CESÀRO SUMMABILITY OF INTEGRALS OF FUZZY-NUMBER-VALUED FUNCTIONS

Year 2018, Volume: 67 Issue: 2, 38 - 49, 01.08.2018

Enes Yavuz Özer Talo Hüsamettin Coşkun

Abstract

In the present study, we have introduced Cesàro summability ofintegrals of fuzzy-number-valued functions and given one-sided Tauberian conditions under which convergence of improper fuzzy Riemann integrals followsfrom Cesàro summability. Also, fuzzy analogues of Schmidt type slow decreaseand Landau type one-sided Tauberian conditions have been obtained

Keywords

Cesàro summability, Tauberian theorems, improper integrals, fuzzynumber-valued function

References

• Fontes, F. G. and Solís, F. J., Iterating the Cesàro Operators, Proc. Amer. Math. Soc. 136(6), –2153 (2008).
• González, M. and León-Saavedra, F., Cyclic behavior of the Cesàro operator on L2(0; 1), Proc. Amer. Math. Soc. 137(6), 2049–2055 (2009).
• Arvanitidis, A. G. and Siskakis, A. G., Cesàro Operators on the Hardy Spaces of the Half- Plane, Canad. Math. Bull. 56, 229–240 (2013).
• Albanese, A. A., Bonet J. and Ricker, W. J., On the continuous Cesàro operator in certain function spaces, Positivity 19, 659–679 (2015).
• Lacruz, M., León-Saavedra F., Petrovic, S. and Zabeti, O., Extended eigenvalues for Cesàro operators, J. Math. Anal. Appl. 429, 623–657 (2015).
• Titchmarsh, E. C., Introduction to the theory of Fourier integrals, Clarendon Press, Oxford (1937).
• Hardy, G. H., Divergent series. Oxford Univ. Press, Oxford (1949).
• Korevaar, J., Tauberian Theory: A Century of Developments. Springer-Verlag, Berlin (2004).
• Móricz, F. and Németh, Z., Tauberian conditions under which convergence of integrals follows from summability (C; 1) over R+. Anal. Math. 26, 53–61 (2000).
• Çanak, ·I. and Totur, Ü., A Tauberian theorem for Cesàro summability of integrals, Appl. Math. Lett. 24, 391–395 (2011).
• Çanak, ·I. and Totur, Ü., Tauberian conditions for Cesàro summability of integrals. Appl. Math. Lett. 24, 891–896 (2011).
• Totur, Ü. and Çanak, ·I., One-sided Tauberian conditions for (C,1) summability method of integrals, Math. Comput. Model. 55, 1813–1818 (2012).
• Çanak, ·I. and Totur, Ü., Alternative proofs of some classical type Tauberian theorems for Cesàro summability of integrals, Math. Comput. Model. 55(3), 1558–1561 (2012).
• Totur, Ü. and Çanak, ·I., On Tauberian conditions for (C,1) summability of integrals, Revista de la Unión Matemática Argentina 54(2), 59–65 (2013).
• Totur, Ü. and Çanak, ·I., On the (C,1) summability method of improper integrals, Appl. Math. Comput. 219(24), 11065–11070 (2013).
• Giang, D. V. and Móricz, F., The strong summability of Fourier transforms, Acta Math. Hungar. 65(4), 403–419 (1994).
• Móricz, F., Strong Cesàro summability and statistical limit of double Fourier integrals, Acta Sci. Math. (Szeged) 71, 159–174 (2005).
• Brown, G., Feng D. and Móricz, F., Strong Cesàro Summability of Double Fourier Integrals, Acta Math. Hungar. 115(1-2), 1–12 (2007).
• Mishra, V. N., Khatri K. and Mishra, L. N., Strong Cesàro Summability of Triple Fourier Integrals, Fasc. Math. 53, 95–112 (2014).
• Dubois, D. and Prade, H., Towards fuzzy diğerential calculus, Fuzzy Set and Syt. 8, 1–7, –116, 225–233 (1982).
• Wu, H., The improper fuzzy Riemann integral and its numerical integration, Inform. Sciences (1–4), 109–137 (1998).
• Anastassiou, G. A., Fuzzy Mathematics: Approximation Theory. Springer-Verlag, Berlin (2010).
• Gong, Z. and Wang, L., The Henstock-Stieltjes integral for fuzzy-number-valued functions, Inform. Sciences 188, 276–297 (2012).
• Ren, X. and Wu, C., The Fuzzy Riemann-Stieltjes Integral, Int. J. Theor. Phys. 52, 2134– (2013).
• Bede, B. and Gal, S. G., Almost periodic fuzzy-number-valued functions, Fuzzy Set Syt. 147, –403 (2004).
• Zadeh, L. A., Fuzzy sets, Inform. Control 8, 29–44 (1965).
• Talo, Ö. and Ba¸sar, F., Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations, Comput. Math. Appl. 58(4), 717–733 (2009).
• Aytar, S. Mammadov, M. and Pehlivan, S., Statistical limit inferior and limit superior for sequences of fuzzy numbers, Fuzzy Set Syt, 157(7), 976–985 (2006).
• Aytar, S. and Pehlivan, S., Statistical cluster and extreme limit points of sequences of fuzzy numbers, Inform. Sci. 177(16), 3290–3296 (2007).
• Li, H. and C. Wu, The integral of a fuzzy mapping over a directed line, Fuzzy Set Syt 158, –2338 (2007).
• Talo, Ö. and Ba¸sar, F., On the Slowly Decreasing Sequences of Fuzzy Numbers, Abstr. Appl. Anal. 2013, 1–7 (2013).
• Goetschel, R. and Voxman, W., Elementary fuzzy calculus, Fuzzy Set Syt 18, 31–43 (1986).