Fixed point problems for generalized contractions with applications

Abstract In this paper, we investigate the conditions on control mappings $\psi ,\varphi :(0,\infty )\rightarrow \mathbb{R}$ ? , ? : ( 0 ? ) ? R that guarantee existence of fixed points mapping $T:X\rightarrow P(X)$ T X P satisfying following inequalities: $$ \psi \bigl(H(Tx,Ty)\bigr)\leq \varphi \bigl(d(x,y)\bigr) \quad \forall x,y\in X, \text{provided } H(Tx,Ty)>0, H x y ? d ? ? provided > and \bigl(A(x,y)\bigr) A where $A(x,y)=\max \{ d(x,y), d(x,Tx), d(y,Ty), (d(x,Ty) +d(Tx,y))/2 \} $ = max { + / 2 } , $(X, d)$ is a metric space. The obtained point results improve many earlier set-valued contractions. As an application, consider solutions FDE.