Motivated by the importance of reaction-diffusion systems in modeling real processes with memory, we are interested in the existence of mild solutions for systems of abstract delay evolution equations subjected to general nonlinear constraints. Wishing to allow the system nonlinearities 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 hybrid character of the systems comes from the different nature of the metrical and topological conditions imposed to the component equations. Also, the assumptions are put in connection with the support of the nonlinear constraints. Two examples are given to illustrate the theory.
Department of Mathematics, Babes–Bolyai University, Cluj-Napoca, Romania
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
nonlinear evolution equation; nonlocal initial condition; delay; Krasnoselskii’s fixed point theorem for a sum of operators
O.-M. Bolojan, R. Precup, Hybrid delay evolution systems with nonlinear constraints, Dynamic Systems and Applications, 27 (2018) no. 4, 773-790, http://doi.org/10.12732/dsa.v27i4.6
Dynamic Systems and Applications
Dynamic Publishers, Inc., Acad. Publishers, Ltd.
google scholar link
 A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994.
 O. Bolojan-Nica, G. Infante, P. Pietramala, Existence results for impulsive systems with initial nonlocal conditions, Math. Model. Anal., 18 (2013), 599–611.
 O. Bolojan, R. Precup, Implicit first order differential systems with nonlocal conditions, Electron. J. Qual. Theory Differ. Equ. 69 (2014), 1–13.
 O. Bolojan, R. Precup, Semilinear evolution systems with nonlinear constraints, Fixed Point Theory 17 (2016), 275–288.
 O. Bolojan-Nica, G. Infante, R. Precup, Existence results for systems with coupled nonlocal initial conditions, Nonlinear Anal. 94 (2014), 231–242.
 O. Bolojan, G. Infante, R. Precup, Existence results for systems with coupled nonlocal nonlinear initial conditions, Math. Bohem. 140 (2015), 371–384.
 A. Boucherif, H. Akca, Nonlocal Cauchy problems for semilinear evolution equations, Dynam. Systems Appl. 11 (2002), 415–420.
 A. Boucherif, R. Precup, On the nonlocal initial value problem for first order differential equations, Fixed Point Theory 4 (2003), 205–212.
 A. Boucherif, R. Precup, Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl. 16 (2007), 507–516.
 M. Burlica, D. Ro¸su, I.I. Vrabie, Abstract reaction-diffusion systems with nonlocal initial conditions, Nonlinear Anal. 94 (2014), 107–119.
 M.-D. Burlica, M. Necula, D. Ro¸su, I.I. Vrabie, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions, Chapman and Hall/CRC Press, 2016.
 L. Byszewski, Theorems about the existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problems, J. Math. Anal. Appl. 162 (1991), 494–505.
 T. Cardinali, R. Precup, P. Rubbioni, A unified existence theory for evolution equations and systems under nonlocal conditions, J. Math.Anal.Appl. 432 (2015), 1039–1057.
 J. Chabrowski, On nonlocal problems for parabolic equations, Nagoya Math. J. 93 (1984), 109–131.
 S.K. Ntouyas, P.Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl. 210 (1997), 679–687.
 W.E. Olmstead, C.A. Roberts, The one-dimensional heat equation with a nonlocal initial condition, Appl. Math. Lett. 10 (1997), 89–94.
 D. O’Regan, R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach, Amsterdam, 2001.
 A. Paicu, I.I. Vrabie, A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal. 72 (2010), 4091–4100.
 P. Pietramala, A note on a beam equation with nonlinear boundary conditions, Bound. Value Probl. 2011, Art. ID 376782, 14 pp.
 C. V. Pao, Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl. 195 (1995), 702–718.
 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.
 R. Precup, D. Trif, Multiple positive solutions of non-local initial value problems for first order differential systems, Nonlinear Anal. 75 (2012), 5961–5970.
 A. Stikonas, A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control 19 (2014), 301–334.
 P.N. Vabishchevich, Non-local parabolic problems and the inverse heat-conduction problem (Russian), Differ. Uravn. 17 (1981), 761–765.
 A. Viorel, Contributions to the Study of Nonlinear Evolution Equations, Ph.D. Thesis, Cluj-Napoca, 2011.
 I.I. Vrabie, C0-Semigroups and Applications, Elsevier, Amsterdam, 2003.
 I.I. Vrabie, Global solutions for nonlinear delay evolution inclusions with nonlocal initial conditions, Set-Valued Var. Anal. 20 (2012), 477–497.
 J.R.L. Webb, G. Infante, Positive solutions of nonlocal initial boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 45–67.
 W. M. Whyburn, Differential equations with general boundary conditions, Bull. Amer. Math. Soc. 48 (1942), 692–704.