Measure pseudoasymptotically Bloch periodic functions in the sense of Stepanov and applications

Youssef Khemili; Mohsen Miraoui; Mounir Ben Salah

doi:10.3842/umzh.v74i4.7339

Youssef Khemili FSS, Sfax University, Tunisia
Mohsen Miraoui IPEIK, Kairouan University, Tunisia
Mounir Ben Salah IPEIK, Kairouan University, Tunisia

DOI: https://doi.org/10.3842/umzh.v74i4.7339

Keywords: Stepanov-like pseudo-asymptotically Bloch periodic functions; positive measure; nonlinear differential equation; fixed point theorem.

Abstract

UDC 517.9

We focus on the measures of Stepanov-like pseudoasymptotically Bloch $\tau$-periodicity and its applications. First, we define a new notion of measures of Stepanov-like pseudoasymptotically Bloch periodic functions and diccuss some of its fundamental properties. Then the obtained results are applied to investigate the existence and uniqueness of the measure Stepanov-like pseudoasymptotically Bloch periodic mild solutions to semilinear delay differential equation in Banach spaces. Finally, an application is presented to illustrate the efficiency of the results.

References

Y. K. Chang, Y. Wei, Pseudo S-asymptotically Bloch type periodic solutions to fractional integro-differential equations with Stepanov-like force terms, Z. Angew. Math. und Phys., 73(77) (2022). DOI: https://doi.org/10.1007/s00033-022-01722-y

E. H. Ait Dads, K. Ezzinbi, M. Miraoui, ($mu$, $nu$)-Pseudo almost automorphic solutions for some non-autonomous differential equations, Int. J. Math., 26, № 11, Article~1550090 (2015). DOI: https://doi.org/10.1142/S0129167X15500901

H. Assel, M. A. Hammami, M. Miraoui, Dynamics and oscillations for some difference and differential equations with piecewise constant arguments, Asian J. Control, 1–9 (2021); https://doi.org/10.1002/asjc.2619. DOI: https://doi.org/10.1002/asjc.2619

F. Bloch, Über die Quantenmechanik der elektronen in Kristallgittern, Z. Physik, 52 (1929). DOI: https://doi.org/10.1007/BF01339455

J. Blot, P. Cieutat, K. Ezzinbi, Measure theory and pseudo almost automorphic functions: new developments and applications, Nonlinear Anal., 75, 2426–2447 (2012). DOI: https://doi.org/10.1016/j.na.2011.10.041

Y. K. Chang, Y. Wei, Pseudo $S$-asymptotically Bloch type periodicity with applications to some evolution equations, Z. Anal. und Anwend., 40, 33–50 (2021). DOI: https://doi.org/10.4171/zaa/1671

C. Cuevas, J. C. Souza, Existence of $S$-asymptotically $omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72, 1683–1689 (2010). DOI: https://doi.org/10.1016/j.na.2009.09.007

T. Diagana, Stepanov-like pseudo-almost periodic functions and their applications to differential equations, Commun. Math. Anal., 3, № 1, 9–18 (2007).

T. Diagana, G. M. N'Guérékata, Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. Anal., 86, 723–733 (2007). DOI: https://doi.org/10.1080/00036810701355018

T. Diagana, G. M. Mophou, G. M. N'Guérékata, Existence of weighted pseudo-almost periodic solutions to some classes of differential equations with $S^p$-weighted pseudo-almost periodic coefficients, Nonlinear Anal., 72, № 1, 430–438 (2010). DOI: https://doi.org/10.1016/j.na.2009.06.077

H. S. Ding, J. Liang, T. J. Xiao, Some properties Stepanov-like almost automorphic functions and applications to abstract evolution equations, Appl. Anal., 88, 1079–1091 (2009). DOI: https://doi.org/10.1080/00036810903156164

Z. B. Fan, J. Liang, T. J. Xiao, Composition of Stepanov-like pseudo almost automorphic functions and applications to nonautomous evolution equations, Nonlinear Anal., 13, 131–140 (2012). DOI: https://doi.org/10.1016/j.nonrwa.2011.07.019

Z. B. Fan, J. Liang, T. J. Xiao, On Stepanov-like (pseudo) almost automorphic functions, Nonlinear Anal., 74, 2853–2861 (2011). DOI: https://doi.org/10.1016/j.na.2011.01.007

H. Gao, K. Wang, F. Wei, X. Ding, Massera-type theorem and asymptotically periodic logistic equations, Nonlinear Anal. Real World Appl., 7, 1268–1283 (2006). DOI: https://doi.org/10.1016/j.nonrwa.2005.11.008

M. F. Hasler, G. M. N'Guérékata, Bloch-periodic functions and some applications, Nonlinear Stud., 21, 21–30 (2014).

H. R. Henriquez, C. Cuevas, A. Caicedo, Asymptotically periodic solutions of neutral partial differential equations with infinite delay, Commun. Pure and Appl. Anal., 12, 2031–2068 (2013). DOI: https://doi.org/10.3934/cpaa.2013.12.2031

M. Miraoui, Measure pseudo almost periodic solutions for differential equations with reflection, Appl. Anal. (2020);

DOI: 10.1080/00036811.2020.1766026. DOI: https://doi.org/10.1080/00036811.2020.1766026

M. Miraoui, $mu$ Pseudo-almost automorphic solutions for some differential equations with reflection of the argument, Numer. Funct. Anal. and Optim., 38, № 3, 376–394 (2017); DOI: https://doi.org/10.1080/01630563.2017.1279175

DOI.org/10.1080/01630563.2017.1279175.

M. Miraoui, T. Diagana, K. Ezzinbi, Pseudo-almost periodic and pseudo-almost automorphic solutions to some evolution equations involving theoretical measure theory, Cubo Math. J., 16, № 02, 01–31 (2014). DOI: https://doi.org/10.4067/S0719-06462014000200001

G. M. N'Guérékata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 68, 2658–2667 (2008). DOI: https://doi.org/10.1016/j.na.2007.02.012

E. R. Oueama-Guengai, G. M. N'Guérékata, On $S$-asymptotically $omega$-periodic and Bloch periodic mild solutions to some fractional differential equations in abstract spaces, Math. Meth. and Appl. Sci., 41, 9116–9122 (2018). DOI: https://doi.org/10.1002/mma.5062

Z. Xia, Weighted pseudo asymptotically periodic mild solutions of evolution equations, Acta Math. Sinica (Engl. Ser.), 31, 1215–1232 (2015). DOI: https://doi.org/10.1007/s10114-015-4727-1