Pseudo-almost automorphic mild solutions to semilinear integral equations in a Banach space. (English) Zbl 1298.45018
The authors investigate the existence of pseudo-almost automorphic solutions to integro-functional equations with infinite delay of the form
\[ x(t)= \int^t_{-\infty} a(t- s)[Ax(s)+ f(s, x(s))]\,ds,\qquad t\in\mathbb{R}, \]
with \(a\in L(\mathbb R_+, \mathbb C)\), a linear operator \(A: D(A)\subseteq X\to x\) on the complex Banach space \(X\), which is the generator of an integral resolvent, and a pseudo-almost automorphic map \(f: \mathbb R\times X\to X\), subject to various technical condition. The problem is naturally transformed into an integral equation of the form
\[ x(t)= \int^t_{-\infty} S(t- s) f(s,x(s))\,ds,\qquad t\in\mathbb{R}, \]
and it is shown by contraction mappings that there exists a solution, which is called a mild solution for the problem. This solution is unique in the class of pseudo-almost automorphic functions. This is the main result, and two more variants are obtained by changing adequately the assumptions on the integral resolvent \((S(t); t\geq 0)\) and on the continuity of \(f\) (for instance, renouncing to the Lipschitz condition).
\[ x(t)= \int^t_{-\infty} a(t- s)[Ax(s)+ f(s, x(s))]\,ds,\qquad t\in\mathbb{R}, \]
with \(a\in L(\mathbb R_+, \mathbb C)\), a linear operator \(A: D(A)\subseteq X\to x\) on the complex Banach space \(X\), which is the generator of an integral resolvent, and a pseudo-almost automorphic map \(f: \mathbb R\times X\to X\), subject to various technical condition. The problem is naturally transformed into an integral equation of the form
\[ x(t)= \int^t_{-\infty} S(t- s) f(s,x(s))\,ds,\qquad t\in\mathbb{R}, \]
and it is shown by contraction mappings that there exists a solution, which is called a mild solution for the problem. This solution is unique in the class of pseudo-almost automorphic functions. This is the main result, and two more variants are obtained by changing adequately the assumptions on the integral resolvent \((S(t); t\geq 0)\) and on the continuity of \(f\) (for instance, renouncing to the Lipschitz condition).
Reviewer: Constantin Corduneanu (Arlington)
MSC:
| 45N05 | Abstract integral equations, integral equations in abstract spaces |
| 44A35 | Convolution as an integral transform |
| 42A85 | Convolution, factorization for one variable harmonic analysis |
| 42A75 | Classical almost periodic functions, mean periodic functions |
Keywords:
pseudo-almost automorphic function; semilinear integral equations; integral resolvent family; fixed pointReferences:
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