Sự thu hút cho các phương trình vi phân bậc phân số và loại ψ-Hilfer

Fractional Calculus and Applied Analysis - Tập 23 - Trang 1188-1207 - 2020
J. Vanterler da C. Sousa1, Mouffak Benchohra2, Gaston M. N’Guérékata3
1Department of Applied Mathematics, IMECC-State University of Campinas, Campinas, Brazil
2Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, Sidi Bel-Abbes, Algeria
3NEERLab, Department of Mathematics, Morgan State University, Baltimore, USA

Tóm tắt

Bài báo này nghiên cứu tính thu hút tổng thể của nghiệm cho phương trình vi phân bậc phân số có liên quan đến đạo hàm phân số ψ-Hilfer và sử dụng định lý điểm cố định Krasnoselskii. Chúng tôi nổi bật một số trường hợp đặc biệt của các kết quả được trình bày ở đây, đặc biệt là liên quan đến Riemann-Liouville, từ đó minh họa cho lớp rộng các đạo hàm phân số mà các kết quả này có thể được áp dụng.

Từ khóa

#phương trình vi phân #bậc phân số #đạo hàm phân số ψ-Hilfer #định lý điểm cố định #tính thu hút

Tài liệu tham khảo

S. Abbas, M. Benchohra, J.R. Graef, Coupled Sytems of Hilfer fractional differential inclusions in Banach spaces. Commun. Pure & Appl. Anal. 17, No 6 (2018), 2479–2493; DOI: 10.3934/cpaa.2018118. S. Abbas, M. Benchohra, N. Hamidi, G. N’Guérékata, Existence and attractivity results for coupled systems of nonlinear Volterra–Stieltjes multidelay fractional partial integral equations. Abstr. Appl. Anal. 2018 (2018), Article ID 8735614, 10 pages; DOI: 10.1155/2018/8735614. S. Abbas, R.P. Agarwal, M. Benchohra, and F. Berhoun, Global attractivity for Volterra type Hadamard fractional integral equations in Fréchet spaces. Demonstr. Math. 51 (2018), 131–140; DOI: 10.1515/dema-2018-0009. S. Abbas, M. Benchohra, N. Hamidi, J. Henderson, Caputo-Hadamard fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 21, No 4 (2018), 1027–1045; DOI: 10.1515/fca-2018-0056; https://www.degruyter.com/view/journals/fca/21/4/fca.21.issue-4.xml. S. Abbas, M. Benchohra, and J. Henderson, Existence and attractivity results for Hilfer fractional differential equations. J. Math. Sci. 243, No 3 (2019), 347–357; DOI: 10.1007/s10958-019-04544-y. S. Abbas and M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses. Appl. Math. Comput. 257 (2015), 190–198; DOI: 10.1016/j.amc.2014.06.073. S. Abbas, M. Benchohra, and J.J. Nieto, Global attractivity of solutions for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes partial integral equations. Electron. J. Qual. Theory Differ. Equ. 2012, No 81 (2012), 1–15; DOI: 10.14232/ejqtde.2012.1.81. S. Abbas, M. Benchohra, A. Petrusel, Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory. Fract. Calc. Appl. Anal. 20, No 2 (2017), 384–398; DOI: 10.1515/fca-2017-0020; https://www.degruyter.com/view/journals/fca/20/2/fca.20.issue-2.xml/view/journals/fca/20/2/fca.20.issue-2.xml. R. Agarwal, S. Hristova, and D. O’Regan, Non-instantaneous impulses in Caputo fractional differential equations. Fract. Calc. Appl. Anal. 20, No 3 (2017), 595–622; DOI: 10.1515/fca-2017-0032; https://www.degruyter.com/view/journals/fca/20/3/fca.20.issue-3.xml/view/journals/fca/20/3/fca.20.issue-3.xml. J. Banaś, D. O’Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order. J. Math. Anal. Appl. 345, No 1 (2008), 573–582; DOI: 10.1016/j.jmaa.2008.04.050. M. Benchohra, Z. Bouteffal, J. Henderson, and S. Litimein, Measure of noncompactness and fractional integro-differential equations with state-dependent nonlocal conditions in Fréchet spaces. AMS Math. 5, No 1 (2019), 15–25; DOI: 10.3934/math.2020002. M. Benchohra, S. Litimein, and J.J. Nieto, Semilinear fractional differential equations with infinite delay and non-instantaneous impulses. J. Fixed Point Theory Appl. 2019 (2019), # 21; DOI: 10.1007/s11784-019-0660-8. T.A. Burton, A fixed point theorem of Krasnoselskii. Appl. Math. Lett. 11 (1998), 85–88; DOI: 10.1016/S0893-9659(97)00138-9. P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Fractional Calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269 (2002), 1–27; DOI: 10.1016/S0022-247X(02)00001-X. F. Chen, J.J. Nieto and Y. Zhou, Global attractivity for nonlinear fractional differential equations. Nonlinear Anal. 13, No 1 (2012), 287–298; DOI: 10.1016/j.nonrwa.2011.07.034. F. Chen and Y. Zhou, Attractivity of fractional functional differential equations. Comput. Math. Appl. 62, No 3 (2011), 1359–1369; DOI: 10.1016/j.camwa.2011.03.062. J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. 23, No 6 (2010), 676–680; DOI: 10.1016/j.aml.2010.02.007. Z. Fan, Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives. Indagationes Math. 25, No 3 (2014), 516–524; DOI: 10.1016/j.indag.2014.01.002. J.K. Hale, Theory of Function Differential Equations. Springer-Verlag, New York (1977). R. Hilfer, Applications of Fractional Calculus in Physics, World Sci., N. Jersey (2000). T.D. Ke, N.N. Quan, Finite-time attractivity for semilinear tempered fractional wave equations. Fract. Calc. Appl. Anal. 21, No 6 (2018), 1471–1492; DOI: 10.1515/fca-2018-0077; https://www.degruyter.com/view/journals/fca/21/6/fca.21.issue-6.xml/view/journals/fca/21/6/fca.21.issue-6.xml. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006). V. Kiryakova, Y. Luchko, Multiple Erdélyi-Kober integrals and derivatives as operators of generalized fractional calculus. In: Handbook of Fractional Calculus with Applications, Chap. 6, Vol. 1, De Gruyter, Berlin (2019), 127–158; DOI: 10.1515/9783110571622-006. J. Losada, J.J. Nieto, and E. Pourhadi, On the attractivity of solutions for a class of multi-term fractional functional differential equations. J. Comput. Appl. Math. 312 (2017), 2–12; DOI: 10.1016/j.cam.2015.07.014. E. de Oliveira, J. Vanterler da C. Sousa, Ulam–Hyers–Rassias stability for a class of fractional integro-differential equations. Results Math. 73, No 3, (2018), # 111; DOI: 10.1007/s00025-018-0872-z. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach. Sci. Publ., London - N. York (1993). J. Vanterler da C. Sousa, K.D. Kucche, E. de Oliveira, On the Ulam-Hyers stabilities of the solutions of ψ-Hilfer fractional differential equation with abstract Volterra operator. Math. Meth. Appl. Sci. 42 (2019), 3021–3032; DOI: 10.1002/mma.5562. J. Vanterler da C. Sousa, E. de Oliveira, On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60 (2018), 72–91; DOI: 10.1016/j.cnsns.2018.01.005. J. Vanterler da C. Sousa, E. de Oliveira, Leibniz type rule: ψ-Hilfer fractional operator. Commun. Nonlinear Sci. Numer. Simul. 77 (2019), 305–311; DOI: 10.1016/j.cnsns.2019.05.003. J. Vanterler da C. Sousa, E. de Oliveira, Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett. 81 (2018), 50–56; DOI: 10.1016/j.aml.2018.01.016. J. Vanterler da C. Sousa, E. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of Hilfer operator. Diff. Equ. & Appl. 11, No 1 (2019), 87–106; DOI: 10.7153/dea-2019-11-02. J. Vanterler da C. Sousa, K. D. Kucche, E. de Oliveira, Stability of ψ-Hilfer impulsive fractional differential equations. Appl. Math. Lett. 88 (2019), 73–80; DOI: 10.1016/j.aml.2018.08.013. J. Vanterler da C. Sousa, E. de Oliveira, On the Ψ-fractional integral and applications. Comput. Appl. Math. 38 No 1, (2019) 4; https://doi.org/10.1007/s40314-019-0774-z. J. Vanterler da C. Sousa, E. de Oliveira, On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator. J. Fixed Point Theory Appl. 20 No 3, (2018), # 96; DOI: 10.1007/s11784-018-0587-5. J. Vanterler da C. Sousa, E. de Oliveira, Fractional order pseudo-parabolic partial differential equations: Ulam–Hyers Stability. Bull. Braz. Math. Soc. 50 (2019), 481–496; DOI: 10.1007/s00574-018-0112-x. J. Vanterler da C. Sousa, E. de Oliveira, On the stability of a hyperbolic fractional partial differential equation. Diff. Equ. Dyn. Sys. 2019 (2019),; DOI: 10.1007/s12591-019-00499-3. J. Vanterler da C. Sousa, E. de Oliveira, Capelas, Fractional order pseudoparabolic partial differential equation: Ulam–Hyers stability. Bull. Braz. Math. Soc., New Series 50, No 2 (2019), 481–496; DOI: 10.1007/s00574-018-0112-x. H.M. Srivastava, D. Kumar, J. Singh, An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model. 45 (2017), 192–204; DOI: 10.1016/j.apm.2016.12.008. Z. Zhang, B. Liu, Existence of mild solutions for fractional evolution equations. J. Fract. Calc. Appl. 2, No 20 (2012), 1–10. Y. Zhou, J.W. He, B. Ahmad, A. Alsaedi, Existence and attractivity for fractional evolution equations. Discrete Dyn. Nat. Soc., (2018) Art. ID 1070713, 9 pp.; DOI: 10.1155/2018/1070713. Y. Zhou, Attractivity for fractional differential equations in Banach space. Appl. Math. Lett. 75 (2018), 1–6; DOI: 10.1016/j.aml.2017.06.008. Y. Zhou, Attractivity for fractional evolution equations with almost sectorial operators. Fract. Calc. Appl. Anal. 21, No 3 (2018), 786–800; DOI: 10.1515/fca-2018-0041; https://www.degruyter.com/view/journals/fca/21/3/fca.21.issue-3.xml/view/journals/fca/21/3/fca.21.issue-3.xml.