Kako citirati
Sažetak
Uvod/cilj: Cilj rada je proučavanje postojanja blagih rešenja problema početne vrednosti Darbuovog problema za parcijalne hiperboličke frakcione diferencijalne jednačine pomoću Kaputovog izvoda sa konačnim kašnjenjem u Frešeovim prostorima.
Metode: U analizi se koriste Darboova teorema o fiksnoj tački i pojam mere nekompaktnosti kako bi se utvrdilo postojanje rešenja svođenjem istraživanja na dokazivanje postojanja i jedinstvenosti fiksnih tačaka odgovarajućih operatora.
Rezultati: Postojanje blagih rešenja posmatranih parcijalnih hiperboličkih frakcionih diferencijalnih jednačina je dokazano pod odgovarajućim uslovima. Primenljivost teorijskih rezultata je ilustrovana primerom.
Zaključak: U radu se razmatraju dovoljni uslovi za utvrđivanje postojanja i jedinstvenosti rešenja Darbuovog problema primenom Darboove teoreme o fiksnoj tački na neograničenom intervalu. Istraživanje pruža metodološki okvir za analizu sličnih problema i doprinosi širem razumevanju blagih rešenja u frakcionom računu i funkcionalnoj analizi. Dobijeni rezultati nude potencijalne primene za istraživače koji se bave frakcionim diferencijalnim jednačinama u beskonačno-dimenzionalnim prostorima.
Ključne reči
frakcioni izvod
blago rešenje
Frešeov prostor
Reference
Abbas, S. and Benchohra, M. (2009). Darboux problem for perturbed partial differential equations of fractional order with finite delay. Nonlinear Analysis: Hybrid Systems, 3, 597-604. Available at: https://doi.org/10.1016/j.nahs.2009.01.001.
Abbas, S. and Benchohra, M. (2012). Fractional order partial hyperbolic differential equations involving Caputo’s derivative. Studia Universitatis Babeş–Bolyai Mathematica, 57(4), 469-479.
Abbas, S., Benchohra, M., and N’Guérékata, G. M. (2012). Topics in Fractional Differential Equations. Springer, New York. Available at: https://doi.org/10.1007/978-1-4614-4036-0.
Appell, J. (1981). Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator. Journal of Mathematical Analysis and Applications, 83, 251-263. Available at: https://doi.org/10.1016/0022-247X(81)90160-9.
Ayerbee Toledano, J. M., Dominguez Benavides, T., and López Acedo, G. (1997). Measures of noncompactness in metric fixed point theory. Operator Theory: Advances and Applications, 99. Birkhäuser, Basel, Boston, Berlin. Available at: https://doi.org/10.1007/978-3-0348-8875-7.
Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J. J. (2012). Fractional Calculus Models and Numerical Methods. World Scientific Publishing, New York. Available at: https://doi.org/10.1142/9789814355212.
Banaei, Sh. (2018). Solvability of a system of integral equations of Volterra type in the Fréchet space $L^p_{loc}(R^+)$ via measure of noncompactness. Filomat, 32, 5255-5263. Available at: https://doi.org/10.2298/FIL1813255B.
Banás, J., Jleli, M., Mursaleen, M., and Samet, B. (2017). Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness. Springer, Singapore. Available at: https://doi.org/10.1007/978-981-10-4565-8.
Banás, J., and Mursaleen, M. (2014). Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, Berlin. Available at: https://doi.org/10.1007/978-3-319-01392-4.
Benchohra, M., and Hellal, M. (2014). Global uniqueness result for fractional partial hyperbolic differential equations with state-dependent delay. Annales Polonici Mathematici, 110(3), 259-281. Available at: https://doi.org/10.4064/ap110-3-2.
Benchohra, M., and Hellal, M. (2014). Perturbed partial fractional order functional differential equations with infinite delay in Fréchet spaces. Nonlinear Dynamics and System Theory, 14(3), 244-257. Available at: https://doi.org/10.1007/s11071-014-1491-5.
Benchohra, M., Henderson, J., and Heris, A. (2017). Fractional partial random differential equations with delay. PanAmerican Mathematical Journal, 27(1), 29-40. Available at: https://doi.org/10.1007/s40815-017-0034-2.
Bothe, D. (1998). Multivalued perturbation of m-accretive differential inclusions. Israel Journal of Mathematics, 108, 109-138. Available at: https://doi.org/10.1007/BF02777780.
Dudek, S. (2017). Fixed point theorems in Fréchet Algebras and Fréchet spaces and applications to nonlinear integral equations. Applied Analysis and Discrete Mathematics, 11, 340-357. Available at: https://doi.org/10.2298/AADM1703340D.
Dudek, S., and Olszowy, L. (2015). Continuous dependence of the solutions of nonlinear integral quadratic Volterra equation on the parameter. Journal of Function Spaces, 2015, Article ID 471235, 9 pages. Available at: https://doi.org/10.1155/2015/471235.
Helal, M. (2020). Perturbed Partial Functional Differential Equations on Unbounded Domains with Finite Delay. Applied Mathematics E-Notes, 20, 82-92. Available at: https://doi.org/10.2478/AMEN-2020-0010.
Hilfer, R. (2000). Applications of Fractional Calculus in Physics. World Scientific, Singapore. Available at: https://doi.org/10.1142/9789812816223.
Kilbas, A. A., and Marzan, S. A. (2005). Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differential Equations, 41, 84-89. Available at: https://doi.org/10.1007/s10625-005-0010-4.
Kuratowski, K. (1930). Sur les espaces complets. Fundamenta Mathematicae, 15, 301-309. Available at: https://doi.org/10.4064/fm-15-1-301-309.
Mönch, H. (1980). Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Analysis: Theory, Methods and Applications, 4, 985-999. Available at: https://doi.org/10.1016/0362-546X(80)90076-6.
Ortigueira, M. D. (2011). Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, 84. Springer, Dordrecht. Available at: https://doi.org/10.1007/978-94-007-0531-3.
Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego. Available at: https://doi.org/10.1016/B978-0-12-558840-9.X5000-3.
Tarasov, V. E. (2010). Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg. Available at: https://doi.org/10.1007/978-3-642-11218-3.
Vityuk, A. N., and Golushkov, A. V. (2004). Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscillations, 7(3), 318-325. Available at: https://doi.org/10.1023/B:NONO.0000045252.92915.0f.
Yosida, K. (1980). Functional Analysis, 6th edition. Springer-Verlag, Berlin. Available at: https://doi.org/10.1007/978-3-662-02410-7.
Vojnotehnički glasnik omogućava otvoreni pristup i, u skladu sa preporukom CEON-a, primenjuje Creative Commons odredbe o autorskim pravima:
Autori koji objavljuju u Vojnotehničkom glasniku pristaju na sledeće uslove:
- Autori zadržavaju autorska prava i pružaju časopisu pravo prvog objavljivanja rada i licenciraju ga Creative Commons licencom koja omogućava drugima da dele rad uz uslov navođenja autorstva i izvornog objavljivanja u ovom časopisu.
- Autori mogu izraditi zasebne, ugovorne aranžmane za neekskluzivnu distribuciju rada objavljenog u časopisu (npr. postavljanje u institucionalni repozitorijum ili objavljivanje u knjizi), uz navođenje da je rad izvorno objavljen u ovom časopisu.
- Autorima je dozvoljeno i podstiču se da postave objavljeni rad onlajn (npr. u institucionalnom repozitorijumu ili na svojim internet stranicama) pre i tokom postupka prijave priloga, s obzirom da takav postupak može voditi produktivnoj razmeni ideja i ranijoj i većoj citiranosti objavljenog rada (up. Efekat otvorenog pristupa).