Random fixed point theorems under weak topology features and application to random integral equations with lack of compa
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Journal of Fixed Point Theory and Applications
Random fixed point theorems under weak topology features and application to random integral equations with lack of compactness Adil El-Ghabi and Mohamed Aziz Taoudi Abstract. We present several random fixed point theorems for random operators with deterministic or stochastic domains. The main assumptions of our results are formulated in terms of the weak topology and an axiomatic definition of the measure of weak noncompactness. The results herein extend in a broad sense some new and classical results in the literature. As an application, we discuss the solvability of a random Hammerstein integral equation with lack of compactness. Mathematics Subject Classification. Primary 47H10, 47H40, 47H08; Secondary 47H30. Keywords. Random fixed point, random operator, deterministic domain, stochastic domain, weak topology, measure of weak noncompactness, weakly countably-condensing, random Hammerstein integral equation.
1. Introduction It is well known that random fixed point theorems are stochastic versions of deterministic fixed point theorems and are useful in the study of various classes of random equations. The study of random fixed point theorems was initiated by the Prague school of probabilists around Spa˘cek [46] and Hanˇs [19,20] in the 1950s and since then has been addressed by many investigators. We quote the contributions by Mukherjea [33], Prakasa Rao [42], Itoh [22], Engl [16,17], Reich [43], Papageorgiou [41], Tan and Yuan [47] and many others. We point out that the random fixed point theorems developed in the literature are not always useful in establishing existence principles for nonlinear random problems with lack of compactness. So, a new theory is needed to complete the picture. In this paper, we develop a new fixed point approach that combines the advantages of the strong topology with the advantages of the weak topology. Such an approach enables us to draw new and meaningful 0123456789().: V,-vol
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A. El-Ghabi and M. A. Taoudi
conclusions about random fixed points for a given random operator and to handle nonlinear random problems with lack of compactness. We underline that the interest of the weak topology stems from the decisive role played by weak compactness in the theory of infinite dimensional linear spaces. In particular, a Banach space X is reflexive if and only if the closed unit ball is weakly compact. We also mention that our operators are allowed to have stochastic domains, which is an extra interesting feature. Finally, we illustrate the applicability and the usefulness of our random fixed point results by establishing a very general existence result for a random integral equation of Hammerstein type. This paper is arranged as follows. In Sect. 1, we fix the notation and present some key tools that will be used to prove our main results. In Sect. 2, we prove some random fixed point theorems for random operators with deterministic domains. Section 3 is mainly devoted to fixed point theorems for random operators with stochastic domains. F
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