Integrodifferential evolution systems with nonlocal initial conditions

3 years ago

Abstract

The paper deals with systems of abstract integrodifferential equations subject to general nonlocal initial conditions. In order to allow the nonlinear terms of the equations to behave independently as much as possible, we use 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 support of the nonlocal initial conditions and the hybrid character of the system. Two examples are given to illustrate the theory.

Authors

Sylvain Koumla
Universite Adam Barka, Abeche, Chad

Radu Precup
Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

integrodifferential equation; nonlinear evolution equation; nonlocal initial condition; delay, Krasnoselskii’s fixed point theorem for a sum of operators.

Paper coordinates

S. Koumla, R. Precup, Integrodifferential evolution systems with nonlocal initial conditions, Studia Univ. Babes-Bolyai Math.65 (2020), 93-108, http://dx.doi.org/10.24193/subbmath.2020.1.08

PDF

http://dx.doi.org/10.24193/subbmath.2020.1.08

About this paper

Journal

Studia Universitatis Babeş-Bolyai Mathematica

Publisher Name

DOI

http://dx.doi.org/10.24193/subbmath.2020.1.08

Print ISSN

0252-1938

Online ISSN

2065-961X
ISSN-L 0252-1938

google scholar link

[1] Baras, P., Hassan, J.C., Veron, L., Compacite de l’operateur definissant la solution d’une equation d’´evolution non homogene, C.R. Acad. Sci. Paris 284(1977), 779-802.
[2] Bolojan-Nica, O., Infante, G., Precup, R., Existence results for systems with coupled nonlocal initial conditions, Nonlinear Anal., 94(2014), 231-242.
[3] Bolojan, O., Infante, G., Precup, R., Existence results for systems with coupled nonlocal nonlinear initial conditions, Math. Bohem., 140(2015), 371-384.
[4] Bolojan, O., Precup, R., Semilinear evolution systems with nonlinear constraints, Fixed Point Theory, 17(2016), 275-288.
[5] Bolojan, O., Precup, R., Hybrid delay evolution systems with nonlinear constraints, Dynam. Systems Appl., 27(2018), 773-790.
[6] Boucherif, A., Akca, H., Nonlocal Cauchy problems for semilinear evolution equations, Dynam. Systems Appl., 11(2002), 415-420.
[7] Boucherif, A., Precup, R., On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4(2003), 205-212.
[8] Boucherif, A., Precup, R., Semilinear evolution equations with nonlocal initial conditions,
Dynam. Systems Appl., 16(2007), 507-516.
[9] Burlica, M.D., Necula, M., Ro¸su, D., Vrabie, I.I., Delay Differential Evolutions Subjected to Nonlocal Initial Conditions, Chapman and Hall/CRC Press, 2016.
[10] Burlica, M., Ro¸su, D., Vrabie, I.I., Abstract reaction-diffusion systems with nonlocal initial conditions, Nonlinear Anal., 94(2014), 107-119.
[11] Byszewski, L., Theorems about the existence and uniqueness of solutions of semilinear
evolution nonlocal Cauchy problems, J. Math. Anal. Appl., 162(1991), 494-505.
[12] Cardinali, T., Precup, R., Rubbioni, P., A unified existence theory for evolution equations and systems under nonlocal conditions, J. Math. Anal. Appl., 432(2015), 1039-1057.
[13] Cazenave, T., Haraux, A., An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998.
[14] Chabrowski, J., On nonlocal problems for parabolic equations, Nagoya Math. J., 93(1984), 109-131.
[15] Cioranescu, N., Sur les conditions lin´eaires dans l’int´egration des equations differentielles ordinaires, Math. Z., 35(1932), 601-608.
[16] Conti, R., Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital., 22(1967), 135-178.
[17] Garcıa-Falset, J., Reich, S., Integral solutions to a class of nonlocal evolution equations, Commun. Contemp. Math., 12(2010), 1032-1054.
[18] Goldstein, J.A., Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.
[19] Infante, G., Maciejewski, M., Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditions, J. London Math. Soc., 94(2016), 859-882.
[20] Jackson, D., Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, J. Math. Anal. Appl., 172(1993), 256-265.
[21] Karakostas, G.L., Tsamatos, P.Ch., Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal., 19(2002), 109-121.
[22] Kerefov, A.A., Nonlocal boundary value problems for parabolic equations, (Russian), Differ. Uravn., 15(1979), 74-78.
[23] Kolmanovskii, V., Myshkis, A., Applied Theory of Functional Differential Equations, Kluwer, Dordrecht, 1992.
[24] Lin, Y., Liu, J.H., Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal., 26(1996), 1023-1033.
[25] Liu, J.H., A remark on the mild solutions of non-local evolution equations, Semigroup Forum, 66(2003), 63-67.
[26] McKibben, M., Discovering Evolution Equations with Applications, Vol. I, Chapman & Hall/CRC, 2011.
[27] Necula, M., Vrabie, I.I., Nonlinear delay evolution inclusions with general nonlocal initial conditions, Ann. Acad. Rom. Sci. Ser. Math., 7(2015), 67-97.
[28] Nica, O., Initial-value problems for first-order differential systems with general nonlocal, Electron. J. Differential Equations, 2012(2012), no. 74, 1-15.
[29] Nica, O., Precup, R., On the nonlocal initial value problem for first order differential systems, Stud. Univ. Babe¸s-Bolyai Math., 56(2011), no. 3, 125-137.
[30] Ntouyas, S.K., Tsamatos, P.Ch., Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210(1997), 679-687.
[31] Olmstead, W.E., Roberts, C.A., The one-dimensional heat equation with a nonlocal initial condition, Appl. Math. Lett., 10(1997), 89-94.
[32] Paicu, A., Vrabie, I.I., A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal., 72(2010), 4091-4100.
[33] Pao, C.V., Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl., 195(1995), 702-718.
[34] Precup, R., The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comp. Modelling, 49(2009), 703-708.
[35] Stikonas, A., A survey on stationary problems, Green’s functions and spectrum of SturmLiouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control, 19(2014), 301-334.
[36] Vabishchevich, P.N., Non-local parabolic problems and the inverse heat-conduction problem, (Russian), Differ. Uravn., 17(1981), 761-765.
[37] Viorel, A., Contributions to the Study of Nonlinear Evolution Equations, Ph.D. Thesis, Cluj-Napoca, 2011.
[38] Vrabie, I.I., C0-Semigroups and Applications, Elsevier, Amsterdam, 2003.
[39] Vrabie, I.I., Global solutions for nonlinear delay evolution inclusions with nonlocal initial conditions, Set-Valued Var. Anal., 20(2012), 477-497.
[40] Webb, G.F., An abstract semilinear Volterra integrodifferential equation, Proc. Amer. Math. Soc., 69(1978), 255-260.
[41] Webb, J.R.L., Infante, G., Positive solutions of nonlocal initial boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl., 15(2008), 45-67.
[42] Whyburn, W.M., Differential equations with general boundary conditions, Bull. Amer. Math. Soc., 48(1942), 692-704.