Abstract
The goal of the present work is to give an existence result for a nonlinear integral equation on time scales by considering the Banach space endowed with its weak topology. More precisely, we obtain the existence of weakly continuous solutions for an integral equation that has on the right hand side the sum of two operators, one of them continuous while the other one satisfies a partial continuity condition and some integrability (in a nonabsolute sense) assumptions.
[1] AGARWAL, R. P.— BOHNER, M.— PETERSON, A.: Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (2001), 535–557. Search in Google Scholar
[2] AVSEC, S.— BANNISH, B.— JOHNSON, B.— MECKLER, S.: The Henstock-Kurzweil delta integral on unbounded time scales, Panamer. Math. J. 16 (2006), 77–98. Search in Google Scholar
[3] BOHNER, M.— GUSEINOV, G. SH.: Riemann and Lebesgue integration. In: Advances in Dynamic Equations on Time Scales (M. Bohner, A. C. Peterson, eds.), Birkhäuser Boston, Boston, MA, 2003, pp. 117–163. http://dx.doi.org/10.1007/978-0-8176-8230-9_510.1007/978-0-8176-8230-9_5Search in Google Scholar
[4] BOHNER, M.— GUSEINOV, G. SH.: Improper integrals on time scales, Dynam. Systems Appl. 12 (2003) 45–65. Search in Google Scholar
[5] BOHNER, M.— GUSEINOV, G. SH.: Line integrals and Green’s formula on time scales, J. Math. Anal. Appl. 326 (2007), 1124–1141. http://dx.doi.org/10.1016/j.jmaa.2006.03.04010.1016/j.jmaa.2006.03.040Search in Google Scholar
[6] BOHNER, M.— PETERSON, A.: Dynamic Equations on Time Scales: An Introduction with Applications, Birkäuser, Boston, MA, 2001. http://dx.doi.org/10.1007/978-1-4612-0201-110.1007/978-1-4612-0201-1Search in Google Scholar
[7] BOHNER, M.— PETERSON, A.: Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. http://dx.doi.org/10.1007/978-0-8176-8230-910.1007/978-0-8176-8230-9Search in Google Scholar
[8] BUGAJEWSKI, D.: On the Volterra integral equation and axiomatic measures of weak noncompactness, Math. Bohem. 126(10) (2001), 183–190. 10.21136/MB.2001.133913Search in Google Scholar
[9] CABADA, A.— VIVERO, D. R.: Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral; application to the calculus of Δ-antiderivatives, Math. Comput. Modelling 43 (2006), 194–207. http://dx.doi.org/10.1016/j.mcm.2005.09.02810.1016/j.mcm.2005.09.028Search in Google Scholar
[10] CAO, S. S.: The Henstock integral for Banach valued functions, Southeast Asian Bull. Math. 16 (1992), 36–40. Search in Google Scholar
[11] CHEW, T. S.: On Kurzweil generalized ordinary differential equations, J. Differential Equations 76 (1988), 286–293. http://dx.doi.org/10.1016/0022-0396(88)90076-910.1016/0022-0396(88)90076-9Search in Google Scholar
[12] CHEW, T. S.— FLORDELIJA, F.: On x′ = f (t,x) and Henstock-Kurzweil integrals, Differential Integral Equations 4 (1991), 861–868. 10.57262/die/1371225020Search in Google Scholar
[13] CICHOŃ, M.: On solutions of differential equations in Banach spaces, Nonlinear Anal. 60 (2005), 651–667. http://dx.doi.org/10.1016/j.na.2004.09.04110.1016/j.na.2004.09.041Search in Google Scholar
[14] CICHOŃ, M.: Weak solutions of differential equations in Banach spaces, Discuss. Math. Differential Incl. 15 (1995) 5–14. Search in Google Scholar
[15] CICHOŃ, M.: On integrals of vector-valued functions on time scales, Comm. Math. Anal. 11 (2011), 94–110. Search in Google Scholar
[16] CICHOŃ, M.— KUBIACZYK, I.— SIKORSKA-NOWAK, A.— YANTIR, A.: Weak solutions for the dynamic Cauchy problem in Banach spaces, Nonlinear Anal. 71 (2009), 2936–2943. http://dx.doi.org/10.1016/j.na.2009.01.17510.1016/j.na.2009.01.175Search in Google Scholar
[17] CORDUNEANU, C.: Abstract Volterra equations and weak topologies. In: Delay Differential Equations and Dynamical Systems (S. Busenberg, M. Martelli, eds.). Lecture Notes in Math. 1475, Springer, Berlin-Heidelberg-New York, 1991, pp. 110–116. http://dx.doi.org/10.1007/BFb008348410.1007/BFb0083484Search in Google Scholar
[18] CRAMER, E.— LAKSHMIKANTHAM, V.— MITCHELL, A. R.: On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal. 2 (1978), 169–177. http://dx.doi.org/10.1016/0362-546X(78)90063-910.1016/0362-546X(78)90063-9Search in Google Scholar
[19] DI PIAZZA, L.— SATCO, B.: A new result on impulsive differential equations involving non-absolutely convergent integrals, J. Math. Anal. Appl. 352 (2009), 954–963. http://dx.doi.org/10.1016/j.jmaa.2008.11.04810.1016/j.jmaa.2008.11.048Search in Google Scholar
[20] FEDERSON, M.— TÁBOAS, P.: Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals, Nonlinear Anal. 50 (2002), 389–407. http://dx.doi.org/10.1016/S0362-546X(01)00769-610.1016/S0362-546X(01)00769-6Search in Google Scholar
[21] FERRERA, J.— GOMEZ, J.— LLAVONA, J. G.: On completion of spaces of weakly continuous functions, Bull. London Math. Soc. 15 (1983), 260–264. http://dx.doi.org/10.1112/blms/15.3.26010.1112/blms/15.3.260Search in Google Scholar
[22] GORDON, R. A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock. Grad. Stud. Math. 4, Amer. Math. Soc., Providence, RI, 1994. 10.1090/gsm/004Search in Google Scholar
[23] GUSEINOV, G. Sh.: Integration on time scales, J. Math. Anal. Appl. 285 (2003), 107–127. http://dx.doi.org/10.1016/S0022-247X(03)00361-510.1016/S0022-247X(03)00361-5Search in Google Scholar
[24] HEIKKILÄ, S.— KUMPULAINEN, M.— SEIKKALA, S.: Convergence theorems for HL integrable vector-valued functions with applications, Nonlinear Anal. 70 (2009), 1939–1955. http://dx.doi.org/10.1016/j.na.2008.02.09310.1016/j.na.2008.02.093Search in Google Scholar
[25] HILGER, S.: Analysis on measure chains — a unified approach, Results Math. 18 (1990), 18–56. http://dx.doi.org/10.1007/BF0332315310.1007/BF03323153Search in Google Scholar
[26] KRASNOSELSKIJ, M. A.— KREIN, S. G.: Theory of ordinary differential equations in Banach spaces, Trudy Sem. Funk. Anal. Voronezh. Univ. 2 (1956), 3–23 (Russian). Search in Google Scholar
[27] KUBIACZYK, I.— SZUFLA, S.: Kneser’s theorem for weak solutions of ordinary differential equations in Banach spaces, Publ. Inst. Math. (Beograd) (N.S.) 46 (1982), 99–103. Search in Google Scholar
[28] KULIK, T.— TISDELL, C.: Volterra Integral Equations on Time Scales: Basic Qualitative and Quantitative Results with Applications to Initial Value Problems on Unbounded Domains, Int. J. Difference Equations 3 (2008), 103–133. Search in Google Scholar
[29] KURZWEIL, J.: Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), 618–646. Search in Google Scholar
[30] KURZWEIL, J.— SCHWABIK, S.: Ordinary differential equations the solution of which are ACG*-functions, Arch. Math. (Brno) 26 (1990), 129–136. Search in Google Scholar
[31] O’REGAN, D.: Integral equations in reflexive Banach spaces and weak topologies, Proc. Amer. Math. Soc. 124 (1996), 607–614. http://dx.doi.org/10.1090/S0002-9939-96-03154-110.1090/S0002-9939-96-03154-1Search in Google Scholar
[32] O’REGAN, D.: Operator equations in Banach spaces relative to the weak topology, Arch. Math. (Basel) 71 (1998), 123–136. http://dx.doi.org/10.1007/s00013005024310.1007/s000130050243Search in Google Scholar
[33] PARK, S.: Generalizations of the Krasnoselskii fixed point theorem, Nonlinear Anal. 49 (2007), 3401–3410. http://dx.doi.org/10.1016/j.na.2006.10.02410.1016/j.na.2006.10.024Search in Google Scholar
[34] PETERSON, A.— THOMPSON, B.: Henstock-Kurzweil delta and nabla integrals, J. Math. Anal. Appl. 323 (2006), 162–178. http://dx.doi.org/10.1016/j.jmaa.2005.10.02510.1016/j.jmaa.2005.10.025Search in Google Scholar
[35] SCHWABIK, S.: The Perron integral in ordinary differential equations, Differential Integral Equations 6 (1993), 836–882. 10.57262/die/1370032239Search in Google Scholar
[36] SIKORSKA-NOWAK, A.: Integrodifferential equations on time scales with Henstock-Kurzweil-Pettis delta integrals, Abstr. Appl. Anal. 2010 (2010), doi:10.1155/2010/836347, 17 pp. 10.1155/2010/836347Search in Google Scholar
[37] SZUFLA, S.: Sets of fixed points of nonlinear mappings in function spaces, Funkcial. Ekvac. 22 (1979), 121–126. Search in Google Scholar
[38] SZUFLA, S.: On the Kneser-Hukuhara property for integral equations in locally convex spaces, Bull. Austral. Math. Soc. 36 (1987), 353–360. http://dx.doi.org/10.1017/S000497270000364610.1017/S0004972700003646Search in Google Scholar
[39] VLADIMIRESCU, C.: Remark on Krasnosel’skii fixed point theorem, Nonlinear Anal. 71 (2009), 876–880. http://dx.doi.org/10.1016/j.na.2008.10.11510.1016/j.na.2008.10.115Search in Google Scholar
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