Solutions to nonlocal evolution equations governed by non-autonomous forms and demicontinuous nonlinearities. (English) Zbl 1498.35560
Summary: We deal with the existence of solutions having \(L^2\)-regularity for a class of non-autonomous evolution equations. Associated with the equation, a general non-local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray-Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity.
MSC:
| 35R09 | Integro-partial differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 35K90 | Abstract parabolic equations |
| 47H11 | Degree theory for nonlinear operators |
References:
| [1] | Achache M., Ouhabaz E.M., Lions’ maximal regularity problem with \(H^{\frac{1}{2}} \)-regularity in time Journal of Differential Equations, 266 6 (2019), 3654-3678. · Zbl 1412.35190 |
| [2] | Arendt, W.; Chill, R.; Fornaro, S.; Poupaud, C., \(L^p\)-maximal regularity for non-autonomous evolution equations, Journal of Differential Equations, 237, 1, 1-26 (2007) · Zbl 1126.34037 · doi:10.1016/j.jde.2007.02.010 |
| [3] | Amann, H., Maximal regularity for nonautonomous evolution equations, Advanced Nonlinear Studies, 4, 4, 417-430 (2004) · Zbl 1072.35103 · doi:10.1515/ans-2004-0404 |
| [4] | Arendt W., Semigroups and evolution equations: functional calculus, regularity and kernel estimates. In: Handbook of Differential Equations: Evolutionary Equations. North-Holland (2002), 1-85. · Zbl 1082.35001 |
| [5] | Arendt W., Batty C.J.K., Hieber M., Neubrander F., Vector-valued Laplace Transforms and Cauchy Problems (second edition), (Birkhäuser Basel, 2011). · Zbl 1226.34002 |
| [6] | Arendt, W.; Monniaux, S., Maximal regularity for non-autonomous Robin boundary conditions, Mathematische Nachrichten, 289, 11-12, 1325-1340 (2016) · Zbl 06639277 · doi:10.1002/mana.201400319 |
| [7] | Auscher, P.; Tchamitchian, P., Square roots of elliptic second order divergence operators on strongly Lipschitz domains: \(L^2\) theory, Journal d’Analyse Mathématique, 90, 1, 1-12 (2003) · Zbl 1173.35420 · doi:10.1007/BF02786549 |
| [8] | Bauschke H.H., Combettes P.L. Convex analysis and monotone operator theory in Hilbert spaces - second ed. (Springer, 2017). · Zbl 1359.26003 |
| [9] | Barroso, CS; Teixeira, EV, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Analysis: Theory, Methods & Applications, 60, 4, 625-650 (2005) · Zbl 1078.47014 · doi:10.1016/j.na.2004.09.040 |
| [10] | Benedetti, I.; Ciani, S., Evolution equations with nonlocal initial conditions and superlinear growth, Journal of Differential Equations, 318, 270-297 (2022) · Zbl 1485.35273 · doi:10.1016/j.jde.2022.02.030 |
| [11] | Benedetti, I.; Loi, NV; Malaguti, L.; Taddei, V., Nonlocal diffusion second order partial differential equations, Journal of Differential Equations, 262, 3 (2017) · Zbl 1353.35297 · doi:10.1016/j.jde.2016.10.019 |
| [12] | Benedetti I., Loi N.V., Malaguti L., Obukhovskii V., An approximation solvability method for nonlocal differential problems in Hilbert spaces. Communications in Contemporary Mathematichs, 1650002 (2016). · Zbl 1378.34083 |
| [13] | Benedetti, I.; Loi, VN; Taddei, V., An approximation solvability method for nonlocal semilinear differential problems in Banach spaces, Discrete & Continuous Dynamical Systems, 37, 6, 2977 (2017) · Zbl 1357.35272 · doi:10.3934/dcds.2017128 |
| [14] | Boyer F. and Fabrie P., Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Applied Mathematical Sciences 183. Springer, 2013. · Zbl 1286.76005 |
| [15] | Boucherif, A.; Precup, R., Semilinear evolution equations with nonlocal initial conditions, Dynamic Systems and Applications, 16, 507-516 (2007) · Zbl 1154.34027 |
| [16] | Brezis H., Operáteurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Elsevier, 1973. · Zbl 0252.47055 |
| [17] | Brezis H., Functional analysis, Sobolev spaces and partial differential equations. Springer Science & Business Media, 2010. · Zbl 1218.46002 |
| [18] | Browder F.E., de Figueiredo D.G., J-monotone nonlinear operators in Banach spaces. In Djairo G. de Figueiredo-Selected Papers (pp. 1-9). Springer, Cham. · Zbl 0148.13602 |
| [19] | Daners, D., Heat kernel estimates for operators with boundary conditions, Mathematische Nachrichten, 217, 1, 13-41 (2000) · Zbl 0973.35087 · doi:10.1002/1522-2616(200009)217:1<13::AID-MANA13>3.0.CO;2-6 |
| [20] | Deng, K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, Journal of Mathematical analysis and applications, 179, 2, 630-637 (1993) · Zbl 0798.35076 · doi:10.1006/jmaa.1993.1373 |
| [21] | Dupuis, P.; Nagurney, A., Dynamical systems and variational inequalities, Annals of Operations Research, 44, 1, 7-42 (1993) · Zbl 0785.93044 · doi:10.1007/BF02073589 |
| [22] | EL-Mennaoui O., Laasri H., A note on the norm-continuity for evolution families arising from non-autonomous forms, Semigroup Forum. Vol. 100. No. 2. Springer US, 2020. · Zbl 1444.34072 |
| [23] | Friesz, TL; Bernstein, D.; Stough, R., Dynamic systems, variational inequalities and control theoretic models for predicting time-varying urban network flows, Transportation Science, 30, 1, 14-31 (1996) · Zbl 0849.90061 · doi:10.1287/trsc.30.1.14 |
| [24] | Furi, M.; Pera, P., A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals, Annales Polonici Mathematici, 47, 331-346 (1987) · Zbl 0656.47052 · doi:10.4064/ap-47-3-331-346 |
| [25] | Goeleven, D.; Brogliato, B., Stability and instability matrices for linear evolution variational inequalities, IEEE Transactions on Automatic Control, 49, 4, 521-534 (2004) · Zbl 1365.49011 · doi:10.1109/TAC.2004.825654 |
| [26] | Garcia-Falset, J.; Muniz-Pérez, O.; Reich, S., Domains of accretive operators in Banach spaces, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 146, 2, 325-336 (2016) · Zbl 1440.47042 · doi:10.1017/S0308210515000451 |
| [27] | Haak, BH; Ouhabaz, EM, Maximal regularity for non-autonomous evolution equations, Mathematische Annalen, 363, 3, 1117-1145 (2015) · Zbl 1327.35220 · doi:10.1007/s00208-015-1199-7 |
| [28] | Haase M., The functional calculus for sectorial operators and similarity methods, Ph.D. Thesis, Ulm (2003) · Zbl 1066.47011 |
| [29] | Leray, J.; Schauder, J., Topologie et équations fonctionnelles, Annales scientifiques de l’École normale supérieure, 51, 45-78 (1934) · JFM 60.0322.02 · doi:10.24033/asens.836 |
| [30] | Koumla, S.; Precup, R., Study on Integrodifferential Evolution Systems with Nonlocal Initial Conditions, Recent Advances in Mathematical Research and Computer Science, 5, 13-27 (2021) · doi:10.9734/bpi/ramrcs/v5/3535F |
| [31] | Lions J. L., Equations differentielles operationnelles: et problémes aux limites. Vol. 111. Springer-Verlag, (2013) · Zbl 0098.31101 |
| [32] | Loi, VL, Method of Guiding Functions for Differential Inclusions in a Hilbert Space, Differential Equations, 46, 10, 1438-1447 (2010) · Zbl 1232.34089 · doi:10.1134/S0012266110100071 |
| [33] | Lu, L.; Liu, Z.; Obukhovskii, V., Second order differential variational inequalities involving anti-periodic boundary value conditions, Journal of Mathematical Analysis and Applications, 473, 2, 846-865 (2019) · Zbl 1415.49008 · doi:10.1016/j.jmaa.2018.12.072 |
| [34] | Lucchetti, R.; Patrone, F., On Nemytskiiś operator and its application to the lower semicontinuity of integral functionals, Indiana University Mathematics Journal, 29, 5, 703-713 (1980) · Zbl 0476.47049 · doi:10.1512/iumj.1980.29.29051 |
| [35] | McIntosh A., On representing closed accretive sesquilinear forms as \((A^{1/2} u, A^{1/2} v)\), Collége de France Seminar, Vol. III (1982), 252-267. · Zbl 0515.47013 |
| [36] | Megginson, R. E., An introduction to Banach space theory, Vol. 183. Springer Science & Business Media (2012) · Zbl 0910.46008 |
| [37] | Moreira, D.; Teixeira, E., On the behavior of weak convergence under nonlinearities and applications, Proceedings of the American Mathematical Society, 133, 6, 1647-1656 (2005) · Zbl 1073.46030 · doi:10.1090/S0002-9939-04-07876-1 |
| [38] | Nagurney A., Ding Z., Projected dynamical systems and variational inequalities with applications. Vol. 2, Springer, (1995) · Zbl 0865.90018 |
| [39] | Nesterov Y., Introductory lectures on convex optimization: A basic course, Vol. 87. Springer Science & Business Media (2003) · Zbl 1086.90045 |
| [40] | Ntouyas S. K., Nonlocal initial and boundary value problems: a survey. In Handbook of differential equations: ordinary differential equations. Vol. 2 (2006), 461-557. North-Holland. · Zbl 1098.34011 |
| [41] | Paicu, A.; Vrabie, II, A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal., 72, 4091-4100 (2010) · Zbl 1200.34068 · doi:10.1016/j.na.2010.01.041 |
| [42] | Pascal, A.; Steve, H.; Michael, L.; Alan, M.; Tchamitchian, Ph, The solution of the Kato square root problem for second order operators on \({\mathbb{R} }^n\), Annals of mathematics, 156, 2, 633-654 (2002) · Zbl 1128.35316 · doi:10.2307/3597201 |
| [43] | Petryshyn, WV, Using degree theory for densely defined A-proper maps in the solvability of semilinear equations with unbounded and noninvertible linear part, Nonlinear Analysis: Theory, Methods & Applications, 4, 2, 259-281 (1980) · Zbl 0444.47046 · doi:10.1016/0362-546X(80)90053-X |
| [44] | Prüss, J.; Schnaubelt, R., Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, Journal of Mathematical Analysis and Applications, 256, 2, 405-430 (2001) · Zbl 0994.35076 · doi:10.1006/jmaa.2000.7247 |
| [45] | Rockafellar, RT, On the maximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society, 149, 1, 75-88 (1970) · Zbl 0222.47017 · doi:10.1090/S0002-9947-1970-0282272-5 |
| [46] | Showalter R. E., Monotone operators in Banach space and nonlinear partial differential equations. Vol. 49. American Mathematical Society, (2013) |
| [47] | Xu, HK; Colao, V.; Muglia, L., Mild solutions of nonlocal semilinear evolution equations on unbounded intervals via approximation solvability method in reflexive Banach spaces, Journal of Mathematical Analysis and Applications, 498, 1 (2021) · Zbl 1481.34080 · doi:10.1016/j.jmaa.2021.124938 |
| [48] | Zeidler E., Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators. Springer Science & Business Media, (2013) |
| [49] | Zeidler E., Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators. Springer Science & Business Media, (2013) |
| [50] | Zhenhai, L.; Lu, L.; Guo, X., Existence results for a class of semilinear differential variational inequalities with nonlocal boundary conditions, Topological Methods in Nonlinear Analysis, 55, 2, 429-449 (2020) · Zbl 1471.49007 |
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