Set Integral Equations in Metric Spaces

Ioana Tişe

Let $P_{cp,cv}(\mathbb{R}^{n})$ be the family of all nonempty compact, convex subsets of $\mathbb{R}^{n}$. We consider the following set integral equations: (1) $X(t) = \int_{a}^{b} K(t,s,X(s))\mathrm{d}s + X_{0}$, (2) $X(t) = \int_{a}^{t} K(t,s,X(s))\mathrm{d}s + X_{0}$, where $K: [a, b] \times [a, b] \times P_{cp,cv}(\mathbb{R}^{n}) \to P_{cp,cv}(\mathbb{R}^{n})$ and $X_{0}\in P_{cp,cv}(\mathbb{R}^{n})$. The purpose of the paper is to study the existence and data dependence of the solutions of the set integral equations (1) and (2), by using a fixed point approach. Our results generalize and extend the results given in [2]. For other similar results see [3] and [4].