuous functions, Electronic J. Qualit. Th. Diff, Eq., 2004, no. 3, 1-7.
[3] C. Bacotiu, Fibre Picard operators on generalized metric spaces, Sem. Fixed Point Theory Cluj-Napoca, 1(2000), 5-8.
[4] D.D. Bainov, S. Hristova, Differential Equations with Maxima, Chapman & Hall/CRC Pure and Applied Mathematics, 2011.
[5] O.-M. Bolojan, Fixed Point Methods for Nonlinear Differential Systems with Nonlocal Conditions, Casa C˘art¸ii de S¸tiint¸˘a, Cluj-Napoca, 2013.
[6] A. Boucherif, R. Precup, On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4(2003), 205-212.
[7] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publ. New York, 2006.
[8] 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, 107-112.
[9] C. Corduneanu, Abstract Volterra equations: a survey, Math. and Computer Model, 32(11-13)(2000), 1503-1528.
[10] E. Egri, On First and Second Order Iterative Functional Differential Equations and Systems, Presa Univ. Clujeana, Cluj-Napoca, 2008.
[11] W.M. Hirsch, C.C. Pugh, Stable manifolds and hyperbolic sets, Proc. Symp. in Pure Math., 14(1970), 133-163.
[12] V. Ilea, D. Otrocol, On the Burton method of progressive contractions for Volterra integral equations, Fixed Point Theory, 21(2020), no. 2, 585-594.
[13] V. Ilea, D. Otrocol, Functional differential equations with maxima, via step by step contraction principle, Carpathian J. Math., 37(2021), no. 2, 195-202.
[14] V. Muresan, Functional Integral Equations, Mediamira, Cluj-Napoca, 2003.
[15] D. Otrocol, I.A. Rus, Functional-differential equations with maxima of mixed type argument, Fixed Point Theory, 9(2008), no. 1, 207-220.
[16] E. De Pascale, L. De Pascale, Fixed points for some norm-obviously contractive operators, Proc. Amer. Math. Soc., 130(2002), no. 11, 3249-3254.
[17] E. De Pascale, P.P. Zabreiko, Fixed point theorems for operators in spaces of continuous functions, Fixed Point Theory, 5(2004), no. 1, 117-129.
[18] A. Petrusel, I.A. Rus, M.A. S¸erban, Fixed points for operators on generalized metric spaces, CUBO – A Mathematical Journal, 10(2008), no. 4, 45-66.
[19] A. Petrusel, I.A. Rus, M.A. S¸erban, Some variants of fibre contraction principle and applications: from existence to the convergence of successive approximations, Fixed Point Theory, 22(2021), no. 2, 795-808.
[20] I.A. Rus, A fibre generalized contraction theorem and applications, Mathematica, 41(1999), no.1, 85-90.
[21] I.A. Rus, Picard operators and applications, Scientiae Mathematicae Japon., 58(2003), no. 1, 191-219.
[22] I.A. Rus, Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 9(2008), no. 1, 293-307.
[23] I.A. Rus, Fibre Picard operators and applications, Stud. Univ. Babe¸s-Bolyai Math., 44(1999), 89-98.
[24] I.A. Rus, Fibre Picard operators on generalized metric spaces and applications, Scripta Sc. Math., 1(1999), 326-334.
[25] I.A. Rus, Generalized Contractions and Applications, Cluj University Press Cluj-Napoca, 2001.
[26] 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.
[27] I.A. Rus, Some variants of contraction principle, generalizations and applications, Stud. Univ. Babes-Bolyai Math., 61(2016), no. 3, 343-358.
[28] I.A. Rus, Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Advances in the Theory of Nonlinear Analysis and its Applications, 3(2019) no. 3, 111-120.
[29] I.A. Rus, Weakly Picard operators and applications, Seminar on Fixed Point Theory, ClujNapoca, 2(2001), 41-58.
[30] I.A. Rus, A. Petrusel, G. Petrusel, Fixed Point Theory, Cluj Univ. Press, Cluj-Napoca, 2008.
[31] I.A. Rus, M.A. Serban, Operators on infinite dimensional Cartesian product, Analele Univ. Vest, Timisoara, Math.-Inf., 48(2010), 253-263.
[32] I.A. Rus, M.A. Serban, Some generalizations of a Cauchy lemma and applications, Topics in Mathematics, Computer Science and Philosophy, Ed. S¸t. Cobza¸s, Cluj University Press, 2008, 173-181.
[33] M.A. Serban, Fibre ϕ−contractions, Stud. Univ. Babes-Bolyai, Math., 44(1999), no. 3, 99-108.
[34] M.A. Serban, Fixed Point Theory for Operators on Cartesian Product, (in Romanian), Cluj University Press, Cluj-Napoca, 2002.
[35] M.A. Serban, I.A. Rus, A, Petrusel, A class of abstract Volterra equations, via weakly Picard operators technique, Math. Ineq. Appl., 13(2010), no. 2, 255-269.
[36] E.S. Zhukovskii, M.J. Alves, Abstract Volterra operators, Russian Mathematics (Iz. VUZ), 52(2008), no. 3, 1-14.