Abstract

Abstract. In this note, we will use a compactness type condition in connection with the weak topology to prove the existence of weakly continuous solutions for a functional integral equation of Volterra-Stieltjes type in nonreflexive Banach spaces. A special case will be considered.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 3
Year: 2019

DOI: 10.12732/ijam.v32i3.11

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References

1. [1] A. Alexiewicz and W. Orlicz, Remarks on Riemann-integration of vector valued functions, Studia Math., 12 (1951), 125-132.
2. [2] A. Ambrosetti, Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Semin. Math. Univ. Padova, 39 (1967), 349-369.
3. [3] J. Bana´s, Some properties of Urysohn-Stieltjes integral operators, Intern. J. Math. and Math. Sci., 21 (1998), 78-88.
4. [4] J. Bana´s, J.R. Rodriguez and K. Sadarangani, On a class of UrysohnStieltjes quadratic integral equations and their applications, J. Comput. Appl. Math., 13 (2000), 35-50.
5. [5] J. Bana´s and J. Dronka, Integral operators of Volterra-Stieltjes type, their properties and applications, Math. Comput. Modelling, 32 (2000), 13211331.
6. [6] J. Bana´s and K. Sadarangani, Solvability of Volterra-Stieltjes operatorintegral equations and their applications, Comput. Math. Appl., 41 (2001), 1535-1544.
7. [7] J. Bana´s, J.C. Mena, Some properties of nonlinear Volterra-Stieltjes integral operators, Comput. Math. Appl., 49 (2005), 1565-1573.
8. [8] J. Bana´s, D. O’Regan, Volterra-Stieltjes integral operators, Math. Comput. Modelling, 41 (2005), 335-344.
9. [9] C.W. Bitzer, Stieltjes-Volterra integral equations, Illinois J. Math., 14 (1970), 434-451.
10. [10] J.M. Ball, Weak continuity properties of mappings and semigroups, Proc. Royal. Soc. Edinbourgh Sect., 72 (1973-1974), 275-280.
11. [11] M. Boudourides, An existence theorem for ordinary differential equations in Banach spaces, Bull. Austral. Math. Soc., 22 (1980), 457-463.
12. [12] E. Cramer, V. Lakshmiksntham and A.R. Mitchell, On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal., 2 (1978), 259-262.
13. [13] M. Cicho´n, Weak solutions of differential equations in Banach spaces, Discuss. Math. Differ. Inc., 15 (1995), 5-14.
14. [14] A.M.A. EL-Sayed, W.G. EL-Sayed and A.A.H. Abd EL-Mowla, VolterraStieltjes integral equation in reflexive Banach spaces, Electr. J. Math. Anal. Appl., 5 (2017), 287-293.
15. [15] A.M.A. EL-Sayed, W.G. EL-Sayed and A.A.H. Abd EL-Mowla, Weak solutions of fractional order differential equations via Volterra-Stieltjes integral operator, J. Math. Appl., 40 (2017), 85-96.
16. [16] R.F. Geitz, Pettis integration, Proc. Amer. Math. Soc., 82 (1981), 81-86.
17. [17] R. Gordon, Riemann integration in Banach spaces, Rocky Mountain J. Math., 3 (1991), 923-949.
18. [18] I. Kubiaczyk, On fixed point theorem for weakly sequentially continuous mappings, Discuss. Math. Differ. Incl., 15 (1995), 15-20.
19. [19] A. Szep, Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar., 6 (1971), 197203.
20. [20] S. Szufla, Kneser’s theorem for weak solutions of ordinary differential equation in reflexive Banach spaces, Bull. Polish Acad. Sci. Math., 26 (1978), 407-413.