Existence, uniqueness and data dependence for the solutions of some integro-differential equations of mixed type in Banach space. (English) Zbl 1062.45006
The author studies existence, uniqueness and data dependence for the solutions of some integro-differential equations of mixed type in Banach space by using Picard and weakly Picard operators’ technique and suitable Bielecki norms.
Reviewer: V. Lakshmikantham (Melbourne/Florida)
MSC:
| 45N05 | Abstract integral equations, integral equations in abstract spaces |
| 45J05 | Integro-ordinary differential equations |
| 45G10 | Other nonlinear integral equations |
Keywords:
integro-differential equations; fixed points; weakly Picard operators; data dependence; Banach space; Bielecki normsReferences:
| [1] | Aman, H.: On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal. 11 (1972), 346 - 384. · Zbl 0244.47046 · doi:10.1016/0022-1236(72)90074-2 |
| [2] | Buic\?a, A.: Existence and continuous dependence of solutions of some functional- differential equations. Preprint. Cluj-Napoca: Babe\?s-Bolyai Univ. (Romania), Seminar on Fixed Point Theory, Preprint 3 (1995), 1 - 13. · Zbl 0894.34059 |
| [3] | Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162 (1991), 494 - 505. · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U |
| [4] | Byszewski, L.: Existence of approximate solution to abstract nonlocal Cauchy problem. J. Appl. Math. & Stoch. Anal. 5 (1992), 363 - 374. · Zbl 0776.34049 · doi:10.1155/S1048953392000303 |
| [5] | Byszewski, L.: Differential and Functional-Differential Problems with Nonlocal Conditions (in Polish; Cracow Univ. Techn. Monograph: Vol. 184). Cracow: Univ. Techn. 1995. · Zbl 0890.35163 |
| [6] | Byszewski, L. and V. Lakshmikantham: Theorem about the existence and unique- ness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40 (1991), 11 - 19. · Zbl 0694.34001 · doi:10.1080/00036819008839989 |
| [7] | Byszewski, L. and A. Tabor: An application of the Kuratowski measure of noncompactness to an investigation of the existence of solutions of an abstract integrodifferential problem. Nonlin. Studies 6 (1999), 111 - 122. V. Mure\?san · Zbl 0959.45005 |
| [8] | Du, S. W. and V. Lakshmikantham: Monotone iterative technique for differ- ential equations in a Banach space. J. Math. Anal. Appl. 87 (1982), 454 -459. · Zbl 0523.34057 · doi:10.1016/0022-247X(82)90134-2 |
| [9] | Guo Dajun: Ordinary Differential Equations in Abstract Spaces (in Chinese) Jiuan Shandong Sci. and Tech. Press 1985. |
| [10] | Guo Dajun: Solutions of nonlinear integrodifferential equations of mixed type in Banach spaces. J. Appl. Math. Simul. 2 (1989), 1 - 11. · Zbl 0681.45011 |
| [11] | Guo Dajun and V. Lakshmikantham: Nonlinear Problems in Abstract Cones. Boston - New York: Acad. Press 1988. · Zbl 0661.47045 |
| [12] | Lakshmikantham, V. and S. Leela: Nonlinear Differential Equations in Abstract Spaces. Oxford et al.: Pergamon Press 1981. · Zbl 0456.34002 |
| [13] | Lakshmikantham, V., Bainov, D. D. and P. S. Simeonov: Theory of Impulsive Differential Equations. Singapore: World Science 1989. |
| [14] | Liu Wei-an: Solutions of nonlinear integrodifferential equations of mixed type with impulses in Banach spaces. Math. Aplicata, Jingyong Shuxue Math. Appl. 9 (1996), 58 - 62. · Zbl 1039.45503 |
| [15] | Mure\?san, V.: Differential Equations with Affine Modification of the Argument (in Romania). Cluj-Napoca (Romania): Transilvania Press 1997. · Zbl 0942.34001 |
| [16] | Petru\?sel, A.: Fredholm-Volterra integral equations and Maia’s theorem. Preprint. Cluj-Napoca (Romania): Babe\?s-Boyai Univ., Seminar on Fixed Point Theory, Preprint 3 (1988), 79 - 82. · Zbl 0659.45003 |
| [17] | Rus,I. A.: Metrical Fixed Point Theorems. Univ. of Cluj-Napoca (Romania) 1979. |
| [18] | Rus, I. A.: Picard mappings: results and problems. Preprint. Cluj-Napoca: Babe\?s-Bolyai Univ., Seminar on Fixed Point Theory, Preprint 6 (1987), 55 -64. |
| [19] | Rus, I. A.: Weakly Picard mappings. Comment. Math. Univ. Carolinae 34 (1993), 769 - 773. · Zbl 0787.54045 |
| [20] | Rus, I. A.: Picard operators and applications. Preprint. Cluj-Napoca (Roma- nia): Babe\?s-Bolyai Univ., Seminar on Fixed Point Theory, Preprint 3 (1996). |
| [21] | Rus, I. A.: Weakly Picard operators and applications. Cluj-Napoca (Romania): Babe\?s-Bolyai Univ., Seminar on Fixed Point Theory 2 (2000), 41 - 57. |
| [22] | Rus, I. A. and S. Mure\?san: Data dependence of the fixed points set of some weakly Picard operators. In: Proc. Itinerant Seminar (Elena Popoviciu editor, Srima Publishing House), Cluj-Napoca (Romania): Babe\?s-Bolyai Univ. 2000, pp. 201 - 207. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
