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Some bounded linear integral operators and linear Fredholm integral equations in the spaces \(H_{\alpha, \delta, \gamma}((a, b) \times (a, b), X)\) and \(H_{\alpha, \delta}((a, b), X)\). (English) Zbl 1381.45007

Summary: The spaces \(H_{\alpha, \delta, \gamma}((a, b) \times (a, b), \mathbb R)\) and \(H_{\alpha, \delta}((a, b), \mathbb R)\) were defined in (Hüseynov (1981), pages 271–277). Some singular integral operators on Banach spaces were examined, [M. R. Dostanić, J. Math. Anal. Appl. 395, No. 2, 496–500 (2012; Zbl 1247.42010); N. Dunford and J. T. Schwartz, Linear operators. Part III: Spectral operators. New York etc.: John Wiley & Sons (1988; Zbl 0635.47003), pages 2419–2426] and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in [E. Babolian and A. Arzhang Hajikandi, J. Comput. Appl. Math. 235, No. 5, 1148–1159 (2011; Zbl 1205.65331); T. Okayama et al., ibid. 234, No. 4, 1211–1227 (2010; Zbl 1191.65185); K. S. Thomas, Math. Comput. 36, 193–205 (1981; Zbl 0466.65078)] by numerical methods. In this paper, we define the sets \(H_{\alpha, \delta, \gamma}((a, b) \times (a, b), X)\) and \(H_{\alpha, \delta}((a, b), X)\) by taking an arbitrary Banach space \(X\) instead of \(\mathbb R\), and we show that these sets which are different from the spaces given in [Dunford, loc. cit.] and (Plamenevskiy (1965)) are Banach spaces with the norms \(||\cdot||_{\alpha, \delta, \gamma}\) and \(||\cdot||_\alpha, \delta\). Besides, the bounded linear integral operators on the spaces \(H_{\alpha, \delta, \gamma}((a, b) \times (a, b), X)\) and \(H_{\alpha, \delta}((a, b), X)\), some of which are singular, are derived, and the solutions of the linear Fredholm integral equations of the form \[ f(s) = \phi(s) + \lambda \int^b_a A(s, t)f(t)\,dt,\quad f(s) = \phi(s) + \lambda \int^b_a A(t, s)f(t)\,dt \] and \[ f(s, t) = \phi(s, t) + \lambda \int^b_a A(s, t)f(t, s)\,dt \] are investigated in these spaces by analytical methods.

MSC:

45B05 Fredholm integral equations
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