Fixed point problems for generalized contractions with applications

Advances in Difference Equations

article

Nazam, Muhammad¹  Park, Choonkil²  Arshad, Muhammad³ 

[1] Department of Mathematics, Allama Iqbal Open University;Research Institute for Natural Sciences, Hanyang University;Department of Mathematics and Statistics, International Islamic University

关键词: Fixed point;    \((\psi;    \varphi )\) -contraction;    Complete metric space;    Applications;   

DOI  :  10.1186/s13662-021-03405-w

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

In this paper, we investigate the conditions on the control mappings$\psi ,\varphi :(0,\infty )\rightarrow \mathbb{R}$ that guarantee the existence of the fixed points of the mapping$T:X\rightarrow P(X)$ satisfying the following inequalities:$$ \psi \bigl(H(Tx,Ty)\bigr)\leq \varphi \bigl(d(x,y)\bigr) \quad \forall x,y\in X, \text{provided that } H(Tx,Ty)>0, $$ and$$ \psi \bigl(H(Tx,Ty)\bigr)\leq \varphi \bigl(A(x,y)\bigr) \quad \forall x,y\in X, \text{provided that } H(Tx,Ty)>0, $$ where$A(x,y)=\max \{ d(x,y), d(x,Tx), d(y,Ty), (d(x,Ty) +d(Tx,y))/2 \} $ , and$(X, d)$ 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.

【 授权许可】

CC BY