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Measure of noncompactness and fractional integro-differential equations with state-dependent nonlocal conditions in Fréchet spaces

1 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P. O. Box 89, 22000, Sidi Bel-Abbes, Algeria
2 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
3 Department of Mathematics, Faculty of Exact Sciences, Mustapha Stambouli University of Mascara, P. O. Box 305, 29000, Algeria
4 Department of Mathematics, Baylor University, Waco, Texas 76798-7328, USA

Special Issues: Initial and Boundary Value Problems for Differential Equations

This paper deals with the existence of mild solutions for non-linear fractional integrodifferential equations with state-dependent nonlocal conditions. The technique used is a generalization of the classical Darbo fixed point theorem for Fréchet spaces associated with the concept of measures of noncompactness. An application of the main result has been included.
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Keywords fractional integro-differential equations; solution operator; mild solution; fixed point; state-dependent nonlocal condition; measure of noncompactness; Fréchet space

Citation: Mouffak Benchohra, Zohra Bouteffal, Johnny Henderson, Sara Litimein. Measure of noncompactness and fractional integro-differential equations with state-dependent nonlocal conditions in Fréchet spaces. AIMS Mathematics, 2020, 5(1): 15-25. doi: 10.3934/math.2020002


  • 1. S. Abbas, M. Benchohra, Advanced Functional Evolution Equations and Inclusions, New York: Springer, 2015.
  • 2. R. P. Agarwal, B. Andradec, G. Siracusa, On fractional integro-differential equations with statedependent delay, Comput. Math. Appl., 63 (2011), 1143-1149.
  • 3. R. R. Akhmerov, M. I. Kamenskii, A. S. Patapov, et al. Measures of Noncompactness and Condensing Operators, Basel: Birkhauser Verlag, 1992.
  • 4. J. C. Alvàrez, Measure of noncompactness and fixed points of nonexpansive condensingmappings in locally convex spaces, Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid, 79 (1985), 53-66.
  • 5. A. Anguraj, P. Karthikeyan, J. J. Trujillo, Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition, Adv. Differ. Equ., 2011 (2011), 690653.
  • 6. W. Arendt, C. Batty, M., Hieber, et al. Vector-Valued Laplace Transforms and Cauchy Problems, Basel: Springer Science & Business Media, 2011.
  • 7. K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1970-1977.    
  • 8. J. Bana$\grave{s}$, K. Goebel, Measures of Noncompactness in Banach Spaces, New York: Marcel Dekker, 1980.
  • 9. M. Benchohra, S. Litimein, Existence results for a new class of fractional integro-differential equations with state dependent delay, Mem. Differ. Equ. Math. Phys., 74 (2018), 27-38.
  • 10. D. Bothe, Multivalued perturbation of m-accretive differential inclusions, Isr. J. Math., 108 (1998), 109-138.    
  • 11. L. Byszewski, Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems, Dynam. Systems Appl., 5 (1996), 595-605.
  • 12. L. Byszewski, H. Akca, Existence of solutions of a semilinear functional differential evolution nonlocal problem, Nonlinear Anal., 34 (1998), 65-72.    
  • 13. C. Cuevas, J. C. de Souza, S-asymptotically ω-periodic solutions of semilinear fractional integrodifferential equations, Appl. Math. Lett., 22 (2009), 865-870.    
  • 14. C. Cuevas, J. C. de Souza, Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72 (2010), 1683-1689.    
  • 15. B. de Andrade, C. Cuevas, S-asymptotically ω-periodic and asymptotically ω-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal., 72 (2010), 3190-3208.    
  • 16. S. Dudek, Fixed point theorems in Fréchet algebras and Fréchet spaces and applications to nonlinear integral equations, Appl. Anal. Discrete Math., 11 (2017), 340-357.    
  • 17. S. Dudek, L. Olszowy, Continuous dependence of the solutions of nonlinear integral quadratic Volterra equation on the parameter, J. Funct. Spaces, 2015 (2015), 471235.
  • 18. K. J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, New York: Springer-Verlag, 2000.
  • 19. H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Amsterdam: North-Holland, 1985.
  • 20. D. J. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, Dordrecht: Kluwer Academic Publishers, 1996.
  • 21. E. Hernandez, On abstract differential equations with state dependent non-local conditions, J. Math. Anal. Appl., 466 (2018), 408-425.    
  • 22. E. Hernandez, D. O'Regan, On state dependent non-local conditions, Appl. Math. Letters, 83 (2018), 103-109.    
  • 23. A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.
  • 24. J. Klamka, Schauders fixed-point theorem in nonlinear controllability problems, Control Cybern., 29 (2000), 153-165.
  • 25. V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge: Cambridge Academic Publishers, 2009.
  • 26. C. Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl., 243 (2000), 278-292.    
  • 27. H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980), 985-999.    
  • 28. L. Olszowy, Solvability of some functional integral equation, Dynam. System. Appl., 18 (2009), 667-676.
  • 29. L. Olszowy, Existence of mild solutions for semilinear nonlocal Cauchy problems in separable Banach spaces, Z. Anal. Anwend., 32 (2013), 215-232.    
  • 30. L. Olszowy, Existence of mild solutions for semilinear nonlocal problem in Banach spaces, Nonlinear Anal., 81 (2013), 211-223.    
  • 31. L. Olszowy, Wędrychowicz S., Mild solutions of semilinear evolution equation on an unbounded interval and their applications, Nonlinear Anal., 72 (2010), 2119-2126.    
  • 32. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer-Verlag, 1983.
  • 33. J. Prüss, Evolutionary Integral Equations and Applications , Birkhüuser Verlag, 2013.
  • 34. R. Wang, D. Chen, On a class of retarded integro-differential equations with nonlocal initial conditions, Comput. Math. Appl., 59 (2010), 3700-3709.    


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