Skip to main content
Erschienen in: Journal of Applied Mathematics and Computing 1-2/2014

01.06.2014 | Original Research

Solvability of fractional order differential systems with general nonlocal conditions

verfasst von: JinRong Wang, Yuruo Zhang

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2014

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper, fractional order nonlinear differential systems with general nonlocal conditions are investigated. The Lipschitz condition, linear and nonlinear growth conditions on the nonlinear terms are divided into two parts on two intervals; fixed point methods, the techniques on matrix and vector norms are used. Existence results for the solutions are derived under the weaken conditions.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Nica, O., Precup, R.: On the nonlocal initial value problem for first order differential systems. Stud. Univ. Babeş–Bolyai, Math. 56, 125–137 (2011) MathSciNet Nica, O., Precup, R.: On the nonlocal initial value problem for first order differential systems. Stud. Univ. Babeş–Bolyai, Math. 56, 125–137 (2011) MathSciNet
2.
Zurück zum Zitat Nica, O.: Nonlocal initial value problems for first order differential systems. Fixed Point Theory 13, 603–612 (2012) MATHMathSciNet Nica, O.: Nonlocal initial value problems for first order differential systems. Fixed Point Theory 13, 603–612 (2012) MATHMathSciNet
3.
Zurück zum Zitat Nica, O.: Initial-value problems for first-order differential systems with general nonlocal conditions. Electron. J. Differ. Equ. 2012(74), 1–15 (2012) MathSciNet Nica, O.: Initial-value problems for first-order differential systems with general nonlocal conditions. Electron. J. Differ. Equ. 2012(74), 1–15 (2012) MathSciNet
4.
Zurück zum Zitat Boucherif, A., Precup, R.: On the nonlocal initial value problem for first order differential equations. Fixed Point Theory 4, 205–212 (2003) MATHMathSciNet Boucherif, A., Precup, R.: On the nonlocal initial value problem for first order differential equations. Fixed Point Theory 4, 205–212 (2003) MATHMathSciNet
5.
Zurück zum Zitat Baleanu, D., Machado, J.A.T., Luo, A.C.-J.: Fractional Dynamics and Control. Springer, Berlin (2012) CrossRefMATH Baleanu, D., Machado, J.A.T., Luo, A.C.-J.: Fractional Dynamics and Control. Springer, Berlin (2012) CrossRefMATH
6.
Zurück zum Zitat Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics (2010) CrossRefMATH Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics (2010) CrossRefMATH
7.
Zurück zum Zitat Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATH Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATH
8.
Zurück zum Zitat Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific, Cambridge (2009) MATH Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific, Cambridge (2009) MATH
9.
Zurück zum Zitat Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) MATH Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) MATH
10.
Zurück zum Zitat Michalski, M.W.: Derivatives of noninteger order and their applications. Dissertationes Mathematicae, CCCXXVIII, Inst. Math., Polish Acad. Sci. (1993) Michalski, M.W.: Derivatives of noninteger order and their applications. Dissertationes Mathematicae, CCCXXVIII, Inst. Math., Polish Acad. Sci. (1993)
11.
Zurück zum Zitat Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATH Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATH
12.
Zurück zum Zitat Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin (2011) Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin (2011)
13.
Zurück zum Zitat Ahmad, B., Nieto, J.J.: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35, 295–304 (2010) MATHMathSciNet Ahmad, B., Nieto, J.J.: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35, 295–304 (2010) MATHMathSciNet
14.
Zurück zum Zitat Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal., TMA 72, 916–924 (2010) CrossRefMATH Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal., TMA 72, 916–924 (2010) CrossRefMATH
15.
Zurück zum Zitat Banaś, J., Rzepka, B.: Monotonic solutions of a quadratic integral equation of fractional order. J. Math. Anal. Appl. 332, 1371–1379 (2007) CrossRefMATHMathSciNet Banaś, J., Rzepka, B.: Monotonic solutions of a quadratic integral equation of fractional order. J. Math. Anal. Appl. 332, 1371–1379 (2007) CrossRefMATHMathSciNet
16.
Zurück zum Zitat Banás, J., O’Regan, D.: On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order. J. Math. Anal. Appl. 345, 573–582 (2008) CrossRefMATHMathSciNet Banás, J., O’Regan, D.: On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order. J. Math. Anal. Appl. 345, 573–582 (2008) CrossRefMATHMathSciNet
17.
Zurück zum Zitat Banaś, J., Zając, T.: Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity. Nonlinear Anal., TMA 71, 5491–5500 (2009) CrossRefMATH Banaś, J., Zając, T.: Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity. Nonlinear Anal., TMA 71, 5491–5500 (2009) CrossRefMATH
18.
Zurück zum Zitat Banás, J.: Measures of noncompactness in the study of solutions of nonlinear differential and integral equations. Cent. Eur. J. Math. 10, 2003–2011 (2012) CrossRefMATHMathSciNet Banás, J.: Measures of noncompactness in the study of solutions of nonlinear differential and integral equations. Cent. Eur. J. Math. 10, 2003–2011 (2012) CrossRefMATHMathSciNet
19.
Zurück zum Zitat Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008) CrossRefMATHMathSciNet Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008) CrossRefMATHMathSciNet
20.
Zurück zum Zitat Chang, Y.K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009) CrossRefMATHMathSciNet Chang, Y.K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009) CrossRefMATHMathSciNet
21.
Zurück zum Zitat Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal., TMA 71, 2724–2733 (2009) CrossRefMATHMathSciNet Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal., TMA 71, 2724–2733 (2009) CrossRefMATHMathSciNet
22.
Zurück zum Zitat Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, 4465–4475 (2010) CrossRefMATHMathSciNet Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, 4465–4475 (2010) CrossRefMATHMathSciNet
23.
Zurück zum Zitat Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal., TMA 73, 3462–3471 (2010) CrossRefMATH Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal., TMA 73, 3462–3471 (2010) CrossRefMATH
24.
Zurück zum Zitat Fec̆kan, M., Zhou, Y., Wang, J.: On the concept and existence of solutions for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 3050–3060 (2012) CrossRefMathSciNet Fec̆kan, M., Zhou, Y., Wang, J.: On the concept and existence of solutions for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 3050–3060 (2012) CrossRefMathSciNet
25.
Zurück zum Zitat Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235 (2012) CrossRefMATHMathSciNet Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235 (2012) CrossRefMATHMathSciNet
26.
Zurück zum Zitat Li, K., Peng, J., Jia, J.: Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. J. Funct. Anal. 263, 476–510 (2012) CrossRefMATHMathSciNet Li, K., Peng, J., Jia, J.: Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. J. Funct. Anal. 263, 476–510 (2012) CrossRefMATHMathSciNet
27.
Zurück zum Zitat Wang, J., Fec̆kan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8, 345–361 (2011) CrossRefMATHMathSciNet Wang, J., Fec̆kan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8, 345–361 (2011) CrossRefMATHMathSciNet
28.
Zurück zum Zitat Wang, J., Zhou, Y., Wei, W.: Optimal feedback control for semilinear fractional evolution equations in Banach spaces. Syst. Control Lett. 61, 472–476 (2012) CrossRefMATHMathSciNet Wang, J., Zhou, Y., Wei, W.: Optimal feedback control for semilinear fractional evolution equations in Banach spaces. Syst. Control Lett. 61, 472–476 (2012) CrossRefMATHMathSciNet
29.
Zurück zum Zitat Wang, J., Fan, Z., Zhou, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl. 154, 292–302 (2012) CrossRefMATHMathSciNet Wang, J., Fan, Z., Zhou, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl. 154, 292–302 (2012) CrossRefMATHMathSciNet
30.
Zurück zum Zitat Kumar, S., Sukavanam, N.: Approximate controllability of fractional order semilinear systems with bounded delay. J. Differ. Equ. 252, 6163–6174 (2012) CrossRefMATHMathSciNet Kumar, S., Sukavanam, N.: Approximate controllability of fractional order semilinear systems with bounded delay. J. Differ. Equ. 252, 6163–6174 (2012) CrossRefMATHMathSciNet
31.
Zurück zum Zitat Bai, C., Fang, J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611–621 (2004) CrossRefMATHMathSciNet Bai, C., Fang, J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611–621 (2004) CrossRefMATHMathSciNet
32.
Zurück zum Zitat Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009) CrossRefMATHMathSciNet Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009) CrossRefMATHMathSciNet
33.
Zurück zum Zitat Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009) CrossRefMATHMathSciNet Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009) CrossRefMATHMathSciNet
34.
Zurück zum Zitat Odibat, Z.M.: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171–1183 (2010) CrossRefMATHMathSciNet Odibat, Z.M.: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171–1183 (2010) CrossRefMATHMathSciNet
35.
Zurück zum Zitat Qian, D., Li, C., Agarwal, R.P., Wong, P.J.Y.: Stability analysis of fractional differential system with Riemann-Liouville derivative. Math. Comput. Model. 52, 862–874 (2010) CrossRefMATHMathSciNet Qian, D., Li, C., Agarwal, R.P., Wong, P.J.Y.: Stability analysis of fractional differential system with Riemann-Liouville derivative. Math. Comput. Model. 52, 862–874 (2010) CrossRefMATHMathSciNet
36.
Zurück zum Zitat Sun, S., Li, Q., Li, Y.: Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations. Comput. Math. Appl. 64, 3310–3320 (2012) CrossRefMATHMathSciNet Sun, S., Li, Q., Li, Y.: Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations. Comput. Math. Appl. 64, 3310–3320 (2012) CrossRefMATHMathSciNet
38.
Zurück zum Zitat Precup, R.: The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Model. 49, 703–708 (2009) CrossRefMATHMathSciNet Precup, R.: The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Model. 49, 703–708 (2009) CrossRefMATHMathSciNet
39.
40.
Zurück zum Zitat Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001) CrossRefMATH Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001) CrossRefMATH
42.
Zurück zum Zitat O’Regan, D., Precup, R.: Theorems of Leray-Schauder Type and Applications. Gordon and Breach, Amsterdam (2001) MATH O’Regan, D., Precup, R.: Theorems of Leray-Schauder Type and Applications. Gordon and Breach, Amsterdam (2001) MATH
Metadaten
Titel
Solvability of fractional order differential systems with general nonlocal conditions
verfasst von
JinRong Wang
Yuruo Zhang
Publikationsdatum
01.06.2014
Verlag
Springer Berlin Heidelberg
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2014
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-013-0719-4

Weitere Artikel der Ausgabe 1-2/2014

Journal of Applied Mathematics and Computing 1-2/2014 Zur Ausgabe

Original Research

On the entropy of LEGO ®

Premium Partner