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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 \[ u(t)=h(t)+\int_0^t G(t,s)f(s,u(s),Tu(s),Su(s))\,ds, \quad t\in J,\tag{1} \] 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 \[ \begin{cases} x'''=f(t,x'',x),\quad 0\leq t\leq1,\\ \alpha_1x(0)+\alpha_2x'(0)=\beta_1x(1)+\beta_2x'(1),\\ x''(0)=x_0,\end{cases}\tag{2} \] 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 \(x'(0) =\beta_1x(1)+\beta_2x'(1)\).

MSC:
45G10 Other nonlinear integral equations
34A34 Nonlinear ordinary differential equations and systems
45N05 Abstract integral equations, integral equations in abstract spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47N20 Applications of operator theory to differential and integral equations
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