Previous |  Up |  Next

# Article

Full entry | Fulltext not available (moving wall 12 months)
Keywords:
nonlinear differential system; nonlocal boundary condition; nonlinear boundary condition; fixed point; vector-valued norm; matrix convergent to zero
Summary:
The purpose of the present paper is to study the existence of solutions to initial value problems for nonlinear first order differential systems subject to nonlinear nonlocal initial conditions of functional type. The approach uses vector-valued metrics and matrices convergent to zero. Two existence results are given by means of Schauder and Leray-Schauder fixed point principles and the existence and uniqueness of the solution is obtained via a fixed point theorem due to Perov. Two examples are given to illustrate the theory.
References:
[1] Agarwal, R. P., Meehan, M., O'Regan, D.: Fixed Point Theory and Applications. Cambridge Tracts in Mathematics 141 Cambridge University Press, Cambridge (2001). DOI 10.1017/CBO9780511543005 | MR 1825411 | Zbl 0960.54027
[2] Aizicovici, S., Lee, H.: Nonlinear nonlocal Cauchy problems in Banach spaces. Appl. Math. Lett. 18 (2005), 401-407. DOI 10.1016/j.aml.2004.01.010 | MR 2124297 | Zbl 1084.34002
[3] Alves, E., Ma, T. F., Pelicer, M. L.: Monotone positive solutions for a fourth order equation with nonlinear boundary conditions. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 3834-3841. DOI 10.1016/j.na.2009.02.051 | MR 2536292 | Zbl 1177.34030
[4] Avalishvili, G., Avalishvili, M., Gordeziani, D.: On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation. Appl. Math. Lett. 24 (2011), 566-571. DOI 10.1016/j.aml.2010.11.014 | MR 2749746 | Zbl 1205.35085
[5] Avalishvili, G., Avalishvili, M., Gordeziani, D.: On integral nonlocal boundary value problems for some partial differential equations. Bull. Georgian Natl. Acad. Sci. (N.S.) 5 (2011), 31-37. MR 2839392 | Zbl 1227.35015
[6] Benchohra, M., Boucherif, A.: On first order multivalued initial and periodic value problems. Dyn. Syst. Appl. 9 (2000), 559-568. MR 1843699 | Zbl 1019.34008
[7] Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics 9 SIAM, Philadelphia (1994). MR 1298430 | Zbl 0815.15016
[8] Bolojan-Nica, O., Infante, G., Pietramala, P.: Existence results for impulsive systems with initial nonlocal conditions. Math. Model. Anal. 18 (2013), 599-611. DOI 10.3846/13926292.2013.865678 | MR 3175667 | Zbl 1301.34017
[9] Bolojan-Nica, O., Infante, G., Precup, R.: Existence results for systems with coupled nonlocal initial conditions. Nonlinear Anal. 94 (2014), 231-242. MR 3120688 | Zbl 1288.34019
[10] Boucherif, A.: First-order differential inclusions with nonlocal initial conditions. Appl. Math. Lett. 15 (2002), 409-414. DOI 10.1016/S0893-9659(01)00151-3 | MR 1902272 | Zbl 1025.34009
[11] Boucherif, A.: Nonlocal Cauchy problems for first-order multivalued differential equations. Electron. J. Differ. Equ. (electronic only) 2002 (2002), Article No. 47, 9 pages. MR 1907723 | Zbl 1011.34002
[12] Boucherif, A.: Differential equations with nonlocal boundary conditions. Proceedings of the Third World Congress of Nonlinear Analysts 47, Part 4, Catania, 2000 Elsevier Oxford Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 47 (2001), 2419-2430. DOI 10.1016/S0362-546X(01)00365-0 | MR 1971648 | Zbl 1042.34518
[13] Boucherif, A., Precup, R.: Semilinear evolution equations with nonlocal initial conditions. Dynam. Systems Appl. 16 (2007), 507-516. MR 2356335
[14] Boucherif, A., Precup, R.: On the nonlocal initial value problem for first order differential equations. Fixed Point Theory 4 (2003), 205-212. MR 2031390 | Zbl 1050.34001
[15] Byszewski, L.: Abstract nonlinear nonlocal problems and their physical interpretation. Biomathematics, Bioinformatics and Applications of Functional Differential Difference Equations H. Akca et al. Akdeniz Univ. Publ. Antalya, Turkey (1999).
[16] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162 (1991), 494-505. DOI 10.1016/0022-247X(91)90164-U | MR 1137634 | Zbl 0748.34040
[17] Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40 (1991), 11-19. DOI 10.1080/00036819008839989 | MR 1121321 | Zbl 0694.34001
[18] Cabada, A.: An overview of the lower and upper solutions method with nonlinear boundary value conditions. Bound. Value Probl. (electronic only) 2011 (2011), Article No. 893753, 18 pages. MR 2719294 | Zbl 1230.34001
[19] Cabada, A., Minhós, F. M.: Fully nonlinear fourth-order equations with functional boundary conditions. J. Math. Anal. Appl. 340 (2008), 239-251. DOI 10.1016/j.jmaa.2007.08.026 | MR 2376151 | Zbl 1138.34008
[20] Cabada, A., Tersian, S.: Multiplicity of solutions of a two point boundary value problem for a fourth-order equation. Appl. Math. Comput. 219 (2013), 5261-5267. DOI 10.1016/j.amc.2012.11.066 | MR 3009485 | Zbl 1294.34016
[21] Deimling, K.: Multivalued Differential Equations. W. de Gruyter Series in Nonlinear Analysis and Applications 1 Walter de Gruyter, Berlin (1992). MR 1189795 | Zbl 0820.34009
[22] Franco, D., O'Regan, D., Perán, J.: Fourth-order problems with nonlinear boundary conditions. J. Comput. Appl. Math. 174 (2005), 315-327. DOI 10.1016/j.cam.2004.04.013 | MR 2106442 | Zbl 1068.34013
[23] Frigon, M.: Applications of the Theory of Topological Transversality to Nonlinear Problems for Ordinary Differential Equations. Diss. Math. (Rozprawy Mat.) 296 French (1990). MR 1075674
[24] Frigon, M., Lee, J. W.: Existence principles for Carathéodory differential equations in Banach spaces. Topol. Methods Nonlinear Anal. 1 (1993), 95-111. MR 1215260 | Zbl 0790.34054
[25] Goodrich, C. S.: On nonlinear boundary conditions satisfying certain asymptotic behavior. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 76 (2013), 58-67. DOI 10.1016/j.na.2012.07.023 | MR 2974249 | Zbl 1264.34030
[26] Goodrich, C. S.: On nonlocal {BVP}s with nonlinear boundary conditions with asymptotically sublinear or superlinear growth. Math. Nachr. 285 (2012), 1404-1421. MR 2959967 | Zbl 1252.34032
[27] Goodrich, C. S.: Positive solutions to boundary value problems with nonlinear boundary conditions. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 417-432. DOI 10.1016/j.na.2011.08.044 | MR 2846811 | Zbl 1237.34153
[28] Han, H.-K., Park, J.-Y.: Boundary controllability of differential equations with nonlocal condition. J. Math. Anal. Appl. 230 (1999), 242-250. DOI 10.1006/jmaa.1998.6199 | MR 1669625 | Zbl 0917.93009
[29] Infante, G.: Nonlocal boundary value problems with two nonlinear boundary conditions. Commun. Appl. Anal. 12 (2008), 279-288. MR 2499284 | Zbl 1198.34025
[30] Infante, G., Minhós, F. M., Pietramala, P.: Non-negative solutions of systems of {ODE}s with coupled boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4952-4960. DOI 10.1016/j.cnsns.2012.05.025 | MR 2960289 | Zbl 1280.34026
[31] Infante, G., Pietramala, P.: Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions. Math. Methods Appl. Sci. 37 (2014), 2080-2090. DOI 10.1002/mma.2957 | MR 3248749 | Zbl 1312.34060
[32] Infante, G., Pietramala, P.: A cantilever equation with nonlinear boundary conditions. Electron. J. Qual. Theory Differ. Equ. (electronic only) 2009 (2009), Article No. 15, 14 pages. MR 2558840 | Zbl 1201.34041
[33] Infante, G., Pietramala, P.: Eigenvalues and non-negative solutions of a system with nonlocal {BC}s. Nonlinear Stud. 16 (2009), 187-196. MR 2527180 | Zbl 1184.34027
[34] Infante, G., Pietramala, P.: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 1301-1310. DOI 10.1016/j.na.2008.11.095 | MR 2527550 | Zbl 1169.45001
[35] Jackson, D.: Existence and uniqueness of solutions to semilinear nonlocal parabolic equations. J. Math. Anal. Appl. 172 (1993), 256-265. DOI 10.1006/jmaa.1993.1022 | MR 1199510 | Zbl 0814.35060
[36] Jankowski, T.: Ordinary differential equations with nonlinear boundary conditions. Georgian Math. J. 9 (2002), 287-294. MR 1916067 | Zbl 1013.34018
[37] Karakostas, G. L., Tsamatos, P. C.: Existence of multiple positive solutions for a nonlocal boundary value problem. Topol. Methods Nonlinear Anal. 19 (2002), 109-121. MR 1921888 | Zbl 1071.34023
[38] Nica, O.: Existence results for second order three-point boundary value problems. Differ. Equ. Appl. 4 (2012), 547-570. MR 3052162 | Zbl 1267.34040
[39] Nica, O.: Initial-value problems for first-order differential systems with general nonlocal conditions. Electron. J. Differ. Equ. (electronic only) 2012 (2012), Article No. 74, 15 pages. MR 2928611 | Zbl 1261.34016
[40] Nica, O.: Nonlocal initial value problems for first order differential systems. Fixed Point Theory 13 (2012), 603-612. MR 3024343 | Zbl 1286.34034
[41] Nica, O., Precup, R.: On the nonlocal initial value problem for first order differential systems. Stud. Univ. Babeş-Bolyai, Math. 56 (2011), 113-125. MR 2869720 | Zbl 1274.34041
[42] Ntouyas, S. K., Tsamatos, P. C.: Global existence for semilinear evolution equations with nonlocal conditions. J. Math. Anal. Appl. 210 (1997), Article No. ay975425, 679-687. DOI 10.1006/jmaa.1997.5425 | MR 1453198 | Zbl 0884.34069
[43] O'Regan, D., Precup, R.: Theorems of Leray-Schauder Type and Applications. Series in Mathematical Analysis and Applications 3 Gordon and Breach Science Publishers, London (2001). MR 1937722 | Zbl 1045.47002
[44] Pietramala, P.: A note on a beam equation with nonlinear boundary conditions. Bound. Value Probl. (electronic only) 2011 (2011), Article No. 376782, 14 pages. MR 2679691 | Zbl 1219.34028
[45] Precup, R.: The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Modelling 49 (2009), 703-708. DOI 10.1016/j.mcm.2008.04.006 | MR 2483674 | Zbl 1165.65336
[46] Precup, R.: Methods in Nonlinear Integral Equations. Kluwer Academic Publishers Dordrecht (2002). MR 2041579 | Zbl 1060.65136
[47] Varga, R. S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics 27 Springer, Dordrecht (2009). MR 1753713 | Zbl 1216.65042
[48] Webb, J. R. L.: A unified approach to nonlocal boundary value problems. Dynamic Systems and Applications 5. Proc. of the 5th International Conf., Morehouse College, Atlanta, 2007 Dynamic Publishers Atlanta (2008), 510-515 G. S. Ladde et al. MR 2468188 | Zbl 1203.34033
[49] Webb, J. R. L., Infante, G.: Semi-positone nonlocal boundary value problems of arbitrary order. Commun. Pure Appl. Anal. 9 (2010), 563-581. DOI 10.3934/cpaa.2010.9.563 | MR 2600449 | Zbl 1200.34025
[50] Webb, J. R. L., Infante, G.: Non-local boundary value problems of arbitrary order. J. Lond. Math. Soc., II. Ser. 79 (2009), 238-258. DOI 10.1112/jlms/jdn066 | MR 2472143 | Zbl 1165.34010
[51] Webb, J. R. L., Infante, G.: Positive solutions of nonlocal boundary value problems involving integral conditions. NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008), 45-67. DOI 10.1007/s00030-007-4067-7 | MR 2408344 | Zbl 1148.34021
[52] Webb, J. R. L., Lan, K. Q.: Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. Methods Nonlinear Anal. 27 (2006), 91-115. MR 2236412 | Zbl 1146.34020
[53] Xue, X.: Existence of semilinear differential equations with nonlocal initial conditions. Acta Math. Sin., Engl. Ser. 23 (2007), 983-988. DOI 10.1007/s10114-005-0839-3 | MR 2319608 | Zbl 1129.34041
[54] Xue, X.: Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces. Electron. J. Differ. Equ. (electronic only) 2005 (2005), Article No. 64, 7 pages. MR 2147144 | Zbl 1075.34051

Partner of