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A quasilinearization method for a class of integro-differential equations with mixed nonlinearities. (English) Zbl 1111.45005
Consider the initial value problem $u'(t)=AN\bigl(t,u(t)\bigr)+ B\int^t_{t_0}K\bigl(t,s,u(s)\bigr)\,ds,\;t_0\leq t\leq t_0+T,\quad u(t_0)=u_0,\tag{*}$ where $$A$$ and $$B$$ are real nonnegative constants and $$N$$ has the representation $N(t,u)=f(t,u)+ g(t,u)+h(t,u),$ where $$f+\varphi$$ is convex for some $$\varphi$$, $$g+ \psi$$ is concave for some $$\psi$$, and $$h$$ is Lipschitzian. By means of the method of lower and upper solutions, the author derives conditions such that there exists a monotone sequence of functions converging uniformly and quadratically to a solution of (*).

##### MSC:
 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations 45L05 Theoretical approximation of solutions to integral equations
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##### References:
 [1] Abd-Ellateef Kamar, A.R.; Drici, Z., Generalized quasilinearization method for systems of nonlinear differential equations with periodic boundary conditions, Dynamics cont. discrete impuls. systems ser. A math. anal., 12, 77-85, (2005) · Zbl 1092.34508 [2] Ahmad, B.; Khan, R.A.; Sivasundaram, S., Generalized quasilinearization method for integro-differential equations, Nonlinear studies, 3, 331-341, (2001) · Zbl 1001.65143 [3] Ahmad, B.; Nieto, J.J.; Shahzad, N., The bellman – kalaba – lakshamikantham quasilinearization method for Neumann problems, J. math. anal. appl., 257, 356-363, (2001) · Zbl 1004.34011 [4] Atici, F.M.; Topal, S.G., The generalized quasilinearization method and three-point boundary value problems on time scales, Appl. math. lett., 18, 577-585, (2005) · Zbl 1075.34006 [5] Brasier, C.M., Rapid evolution of introduced plant pathogens via interspecific hybridization, Bioscience, 2, 123-133, (2001) [6] Buica, A., Quasilinearization method for nonlinear elliptic boundary value problems, J. optim. theory appl., 124, 323-338, (2005) · Zbl 1125.35036 [7] Cabada, A.; Nieto, J.J., Quasilinearization and rate of convergence for higher order nonlinear periodic boundary value problems, J. optim. theory appl., 108, 97-107, (2001) · Zbl 0976.34015 [8] M. El-Gebeily, D. O’Regan, A quasilinearization method for a class of second order singular nonlinear differential equations with nonlinear boundary conditions, Nonlinear Anal. Real World Appl., to appear. [9] Hadeler, K.P., Reaction transport systems in biological modeling, (), 95-150 · Zbl 1002.92506 [10] Hubalek, Z.; Halouzka, J., West nile fever—a reemerging mosquito-Born viral disease in Europe, Emerg. infec. dis., 5, 643-650, (1999) [11] Jankowski, T., Quadratic approximation of solutions for differential equations with nonlinear boundary conditions, Comput. math. appl., 47, 1619-1626, (2004) · Zbl 1097.34008 [12] R.A. Khan, J.J. Nieto, A. Torres, Approximation and rapid convergence of solutions for periodic nonlinear problems with an application to a nonlinear biomathematical model of blood flow in intracranial aneurysms, preprint. [13] Kot, M.; Lewis, M.A.; Driessche, P.V., Dispersal data and the spread of invading organisms, Ecology, 7, 2027-2042, (1996) [14] Kot, M.; Schaffer, W.M., Discrete-time growth-dispersal models, Math. biosci., 1, 109-136, (1986) · Zbl 0595.92011 [15] Lakshmikantham, V.; Malek, S., Generalized quasilinearization, Nonlinear world, 1, 59-63, (1994) · Zbl 0799.34012 [16] Lakshmikantham, V.; Rao, M.R.M., Theory of integro-differential equations, (1995), Gordon & Breach London [17] Lakshmikantham, V.; Vatsala, A.S., Generalized quasilinearization for nonlinear problems, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0997.34501 [18] Lakshmikantham, V.; Vatsala, A.S., Generalized quasilinearization versus Newton’s method, Appl. math. comput., 164, 523-530, (2005) · Zbl 1078.65037 [19] Mandelzweig, V.B.; Tabakin, F., Quasilinearization approach to nonlinear problems in physics with application to nonlinear odes, Comput. phys. commun., 141, 268-281, (2001) · Zbl 0991.65065 [20] Medlock, J.; Kot, M., Spreading disease: integro-differential equations old and new, Math. biosci., 2, 201-222, (2003) · Zbl 1036.92030 [21] Murray, J.D., Mathematical biology, (1993), Springer New York · Zbl 0779.92001 [22] Nieto, J.J., Quadratic approximation of solutions for ordinary differential equations, Bull. austral. math. soc., 55, 161-168, (1997) · Zbl 0889.34017 [23] Neito, J.J., Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions, Proc. am. math. soc., 125, 2599-2604, (1997) · Zbl 0884.34011 [24] Nieto, J.J.; Torres, A., A nonlinear biomathematical model for the study of intracranial aneurysms, J. neurol. sci., 177, 18-23, (2000) [25] Nikolov, S.; Stoytchev, S.; Torres, A.; Nieto, J.J., Biomathematical modeling and analysis of blood flow in an intracranial aneurysms, Neurol. res., 25, 497-504, (2003) [26] Yermachenko; Sadyrbaev, F., Quasilinearization and multiple solutions of the emden – fowler type equation, Math. model. anal., 10, 41-50, (2005) · Zbl 1084.34020
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