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Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. (English) Zbl 1080.45005
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 $x'(0) =\beta_1x(1)+\beta_2x'(1)$.

45G10Nonsingular nonlinear integral equations
34A34Nonlinear ODE and systems, general
45N05Abstract integral equations, integral equations in abstract spaces
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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