On some qualitative properties of mild solutions of nonlocal semilinear functional differential equations

• Autorzy: Jain, R. S. , Dhakne, M. B.
• Języki publikacji: EN

Abstrakty

EN

In the present paper, we investigate the qualitative properties such as existence, uniqueness and continuous dependence on initial data of mild solutions of first and second order nonlocal semilinear functional differential equations with delay in Banach spaces. Our analysis is based on semigroup theory and modified version of Banach contraction theorem.

Słowa kluczowe

PL

istnienie; niepowtarzalność; równanie różniczkowe; punkt stały; teoria półgrupy

EN

existence; uniqueness; functional differential equation; fixed point; semigroup theory

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2014
• Tom: Vol. 47, nr 4
• Strony: 854--865
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Jain, R. S.

• School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded-431606, India,

autor

Dhakne, M. B.

• Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431004, India,

Bibliografia

• [1] L. Byszewski, H. Akca, On mild solution of a semilinear functional differential evolution nonlocal problem, J. Appl. Math. Stochastic Anal. 10(3) (1997), 265–271.
• [2] L. Byszewski, H. Akca, Existence of solutions of a semilinear functional evolution nonlocal problem, Nonlinear Anal. 34 (1998), 65–72.
• [3] L. Byszewski, V. Lakshamikantham, Theorems about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1990), 11–19.
• [4] C. Corduneanu, Integral Equations and Stability of Feedback System, Academic Press, New York, 1973.
• [5] H. O. Fattorini, Second Order Linear Differntial Equations in Banach Spaces, North Holland Mathematics Studies, Vol. 108, 1985.
• [6] W. E. Fitzgibbon, Global existence and boundedness of solutions of the extensible beam equation, SIAM J. Math. Anal. 13 (1982), 739–745.
• [7] J. K. Hale, Theory of Functional Differential Equations, Springer, 1977.
• [8] Y. Lin, J. H. Liu, Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal. 26 (1996), 1023–1033.
• [9] S. K. Ntouyas, Initial and boundary value problems for functional differential equations via the topological transversality method: A survey, Bull. Greek Math. Soc. 40 (1998), 3–41.
• [10] S. K. Ntouyas, Nonlocal initial and boundary value problems: A survey, Handbook of Differential Equations: Ordinary Differential Equations, Vol. 2, 2006, 461–557.
• [11] B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, 1998.
• [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983.
• [13] A. H. Siddiqi, Functional Analysis with Applications, Tata McGraw-Hill Publishing ltd, New Delhi, 1986.
• [14] H. L. Tidke, M. B. Dhakne, On abstract nonlinear differential equations of second order, Advances in Differential Equations and Control Processes 3(1) (2009), 33–39.
• [15] C. C. Travis, G. F. Webb, Second order differential equations in Banach spaces, Proc. Int. Symp. on non equations in abstract spaces, Academic Press, New York, 1978, 331–361.
• [16] C. C. Travis, G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar. 32 (1978), 76–96.
• [17] C. C. Travis, G. F. Webb, An abstract second order semilinear Volterra integro differential equations, SIAM J. Math. Anal. 10 (1979), 412–424.
• [18] Y. Quang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.