In this paper, we introduce a generalized (Formula presented.) -implicit locally contractive condition and give some examples to support it and show its significance in fixed point theory. We prove that the mappings satisfying the generalized (Formula presented.) -implicit locally contractive condition admit a common fixed point, where the ordered multiplicative (Formula presented.) -metric space is chosen as the underlying space. The obtained fixed point theorems generalize many earlier fixed point theorems on implicit locally contractive mappings. In addition, some nontrivial and interesting examples are provided to support our findings. To demonstrate the originality of our new main result, we apply it to show the existence of solutions to a system of nonlinear—Volterra type—integral equations.
- closed ball
- integral equations
- locally generalized Δ-implicit contraction
- ordered complete multiplicative G-metric space