Existence results for some neutral functional integrodifferential equations with bounded delay

3 years ago

Abstract

In this paper, we study a class of neutral functional integrodifferential equations with finite delay in Banach spaces. We are interested in the global existence, uniqueness of mild solutions with values in the Banach space and in its subspace $D(A)$ . The results are based on Banach’s and Schauder’s fixed point theorems and on the technique of equivalent norms. As an application, we consider a diffusion neutral functional integrodifferential equation

Authors

Sylvain Koumla
Department of Mathematics, Faculty of Science and Techniques, Adam Barka University, Abéché, Chad

Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Abdou Sene
Department of Mathematics, Virtual University of Dakar, Senegal

Keywords

Mild solution; functional integrodifferential equation; neutral equation; semigroup of bounded linear operators; infinitesimal generator; finite delay

Paper coordinates

S. Koumla, R. Precup, A. Sene, Existence results for some neutral functional integrodifferential equations with bounded delay, Turk. J. Math. 43 (2019), no. 4, 1809-1822, https://doi.org/10.3906/mat-1807-37

PDF

https://journals.tubitak.gov.tr/cgi/viewcontent.cgi?article=1440&context=math

About this paper

Journal

Turkish Journal of Mathematics

Publisher Name

TUBITAK

DOI

https://doi.org/10.3906/mat-1807-37

Print ISSN

Online ISSN

13000098

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[1] Adimy M, Ezzinbi K. A class of linear partial neutral functional differential equations with nondense domain. Journal of Differential Equations 1998; 147 (2): 285-332. doi: 10.1006/jdeq.1998.3446
[2] Adimy M, Ezzinbi K. Strict solutions of nonlinear hyperbolic neutral functional differential equations. Applied Mathematics Letters 1999; 12 (1): 107-112. doi: 10.1016/S0893-9659(98)00134-7
[3] Alia M, Ezzinbi K, Koumla S. Mild solutions for some partial functional integrodifferential equations with statedependent delay. Discussiones Mathematicae. Differential Inclusions, Control and Optimization 2017; 37: 173-186. doi: 10.7151/dmdico.1197
[4] Barbu V. Nonlinear Volterra equations in a Hilbert space. SIAM Journal on Mathematical Analysis 1975; 6 (4): 728-741. doi: 10.1137/0506064
[5] Bolojan OM, Precup R. Hybrid delay evolution systems with nonlinear constraints. Dynamic Systems and Applications 2018; 27 (4): 773-790. doi: 10.12732/dsa.v27i4.6
[6] Bunoiu R, Precup R. Vectorial approach to coupled nonlinear Schrödinger systems under nonlocal Cauchy conditions. Applicable Analysis 2016; 95 (4): 731-747. doi: 10.1080/00036811.2015.1028921
[7] Cardinali T, Precup R, Rubbioni P. A unified existence theory for evolution equations and systems under nonlocal conditions. Journal of Mathematical Analysis and Applications 2015; 432 (2): 1039-1057. doi: 10.1016/j.jmaa.2015.07.019
[8] Coleman BD, Gurtin ME. Equipresence and constitutive equations for rigid heat conductors. Journal of Applied Mathematics and Physics (ZAMP) 1967; 18 (2): 199-208.
[9] Hernandez E, Henriquez HR. Existence results for partial neutral functional differential equations with unbouned delay. Journal of Mathematical Analysis and Applications 1998; 221 (2): 452-475. doi: 10.1006/jmaa.1997.5875
[10] Hernandez E, Henriquez HR. Existence of periodic solutions for partial neutral functional differential equations with unbouned delay. Journal of Mathematical Analysis and Applications 1998; 221 (2): 499-522. doi: 10.1006/jmaa.1997.5899
[11] Ezzinbi K, Ghnimi S. Existence and regularity of solutions for neutral partial functional integrodifferential equations.
Nonlinear Analysis: Real World Applications 2010; 11 (4): 2335-2344. doi: 10.1016/j.nonrwa.2009.07.007
[12] Ezzinbi K, Toure H, Zabsonre I. Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces. Nonlinear Analysis 2009; 70 (7): 2761-2771. doi: 10.1016/j.na.2008.04.001
[13] Ezzinbi K, Koumla S. An abstract partial functional integrodifferential equations. Advances in Fixed Point Theory 2016; 6 (4): 469-485.
[14] Ezzinbi K, Koumla S, Sene A. Existence and regularity for some partial functional integrodifferential equations with infinite delay. Journal of Semigroup Theory and Applications 2016; 2016: Article ID 6.
[15] Goldstein JA. Semigroups of Linear Operators and Applications. New York, USA: Oxford University Press, 1985.
[16] Gurtin ME, Pipkin AC. A general theory of heat conduction with finite wave speeds. Archive for Rational Mechanics and Analysis 1968; 31 (2): 113-126. doi: 10.1007/BF00281373
[17] Hale JK, Verduyn Lunel SM. Introduction to Functional Differential Equations. New York, USA: Springer, 1993.
[18] Hale JK. Partial neutral functional differential equations. Revue Roumaine de Mathématiques Pures et Appliquées 1994; 39: 339-344.
[19] Hale JK. Coupled oscillators on a circle. Resenhas IME-USP 1994; 1 (4): 441-457.
[20] Koumla S, Ezzinbi K, Bahloul R. Mild solutions for some partial functional integrodifferential equations with finite delay in Fréchet spaces. SeMA Journal 2017; 74 (4): 489-501.
[21] Kato T. Perturbation Theory for Linear Operators. Berlin, Germany: Springer, 1966.
[22] Cheng-Lien Lang, Jung-Chan Chang. Local existence for nonlinear Volterra integrodifferential equations with infinite delay. Nonlinear Analysis 2008; 68 (10): 2943-2956. doi: 10.1016/j.na.2007.02.038
[23] Ntouyas SK. Global existence for neutral functional integrodifferential equations. Nonlinear Analysis 1997; 30 (4): 2133-2142. doi: 10.1016/S0362-546X(97)00267-8
[24] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Berlin, Germany: Springer, 1983.
[25] Da Prato G, Iannelli M. Existence and regularity for a class of integrodifferential equations of parabolic type. Journal of Mathematical Analysis and Applications 1985; 112 (1): 36-55. doi: 10.1016/0022-247X(85)90275-6
[26] Precup R. The nonlinear heat equation via fixed point principles. Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity 2006; 4: 111-127.
[27] Precup R. Linear and Semilinear Partial Differential Equations. Berlin, Germany: De Gruyter, 2013.
[28] Vrabie II. C0 -Semigroups and Applications. Amsterdam, The Netherlands: Elsevier, 2003.
[29] Webb GF. An abstract semilinear Volterra integrodifferential equation. Proceedings of the American Mathematical Society 1978; 69 (2): 255-260. doi: 10.2307/2042608
[30] Wu J, Xia H. Rotating waves in neutral partial functional differential equations. Journal of Dynamics and Differential Equations 1999; 11 (2): 209-238. doi: 10.1023/A:1021973228398
[31] Wu J, Xia H. Self-sustained oscillations in a ring array of coupled lossless transmission lines. Journal of Differential Equations 1996; 124 (1): 247-278. doi: 10.1006/jdeq.1996.0009
[32] Wu J. Theory and Applications of Partial Functional Differential Equations. New York, USA: Springer, 1996.