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Abstract weighted pseudo almost automorphic functions, convolution invariance and neutral integral equations with applications. (English) Zbl 1442.43005

The authors investigate the following abstract neutral integral equation on a Banach space \(\mathbb{X}\): \[\begin{array}{lll} u(t)&=&f_0(t,u(t),u(h_0(t)))\\ & & +\int^t_{-\infty}R_1(t,s)f_1(s,u(s),u(h_1(s)))ds\\ & & +\int^{\infty}_tR_2(t,s)f_2(s,u(s),u(h_1(s)))ds, \quad t\in \mathbb{R}, \end{array} \] where for each \(i = 0, 1, 2\), \(h_i: \mathbb{R}\to \mathbb{R}\) is surjective and continuous, and \(f_i: \mathbb{R}\times \mathbb{X} \times \mathbb{X} \to \mathbb{X}\), and for each \(i = 1, 2,\) \(R_i:\mathbb{R} \times \mathbb{R} \to \mathcal{B}(\mathbb{X})\). They present some new sufficient conditions for the existence and uniqueness of weighted pseudo almost automorphic solution of the abstract neutral integral equation.

MSC:

43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
34G20 Nonlinear differential equations in abstract spaces
45G10 Other nonlinear integral equations
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