Existence and uniqueness of solution of differential equation of second order in cone metric spaces

• Autorzy: Tidke H. L. , Dhakne M. B.
• Języki publikacji: EN

Abstrakty

EN

In this paper we investigate the existence and uniqueness for fractional and non-fractional dierential equations in cone metric spaces. The result is obtained by using the some extensions of Banach's contraction principle in complete cone metric space, fractional calculus and the theory of strongly continuous cosine family.

Słowa kluczowe

EN

cone metric space; fixed point; fractional calculus; cosine family

PL

przestrzeń metryczna; punkt stały

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2010
• Tom: Nr 45
• Strony: 121--131
• Opis fizyczny: Bibliogr. 33 poz.

Twórcy

autor

Tidke H. L.

autor

Dhakne M. B.

• Department of Mathematics Maharashtra University Jalgaon-425 001, India,

Bibliografia

• [1] Abbas M., Jungck G., Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341(1)(2008), 416-420.
• [2] Abd El-Salam Sh.A., El-Sayed A.M.A., On the stability of some fractional order non-autonomous systems, Electronic Journal of Qualitative Theory of Differential Equations, 6(2007), 1-14.
• [3] Azam A., Arshad M., Common fixed points of generalized contractive maps in cone metric spaces, Bull. Iranian Math. Soc., (2009) (in press).
• [4] Bashir A., Some existence results for boundary value problems of fractional semilinear evolution equations, Electronic Journal of Qualitative Theory of Differential Equations, 28(2009), 1-7.
• [5] Balchandran K., Park J.Y., Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Analysis, 71(2009), 4471-4475.
• [6] Banas J., Solutions of a functional integral equation in BC(R+), International Mathematical Forum, 1(24)(2006), 1181-1194.
• [7] Choudhury B.S., Metiya N., Fixed points of weak contractions in cone metric spaces, Nonlinear Analysis, (2009) doi:10.1016/j.na.2009.08.040.
• [8] Fattorini H.O., Second Order Linear Di_erential Equations in Banach Spaces, North-Holland Mathematics Studies, 108, North-Holland, Amsterdam, 1985.
• [9] Haghi R.H., Rezapour Sh., Fixed points of multifunctions on regular cone metric spaces, Expo. Math., (2009) doi: 10.1016 /j.exmath.2009.04.001.
• [10] Hilfer R., Application of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [11] Huang L.G., Zhang X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332(2)(2007), 1468-1476.
• [12] Ilic D., Rakocevic V., Common fixed points for maps on cone metric space, J. Math. Anal. Appl., 341(2)(2008), 876-882.
• [13] Jaradat O.K., Al-Omari A., Momani S., Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Analysis, 69(2008), 3153-3159. 130 Haribhau L. Tidke and Machindra B. Dhakne
• [14] Jankovic S., Kadelburg Z., Radonevic R., Rhoades B.E., Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces, Fixed Point Theory and Applications, (2009), 16 pages, Article ID 761086, doi: 10.1155 /2009/761086.
• [15] Kadelburg Z., Radonevic S., Rosic B., Strict contractive conditions and common fixed point theorems in cone metric spaces, Fixed Point Theory and Applications, (2009), 14 pages, Article ID 173838, doi: 10.1155 /2009/173838.
• [16] Karoui A., On the existence of continuous solutions of nonlinear integral equations, Applied Mathematics Letters, 18(2005), 299-305.
• [17] 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.
• [18] Kisyński J., On second order Cauchy's problem in a Banach space, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronm. Phys., 18(1970), 371-374.
• [19] Kwong M.K., On Krasnoselskii's cone fixed point theorems, Fixed Point Theory and Applications, Volume 2008, Article ID 164537, 18pages.
• [20] Kosmatov N., Solutions to a class of nonlinear differential equations of fractional order, Electronic Journal of Qualitative Theory of Differential Equations, 20(2009), 1-10.
• [21] Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley Interscience Publication, John Wiley and Sons, New York, NY, USA, 1993.
• [22] Pachpatte B.G., Applications of the Leray-Schauder Alternative to some Volterra integral and integrodifferential equations, Indian J. Pure Appl. Math., 26(12)(1995), 1161-1168.
• [23] Pazy A., Semigroups of Linear Operators and applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
• [24] Podlubny I., Fractional Differential Equations, Vol. 198 of Mathematics in Science and Engineering, San Diego, Academic Press, Calif, USA, 1999.
• [25] Radonevic S., Common fixed points under contractive conditions in cone metric spaces, Computer and Math. with Appl., (2009) doi:10.1016/j.camwa.2009.07.035.
• [26] Raja P., Vaezpour S.M., Some extensions of Banach's contraction principle in complete cone metric spaces, Fixed Point Theory and Applications, Volume 2008, Article ID 768294, 11 pages.
• [27] Rezapour Sh., Hamlbarani R., Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings", J. Math. Anal. Appl., 345(2008), 719-724.
• [28] Rezapour Sh., Haghi R.H., Fixed point of multifunctions on cone metric spaces, Numer. Funct. Anal. and Opt., 30(7-8)(2009), 825-832.
• [29] Rezapour Sh., Haghi R.H., Two results about fixed point of multifunctions, Bull. Iranian Math. Soc., (2009)(in press).
• [30] Tidke H.L., Dhakne M.B., On abstract nonlinear di_erential equations of second order, Advances in Differential Equations and Control Processes, 3(1)(2009), 33-39.
• [31] Travis C.C., Webb G.F., Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston J. Math., 3(4)(1977), 555-567.
• [32] Travis C.C., Webb G.F., Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungaricae, 32(1978), 76-96.
• [33] Wlodarczyk K., Plebaniak R., Obczynski C., Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasiasyptotic contractions in cone uniform spaces, Computer and Math. with Appl., (2009) doi:10.1016/j.camwa.2009.07.035.