On Solutions To Open Problems And Volterra-Hammerstein Non-linear Integral Equation

Taking into account the fact that the contractive conditions carry out the magnificent role in es-tablishing coincidence and common fixed points, we introduce generalized condition (B) for self maps in g-metric spaces and utilize it to establish a unique fixed point, a unique common fixed point, and a unique coincidence point. Conclusively, we deal with two questions about the survival of a fixed point in Abbas et al. [M. Abbas, G. V. R. Babu, and G. N. Alemayehu, On common fixed points of weakly compatible mappings satisfying generalized condition, Filomat 25 (2011), 9-19] and on the survival of contractive condition which assure the fixed point at the discontinuity of a map in Rhoades [B. E. Rhoades, Con-tractive definitions and continuity, Fixed Point Theory and its Applications (Berkeley 1986), Contemp. Math. (Amer. Math. Soc.), 72 (1988), 233-245]. Further, we introduce circle, fixed circle, common fixed circle, and u0-generalized condition (B) via g-metric to establish fixed circle and common fixed circle theorems. Also, we give examples and an application to solve Volterra-Hammerstein non-linear integral equation in order to demonstrate the significance of obtained results.