Some bounded linear integral operators and linear Fredholm integral equations in the spaces \(H_{\alpha, \delta, \gamma}((a, b) \times (a, b), X)\) and \(H_{\alpha, \delta}((a, b), X)\)
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, [\textit{M. R. Dostanić}, J. Math. Anal. Appl. 395, No. 2, 496--500 (2012; Zbl 1247.42010); \textit{N. Dunford} and \textit{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 [\textit{E. Babolian} and \textit{A. Arzhang Hajikandi}, J. Comput. Appl. Math. 235, No. 5, 1148--1159 (2011; Zbl 1205.65331); \textit{T. Okayama} et al., ibid. 234, No. 4, 1211--1227 (2010; Zbl 1191.65185); \textit{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.
scientific article; zbMATH DE number 6209367
scientific article