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

Yıl 2021, Cilt: 23 Sayı: 2, 608 - 619, 04.07.2021

Müfit Şan

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.

Anahtar Kelimeler

Kesirli mertebeden türevli denklemler, Riemann-Liouville türevi, varlık ve teklik, Lebesgue uzayları

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

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.

Anahtar Kelimeler

Fractional differential equations, Riemann-Liouville derivative, existence and uniqueness, 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).