Adjusted Hardy-Rogers-Type Result Generalization

The adjusted Hardy-Rogers result generalization for the fixed point is demonstrated in this study, validating our results utilizing an application.

The existence and uniqueness of a point ξ ∈ X, such that T: X → X, is a contraction mapping where X is a complete metric space was proved by Banach [1].

$d\mathrm{<mo\; stretchy="false">(</mo>}f\xi \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}\le \alpha d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo>,}$ (1)

for all ξ, ζ ∈ X and α [0,1). Kannan [2] developed (1) as

$d\mathrm{<mo\; stretchy="false">(</mo>}f\xi \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}\le \alpha \mathrm{<mo\; stretchy="false">[</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}f\xi \mathrm{,}\xi \mathrm{<mo\; stretchy="false">)</mo>}+d\mathrm{<mo\; stretchy="false">(</mo>}f\zeta \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>,}$ (2)

for all ξ, ζ ∈ X and $\alpha \in \mathrm{<mo\; stretchy="false">(</mo>0,}\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}$ . Reich in [3] generalized (2) as

$d\mathrm{<mo\; stretchy="false">(</mo>}f\xi \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}\le \mathrm{<mo\; stretchy="false">[</mo>}{\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}f\xi \mathrm{,}\xi \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}f\zeta \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>,}$ (3)

for all ξ, ζ ∈ X such that η1 + η₂ + η₃ < 1. Then f has a unique fixed point in X.

In the same direction, Hardy and Rogers in [4] introduced the following

Theorem 1.1

Let (X, d) be a metric space and f a self mapping of X satisfies

$d\mathrm{<mo\; stretchy="false">(</mo>}f\xi \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}f\xi \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{,}f\xi \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo>,}$ (4)

for ξ, ζ ∈ X where η₁, η₂, η₃, η₄, η₅ are non-negative and we set α = η₁ + η₂ + η₃ + η₄ + η₅. Then,

If X is complete metric space and α1, f has a unique fixed point.

If (4) is adjusted to the condition ξ ≠ ζ implies

$d\mathrm{<mo\; stretchy="false">(</mo>}f\xi \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}{\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}f\xi \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}f\zeta \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{,}f\xi \mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo>.}$ (5)

Such that X is a compact with continuous mapping and α + 1, then f has a unique fixed point.

Recently, many of the Hardy-Rogers-type notions have been developed. From these studies, we refer to Rangama [5] established the existence of the Hardy-Rogers-type common fixed point in 2-metric space. With respect to the aiding function, Chifu [6] provided a few fixed point theorems in b-metric space utilizing the Hardy-Rogers type. New Hardy-Rogers-type results have been provided by Patil, et al. [7]. Victoria [8] obtained the P-proximate cyclic contraction in the uniform spaces utilizing the Hardy-Rogers type. Using partially ordered partial metric space, Abbas [9] developed a few fixed point theorems for the Hardy-Rogers type. The common fixed point theorem for T-Hardy-Rogers contraction mapping in a cone metric space was established by Rhymend, et al. in [10]. Saipara generalized some fixed point theorems for Hardy-Rogers-type in metric-like space [11]. Raghavendran, et al. [12] included a recent article relevant to the focused topic.

We will introduce and prove the adjusting generalization of the Hardy-Rogers type as

Theorem 2.1

Let {f_a} be a family continuous self-mappings in a complete metric space X, suppose that

$+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo>}$ (6)

for every ξ, ζ ∈ X, ξ ≠ ζ and ${\eta }_1\mathrm{,}{\eta }_2\mathrm{,}{\eta }_3\mathrm{,}{\eta }_4\mathrm{,}{\eta }_5\in {\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^5}{\eta }_{i}1.$ Then f_α (ξ) has a unique fixed point u₁ ∈ X.

Proof. For ${\xi }_0\mathrm{,}{\zeta }_0\in X$ take $f_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{n-1}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\xi }_n\mathrm{,}{g}_{\beta }\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{n-1}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\zeta }_n\mathrm{,}$

$d\mathrm{<mo\; stretchy="false">(</mo>}{x}_k\mathrm{,}{y}_k\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{g}_{\beta }\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{,}g\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}g\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{<mo\; stretchy="false">)</mo>}$

$+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{\zeta }_{k-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}k\in \mathbb{N}\mathrm{.}$ (7)

So,

${\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{x}_k\mathrm{,}{y}_k\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }{}_1\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{f}_{\alpha }{}_2\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$\le {\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}\mathrm{<mo\; stretchy="false">[</mo>}{\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{\xi }_k\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{,}{\zeta }_k\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{\zeta }_k\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{,}{\xi }_k\mathrm{<mo\; stretchy="false">)</mo>}$

$+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{\zeta }_{k-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>}$

$\le \mathrm{<mo\; stretchy="false">[</mo>}{\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_0\mathrm{,}{\xi }_n\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_0\mathrm{,}{\zeta }_n\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_3{\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{\zeta }_k\mathrm{<mo\; stretchy="false">)</mo>}+{\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{,}{\xi }_k\mathrm{<mo\; stretchy="false">)</mo>}$

$+{\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{\zeta }_{k-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>.}$

Also,

${\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k+1}\mathrm{,}{y}_k\mathrm{<mo\; stretchy="false">)</mo>}\le \mathrm{<mo\; stretchy="false">[</mo>}{\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_1\mathrm{,}{\xi }_n\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_0\mathrm{,}{\zeta }_n\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_3{\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_k\mathrm{,}{\zeta }_k\mathrm{<mo\; stretchy="false">)</mo>}+{\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_{k-1}\mathrm{,}{\xi }_k\mathrm{<mo\; stretchy="false">)</mo>}$

$+{\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k-1}\mathrm{,}{\zeta }_{k-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>,}$

and,

${\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_k\mathrm{,}{\xi }_{k+1}\mathrm{<mo\; stretchy="false">)</mo>}\le \mathrm{<mo\; stretchy="false">(</mo>}{\eta }_1+{\eta }_5\mathrm{<mo\; stretchy="false">)</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_0\mathrm{,}{\xi }_n\mathrm{<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>}{\eta }_2+{\eta }_3\mathrm{<mo\; stretchy="false">)</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_1\mathrm{,}{\xi }_{n+1}\mathrm{<mo\; stretchy="false">)</mo>.}$

Then,

${\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_k\mathrm{,}{\xi }_{k+1}\mathrm{<mo\; stretchy="false">)</mo>}\le {\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_k\mathrm{,}{y}_k\mathrm{<mo\; stretchy="false">)</mo>}\le {\displaystyle \sum _{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{k+1}\mathrm{,}{y}_k\mathrm{<mo\; stretchy="false">)</mo>}\infty \mathrm{.}$ (8)

Therefore, ${\displaystyle {\sum }_{k\mathrm{<mo>=</mo>1}}^n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_k\mathrm{,}{\xi }_{k+1}\mathrm{<mo\; stretchy="false">)</mo>}\to 0ask\to \infty$ , hence {XK} is a Cauchy sequence. Also, {YK} is a Cauchy sequence in X, and since X is a complete metric space, there exists a common fixed point in X.

Suppose that,

$u_1\mathrm{<mo>=</mo>}\underset{n\to \infty }{{\displaystyle \lim }}{\xi }_n\mathrm{,}{u}_2\mathrm{<mo>=</mo>}\underset{n\to \infty }{{\displaystyle \lim }}{\zeta }_n\mathrm{,}\forall u_1\mathrm{,}{u}_2\in X\mathrm{,}$

we get,

$d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_n\mathrm{,}{u}_1\mathrm{<mo\; stretchy="false">)</mo>}\to \mathrm{0,}n\to \infty \mathrm{,}$

$d\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_n\mathrm{,}{u}_1\mathrm{<mo\; stretchy="false">)</mo>}\to \mathrm{0,}n\to \infty \mathrm{.}$

since, fα, gβ are continuous mappings we obtained,

$d\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_n\mathrm{<mo\; stretchy="false">)</mo>,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\to \mathrm{0,}n\to \infty \mathrm{,}$

$d\mathrm{<mo\; stretchy="false">(</mo>}{g}_{\beta }\mathrm{<mo\; stretchy="false">(</mo>}{\zeta }_n\mathrm{<mo\; stretchy="false">)</mo>,}{g}_{\beta }\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\to \mathrm{0,}n\to \infty \mathrm{.}$

We have

$d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo>=</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }{}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }{}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_{\alpha }{}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>,}{u}_1\mathrm{<mo\; stretchy="false">)</mo>}$

$\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>}{\eta }_1+{\eta }_2+{\eta }_5\mathrm{<mo\; stretchy="false">)</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>.}$

Hence, $f_{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_1$ .

likewise, we can prove that gβ (u₂) = u₂. Now, we will prove that u₁ is a common fixed point of fα and gβ, as

$d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_2\mathrm{<mo\; stretchy="false">)</mo>}\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_{\alpha }{}_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{,}{f}_{\alpha }{}_2\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_{\alpha }{}_2\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{,}{f}_{\alpha }{}_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_2\mathrm{<mo\; stretchy="false">)</mo>}$

$\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>}{\eta }_3+{\eta }_4+{\eta }_5\mathrm{<mo\; stretchy="false">)</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_2\mathrm{<mo\; stretchy="false">)</mo>.}$

Consider u₃ ∈ X such that it can be used to demonstrate the uniqueness of u₁.

$f_{\alpha }{}_{\mathrm{<mo\; stretchy="false">(</mo>}}{u}_3\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_3\mathrm{,}and{g}_{\beta }\mathrm{<mo\; stretchy="false">(</mo>}{u}_3\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_3\mathrm{.}$

Therefore

$d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_3\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}{f}_{\alpha }{}_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>,}{f}_{\alpha }{}_3\mathrm{<mo\; stretchy="false">(</mo>}{u}_3\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_{\alpha }{}_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_3\mathrm{,}{f}_{\alpha }{}_2\mathrm{<mo\; stretchy="false">(</mo>}{u}_3\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_{\alpha }{}_2\mathrm{<mo\; stretchy="false">(</mo>}{u}_3\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_3\mathrm{,}{f}_{\alpha }{}_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_3\mathrm{<mo\; stretchy="false">)</mo>}$

$\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>}{\eta }_3+{\eta }_4+{\eta }_5\mathrm{<mo\; stretchy="false">)</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_3\mathrm{<mo\; stretchy="false">)</mo>.}$

Hence,

$u_1\mathrm{<mo>=</mo>}{u}_2\mathrm{<mo>=</mo>}{u}_3\mathrm{.}$

Thus, u₁ is the unique fixed point of fα and gβ.

Theorem 2.1 can be stated as follows:

Theorem 2.2

Let fk be a self-mappings on X, such that $f_k\mathrm{<mo\; stretchy="false">(</mo>}{z}_k\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{z}_k\mathrm{,}\forall \xi \in Xand{z}_k\in X\forall k$ respectively, such that

$d\mathrm{<mo\; stretchy="false">(</mo>}{f}_k\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{<mo\; stretchy="false">)</mo>,}{f}_k\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}{f}_k\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{,}{f}_k\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}{f}_k\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{,}{f}_k\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo>.}$ (9)

For all ξ, ζ ∈ X, ξ ≠ ζ and ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^5}{\eta }_{i}1.$

Proof. Theorem 2.1 may be proven using the same way used to prove Theorem 2.2.

Our main result has corollaries, we leave their proof for the reader.

Corollary 2.3

Let X be a complete metric space and let f : X → \mathbb{R} a continuous self-mapping on X, let f satisfying (4) for all ξ, ζ ∈ X, ξ ≠ ζ and for some η₁, η₂, η₃, η₄, η₅ ∈ [0,1) such that

Corollary 2.4

Let X be a complete metric space and let f, g are two continuous self-mappings on X satisfying

$d\mathrm{<mo\; stretchy="false">(</mo>}f\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{<mo\; stretchy="false">)</mo>,}g\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}f\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{,}g\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}g\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}\zeta \mathrm{,}f\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}\xi \mathrm{,}\zeta \mathrm{<mo\; stretchy="false">)</mo>}$ (10)

for all ξ, ζ ∈ X, ξ ≠ ζ and for some η₁, η₂, η₃, η₄, η₅ ∈ [0,1) such that ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^5}{\eta }_{i}1.$ Then f and g have a unique fixed point.

The existence and uniqueness of a common fixed point of two mappings that are not necessarily continuous can be investigated using our findings by introducing the next theorem [13-15].

Theorem 2.5

Let fα₁, fα₂ be two self-mappings on a complete metric space X, satisfies

Proof. Take ${\xi }_n\mathrm{<mo>=</mo>}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{n-1}\mathrm{<mo\; stretchy="false">)</mo>,}{\xi }_n\mathrm{<mo>=</mo>}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{n-1}\mathrm{<mo\; stretchy="false">)</mo>}andf{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{n-1}\mathrm{<mo\; stretchy="false">)</mo>}\ne f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{n-1}\mathrm{<mo\; stretchy="false">)</mo>,}{\xi }_n\ne {\xi }_{n-1}\mathrm{,}\forall n\in \mathbb{N}\mathrm{.}$ Therefore,

$d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n+1}\mathrm{,}{\xi }_{2n}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{<mo\; stretchy="false">)</mo>,}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$\le {\eta }_1\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{,}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_2\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n-1}\mathrm{,}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_3\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{,}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}{\xi }_{2n-1}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$+{\eta }_4\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}{\xi }_{2n-1}\mathrm{,}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_5\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{,}{\xi }_{2n-1}\mathrm{<mo\; stretchy="false">)</mo>}$

$\mathrm{<mo>=</mo>}{\eta }_1\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{,}{\xi }_{2n+1}\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_2\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n-1}\mathrm{,}{\xi }_{2n}\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_3\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{,}{\xi }_{2n}\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_4\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n-1}\mathrm{,}{\xi }_{2n+1}\mathrm{<mo\; stretchy="false">)</mo>}$

$+{\eta }_5\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{,}{\xi }_{2n-1}\mathrm{<mo\; stretchy="false">)</mo>.}$

So, we have

$d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n+1}\mathrm{,}{\xi }_{2n}\mathrm{<mo\; stretchy="false">)</mo>}\le \mathrm{<mo\; stretchy="false">(</mo>}\frac{{\eta }_2+{\eta }_4+{\eta }_5}{1-{\eta }_2-{\eta }_4}\mathrm{<mo\; stretchy="false">)</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n}\mathrm{,}{\xi }_{2n-1}\mathrm{<mo\; stretchy="false">)</mo>.}$ (11)

From (11) we obtain

$d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_{2n+1}\mathrm{,}{\xi }_{2n}\mathrm{<mo\; stretchy="false">)</mo>}\le {\mathrm{<mo\; stretchy="false">(</mo>}\frac{{\eta }_2+{\eta }_4+{\eta }_5}{1-{\eta }_2-{\eta }_4}\mathrm{<mo\; stretchy="false">)</mo>}}^{2n}d\mathrm{<mo\; stretchy="false">(</mo>}{\xi }_1\mathrm{,}{\xi }_0\mathrm{<mo\; stretchy="false">)</mo>.}$ (12)

We get

$f{\alpha }_{1}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}f{\alpha }_{2}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}f{\alpha }_{1}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}\underset{k\to \infty }{{\displaystyle \lim }}{\xi }_{n_k}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\underset{k\to \infty }{{\displaystyle \lim }}{\xi }_{n_{k+1}}\mathrm{<mo>=</mo>}{u}_1\mathrm{.}$

Let u1 is a fixed point of f₁ fα₂ such that fα₁ fα₂ (u₁)= u₁. Now, we must show that fα₁ (u₁)= u₁ and fα₂ (u₁)= u₁. For that we let fα₁ (u₁) ≠ u₁ and fα₂ (u₁) ≠ u₁. Then,

$d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo>=</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}f{\alpha }_{2}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>,}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>,}f{\alpha }_{2}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{f}_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>}$

$+{\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>,}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>0.}$

Hence,

u₁ is a fixed point of fα₁. Similarly we can get fα₂ (u₁)= u₁. This indicates that fα₁ and fα₂ have a common fixed point in X. That was proof of existence.

As for proving uniqueness, let's suppose u₂ ∈ X, u₂ ≠ u₁ be another fixed point of f₁ and fα₂. Then

$d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_2\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>,}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$\le {\eta }_{1}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{2}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{,}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+{\eta }_{3}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}f{\alpha }_2\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{<mo\; stretchy="false">)</mo>}+$

${\eta }_{4}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_2\mathrm{,}f{\alpha }_1\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">)</mo>}+{\eta }_{5}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_2\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}{\eta }_3+{\eta }_4+{\eta }_5\mathrm{<mo\; stretchy="false">)</mo>}d\mathrm{<mo\; stretchy="false">(</mo>}{u}_1\mathrm{,}{u}_2\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>0.}$

We have demonstrated a uniqueness and completed proof of the theorem.

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