Positive solution for nonlinear fractional differential equation with integral boundary value condition

• Autorzy: Zatla Y. T. , Daudi-Merzagui N.

Identyfikatory

DOI

0.1515/fascmath-2018-0024

• Języki publikacji: EN

Abstrakty

EN

In this paper, we consider a fractional differential equation, with integral boundary conditions, when the nonlinearities are sign changing. Our approach is based on the Krasnoselskii theorem in double cones. We generalize some recent results.

Słowa kluczowe

EN

fractional differential equation; integral boundary value condition; positive solution; fixed point theorem; cones

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2018
• Tom: Nr 61
• Strony: 163--177
• Opis fizyczny: Bibliogr. 43 poz.

Twórcy

autor

Zatla Y. T.

• Department of Mathematics University of Tlemcen 13000 Tlemcen, Algeria

autor

Daudi-Merzagui N.

• Department of Mathematics University of Tlemcen 13000 Tlemcen, Algeria

Bibliografia

• [1] Adomian G., Adomian G.E., Cellular systems and aging models, Comput. Math. Appl., 11(1985), 283-291.
• [2] Ahmad B., Ntouyas S.K., Tariboon J., Fractional differential equations with nonlocal integral and integer-fractional-order Neumann type boundary conditions, Mediterr. J. Math., 2015(2015), 1-17.
• [3] Ali A., Shah K., Khan R.A., Existence of positive solution to a class of boundary value problems of fractional differential equations, Computational Methods for Differential Equations, 4(1)(2016), 19-29.
• [4] Babakhani A., Daftardar-Gejji V., Existence of positive solutions of nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 278(2003), 434-442.
• [5] Bai Z.B., Lu H.S., Positive solutions of boundary value problem problems of nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 311(2005), 495-505.
• [6] Baleanu D., Agarwal R.P., Khan H., Khan R.A., Jafari H., On the existence of solution for fractional differential equations of order 3 < δ1 ≤ 4, Advances in Difference Equations, 362(2015).
• [7] Bensebaa S., Guezane-Lakoud A., Existence of positive solutions for boundary value problem of nonlinear fractional differential equation, Appl. Math. Inf. Sci., 2(2016), 519-525.
• [8] Blayneh K.W., Analysis of age structured host-parasitoid model, Far East J. Dyn. Syst., 4(2002), 125-145.
• [9] Cabada A., Cid J.A., Infante G., New criteria for the existence of non-trivial fixed points in cones, Fixed Point Theory and Applications, 125(2013).
• [10] Cabada A., Dimitrijevic S., Tomovic Т., Aleksic S., The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions, Mathematical Methods in the Applied Sciences, First published: 25 July 2016. Online Version of Record published before inclusion in an issue.
• [11] Cabada A., Hamdi Z., Nonlinear fractional differential equations with integral boundary value conditions, Applied Mathematics and Computation, 228(2014), 251-257.
• [12] Cabada A., Wang G., Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389(1)(2012), 403-411.
• [13] Chen Y., Tang X., Positive solutions of fractional differential equations at resonance on the half-line, Bound. Value Probl., 64(2012).
• [14] Chen Z., Xu F., Multiple positive solutions for nonlinear second-order m-point boundary-value problems with sign changing nonlinearities, Electronic Journal of Differential Equations, 2008(2008), 1-12.
• [15] Diethelm K., Freed A.D., On the solution of non-linear fractional order differential equations used in the modeling of viscoplasticity, in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, F. Keil, W. Mackens, H. Voss, and J. Werther, eds. Springer-Verlag, Heidelberg, (1999), 217-224.
• [16] Franco D., Infante G., Peran J., A new criterion for the existence of multiple solutions in cones, Proceedings of the Royal Society of Edinburgh: Section A, 142(2012), 1043-1050.
• [17] Gaul L., Klein Р., Kempfle S., Damping description involving fractional operators, Mech. Systems Signal Processing, 5(1991), 81-88.
• [18] Ge F., Kou C., Stability analysis by Krasnoselskiis fixed point theorem for nonlinear fractional differential equations, Applied Mathematics and Computation, 257(2015), 308-316.
• [19] Ge W.G., Ren J.L., Fixed point theorems in double cones and their applications to nonlinear boundary value problems, Chinese Annals of Mathematics, 27(2006), 155-168.
• [20] Glockle W.G., Nonnenmacher T.F., A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68(1995), 46-53.
• [21] Guo Y., Ge W., Dong S., Two positive solutions for second order three point boudndary value problems with sign change nonlinearities, Acta Math. Appl. Sinica, 27(2004), 522-529.
• [22] Hilfer R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [23] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
• [24] Liu X., Lin L., Fang H., Existence of positive solutions for nonlocal boundary value problem of fractional differential equation, Centr. Eur. J. Phys., 11(10)(2013), 1423-1432.
• [25] Liu J., Zhao Z., Multiple positive solutions for second order three point boundary value problems with sign changing nonlinearities, Electronic Journal of Differential Equations, 152(2012), 1-7.
• [26] Magin R., Fractional calculus in bioengineering, Critical Reviews in Biomedical Engineering, 32(1)(2004), 1-104.
• [27] Merzagui N., Tabet Y., Existence of multiple positive solutions for a nonlocal boundary value problem with sign changing nonlinearities, Filomat, 27(2013), 487-499.
• [28] Metzler R., Joseph K., Boundary value problems for fractional diffusion equations, Physica. A., 278(2000), 107-125.
• [29] Miller K.S., Ross B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
• [30] Nyamoradi N., Baleanu D., Agarwal R.P., Existence and uniqueness of positive solutions to fractional boundary value problems with nonlinear boundary conditions, Advances in Difference Equations, 266(2013).
• [31] Oldham K.B., Fractional differential equations in electrochemistry, Advances in Engineering Software, 41(2010), 9-12.
• [32] Oldham K.B., Spanier J., The Fractional Calculus, Academic Press, New York, London, 1974.
• [33] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
• [34] Su X., Jia М., Li М., The existence and nonexistence of positive solutions for fractional differential equations with nonhomogeneous boundary conditions, Advances in Difference Equations, 30(2016), 1-24.
• [35] Torres C., Mountain pass solution for a fractional boundary value problem, Journal of Fractional Calculus and Applications, 5(1)(2014), 1-10.
• [36] Ervin J.V., Roop J.P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations, 22(2006), 58-76.
• [37] Vintagre B., Podlybni I., Hernandez A., Feliu V., Some approximations of fractional order operators used in control theory and applications, Fractional Calculus and Applied Analysis, 3(3)(2000), 231-248.
• [38] Wang Y., Liu L., Wu Y., Positive solutions of a fractional boundary value problem with changing sign nonlinearity, Abstract and Applied Analysis, 1(2012), 1-12.
• [39] Wu Т., Zhang X., Lu Y., Solutions of Sign-Changing fractional differential equation with the fractional derivatives, Boundary Value Problems, 2(2012), 1-16.
• [40] Yang L., Application of Avery-Peterson fixed point theorem to nonlinear boundary value problem of fractional differential equation with the Caputo derivative, Communications in Nonlinear Science and Numerical Simulation, 17(2012), 4576-4584.
• [41] Yanping G., Weigao G., Ying G., Twin positive solutions for higher order m-point boundary value problems with sign changing nonlinearities, Applied Mathematics and Computation, 146(2003), 299-311.
• [42] Zhang X., Wang L., Sun Q., Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter, Applied Mathematics and Computation, 226(2014), 708-718.
• [43] Zhang X., Zhong Q., Multiple positive solutions for nonlocal boundary value problems of singular fractional differential equations, Boundary Value Problems, 65(2016), 1-11.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).