Study on Integrodifferential Evolution Systems with Nonlocal Initial Conditions


The work is concerned with systems of abstract integrodifferential equations with general nonlocal initial conditions. To allow the nonlinear terms of the equations to behave as independently as possible, we employ a vector approach based on matrices, vector-valued norms, and a vector version of Krasnoselskii’s fixed point theorem for a sum of two operators. The assumptions take into account the system’s hybridity and the support for nonlocal initial conditions. To demonstrate the principle, two examples are given.


Sylvain Koumla

Radu Precup
Babes-Bolyai University,
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy


Integrodifferential equations; nonlinear evolution equation; nonlocal initial condition; delay; krasnoselskii’s fixed point theorem for a sum of operators



Cite this paper as:

S. Koumla and R. Precup, Study on Integrodifferential Evolution Systems with Nonlocal Initial Conditions, Recent Advances in Mathematical Research and Computer Science, vol. 5, 2021, pp. 13-27,

About this paper

Print ISSN

Not available yet.

Online ISSN

Not available yet.

Google Scholar Profile

[1] Kolmanovskii V, Myshkis A., Applied Theory of Functional Differential Equations, Kluwer, DorDrecht; 1992,
[2] Cioranescu N., Sur les conditions lineaires dans l’integration des equations differentielles ordinaires, Math. Z. 1932;35:601–608.
[3] Whyburn WM., Differential equations with general boundary conditions, Bull. Amer. Math. Soc. 1942;48:692–704,
[4] Conti R., Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital. 1967;22:135–178,
[5] Bolojan-Nica O, Infante G, Precup R., Existence results for systems with coupled nonlocal initial conditions, Nonlinear Anal. 2014;94:231–242,
[6] Bolojan O, Infante G, Precup R., Existence results for systems with coupled nonlocal nonlinear initial conditions, Math. Bohem. 2015;140:371–384,
[7] Boucherif A, Precup R., On the nonlocal initial value problem for first order differential equations, Fixed Point Theory. 2003;4:205–212,
[8] Garcıa-Falset J, Reich S., Integral solutions to a class of nonlocal evolution equations, Commun. Contemp. Math. 2010;12:1032–1054
[9] Karakostas GL, Tsamatos PCh., Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal. 2002;19:109–121.
[10] Nica O., Initial-value problems for first-order differential systems with general nonlocal conditions, Electron. J. Differential Equations. 2012;2012(74):1–15,
[11] Nica O, Precup R., On the nonlocal initial value problem for first order differential systems, Stud. Univ. Babes-Bolyai Math. 2011;56(3):125–137,,URL%3A,,-INITIAL%2DVALUE
[12] Webb JRL, Infante G., Positive solutions of nonlocal initial boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl. 2008;15:45–67. [13] Stikonas A., A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control. 2014;19:301–334,
[14] Kerefov A.A., Nonlocal boundary value problems for parabolic equations (Russian), Differ. Uravn. 1979;15:74–78.
[15] Vabishchevich PN., Non-local parabolic problems and the inverse heat-conduction problem (Russian), Differ. Uravn. 1981;17:761–765.
[16] Chabrowski J., On nonlocal problems for parabolic equations, Nagoya Math. J. 1984;93:109–131.
[17] Pao CV., Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl. 1995;195:702–718,
[18] Olmstead WE, Roberts CA., The one-dimensional heat equation with a nonlocal initial condition, Appl. Math. Lett. 1997;10:89–94.
[19] McKibben M., Discovering Evolution Equations with Applications, Chapman &Hall/CRC. 2011;I.
[20] Byszewski L., Theorems about the existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problems, J. Math. Anal. Appl. 1991;162:494–505,
[21] Jackson D., Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, J. Math. Anal. Appl. 1993;172;256–265,
[22] Lin Y, Liu JH., Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal. 1996;26:1023–1033.
[23] Bolojan O, Precup R., Semilinear evolution systems with nonlinear constraints, Fixed Point Theory. 2016;17:275–288.
[24] Boucherif A, Akca H., Nonlocal Cauchy problems for semilinear evolution equations, Dynam. Systems Appl. 2002;11:415–420.
[25] Boucherif A, Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl. 2007;16:507–516.
[26] Burlica M, Rosu D, Vrabie II.,  Abstract reaction-diffusion systems with nonlocal initial conditions, Nonlinear Anal. 2014;94:107–119,
[27] Cardinali T, Precup R, Rubbioni P., A unified existence theory for evolution equations and systems under nonlocal conditions, J. Math. Anal. Appl. 2015;432:1039–1057,
[28] Infante G, Maciejewski M., Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditions, J. London Math. Soc. 2016;94:859–882,
[29] Liu JH., A remark on the mild solutions of non-local evolution equations, Semigroup Forum. 2003;66:63– 67.
[30] Necula M, Vrabie II., Nonlinear delay evolution inclusions with general nonlocal initial conditions, Ann. Acad. Rom. Sci. Ser. Math. 2015;7:67–97.
[31] Ntouyas SK, Tsamatos PCh., Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl. 1997;210:679–687.
[32] Paicu A, Vrabie II., A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal. 2010;72:4091–4100.
[33] Vrabie II., Global solutions for nonlinear delay evolution inclusions with nonlocal initial conditions, SetValued Var. Anal. 2012;20:477–497.
[34] Kumar K, Karnatak M, Kumar R., Nonlinear Integrodifferential Equations with Time Varying Delay. Advances in the Theory of Nonlinear Analysis and its Application.2021;5(3):433-444,
[35] Yang H, Zhao Y., Existence and optimal controls of non-autonomous impulsive integro-differential evolution equation with nonlocal conditions. Chaos, Solitons & Fractals. 2021 Jul 1;148:111027,
[36] Munusamy K, Ravichandran C, Nisar KS, Ghanbari B., Existence of solutions for some functional integrodifferential equations with nonlocal conditions. Mathematical Methods in the Applied Sciences. 2020 Nov 30;43(17):10319-31,
[37] Burlica MD, Necula M, Rosu D, Vrabie II., Delay Differential Evolutions Subjected to Nonlocal Initial Conditions, Chapman and Hall/CRC Press; 2016,
[38] Bolojan O, Precup R., Hybrid delay evolution systems with nonlinear constraints, Dynam. Systems Appl., to appear.
[39] Webb GF., An abstract semilinear Volterra integrodifferential equation, Proc. Amer. Math. Soc. 1978;69:255–260.
[40] Viorel A., Contributions to the Study of Nonlinear Evolution Equations, Ph.D. Thesis, Cluj-Napoca; 2011.
[41] Precup R., The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comp. Modelling. 2009;49:703–708,
[42] Baras P, Hassan JC, Veron L., Compacite de l’operateur definissant la solution d’une equation d’evolution non homogene, C. R. Acad. Sci. Paris. 1977;284:779–802.
[43] Cazenave T, Haraux A., An Introduction to Semilinear Evolution Equations, Oxford University Press, New York; 1998.
[44] Goldstein JA., Semigroups of Linear Operators and Applications, Oxford University Press, New York; 1985.
[45] Vrabie, C0-Semigroups and Applications, Elsevier, Amsterdam; 2003.

Related Posts