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Nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition. (English) Zbl 1265.45012
The aim of this paper is to study the global existence of solutions of the equation $\frac {\text{d}}{\text{d}t}[x(t)-u(t,x(t))]=Ax(t)+f\left (t,x(t),\int ^{t}_{0}k(t,s,x(s))\text{d}s,\int ^{b}_{0}h(t,s,x(s))\text{d}s\right )$ satisfying the initial data $$x(0)=x_0+g(x)$$. In order to study the global existence of mild solutions to the above problem, the authors use a theorem whose proof is based on an application of the Leray-Schauder alternative. This theorem ensures simultaneously the global existence of solutions and the maximal interval of existence, a fact which is somewhat remarkable. The preceeding results of those two authors were gradually obtained and the result of this paper represents a further work of the previous authors’ research. The existence and uniqueness of mild and strong solutions of a nonlinear Volterra integro-differential equation with nonlocal condition (the analysis is based on semigroups theory and the Banach fixed point theorem, and inequalities were established by T. H. Gronwall and B. G. Pachpatte) were proved in [Demonstr. Math. 43, No. 3, 643–652 (2010; Zbl 1228.45009)]. The existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integro-differential equation with nonlocal condition (the results was obtained by using the Schauder fixed point theorem and the theory of semigroups) were proved in [the authors, Tamkang J. Math. 41, No. 4, 361–373 (2010; Zbl 1256.45001)].
The paper gives a theoretical method, which makes it possible to obtain some results on the global existence of mild solutions of some nonlinear integro-differential equations (with nonlocal condition).
##### MSC:
 45G10 Other nonlinear integral equations 45N05 Abstract integral equations, integral equations in abstract spaces 47G20 Integro-differential operators 45J05 Integro-ordinary differential equations 45B05 Fredholm integral equations 45D05 Volterra integral equations
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##### References:
 [1] K. Balachandran, M. Chandrasekaran: Existence of solutions of nonlinear integrodifferential equation with nonlocal condition. J. Appl. Math. Stochastic Anal. 10 (1997), 279–288. · Zbl 0986.45005 · doi:10.1155/S104895339700035X [2] K. Balachandran: Existence and uniqueness of mild and strong solutions of nonlinear integrodifferential equations with nonlocal condition. Differ. Equ. Dyn. Syst. 6 (1998), 159–165. · Zbl 0985.45009 [3] L. Byszewski: 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] L. Byszewski: Existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. Zesz. Nauk. Politech. Rzesz. 121, Mat. Fiz. 18 (1993), 109–112. · Zbl 0858.34045 [5] M.B. Dhakne, S.D. Kendre: On a nonlinear Volterra integrodifferential equation in Banach spaces. Math. Inequal. Appl. 9 (2006), 725–735. · Zbl 1108.45009 [6] M.B. Dhakne, S.D. Kendre: On an abstract nonlinear integrodifferential equation. Proceedings of Second International Conference on Nonlinear Systems, December 2007. Bulletin of Marathwada Mathematical Society 8 (2007), 12–22. · Zbl 1097.45009 [7] J. Dugundji, A. Granas: Fixed Point Theory. I: Monografie Matematyczne. PWN, Warszawa, 1982. [8] S.K. Ntouyas, P.C. Tsamatos: Global existence for semilinear evolution equations with nonlocal conditions. J. Math. Anal. Appl. 210 (1997), 679–687. · Zbl 0884.34069 · doi:10.1006/jmaa.1997.5425 [9] S.K. Ntouyas, P.C. Tsamatos: Global existence for second-order semilinear ordinary and delay integrodifferential equations with nonlocal conditions. Appl. Anal. 67 (1997), 245–257. · Zbl 0906.35110 · doi:10.1080/00036819708840609 [10] B.G. Pachpatte: Applications of the Leray-Schauder alternative to some Volterra integral and integrodifferential equations. Indian J. Pure Appl. Math. 26 (1995), 1161–1168. · Zbl 0852.45012 [11] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983. · Zbl 0516.47023 [12] H. L. Tidke, M.B. Dhakne: On global existence of solutions of abstract nonlinear mixed integrodifferential equation with nonlocal condition. Commun. Appl. Nonlinear Anal. 16 (2009), 49–59. · Zbl 1179.45021 [13] H. L. Tidke: Existence of global solutions to nonlinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions. Electron. J. Differ. Equ., paper No. 55 2009 (2009), 1–7. [14] A. Winter: The nonlocal existence problem for ordinary differential equations. Am. J. Math. 67 (1945), 277–284. · Zbl 0063.08284 · doi:10.2307/2371729
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