Some Generalizations of Fixed Point Theorems in Partially Ordered Metric Spaces and Applications to Partial Differential Equations with Uncertainty

Some generalized contractions using altering distances in partially ordered metric spaces are investigated and their applications to fuzzy partial differential equations are considered. Starting from the Banach contraction principle, our theorems presented here generalize, extend, and improve different results existing in the literature on the existence of coincidence points for a pair of mappings. In terms of their applicability, this might constitute the first paper dealing with the solvability of fuzzy partial differential equations from the point of view of considering the structure of the fuzzy number space as a partially ordered space. Under the generalized contractive-like property over comparable items, which is weaker than the Lipschitz condition, we show that the existence of just a lower or an upper solution is enough to prove the existence and uniqueness of two types of fuzzy solutions in the sense of gH-differentiability.

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