Nguyen Dat THUC, Le Thi Phuong NGOC, Nguyen Thanh LONG

Solvability, stability, smoothness and compactness of the set of solutions for a nonlinear functional integral equation

This paper is devoted to the study of the following nonlinear functional integral equation f(x) = ∑q i=1 αi(x)f(τi(x)) + ∫ σ1(x) 0 Ψ ( x, t, f(σ2(t)), ∫ σ3(t) 0 f(s)ds) dt + g(x), ∀x ∈ [0, 1], (E) where τi, σ1, σ2, σ3 : [0, 1] → [0, 1]; αi, g : [0, 1] → R; Ψ : [0, 1] × [0, 1] × R 2 → R are the given continuous functions and f : [0, 1] → R is an unknown function. First, two sufficient conditions for the existence and some properties of solutions of Eq. (E) are proved. By using Banach’s fixed point theorem, we have the first sufficient condition yielding existence, uniqueness and stability of the solution. By applying Schauder’s fixed point theorem, we have the second sufficient condition for the existence and compactness of the solution set. An example is also given in order to illustrate the results obtained here. Next, in the case of Ψ ∈ C 2 ([0, 1] × [0, 1] × R 2 ; R), we investigate the quadratic convergence for the solution of Eq. (E). Finally, the smoothness of the solution depending on data is established.

PDF

___

[1] Abdou MA, El-Sayed W.G, Deebs E.I. A solution of a nonlinear integral equation. Applied Mathematics and Computation 2005; 160 (1): 1-14. doi: 10.1016/S0096-3003(03)00613-1
[2] Abdou MA, Badr AA, El-Kojok MM. On the solution of a mixed nonlinear integral equation. Applied Mathematics and Computation 2011; 217 (12): 5466-5475. doi: 10.1016/j.amc.2010.12.016
[3] Avramescu C, Vladimirescu C. Asymptotic stability results for certain integral equations. Electronic Journal of Differential Equations 2014; 126: 1-10.
[4] Corduneanu C. Integral Equations and Applications. NY, USA: Cambridge University Press, 1991.
[5] Danh PH, Ngoc LTP, Long NT. On a nonlinear functional integral equation with variable delay. Journal of Abstract Differential Equations and Applications 2014; 5 (1): 35-51.
[6] Deimling K. Nonlinear functional analysis. New York, NY, USA: Springer, 1985.
[7] Dhage BC, Ntouyas SK. Existence results for nonlinear functional integral equations via a fixed point theorem of Krasnoselskii-Schaefer type. Nonlinear Studies 2002; 9 (3): 307-317.
[8] Dhage BC. On some nonlinear alternatives of Leray-Schauder type and functional interal equations. Archivum Mathematicum (BRNO) 2006; 42: 11-23.
[9] Dhakne MB, Lamb GB. On an abstract nonlinear second order integrodifferential equation. Journal of Function Spaces and Applications 2007; 5 (2): 167-174.
[10] Kostrzewski T. Existence and uniqueness of BC[a, b] solutions of nonlinear functional equation. Demonstratio Mathematica 1993; 26: 61-74.
[11] Kostrzewski T. BC-solutions of nonlinear functional equation. A nonuniqueness case. Demonstratio Mathematica 1993; 26: 275-285.
[12] Liu Z, Kang SM, Ume JS. Solvability and asymptotic stability of a nonlinear functional-integral equation. Applied Mathematics Letters 2011; 24 (6): 911-917.
[13] Lupa M. On solutions of a functional equation in a special class of functions. Demonstratio Mathematica 1993; 26: 137-147.
[14] Long NT, Nghia NH, Khoi NK, Ruy DV. On a system of functional equations. Demonstratio Mathematica 1998; 31: 313-324. doi: 10.1515/dema-1998-0209
[15] Long NT, Nghia NH. On a system of functional equations in a multi-dimensional domain. Zeitschrift für Analysis und ihre Anwendungen 2000; 19: 1017-1034. doi: 10.4171/ZAA/995
[16] Long NT. Linear approximation and asymptotic expansion associated with the system of functional equations. Demonstratio Mathematica 2004; 37 (2): 349-362. doi: 10.1515/dema-2004-0212
[17] Ngoc LTP, Long NT. Applying a fied point theorem of Krasnosel’skii type to the existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation. Nonlinear Analysis: Theory, Methods & Applications 2011; 74 (11): 3769-3774. doi: 10.1016/j.na.2011.03.021
[18] Ngoc LTP, Long NT. On a nonlinear Volterra-Hammerstein integral equation in two variables. Acta Mathematica Scientia 2013; 33B (2): 484-494. doi: 10.1016/S0252-9602(13)60013-2
[19] Ngoc LTP, Dung HTH, Danh PH, Long NT. Linear approximation and asymptotic expansion associated with the system of nonlinear functional equations. Demonstratio Mathematica 2014; 47 (1): 103-124. doi: 10.2478/dema2014-0008
[20] Ngoc LTP, Long NT. Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation in N variables. Mathematische Nachrichten 2015; 288 (5-6): 633-647. doi: 10.1002/mana.201300065
[21] Ngoc LTP, Long NT. A continuum of solutions in a Fréchet space of a nonlinear functional integral equation in N variables. Mathematische Nachrichten 2016; 289 (13): 1665-1679. doi: 10.1002/mana.201500008
[22] Purnaras IK. A note on the existence of solutions to some nonlinear functional integral equations. Electronic Journal of Qualitative Theory of Differential Equations 2006; 2006 (17): 1-24. doi: 10.14232/ejqtde.2006.1.17
[23] Zeidler E, Nonlinear Functional Analysis and its Applications. Part I. Berlin, Germany: Springer-Verlag, 1985.