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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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