We study some common results in C*-algebra-valued $S_b$-metric spaces. We also present an interesting application of an existing and unique result for one type of integral equation.

We discuss about the generalized $F$-contraction mappings in partially ordered metric spaces. For this, we first introduce the notion of ordered weakly $F$-contraction mapping. We also present some fixed point results about this type of mapping in partially ordered metric spaces. Next, we introduce the notion of $acute{mathrm{C}}$iri$acute{mathrm{c}}$ type generalized ordered weakly $F$-contraction mapping. We also prove some fixed point results about this notion in partially ordered metric spaces. We also provide an example to support our results. In fact, this example shows that our main theorem is a genuine generalization in the area of the generalized $F$-contraction mappings in partially ordered metric spaces.

Introduction/purpose: The aim of the paper is to establish some coupled fixed point results in C*-algebra-valued b-metric spaces. Moreover, the obtained results are used to define the sufficient conditions for the existence of the solutions of some classes of integral equations. Methods: The method of coupled fixed points gives the sufficient conditions for the existence of the solution of some classes of integral equations. Results: New results were obtained on coupled fixed points in C*-algebra-valued b-metric space. Conclusion: The obtained results represent a contribution in the fixed point theory and open new possibilities of application in the theory of differential and integral equations.

We study some coupled fixed point theorems in C*-algebra-valued Sb-metric spaces. As applications, existence and uniqueness results for one type of integral equation x(t) = fE (K1(t, s) + K2(t, s))(f(s, x(s)) + g(s, x(s)))ds + h(t), t ∈ E where E is the Lebesque measurable set and m(E) < +∞, and under some other conditions are given.

By combining the concept of orthogonality and the Geraghty type contraction, we give some fixed point results in the class of $ O $-metric spaces. Our obtained results extend the existing results in the literature. We also resolve an ordinary type differential equation.