## Abstract

The purpose of the present paper is to study the existence of solutions to semilinear evolution systems with nonlinear constraints. We establish new existence results using the ﬁxed point principles of Perov and Schauder, combined with the technique that uses matrices with the spectral radius less than one and vector-valued norms. This vectorial approach is fruitful for the treating of systems in general and allows the system nonlinearities to behave independently as much as possible. Moreover, the constants from the Lipschitz or growth conditions are put into connection with the support of the nonlinear operators expressing the constraints. The paper extends and complements previous results from the literature

## Authors

**Octavia Bolojan
**Department of Mathematics, Babes-Bolyai University Cluj-Napoca, Romania

**Radu Precup**

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

## Keywords

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## Paper coordinates

O. Bolojan, R. Precup, *Semilinear evolution systems with nonlinear constraints*, Fixed Point Theory 17 (2016) no. 2, 275-288.

## About this paper

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Fixed Point Theory

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[1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994.

[2] O. Bolojan-Nica, G. Infante, P. Pietramala, Existence results for impulsive systems with initial nonlocal conditions, Math. Model. Anal., 18(2013), 599–611.

[3] O. Bolojan-Nica, G. Infante, R. Precup, Existence results for systems with coupled nonlocal initial conditions, Nonlinear Anal., 94(2014), 231–242.

[4] O. Bolojan, R. Precup, Implicit ﬁrst order diﬀerential systems with nonlocal conditions, Electron. J. Qual. Theory Diﬀer. Eq., 69(2014), 1–13.

[5] O. Bolojan-Nica, G. Infante, R. Precup, Existence results for systems with coupled nonlocal nonlinear initial conditions, Math. Bohemica, 140(2015), 371-384

[5] O. Bolojan-Nica, G. Infante, R. Precup, Existence results for systems with coupled nonlocal nonlinear initial conditions, Math. Bohemica, 140(2015), 371-384

[6] A. Boucherif, Diﬀerential equations with nonlocal boundary conditions, Nonlinear Anal., 47(2001), 2419–2430.

[7] A. Boucherif, H. Akca, Nonlocal Cauchy problems for semilinear evolution equations, Dynam. Systems Appl., 11(2002), 415–420.

[8] A. Boucherif, R. Precup, On the nonlocal initial value problem for ﬁrst order diﬀerential equations, Fixed Point Theory, 4(2003), 205–212.

[9] A. Boucherif, R. Precup, Semilinear evolution equations with nonlocal initial conditions, Dynamic Systems Appl., 16(2007), 507–516.

[10] T. Cazenave, A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998.

[11] D. Franco, D. O’Regan, J. Peran, Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math., 174(2005), 315–327.

[12] J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.

[13] G. Infante, Positive solutions of diﬀerential equations with nonlinear boundary conditions, Discrete Contin. Dyn. Syst. (Suppl.), (2003), 432–438.

[14] G. Infante, Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Anal., 12(2008), 279–288.

[15] G.L. Karakostas, P.Ch. Tsamatos, Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal., 19(2002), 109–121.

[16] D. Jackson, Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, J. Math. Anal. Appl., 172(1993), 256–265.

[17] J.H. Liu, A remark on the mild solutions of non-local evolution equations, Semigroup Forum, 66(2003), 63–67.

[18] O. Nica, R. Precup, On the nonlocal initial value problem for ﬁrst order diﬀerential systems, Stud. Univ. Babes-Bolyai Math., 56 (2011), no. 3, 125–137.

[19] O. Nica, Initial-value problems for ﬁrst-order diﬀerential systems with general nonlocal conditions, Electron. J. Diﬀerential Equations, 2012(2012), no. 74, 1–15.

[20] O. Nica, Nonlocal initial value problems for ﬁrst order diﬀerential systems, Fixed Point Theory, 13(2012), 603–612.

[21] M. Necula, I.I. Vrabie, Nonlinear delay evolution inclusions with general nonlocal initial conditions, Ann. Acad. Rom. Sci. Ser. Math., 7(2015), 67–97.

[22] S.K. Ntouyas, P.Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210(1997), 679–687.

[23] A. Paicu, I.I. Vrabie, A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal., 72(2010), 4091–4100.

[24] P. Pietramala, A note on a beam equation with nonlinear boundary conditions, Bound. Value Probl. (2011), Art. ID 376782, 14 pp.

[25] R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht-Boston-London, 2002.

[26] R. Precup, A. Viorel, Existence results for systems of nonlinear evolution equations, Int. J. Pure Appl. Math., 47(2)(2008), 199–206.

[27] R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comp. Modelling, 49(2009), 703–708.

[28] R. Precup, D. Trif, Multiple positive solutions of non-local initial value problems for ﬁrst order diﬀerential systems, Nonlinear Anal., 75(2012), 5961–5970.

[29] F. Robert, Matrices non-negatives et normes vectorielles, Universite Scientiﬁque et Medicale, Lyon, 1973.

[30] R.S. Varga, Matrix Iterative Analysis, Springer, Berlin, 2000.

[31] I.I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc., 109(1990), 653–661.

[32] I.I. Vrabie, C0-Semigroups and Applications, Elsevier, Amsterdam, 2003.

[33] I.I. Vrabie, Global solutions for nonlinear delay evolution inclusions with nonlocal initial con ditions, Set-Valued Var. Anal., 20(2012), 477–497.

[34] I.I. Vrabie, Semilinear delay evolution equations with nonlocal initial conditions, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, (A. Favini, G.Fragnelli and R. Mininni – Eds.), Springer INdAM Series, 10(2014), 419–435.

[35] J.R.L. Webb, G. Infante, Positive solutions of nonlocal initial boundary value problems involving integral conditions, NoDEA Nonlinear Diﬀ. Eq. Appl., 15(2008), 45–67.

[36] J.R.L. Webb, G. Infante, Semi-positone nonlocal boundary value problems of arbitrary order, Commun. Pure Appl. Anal., 9(2010), 563–581.

[37] X. Xue, Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces, Electron. J. Diﬀ. Eq., 2005(2005), no. 64, 1–7.

[38] X.M. Xue, Existence of semilinear diﬀerential equations with nonlocal initial conditions, Acta Math. Sin. (Engl. Ser.), 23(6)(2007), 983–988