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Existence Theorem for a Nonlinear Functional Integral Equation and an Initial Value Problem of Fractional Order in L<sub>1</sub>(R<sub>+</sub>)
Mathematics Department, Faculty of Science and Education, Taif University, Al-Khurmah Branch, Taif, KSA
Mathematics Department, Faculty of Science and Education, Taif University, Al-Khurmah Branch, Taif, KSA
Mathematics Department, Faculty of Science and Education, Taif University, Al-Khurmah Branch, Taif, KSA
- 1 Mathematics Department, Faculty of Science and Education, Taif University, Al-Khurmah Branch, Taif, KSA
- 2 Mathematics Department, Faculty of Science and Education, Taif University, Al-Khurmah Branch, Taif, KSA
- 3 Mathematics Department, Faculty of Science and Education, Taif University, Al-Khurmah Branch, Taif, KSA
Applied Mathematics·Volume 04 (2013)·Pages 402–409·Published 22 February 2013·DOI10.4236/am.2013.42060
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Abstract
The aim of this paper is to study the existence of integrable solutions of a nonlinear functional integral equation in the space of Lebesgue integrable functions on unbounded interval, L 1 (R + ). As an application we deduce the existence of solution of an initial value problem of fractional order that be studied only on a bounded interval. The main tools used are Schauder fixed point theorem, measure of weak noncompactness, superposition operator and fractional calculus.
KeywordsNonlinear Functional Integral EquationVolterra OperatorMeasure of Weak NoncompactnessFractional CalculusSchauder Fixed Point Theorem
- J. Banas and Z. Knap, “Integrable Solutions of a Functional-Integral Equation,” Revista Matemática de la Universidad Complutense de Madrid, Vol. 2, No. 1, 1989, pp. 31-38.
- J. Banas and A. Chlebowicz, “On Existence of Integrable Solutions of a Functional Integral Equation under Carathéodory Conditions,” Nonlinear Analysis, Vol. 70, No. 9, 2009, pp. 3172-3179.
- G. Emmanuele, “Integrable Solutions of a Functional Integral Equation,” Journal of Integral Equations and Applications, Vol. 4, No. 1, 1992, pp. 89-94. doi:10.1216/jiea/1181075668
- P. P. Zabrejko, A. I. Koshelev, M. A. Krasnosel’skii, S. G. Mikhlin, L. S. Rakovshchik and V. J. Stecenko, “Integral Equations,” Noordhoff, Leyden, 1975.
- A. M. A. El-Sayed, “Nonlinear Functional Differential Equations of Arbitrary Orders,” Nonlinear Analysis, Vol. 33, No. 2, 1998, pp. 181-186. doi:10.1016/S0362-546X(97)00525-7
- A. M. A. El-Sayed, N. Sherif and I. A. Ibrahim, “On a Mixed Type Integral Equation and Fractional Order Functional Differential Equations,” Commentationes Mathematicae. Prace Matematyczne, Vol. 45, No. 2, 2005, pp. 237-247.
- I. A. Ibrahim, T. S. Amer and Y. M. Abo Essa, “Integrable Solutions of Initial Value Problems of Fractional Order,” Far East Journal of Mathematical Sciences, Vol. 62, No. 1, 2012, pp. 97-123.
- J. Appel, “Implicit Functions, Nonlinear Integral Equations and the Measure of Noncompactness of the Superposition Operator,” Journal of Mathematical Analysis and Applications, Vol. 83, No. 1, 1981, pp. 251-263. doi:10.1016/0022-247X(81)90261-4
- M. A. Krasnosel’skii, P. P. Zabrejko, J. I. Pustyl’nik and P. J. Sobolevskii, “Integral Operators in Spaces of Summable Functions,” Noordhoff, Leyden, 1976.
- R. Pluciennik, “On Some Properties of the Superposition Operator in Generalized Orlicz Spaces of Vector-Valued Functions,” Commentationes Mathematicae. Prace Matematyczne, Vol. 25, No. 2, 1985, pp. 321-337.
- K. Carathéodory, “Vorlesungen über Reele Funktionen,” De Gruyter, Leipzig, 1918.
- J. Appel and P. P. Zabrejko, “Nonlinear Superposition Operators,” In: Cambridge Tracts in Mathematics, Vol. 95, Cambridge University Press, Cambridge, 1990.
- G. Scorza Dragoni, “Un Teorema Sulle Funzioni Continue Rispetto ad une e Misarubili Rispetto ad Un’altra Variable,” Rendiconti del Seminario Matematico della Università di Padova, 1948, pp. 102-106.