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Weak solution for fractional order integral equations in reflexive Banach spaces. (English) Zbl 1111.26011
Summary: We define the fractional order Pettis-integral operator in reflexive Banach spaces and we investigate the properties of such operator. A fixed point theorem is used to establish an existence result for the nonlinear Pettis-fractional order integral equation of the following type \[ x(t)=g(t)+\lambda \, I^{\alpha }f\bigl (t,x(t)\bigr )\,, \qquad t\in [0,1]\,,\;\;0<\alpha <1\,. \] Moreover, the existence of a solution for the Cauchy problem \[ \frac {d\!x}{d\!t} = f\bigl (t,\operatorname {D}^{\beta }\!x(t)\bigr ), \qquad t\in [0,1]\,,\^^M\;0< \beta <1\,,\;\;x(0) = x_0\,, \] is proved.

MSC:
26A33 Fractional derivatives and integrals
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47G10 Integral operators
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References:
[1] AL-ABEDEEN A. Z.-ARORA H. L.: A global existence and uniqueness theorem for ordinary differential equations of generalized order. Canad. Math. Bull. 21 (1978), 267-271. · Zbl 0397.34003
[2] ARINO O.-GAUTIER S.-PENOT T. P. : A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations. Funkcial. Ekvac 27 (1984), 273-279. · Zbl 0599.34008
[3] BASSAM M.: Some existence theorems on differential equations of generalized order. J. Reine Angew. Math. 218 (1965), 70-78. · Zbl 0156.30804
[4] CICHOŃ M.: Weak solutions of ordinary differential equations in Banach spaces. Discuss. Math. Differ. Inch Control Optim. 15 (1995), 5-14. · Zbl 0829.34051
[5] CICHOŃ M.-EL-SAYED A. M.-SALEM H. A. H.: Existence theorem for nonlinear functional integral equations of fractional orders. Comment. Math. Prace Mat. 41 (2001), 59-67. · Zbl 1011.45002
[6] CRAMER E.-LAKSHMIKANTHAM V.-MITCHELL A. R.: On the existence of weak solutions of differential equations in nonreflexive Banach spaces. Nonlinear Anal. 2 (1978), 259-262. · Zbl 0379.34041
[7] DIESTEL J.-UHL J. J., Jr.: Vector Measures. Math. Surveys Monogr. 15, Amer. Math. Soc, Providence, R.I., 1977. · Zbl 0369.46039
[8] EDGAR G. A.: Geometry and the Pettis-integral. Indiana Univ. Math. J. 26 (1977), 663-677.
[9] EDGAR G. A.: Geometry and the Pettis-integral II. Indiana Univ. Math. J. 28 (1979), 559-579. · Zbl 0418.46034
[10] EL-SAYED A. M.-EL-SAYED W. G.-MOUSTAFA O. L.: On some fractional functional equations. Pure. Math. Appl. 6 (1995), 321-332. · Zbl 0867.49001
[11] EL-SAYED A. M. A.: Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33 (1998), 181-186. · Zbl 0934.34055
[12] EL-SAYED A. M. A.-IBRAHIM A. G.: Set-valued integral equations of arbitrary (fractional) order. Appl. Math. Comput. 118 (2001), 113-121. · Zbl 1024.45003
[13] GEITZ R. F.: Pettis integration. Proc Amer. Math. Soc 82 (1981), 81-86. · Zbl 0506.28007
[14] GEITZ R. F.: Geometry and the Pettis integration. Trans. Amer. Math. Soc. 269 (1982), 535-548. · Zbl 0498.28005
[15] HADID S. B. : Local and global existence theorem on differential equation on non integral order. Math. Z. 7 (1995), 101-105. · Zbl 0839.34003
[16] HILLE E.-PHILLIPS R. S.: Functional Analysis and Semi-groups. Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc, Providence, R.I., 1957.
[17] KNIGHT W. J.: Solutions of differential equations in B-spaces. Duke Math. J. 41 (1974), 437-442. · Zbl 0288.34063
[18] KUBIACZYK I.-SZUFLA S.: Kneser’s theorem for weak solutions of ordinary differential equations in Banach spaces. Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 99-103. · Zbl 0516.34058
[19] MILLER K. S.-ROSS B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley, New York, 1993. · Zbl 0789.26002
[20] MITCHELL A. R.-SMITH, CH.: An existence theorem for weak solutions of differential equations in Banach spaces. Nonlinear Equations in Abstract Spaces. Proc. Int. Symp., Arlington 1977, 1978, pp. 387-403.
[21] O’REGAN D.: Fixed point theory for weakly sequentially continuous mapping. Math. Comput. Modeling 27 (1998), 1-14. · Zbl 1185.34026
[22] PETTIS B. J.: On integration in vector spaces. Trans. Amer. Math. Soc. 44 (1938), 277-304. · Zbl 0019.41603
[23] PHILLIPS R. S.: Integration in a convex linear topological space. Trans. Amer. Math. Soc. 47 (1940), 114-115. · Zbl 0022.31902
[24] PODLUBNY I.-EL-SAYED A. M. A.: On two definitions of fractional calculus. Preprint UEF-03-96, Slovak Academy of Sciences, Institute of Experimental Phys., 1996.
[25] PODLUBNY I.: Fractional Differential Equation. Acad. Press, San Diego-New York-London, 1999. · Zbl 0924.34008
[26] SALEM H. A. H.-EL-SAYED A. M. A.-MOUSTAFA O. L.: Continuous solutions of some nonlinear fractional order integral equations. Comment. Math. Prace Mat. 42 (2002), 209-220. · Zbl 1033.45003
[27] SALEM H. A. H.-VÄTH M.: An abstract Gronwall lemma and application to global existence results for functional differential and integral equations of fractional order. J. Integral Equations Appl. 16 (2004), 411-439. · Zbl 1080.45006
[28] SAMKO S.-KILBAS A.-MARICHEV O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Sci. PubL, New York, 1993. · Zbl 0818.26003
[29] VÄTH M.: Ideal spaces. Lecture Notes in Math. 1664, Springer, Berlin-Heidelberg, 1997. · Zbl 0896.46018
[30] SZEP A.: Existence theorem for weak solutions of differential equations in Banach spaces. Studia Sci. Math. Hungar. 6 (1971), 197-203. · Zbl 0238.34100
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