A More General Existence Theorem for Weak Solutions in Banach Spaces
DOI:
https://doi.org/10.58987/8nzmrz73Keywords:
Cauchy problem, Volterra-Stieltjes integral, weak continuity, Measure of weakAbstract
This paper contains a Kubiaczyks fixed point theorem for weakly continuous solutions of a Volterra-Stieltjes integral equation in nonreflexive Banach spaces. The Kubiaczyk's fixed point theorem is applied to establish the existence of a solution. A compactness type condition in connection with the weak topology is used. The existence is proved by defining an integral operator F which is shown to be weakly sequentially continuous on a closed, convex, and equicontinuous subset Q of weakly continuous functions Cw. The measure of weak noncompactness β is a crucial tool in this proof. As an application, the existence of a weakly continuous solution for Cauchy problems in Banach spaces is deduced. It is also shown that a fractional integral equation in the sense of weakly Riemann is a special case of the integral equation of Volterra -Stieltjes type. This leads to the existence of weak solutions in Cw for Cauchy problems involving the fractional Caputo weak derivative.
References
[1] J. Banaś, “Some properties of Urysohn-Stieltjes integral operators,” Intern. J. Math. And Math. Sci, Vol. 21, 78-88, 1998.
[2] J. Bana´s, J.R. Rodriguez, K. Sadarangani, “On a class of Urysohn-Stieltjes quadratic integral equations and their applications,” J. Comput. Appl. Math, Vol. 113(1-2), 35-50, 2000.
[3] J. Bana´s, J. Dronka, “Integral operators of Volterra-Stieltjes type, their properties and applications,” Math. Comput. Modelling, Vol. 32, 1321-1331, 2000.
[4] J. Bana´s, K. Sadarangani, “Solvability of Volterra-Stieltjes operator-integral equautions and their applications,” Comput Math. Appl. Vol. 41, 1535-1544, 2001.
[5] J. Bana´s, J.C. Mena, “Some Properties of Nonlinear Volterra-Stieltjes Integral Operators,” Comput Math. Appl. Vol. 49, 1565-1573, 2005.
[6] J. Bana´s, D. O’Regan, “Volterra-Stieltjes integral operators,” Math. Comput. Modelling, Vol. 41, 335-344, 2005.
[7] J. Bana´s, T. Zajac, “A new approach to the theory of functional integral equations of fractional order,” J. Math. Anal. Appl, Vol. 375, 375-387, 2011.
[8] C.W. Bitzer, “Stieltjes-Volterra integral equations”, Illinois J. Math, Vol. 14, 434-451,1970.
[9] A. Szep, “Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces,” Studia Sci. Math. Hungar, Vol. 6, 197-203, 1971.
[10] H.A.H. Salem, A.M.A. El-Sayed, “A note on the fractional calculus in Banach spaces,” Studia Scientiarum Mathematicarum Hungarica, Vol. 42(2), 115-130, 2005.
[11] M. Boudourides, “An existence theorem for ordinary differential equations in Banach spaces,” Bull Austral Math Soc. Vol. 22, 457-63, 1980.
[12] E. Cramer, V. Lakshmiksntham, A.R. Mitchell, “On the existence of weak solutions of differential equations in nonreflexive Banach spaces,” Nonlinear Anal, Vol. 2, 259-262, 1978.
[13] R.P. Agarwal, V. Lupulescu, D. O'Regan. G.U. Rahman, “Nonlinear fractional differential equations in nonreflexive Banach spaces and fractional calculus,” Advances in Difference Equations, Vol. 2015, 112, 2015.
[14] R.P. Agarwal, V. Lupulescu, D. O’Regan, G.U. Rahman, “Weak solutions for fractional differential equations in nonreflexive Banach spaces via Riemann-Pettis integrals,” Math. Nachr, Vol. 4, 395-409, 2016.
[15] A.M.A. EL-Sayed, W.G. EL-Sayed, A.A.H. Abd EL-Mowla, “Volterra-Stieltjes integral equation in reflexive Banach spaces,” Electronic Journal of Mathematical Analysis and Applications, Vol. 5(1), 287-293, 2017.
[16] A.M.A. EL-Sayed, W.G. EL-Sayed, A.A.H. Abd EL-Mowla, “Weak solutions of fractional order differential equations via Volterra-Stieltjes integral operator,” Journal of Mathematics and Applications, Vol. 40, 85-96, 2017.
[17] J.M. Ball, “Weak continuity properties of mappings and semigroups,” Proc. Royal. Soc. Edinbourgh Sect, Vol. 72, 275-280, 1975.
[18] S. Szufla, “Kneser’s theorem for weak solutions of ordinary differential equation in reflexive Banach spaces,” Bull. Polish Acad. Sci. Math, Vol. 26, 407-413, 1978.
[19] A. Ambrosetti, “Un teorema di esistenza per le equazioni differenziali negli spazi di Banach,” Rend. Semin. Mat. Univ. Padova, Vol. 39, 349-369, 1967.
[20] M. Cichon´, “Weak solutions of differential equations in Banach spaces,” Discuss. Math. Differ. Inc, Vol. 15, 5-14, 1995.
[21] I. Kubiaczyk, “On fixed point theorem for weakly sequentially continuous mappings,” Discuss. Math. Differ. Incl, Vol. 15, 15-20, 1995.
[22] R.F. Geitz, “Pettis integration,” Proc. Amer. Math. Soc., Vol. 82, 81-86, 1981.
[23] A. Alexiewicz, W. Orlicz, “Remarks on Riemann-integration of vector valued functions,” Studia Math, Vol. 12, 125-132, 1951.
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