Integral solution of a class of nonlinear integral equations.(English)Zbl 1285.45004

Let $$f_i:[0,+\infty)\times\mathbb R\to \mathbb R$$, $$\phi_i:[0,+\infty)\to [0,+\infty)$$ for $$i\in \{1,2,3\}$$, $$k:[0,+\infty)\times [0,+\infty)\to \mathbb{R}$$, $$x:[0,+\infty)\to\mathbb{R}$$ and the nonlinear integral equation $x(t)= f_1(t,x(\phi_1(t)))+ f_2\Biggl(t, \int^{\phi_2(t)}_0 k(t,s) f_3(s,x(\phi_3(s)))\,ds\Biggr),\;t\geq 0.\tag{1}$ If $$A\subset\mathbb{R}$$ is a Lebesgue measurable subset of $$\mathbb{R}$$, consider the space $$(L^1(A),\|\cdot\|)$$ of Lebesgue integrable functions on $$A$$, where $(\forall x: A\to\mathbb{R})(x\in L^1(A))\Biggl(\| x\|=\int_A |x(t)|\,dt\Biggr).$ The authors prove the existence of integral solutions $$x$$ of the equation (1) under the following hypotheses:
$$(\text{H}_1)$$ the functions $$f_i$$, $$i\in \{1,2,3\}$$, satisfy the Carathéodory condition $\begin{gathered} (\forall i\in \{1,2,3\})(\exists a_i\in L^1[0,+\infty))(\exists b_i\in (0,+\infty))(\forall(t,x)\in [0,+\infty)\times\\ L^1[0,+\infty))(|f_i(t,x(t))|\leq a_i(t)+ b_i|x(t)|)\end{gathered}$ and $$(\forall(t,s)\in [0,+\infty)\times \mathbb{R})(s\mapsto f(t,s))$$ is a contraction mapping,
$$(\text{H}_2)$$ the function $$k$$ satisfies the Carathéodory condition, the linear Volterra integral operator $$K: L^1[0,+\infty)\to L^1[0,+\infty)$$ defined by $(\forall t\in [0,+\infty))\Biggl(Ky(t)= \int^{\phi_2(t)}_0 x(t,s) y(s)\,ds\Biggr)$ transforms the space $$L^1[0,+\infty)$$ into itself and $(\forall(t,s)\in [0,+\infty)\times [0,+\infty))(0\leq s\leq t)\Biggl(\text{ess\,sup}_{s\geq 0}\;\int^{+\infty}_s k(t,s)\,dt<+\infty\Biggr),e$ $$(\text{H}_3)$$ the functions $$\phi_i$$, $$i\in \{1,2,3\}$$, are absolutely continuous, $(\forall i\in \{1,2,3\})\Biggl(\lim_{t\to+\infty}\, \phi_i(t)=+\infty\Biggr),$ $$\phi_2$$ is increasing and $(\forall i\in \{1,2,3\})(\exists\alpha_i\geq 0)(\forall t\in [0,+\infty)) (\phi_i(t)\geq \alpha_i),$ $$(\text{H}_4)$$ $$b_1 \alpha^{-1}_1+ b_2 b_3 \alpha^{-1}_3\| K\|< 1$$, where $$\| K\|$$ denotes the norm of the operator $$K$$ in $$L^1[0,+\infty)$$.
Under the hypotheses $$(\text{H}_1)$$–$$(\text{H}_4)$$, the equation (1) has at least a solution $$x\in L^1[0,+\infty)$$.
The authors give an example which can be treated by the theorem written above.

MSC:

 45G10 Other nonlinear integral equations 45D05 Volterra integral equations
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