Weak Solutions for a Coupled System of Partial Pettis Hadamard Fractional Integral Equations

Abstract

In this paper we investigate the existence of weak solutions under the Pettis integrability assumption for a coupled system of partial integral equations via Hadamard’s fractional integral, by applying the technique of measure of weak noncompactness and Mönch’s fixed point theorem.

Keywords

• Functional integral equation,coupled system,partial Pettis Hadamard,fractional integral measure of weak noncompactness,weak solution

Kaynakça

1. S. Abbas, M. Benchohra and G.M. N’Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012.
2. S. Abbas, M. Benchohra and G.M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
3. S. Abbas, M. Benchohra and A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Frac. Calc. Appl. Anal. 15 (2012), 168–182.
4. R.R. Akhmerov, M.I. Kamenskii, A.S. Patapov, A.E. Rodkina and B.N. Sadovskii, Measures of Noncompactness and Condensing Operators. Birkhauser Verlag, Basel, 1992.
5. J.C. Alvàrez, Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces, Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid 79 (1985), 53–66.
6. J. Bana`s and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.
7. M. Benchohra, J.R. Graef and F-Z. Mostefai, Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces, Electron. J. Qual. Theory Differ. Equ. 54 (2010), 1–10.
8. M. Benchohra, J. Graef and F-Z. Mostefai,Weak solutions for boundary-value problems with nonlinear fractional differential inclusions, Nonlinear Dyn. Syst. Theory 11 (3) (2011), 227–237.

9. M. Benchohra, J. Henderson and F-Z. Mostefai, Weak solutions for hyperbolic partial fractional differential inclusions in Banach spaces, Comput. Math. Appl. 64 (2012), 3101–3107.
10. M. Benchohra, J. Henderson and D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal. 12 (4) (2008), 419–428.
11. M. Benchohra and F-Z. Mostefai, Weak solutions for nonlinear differential equations with integral boundary conditions in Banach spaces, Opuscula Math. 32 (1) (2012), 31–40.
12. M. Benchohra, J.J. Nieto and D. Seba, Measure of noncompactness and hyperbolic partial fractional differential equations in Banach spaces, PanAmer. Math. J. 20 (3) (2010), 27–37.
13. D. Bugajewski and S. Szufla, Kneser’s theorem for weak solutions of the Darboux problem in a Banach space, Nonlinear Anal. 20 (2) (1993), 169–173.
14. P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269 (2002), 1–27.
15. P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270 (2002), 1–15.
16. M.A. Darwish, On integral equations of UrysohnâASVolterra type, Appl. Math. Comput. 136 (1) (2003), 93–98.
17. M.A. Darwish, J. Henderson and D. O’Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument, Bull. Korean Math. Soc. 48 (3) (2011), 539–553.
18. M.A. Darwish and S.K. Ntouyas, On a quadratic fractional HammersteinâASVolterra integral equation with linear modification of the argument, Nonlinear Anal. 74 (11) (2011), 3510–3517.
19. M.A. Darwish, On a perturbed functional integral equation of Urysohn type, Appl. Math. Comput. 218 (2012), 8800- 8805.
20. M.A. Darwish and J. Henderson, Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type, Rocky Mountain J. Math. 42 (2) (2012), 545–566.
21. M.A. Darwish and J. Banas, Existence and characterization of solutions of nonlinear Volterra-Stieltjes integral equations in two vriables, Abstr. Appl. Anal. 2014, Art. ID 618434, 11 pp.
22. F.S. De Blasi, On the property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259–262.
23. D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, 1996.
24. J. Hadamard, Essai sur l’étude des fonctions données par leur développment de Taylor, J. Pure Appl. Math. 4 (8) (1892), 101–186.
25. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
26. A.A. Kilbas, Hari M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam, 2006.
27. K. Latrach and M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L1 spaces. Nonlinear Anal. 66 (2007), 2325–2333.
28. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
29. A. R. Mitchell and Ch. Smith, Nonlinear Equations in Abstract Spaces. In: Lakshmikantham, V. (ed.) An existence theorem for weak solutions of differential equations in Banach spaces, pp. 387âAS403. Academic Press, New York (1978)
30. H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4 (1980), 985–999.
31. H. Mönch and G.F. Von Harten, On the Cauchy problem for ordinary differential equations in Banach spaces, Archiv. Math. Basel 39 (1982), 153–160.
32. D. O’Regan, Fixed point theory for weakly sequentially continuous mapping, Math. Comput. Model. 27 (5) (1998), 1–14.
33. D. O’Regan, Weak solutions of ordinary differential equations in Banach spaces, Appl. Math. Lett. 12 (1999), 101–105.
34. A. Petrusel, G. Petrusel, A study of a general system of operator equations in b-metric spaces via the vector approach in fixed point theory. J. Fixed Point Theory Appl. 19 (2017), 1793-1814.
35. B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277–304.
36. S. Pooseh, R. Almeida, and D. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative. Numer. Funct. Anal. Optim. 33 (3) (2012), 301–319.
37. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
38. S. Szufla, On the application of measure of noncompactness to existence theorems, Rend. Sem. Mat. Univ. Padova 75 (1986), 1–14.
39. M.A. Taoudi, Integrable solutions of a nonlinear functional integral equation on an unbounded interval, Nonlinear Anal. 71 (2009), 4131–4136.
40. V.E. Tarasov, Fractional dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010.
41. A.N. Vityuk, On solutions of hyperbolic differential inclusions with a nonconvex right-hand side. (Russian) Ukran. Mat. Zh. 47 (4) (1995), 531–534; translation in Ukrainian Math. J. 47 (4) (1996), 617–621.
42. A.N. Vityuk and A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (2004), 318–325.