Fixed point problems for generalized contractions with applications

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