Glasnik Matematicki, Vol. 51, No. 2 (2016), 413-430.

EXISTENCE RESULTS FOR SOME PARTIAL INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

Khalil Ezzinbi, Saifeddine Ghnimi and Mohamed-Aziz Taoudi

Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, BP 2390 Marrakech, Morocco
e-mail: ezzinbi@uca.ma

Department of Mathematics, Faculty of Sciences, University of Gafsa, B.P. 2112 Gafsa, Tunisia
e-mail: ghnimisaifeddine@yahoo.fr

Cadi Ayyad University, National School of Applied Sciences, Marrakech, Morocco
e-mail: a.taoudi@uca.ma

Abstract.   In this work, we study the existence of mild solutions for a class of partial integrodifferential equations with nonlocal conditions. Our analysis uses the resolvent operator theory and relies on a new fixed point theorem of Sadovskii-Krasnosel'skii type. Our results improve and complement several earlier related works. Some examples are provided to illustrate the theoretical results.

2010 Mathematics Subject Classification.   45K05, 47H10, 47D06.

Key words and phrases.   Partial integrodifferential equation, nonlocal conditions, mild solution, fixed point theorem, resolvent operator, semigroup operator.

Full text (PDF) (free access)

DOI: 10.3336/gm.51.2.09

References:

  1. J. Banaś and K. Goebel, Mesure of noncompactness in Banach spaces, Marcel Dekker, New York, 1980.
    MathSciNet    

  2. D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109-138.
    MathSciNet     CrossRef

  3. L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505.
    MathSciNet     CrossRef

  4. L. Byszewski, Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem, Zeszyty Nauk. Politech. Rzeszowskiej Mat. Fiz. 18 (1993), 109-112.
    MathSciNet    

  5. L. Byszewski and V. Lakshmikantham, 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.
    MathSciNet     CrossRef

  6. L. Byszewski and H. Akca, Existence of solutions of a semilinear functional-differential evolution nonlocal problem, Nonlinear Anal. 34 (1998), 65-72.
    MathSciNet     CrossRef

  7. P. Cannarsan and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms, NoDEA Nonlinear Differential Equations Appl. 10 (2003), 399-430.
    MathSciNet     CrossRef

  8. G. Chen and R. Grimmer, Semigroups and integral equations, J. Integral Equations 2 (1980), 133-154.
    MathSciNet    

  9. E. N. Chukwu, Differential models and neutral systems for controlling the wealth of nations, World Scientific Publishing, River Edge, 2001.
    MathSciNet    

  10. W. Desch, R. Grimmer and W. Schappacher, Some considerations for linear integro-differential equations, J. Math. Anal. Appl. 104 (1984), 219-234.
    MathSciNet     CrossRef

  11. K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179 (1993), 630-637.
    MathSciNet     CrossRef

  12. J. P. C. Dos Santos, S. M. Guzzo and M. N. Rabelo, Asymptotically almost periodic solutions for abstract partial neutral integro-differential equation, Adv. Difference Equ. 2010, Art. ID 310951, 26 pp.
    MathSciNet    

  13. K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000.
    MathSciNet    

  14. K. Ezzinbi and J. H. Liu, Nondensely defined evolution equations with nonlocal conditions, Math. Comput. Modelling 36 (2002), 1027-1038.
    MathSciNet     CrossRef

  15. K. Ezzinbi, S. Ghnimi and M. A. Taoudi, Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay, Nonlinear Anal. Hybrid Syst. 4 (2010), 54-64.
    MathSciNet     CrossRef

  16. K. Ezzinbi and X. Fu, Existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal. 57 (2004), 1029-1041.
    MathSciNet     CrossRef

  17. K. Ezzinbi, X. Fu and K. Hilal, Existence and regularity in the α-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal. 67 (2007), 1613-1622.
    MathSciNet     CrossRef

  18. K. Ezzinbi and M. A. Taoudi, Sadovskii-Krasnosel'skii type fixed point theorems in Banach spaces with application to evolution equations, J. Appl. Math. Comput. 49 (2015), 243-260.
    MathSciNet     CrossRef

  19. X. Fu and K. Ezzinbi, Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal. 54 (2003), 215-227.
    MathSciNet     CrossRef

  20. R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982), 333-349.
    MathSciNet     CrossRef

  21. G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations, Cambridge University Press, Cambridge, 1990.
    MathSciNet     CrossRef

  22. M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Rational Mech. Anal. 31 (1968), 113-126.
    MathSciNet     CrossRef

  23. E. Hernández, M. Pierri and A. Prokopczyk, On a class of abstract neutral functional differential equations, Nonlinear Anal. 74 (2011), 3633-3643.
    MathSciNet     CrossRef

  24. J. Liang, J. Liu and T. J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal. 57 (2004), 183-189.
    MathSciNet     CrossRef

  25. J. Liang and T. J. Xiao, Semilinear integrodifferential equations with nonlocal initial conditions, Comput. Math. Appl. 47 (2004), 863-875.
    MathSciNet     CrossRef

  26. L. Liu, F. Guo, C. Wu and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), 638-649.
    MathSciNet     CrossRef

  27. C. Lizama and J. C. Pozo, Existence of mild solutions for a semilinear integrodifferential equation with nonlocal initial conditions, Abstr. Appl. Anal. 2012 (2012), Art. ID 647103, 15 pp.
    MathSciNet    

  28. A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal. 21 (1990), 1213-1224.
    MathSciNet     CrossRef

  29. R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math. 35 (1977/78), 1-19.
    MathSciNet    

  30. R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), 313-332.
    MathSciNet     CrossRef

  31. H. Monch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), 985-999.
    MathSciNet     CrossRef

  32. S. Ntouyas and P. Tsamotas, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl. 210 (1997), 679-687.
    MathSciNet     CrossRef

  33. J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187-204.
    MathSciNet    

  34. J. Pruss, Evolutionary integral equations and applications, Birkhäuser, Basel, 1993.
    MathSciNet     CrossRef

  35. J. Sun and X. Zhang, A fixed point theorem for convex-power condensing operators and its applications to abstract semilinear evolution equations, Acta Math. Sinica (Chin. Ser.) 48 (2005), 439-446.
    MathSciNet    

  36. L. Zhu and G. Li, Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces, Nonlinear Anal. 74 (2011), 5133-5140.
    MathSciNet     CrossRef
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