| Abstract: |
| Let $X$ and $Y$ be two arbitrary sets, $P\subset X \times Y $ and $S: X \rightarrow Y$ . We consider
the problem of finding an element $u \in X$ such that $(u, Su) \in P$. We prove that the existence
of the solution to this problem is obtained, provided that an associated operator $\Lambda$ has a fixed
point. Moreover, under an additional assumption, the solution is unique if and only if the fixed
point is unique. Then, in the framework of metric spaces $X$ and $Y$, we provide necessary and
sufficient conditions for the convergence of an arbitrary sequence $\{u_n\} \subset X$ to the solution $u$.
We also show some applications of our results in the study of nonlinear boundary value problems
for partial differential equations. |
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