An observation on F -weak contractions and discontinuity at the fixed point with an application
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Journal of Fixed Point Theory and Applications
An observation on F -weak contractions and discontinuity at the fixed point with an application Waleed M. Alfaqih , Mohammad Imdad and Rqeeb Gubran Abstract. In this paper, we have some observations on F -weak contractions due to Wardowski and Van Dung (Demonstr Math 47(1):146–155, 2014). Our observations lead us to introduce the notion of F ∗ -weak contractions and utilize the same to prove some fixed point results. The proven results give an affirmative answer to certain open questions raised by Kannan (Bull Calcutta Math Soc 60:71–76, 1968) and Rhoades (Contemp. Math. 72:233–245, 1988) on the existence of contractive definitions not forcing the continuity at the fixed point. Some illustrative examples are also given. As an application, we investigate the existence and uniqueness of the solution of an integral equation of Volterra type. Mathematics Subject Classification. 47H10, 54H25. Keywords. F -contraction, F -weak contraction, fixed point, discontinuity at the fixed point, integral equation.
1. Introduction and preliminaries In 1922, Stefan Banach in his Ph.D. thesis [1] introduced the celebrated Banach contraction principle which asserts that every self-mapping S defined on a complete metric space (X, d) satisfying (for all x, y ∈ X) d(Sx, Sy) ≤ λd(x, y),
where λ ∈ (0, 1)
(1.1)
has a unique fixed point and, for each x ∈ X, the sequence {S n x} converges to the fixed point of S. Here it is easy to see that S is continuous mapping on M . To study the existence of fixed points for discontinuous mappings, Kannan [7] introduced a weaker contraction condition and proved a very interesting fixed point result which states that every self-mapping S defined on a 0123456789().: V,-vol
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complete metric space (X, d) satisfying (for all x, y ∈ X) d(Sx, Sy) ≤ λ d(x, Sx) + d(y, Sy) ,
where β ∈
0,
1 2
(1.2)
has a unique fixed point. Mappings satisfying (1.2) are called Kannan type mappings. Rhoades [10] compared 250 contractive definitions (including (1.2)) and he showed that though most of the contractive definitions do not force the mapping to be continuous on the entire domain, all of them force the mapping to be continuous at the fixed point. Motivated by his observations, Rhoades [11] formulated an interesting open question as follows: Open Question 1.1. Whether there exists a contractive definition which is strong enough to ensure the existence and uniqueness of a fixed point which does not force the mapping to be continuous at the underlying fixed point. The first answer of the Open Question 1.1 appeared after more than a decade by Pant [8]. On the other hand, Wardowski [14] introduced a new class of auxiliary functions and utilized the same to define F -contractions as follows. Definition 1.1. [14] Let F be the family of all functions F : (0, ∞) → R satisfying the following conditions: F1 : F is strictly increasing, i.e., for all α, β ∈ (0, ∞), α < β ⇒ F (α) < F (β), F2 : for every sequence {βn } ⊂ (0, ∞), limn→∞ F (βn ) = −∞ ⇔ limn→
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