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Riemann-Liouville tip kesirli türevli lineer olmayan denklemlerin bazı sınıfları için teklik teoremleri

Yıl 2021, Cilt: 23 Sayı: 2, 608 - 619, 04.07.2021
https://doi.org/10.25092/baunfbed.893685

Öz

Bu çalışmada. sağ yan fonksiyonları birinci değişkenlerine göre singülerliğe sahip ve başlangıç koşulu homojen olmayan Riemann-Liouville kesirli diferansiyel denklemlerinin bazı sınıfları göz önüne alınmıştır. İlk önce, bu başlangıç-değer probleminin bir lokal sürekli çözümünün varlığını hangi koşular altında gerçekleştiği kısaca ifade edilmiştir. Daha sonra ise, sırasıyla Krasnosel’skii-Krein, Kooi, Roger ve Banaś-Rivero tiplerinde teklik teoremleri ortaya çıkarılmıştır. Bu teoremler daha önceden elde edilen sonuçları geliştirken, bu teoremlerin ispatları için, daha önceden var olan teknikler Lebesgue uzaylarının araçları ile zenginleştirilmiştir.

Kaynakça

  • Baleanu, D., Fernandez, A., On fractional operators and their classifications, Mathematics, 7(9), 830, (2019).
  • Kilbas, A. A. A., Srivastava, H.M., Trujillo, J.J., Theory and applications of fractional differential equations, Vol. 204, Elsevier Science Limited, (2006).
  • Miller, K.S., Ross, B., An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley Sons Inc., New York, (1993).
  • Ortigueira, M.D., Machado, J.A.T. "What is a fractional derivative?.", Journal of Computational Physics, 293 (2015): 4-13.
  • Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, (1999).
  • Samko, S.G., Kilbas A.A.A, Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, (1993).
  • Bilgici, S.S., Şan, M., Existence and Uniqueness Results for a nonlinear fractional differential equations of order . TWMS Journal of Applied and Engineering Mathematics, (2020). (accepted)
  • Delbosco, D., and Luigi R., Existence and uniqueness for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 204.2 609-625, (1996).
  • Lan, K., Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations. Proceedings of the American Mathematical Society, 148(12), 5225-5234, (2020).
  • Şan, M., Complex variable approach to the analysis of a fractional differential equation in the real line, Comptes Rendus Mathematique, 356, 3, 293-300, (2018).
  • Şan, M., Sert, U., Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero, Hacettepe Journal of Mathematics and Statistics, 49 (5), 1718 – 1725, (2020).
  • Wu, C., Liu, X, The continuation of solutions to systems of Caputo fractional order differential equations, Fractional Calculus and Applied Analysis, 23(2), 591-599, (2020).
  • Yörük, F., Bhaskar, T. G., Agarwal, R. P., New uniqueness results for fractional differential equations, Applicable Analysis, 92, No 2, 259-269, (2013).
  • Krasnosel'skii, M.A. Krein, S.G., Nonlocal existence theorems and uniqueness theorems for systems of ordinary differential equations, Doklady Akademii Nauk SSSR (N.S.) 102, 13-16, (1955).
  • Krasnosel’skii, M.A. Krein, S.G., On a class of uniqueness theorems for the equation y' = f(x,y), Uspekhi Matematicheskikh Nauk, 11, 209-213, (1956).
  • Agarwal, R. P., Lakshmikantham, V., Uniqueness and nonuniqueness criteria for ordinary differential equations, Vol. 6, World Scientific, 1993.
  • Lakshmikantham, V., Leela, S., Nagumo-type uniqueness result for fractional differential equations, Nonlinear Analysis, 71, 7-8, 2886-2889, (2009).
  • Lakshmikantham, V., Leela, S., A Krasnoselskii–Krein-type uniqueness result for fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 71, 7-8, 3421-3424, (2009).
  • Kooi, O. The method of successive approximations and a uniqueness theorem of Krasnoselskii and Krein in the theory of differential equations, Indagationes Mathematicae, 20, 322-327, (1958).
  • Rogers, T., On Nagumo's condition, Canadian Mathematical Bulletin, 15, 609-611, (1972).
  • Banaś, J., Rivero, J., Remarks concerning J. Witte's theorem and its applications, Commentationes Mathematicae Universitatis Carolinae, 28(1), 23-31, (1987).

Uniqueness theorems for some classes of nonlinear fractional differential equations in the Riemann-Liouville sense

Yıl 2021, Cilt: 23 Sayı: 2, 608 - 619, 04.07.2021
https://doi.org/10.25092/baunfbed.893685

Öz

In this study, some classes of Riemann-Liouville fractional differential equations with right-hand side functions having a singularity with respect to their first variable and with a nonhomogeneous initial condition are considered. First, it is briefly stated that under which conditions the existence of a local continuous solution of this initial value problem occurs. Later, uniqueness theorems were developed in types of Krasnosel’skii-Krein, Kooi, Roger and Banaś-Rivero, respectively. These theorems improve the previously obtained results, and for their proofs pre-existing techniques are enriched by the tools of Lebesgue spaces.

Teşekkür

We would like to thank the referees for their valuable comments eliminating deficiencies of the manuscript.

Kaynakça

  • Baleanu, D., Fernandez, A., On fractional operators and their classifications, Mathematics, 7(9), 830, (2019).
  • Kilbas, A. A. A., Srivastava, H.M., Trujillo, J.J., Theory and applications of fractional differential equations, Vol. 204, Elsevier Science Limited, (2006).
  • Miller, K.S., Ross, B., An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley Sons Inc., New York, (1993).
  • Ortigueira, M.D., Machado, J.A.T. "What is a fractional derivative?.", Journal of Computational Physics, 293 (2015): 4-13.
  • Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, (1999).
  • Samko, S.G., Kilbas A.A.A, Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, (1993).
  • Bilgici, S.S., Şan, M., Existence and Uniqueness Results for a nonlinear fractional differential equations of order . TWMS Journal of Applied and Engineering Mathematics, (2020). (accepted)
  • Delbosco, D., and Luigi R., Existence and uniqueness for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 204.2 609-625, (1996).
  • Lan, K., Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations. Proceedings of the American Mathematical Society, 148(12), 5225-5234, (2020).
  • Şan, M., Complex variable approach to the analysis of a fractional differential equation in the real line, Comptes Rendus Mathematique, 356, 3, 293-300, (2018).
  • Şan, M., Sert, U., Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero, Hacettepe Journal of Mathematics and Statistics, 49 (5), 1718 – 1725, (2020).
  • Wu, C., Liu, X, The continuation of solutions to systems of Caputo fractional order differential equations, Fractional Calculus and Applied Analysis, 23(2), 591-599, (2020).
  • Yörük, F., Bhaskar, T. G., Agarwal, R. P., New uniqueness results for fractional differential equations, Applicable Analysis, 92, No 2, 259-269, (2013).
  • Krasnosel'skii, M.A. Krein, S.G., Nonlocal existence theorems and uniqueness theorems for systems of ordinary differential equations, Doklady Akademii Nauk SSSR (N.S.) 102, 13-16, (1955).
  • Krasnosel’skii, M.A. Krein, S.G., On a class of uniqueness theorems for the equation y' = f(x,y), Uspekhi Matematicheskikh Nauk, 11, 209-213, (1956).
  • Agarwal, R. P., Lakshmikantham, V., Uniqueness and nonuniqueness criteria for ordinary differential equations, Vol. 6, World Scientific, 1993.
  • Lakshmikantham, V., Leela, S., Nagumo-type uniqueness result for fractional differential equations, Nonlinear Analysis, 71, 7-8, 2886-2889, (2009).
  • Lakshmikantham, V., Leela, S., A Krasnoselskii–Krein-type uniqueness result for fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 71, 7-8, 3421-3424, (2009).
  • Kooi, O. The method of successive approximations and a uniqueness theorem of Krasnoselskii and Krein in the theory of differential equations, Indagationes Mathematicae, 20, 322-327, (1958).
  • Rogers, T., On Nagumo's condition, Canadian Mathematical Bulletin, 15, 609-611, (1972).
  • Banaś, J., Rivero, J., Remarks concerning J. Witte's theorem and its applications, Commentationes Mathematicae Universitatis Carolinae, 28(1), 23-31, (1987).
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Araştırma Makalesi
Yazarlar

Müfit Şan Bu kişi benim 0000-0001-6852-1919

Yayımlanma Tarihi 4 Temmuz 2021
Gönderilme Tarihi 8 Kasım 2020
Yayımlandığı Sayı Yıl 2021 Cilt: 23 Sayı: 2

Kaynak Göster

APA Şan, M. (2021). Uniqueness theorems for some classes of nonlinear fractional differential equations in the Riemann-Liouville sense. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 23(2), 608-619. https://doi.org/10.25092/baunfbed.893685
AMA Şan M. Uniqueness theorems for some classes of nonlinear fractional differential equations in the Riemann-Liouville sense. BAUN Fen. Bil. Enst. Dergisi. Temmuz 2021;23(2):608-619. doi:10.25092/baunfbed.893685
Chicago Şan, Müfit. “Uniqueness Theorems for Some Classes of Nonlinear Fractional Differential Equations in the Riemann-Liouville Sense”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 23, sy. 2 (Temmuz 2021): 608-19. https://doi.org/10.25092/baunfbed.893685.
EndNote Şan M (01 Temmuz 2021) Uniqueness theorems for some classes of nonlinear fractional differential equations in the Riemann-Liouville sense. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 23 2 608–619.
IEEE M. Şan, “Uniqueness theorems for some classes of nonlinear fractional differential equations in the Riemann-Liouville sense”, BAUN Fen. Bil. Enst. Dergisi, c. 23, sy. 2, ss. 608–619, 2021, doi: 10.25092/baunfbed.893685.
ISNAD Şan, Müfit. “Uniqueness Theorems for Some Classes of Nonlinear Fractional Differential Equations in the Riemann-Liouville Sense”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 23/2 (Temmuz 2021), 608-619. https://doi.org/10.25092/baunfbed.893685.
JAMA Şan M. Uniqueness theorems for some classes of nonlinear fractional differential equations in the Riemann-Liouville sense. BAUN Fen. Bil. Enst. Dergisi. 2021;23:608–619.
MLA Şan, Müfit. “Uniqueness Theorems for Some Classes of Nonlinear Fractional Differential Equations in the Riemann-Liouville Sense”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, c. 23, sy. 2, 2021, ss. 608-19, doi:10.25092/baunfbed.893685.
Vancouver Şan M. Uniqueness theorems for some classes of nonlinear fractional differential equations in the Riemann-Liouville sense. BAUN Fen. Bil. Enst. Dergisi. 2021;23(2):608-19.