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Existence results for Urysohn integral inclusions involving the Henstock integral. (English) Zbl 1123.45004

The author presents two existence results for the following Urysohn integral inclusion
\[ x(t)\in (H)\int_{0}^{1}F(t,s,x(s))\,ds, \]
where \(F:[0,1]\times [0,1]\times X\to {\mathcal P}_{0}(X)\), \(X\) is a separable Banach space, and \({\mathcal P}_{0}(X)\) denotes the family of nonempty subsets of \(X\). Here the set-valued integral involved being the Henstock integral. In particular, the existence of continuous solutions for integral inclusions of Volterra and Hammerstein type is obtained. These results are obtained upon an application of a set-valued variant of Monch’s fixed point theorem.

MSC:

45G10 Other nonlinear integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
26E25 Set-valued functions
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[1] Ambrosetti, A., Un teorema di existenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Univ. Padova, 39, 349-360 (1967) · Zbl 0174.46001
[2] Cao, S. S., The Henstock integral for Banach-valued functions, SEA Bull. Math., 16, 35-40 (1992) · Zbl 0749.28007
[3] Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., vol. 580 (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0346.46038
[4] Chew, T. S.; Flordeliza, F., On \(x^\prime = f(t, x)\) and Henstock-Kurzweil integrals, Differential Integral Equations, 4, 4, 861-868 (1991) · Zbl 0733.34004
[5] Cichón, M.; Kubiaczyk, I.; Sikorska, A., The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem, Czechoslovak Math. J., 54, 129, 279-289 (2004) · Zbl 1080.34550
[6] Di Piazza, L., Kurzweil-Henstock type integration on Banach spaces, Real Anal. Exchange, 29, 2, 543-556 (2003/2004) · Zbl 1083.28007
[7] Di Piazza, L.; Musial, K., A decomposition theorem for compact-valued Henstock integral, Monatsh. Math., 148, 2, 119-126 (2006) · Zbl 1152.28016
[8] Di Piazza, L.; Musial, K., Set-valued Kurzweil-Henstock-Pettis integral, Set-Valued Anal., 13, 2, 167-179 (2005) · Zbl 1100.28008
[9] Federson, M.; Bianconi, R., Linear integral equations of Volterra concerning Henstock integrals, Real Anal. Exchange, 25, 1, 389-418 (1999/2000) · Zbl 1015.45001
[10] Federson, M.; Táboas, P., Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals, Nonlinear Anal., 50, 3, 389-407 (2002) · Zbl 1011.34070
[11] Fremlin, D. H.; Mendoza, J., On the integration of vector-valued functions, Illinois J. Math., 38, 1, 127-147 (1994) · Zbl 0790.28004
[12] Gordon, R. A., The Integrals of Lebesgue, Denjoy, Perron and Henstock, Grad. Stud. in Math., vol. 4 (1994), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0807.26004
[13] Heinz, H. P., On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7, 1351-1371 (1983) · Zbl 0528.47046
[14] Michael, E., Continuous selection I, Ann. of Math., 63, 361-381 (1956)
[15] Mitchell, A. A.; Smith, C., An existence theorem for weak solutions of differential equations in Banach spaces, (Nonlinear Equations in Abstract Spaces. Nonlinear Equations in Abstract Spaces, Proc. Internat. Sympos., Univ. Texas, Arlington, TX, 1977 (1978), Academic Press: Academic Press New York), 387-403
[16] Mönch, H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4, 985-999 (1980) · Zbl 0462.34041
[17] O’Regan, D.; Precup, R., Fixed point theorems for set-valued maps and existence principles for integral inclusions, J. Math. Anal. Appl., 245, 594-612 (2000) · Zbl 0956.47026
[18] O’Regan, D.; Precup, R., Existence criteria for integral equations in Banach spaces, J. Inequal. Appl., 6, 77-97 (2001) · Zbl 0993.45011
[19] Satco, B., Second order three boundary value problem in Banach spaces via Henstock and Henstock-Kurzweil-Pettis integral, J. Math. Anal. Appl., 332, 919-933 (2007) · Zbl 1127.34033
[20] Satco, B., Volterra integral inclusions via Henstock-Kurzweil-Pettis integral, Discuss. Math. Differ. Incl. Control Optim., 26, 87-101 (2006) · Zbl 1131.45001
[21] Schwabik, S., The Perron integral in ordinary differential equations, Differential Integral Equations, 6, 4, 863-882 (1993) · Zbl 0784.34006
[22] Sikorska-Nowak, A., Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals, Demonstratio Math., 35, 1, 49-60 (2002) · Zbl 1011.34066
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