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Zbl 1080.45005
Liu, Lishan; Guo, Fei; Wu, Congxin; Wu, Yonghong
Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. (English)
[J] J. Math. Anal. Appl. 309, No. 2, 638-649 (2005). ISSN 0022-247X

The following Volterra type integral equation is considered $$u(t)=h(t)+\int_0^t G(t,s)f(s,u(s),Tu(s),Su(s))\,ds, \quad t\in J,\tag1$$ where $J=[0,a]$, $$ Tu(t)=\int_0^t k(t,s)u(s)\,ds ,\quad Tu(t)=\int_0^a h(t,s)u(s)\,ds, \quad t\in J,$$ and $k$, $h$, $f$ are continuous kernels. Existence of a global solution is studied with the help of a fixed point theorem, generalizing Darbo's fixed point theorem. As a particular application, the authors establish the existence of a global solution to the following initial value problem for a nonlinear ordinary differential equation $$\cases x'''=f(t,x'',x),\quad 0\leq t\leq1,\\ \alpha_1x(0)+\alpha_2x'(0)=\beta_1x(1)+\beta_2x'(1),\\ x''(0)=x_0,\endcases\tag2$$ and a similar problem for $ x'''=f(t,x'',x',x)$ when the first initial value condition in (2) is replaced by the following one $x'(0) =\beta_1x(1)+\beta_2x'(1)$.
[Roland Duduchava (Tbilisi)]

MSC 2000:: *45G10 Nonsingular nonlinear integral equations
34A34 Nonlinear ODE and systems, general
45N05 Integral equations in abstract spaces
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47N20 Appl. of operator theory to differential and integral equations

Keywords: nonlinear Volterra type integral equation; measure of noncompactness; Darbo's fixed point theorem; global solution; initial value problem

Cited in: Zbl 1169.45300

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