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Ukrainian Mathematical Journal

, Volume 70, Issue 1, pp 142–163 | Cite as

Existence of Global Solutions for Some Classes of Integral Equations

  • T. Jabeen
  • R. P. Agarwal
  • V. Lupulescu
  • D. O’Regan
Article
  • 22 Downloads

We study the existence of Lp-solutions for a class of Hammerstein integral equations and neutral functional differential equations involving abstract Volterra operators. Using compactness-type conditions, we establish the global existence of solutions. In addition, a global existence result for a class of nonlinear Fredholm functional integral equations involving abstract Volterra equations is given.

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References

  1. 1.
    R. P. Agarwal, Y. Zhou, J. R. Wang, and X. Luo, “Fractional functional differential equations with causal operators in Banach spaces,” Math. Comput. Model., 54, 1440–1452 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R. P. Agarwal, S. Arshad, V. Lupulescu, and D. O’Regan, “Evolution equations with causal operators,” Differ. Equat. Appl., 7, No. 1, 15–26 (2015).MathSciNetzbMATHGoogle Scholar
  3. 3.
    R. P. Agarwal and D. O’Regan, Infinite Interval Problems for Differential, Differential, and Integral Equations, Kluwer AP, New York (2001).CrossRefzbMATHGoogle Scholar
  4. 4.
    R. Ahangar, “Nonanticipating dynamical model and optimal control,” Appl. Math. Lett., 2, 15–18 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    R. Akmerov, M. Kamenskii, A. Potapov, and B. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhäuser, Basel (1992).CrossRefGoogle Scholar
  6. 6.
    J. Banaś, “Integrable solutions of Hammerstein and Urysohn integral equations,” J. Austral. Math. Soc., 46, 61–68 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    T. A. Barton and I. K. Purnaras, “Lp-solutions of singular integrodifferential equations,” J. Math. Anal. Appl., 386, 830–841 (2012).MathSciNetCrossRefGoogle Scholar
  8. 8.
    T. A. Barton and B. Zhang, “Lp-solutions of fractional differential equations,” Nonlin. Stud., 19, 161 (2012).Google Scholar
  9. 9.
    D. M. Bedivan and D. O’Regan, “The set of solutions for abstract Volterra equations in L p([0,a],R m),” Appl. Math. Lett., 12, 7–11 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    C. Corduneanu, Functional Equations with Causal Operators, Taylor & Francis, London; New York (2002).CrossRefzbMATHGoogle Scholar
  11. 11.
    C. Corduneanu, “A modified LQ-optimal control problem for causal functional differential equations,” Nonlinear Dynam. Syst. Theory, 4, 139–144 (2004).MathSciNetzbMATHGoogle Scholar
  12. 12.
    C. Corduneanu and M. Mahdavi, “Neutral functional equations with causal operators on a semiaxis,” Nonlin. Dynam. Syst. Theory, 8, 339–348 (2008).zbMATHGoogle Scholar
  13. 13.
    M. A. Darwish and A. A. El-Bary, “Existence of fractional integral equation with hysteresis,” Appl. Math. Comput., 176, 684–687 (2006).MathSciNetzbMATHGoogle Scholar
  14. 14.
    Z. Drici, F. A. McRae, and J. V. Devi, “Differential equations with causal operators in a Banach space,” Nonlin. Anal., 62, 301–313 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    R. E. Edwards, Functional Analysis—Theory and Applications, Holt, Rinehart, and Winston, Inc., New York (1965).zbMATHGoogle Scholar
  16. 16.
    G. Emmanuele, “About the existence of integrable solution of a functional-integral equation,” Rev. Mat. Comput., 4, No. 1 (1991).Google Scholar
  17. 17.
    M. I. Gil’, “Explicit stability conditions for neutral-type vector functional differential equations. A survey,” Surv. Math. Appl., 9, 1–54 (2014).MathSciNetzbMATHGoogle Scholar
  18. 18.
    M. I. Gil’, Stability of Neutral Functional Differential Equations, Atlantis Press, Paris (2014).Google Scholar
  19. 19.
    G. Gripenberg, S. O. Londen, and O. Staffans, Volterra Integral and Functional Equations, Cambridge Univ. Press (1990).Google Scholar
  20. 20.
    D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer AP, Dordrecht (1996).CrossRefzbMATHGoogle Scholar
  21. 21.
    H. P. Heinz, “On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions,” Nonlin. Anal., 7, 1351–1371 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    E. Hernández and K. Balachandran, “Existence results for abstract degenerate neutral functional differential equations,” Bull. Austral. Math. Soc., 81, 329–342 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    E. Hernández, M. Pierri, and A. Prokopczyk, “On a class of abstract neutral functional differential equations,” Nonlin. Anal., 74, 3633–3643 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    E. Hernández, D. O’Regan, and M. A. Ben, “On a new class of abstract integral equations and applications,” Appl. Math. Comput., 219, No. 4, 2271–2277 (2012).MathSciNetzbMATHGoogle Scholar
  25. 25.
    A. Ilchmann, E. P. Ryan, and C. J. Sangwin, “Systems of controlled functional differential equations and adaptive tracking,” SIAM J. Control Optim., 40, No. 6, 1746–1764 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    F. Isaia, “On a nonlinear integral equation without compactness,” Acta Math. Univ. Comenian. (N.S.), 75, No. 2, 233–240 (2006).MathSciNetzbMATHGoogle Scholar
  27. 27.
    M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin–New York (2001).Google Scholar
  28. 28.
    M. Kisielewicz, “Multivalued differential equations in separable Banach spaces,” J. Optim. Theory Appl., 37, No. 2, 231–249 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    C. Kuratowski, “Sur les espaces complets,” Fund. Math., 51, 301–309 (1930).CrossRefzbMATHGoogle Scholar
  30. 30.
    M. Kwapisz, “Remarks on the existence and uniqueness of solutions of Volterra functional equations in L p spaces,” J. Integral Equat. Appl., 3, No. 3 (1991).Google Scholar
  31. 31.
    M. Kwapisz, “Bielecki’s method, existence and uniqueness results for Volterra integral equations in L p space,” J. Math. Anal. Appl., 154, 403–416 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    V. Lakshmikantham, S. Leela, Z. Drici, and F. A. McRae, “Theory of causal differential equations,” Atlantis Stud. Math. Eng. Sci., Vol. 5 (2010).Google Scholar
  33. 33.
    J. Liang, S. Yan-H., R. P. Agarwal, and T.-W. Huang, “Integral solution of a class of nonlinear integral equations,” Appl. Math. Comput., 219, 4950–4957 (2013).MathSciNetzbMATHGoogle Scholar
  34. 34.
    V. Lupulescu, “Causal functional differential equations in Banach spaces,” Nonlin. Anal., 69, 4787–4795 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    V. Lupulescu, “On a class of functional differential equations in Banach spaces,” Electron. J. Qual. Theory Differ. Equat., 64, 1–17 (2010).MathSciNetzbMATHGoogle Scholar
  36. 36.
    D. Mamrilla, “On L p-solutions of nth-order nonlinear differential equations,” Cӑas. Peštov. Mat., 113, 363–368 (1988).MathSciNetzbMATHGoogle Scholar
  37. 37.
    A. A. Martynyuk and Yu. A. Martynyuk-Chernienko, “Analysis of the set of trajectories of nonlinear dynamics: equations with causal robust operator,” Different. Equat., 51, No. 1, 11–22 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    V. Obukhovskii and P. Zecca, “On certain classes of functional inclusions with causal operators in Banach spaces,” Nonlin. Anal., 74, 2765–2777 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    L. Olszowy, “A family of measures of noncompactness in the space \( {L}_{\mathrm{loc}}^1\left(R+\right) \) and its application to some nonlinear Volterra integral equations,” Mediterr. J. Math., 11, 687–701 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel (1993).CrossRefzbMATHGoogle Scholar
  41. 41.
    D. O’Regan and R. Precup, “Existence criteria for integral equations in Banach spaces,” J. Inequal. Appl., 6, 77–97 (2001).MathSciNetzbMATHGoogle Scholar
  42. 42.
    E. P. Ryan and C. J. Sangwin, “Controlled functional differential equations and adaptive tracking,” Systems Control Lett., 47, 365–374 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    S. Szufla, “Existence for L p-solutions of integral equations in Banach spaces,” Publ. Inst. Math. (Beograd) (N.S.), 54, 99–105 (1986).zbMATHGoogle Scholar
  44. 44.
    S. Szufla, “Appendix to the paper: “An existence theorem for the Urysohn integral equation in Banach spaces,” Comment. Math. Univ. Carolin., 25, 763–764 (1984).Google Scholar
  45. 45.
    R. J. Taggart, Evolution Equations and Vector-Valued L p Spaces: PhD Thesis, New South Wales University (2004).Google Scholar
  46. 46.
    A. N. Tikhonov, “Functional Volterra-type equations and their applications to some problems of mathematical physics,” Bull. Mosk. Gos. Univ. Sek. A, 1, No. 8, 1–25 (1938).Google Scholar
  47. 47.
    L. Tonelli, “Sulle equazioni funzionali di Volterra,” Bull. Calcutta Math. Soc., 20, 31–48 (1930).zbMATHGoogle Scholar
  48. 48.
    F. Wang, “A fixed-point theorem for nonautonomous type superposition operators and integrable solutions of a general nonlinear functional integral equation,” J. Inequal. Appl., 2014 (2014).Google Scholar
  49. 49.
    J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York (1972).zbMATHGoogle Scholar
  50. 50.
    H. Zhu, “On a nonlinear integral equation with contractive perturbation,” Adv. Difference Equat., 2011 (2011), Article ID 154742, 10 p., doi: https://doi.org/10.1155/2011/154742.
  51. 51.
    E. S. Zhukovskii and M. J. Alves, “Abstract Volterra operators,” Russian Math. (Izv. Vuzov), 52, 1–14 (2008).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • T. Jabeen
    • 1
  • R. P. Agarwal
    • 2
  • V. Lupulescu
    • 3
  • D. O’Regan
    • 4
  1. 1.Abdus Salam School of Mathematical SciencesGovernment College UniversityLahorePakistan
  2. 2.Texas A&M University-KingvsilleKingsvilleUSA
  3. 3.Constantin Brancusi UniversityTargu-JiuRomania
  4. 4.School of Mathematics, Statistics & Applied MathematicsNational University of IrelandGalwayIreland

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