Weakly compatible and quasi-contraction results in fuzzy cone metric spaces with application to the Urysohn type integral equations

• Published in: Advances in difference equations Vol. 2020; no. 1; pp. 1 - 16
• Main Authors: Jabeen, Shamoona, Ur Rehman, Saif, Zheng, Zhiming, Wei, Wei
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 10.06.2020; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Banach spaces; Coincidence point; Common fixed point; Contraction conditions; Difference and Functional Equations; Existence theorems; Fixed points (mathematics); Functional Analysis; Fuzzy cone metric space; Generalized method of moments; Integral equations; Mathematical analysis; Mathematics; Mathematics and Statistics; Metric space; Monte Carlo simulation; Ordinary Differential Equations; Partial Differential Equations; Weakly compatible mappings; Contraction conditions; 47H10; Weakly compatible mappings; Common fixed point; Coincidence point; Fuzzy cone metric space; 54H25
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-020-02743-5

In this paper, we present some weakly compatible and quasi-contraction results for self-mappings in fuzzy cone metric spaces and prove some coincidence point and common fixed point theorems in the said space. Moreover, we use two Urysohn type integral equations to get the existence theorem for common solution to support our results. The two Urysohn type integral equations are as follows: x ( l ) = ∫ 0 1 K 1 ( l , v , x ( v ) ) d v + g ( l ) , y ( l ) = ∫ 0 1 K 2 ( l , v , y ( v ) ) d v + g ( l ) , where l ∈ [ 0 , 1 ] and x , y , g ∈ E , where E is a real Banach space and K 1 , K 2 : [ 0 , 1 ] × [ 0 , 1 ] × R → R .