*Immediate Consequences*

In this subsection, we list a few consequences of our main result. These corollaries also indicate how we can conclude further consequences.

If we let *<sup>α</sup>*(*<sup>x</sup>*, *Tx*) = 1 for all *x* ∈ *X*, we ge<sup>t</sup> the following results:

**Theorem 4.** *Let T be a self-mapping on a metric space* (*<sup>X</sup>*, *d*)*. Suppose that β* ∈ [0, *α*] Λ ≥ 0 *and α* ≥ 1 *are fixed constants. A self-mapping T possesses a unique fixed point if* 12*d*(<sup>κ</sup>, T κ) ≤ *d*(<sup>κ</sup>, *y*) *implies*

$$d(\mathcal{T} \varkappa, \mathcal{T}y) \le P(\varkappa, y)$$

*where*

$$P(\varkappa, y) := (1 - \varepsilon) \max \left\{ d(\varkappa, y), d(\varkappa, \mathcal{T} \varkappa), d(y, \mathcal{T}y), \frac{1}{2} \left[ d(\varkappa, \mathcal{T}y) + d(y, \mathcal{T} \varkappa) \right] \right\},$$

$$+ \Lambda(\varepsilon)^{a} \psi(\varepsilon) \left[ 1 + \|\varkappa\| + \|y\| + \|\mathcal{T}\varkappa\| + \|\mathcal{T}y\| \right]^{\beta}.$$

*for all* κ, *y* ∈ *X and for every ε* ∈ [0, 1]*.*

Let (*<sup>X</sup>*, ) be a partially ordered set and *d* be a metric on *X*. We say that (*<sup>X</sup>*, , *d*) is regular if for every nondecreasing sequence {<sup>κ</sup>*n*} ⊂ *X* such that κ*n* → *x* ∈ *X* as *n* → <sup>∞</sup>, there exists a subsequence {<sup>κ</sup>*n*(*k*)} of {<sup>κ</sup>*n*} such that <sup>κ</sup>*n*(*k*) *x* for all *k*.

**Theorem 5.** *Let* (*<sup>X</sup>*, ) *be a partially ordered set and d be a metric on X such that* (*<sup>X</sup>*, *d*) *is complete. Let T* : *X* → *X be a nondecreasing mapping with respect to . Suppose that β* ∈ [0, *α*] Λ ≥ 0 *and α* ≥ 1 *are fixed constants such that the self-mapping T satisfies the following condition:* 12 *d*(<sup>κ</sup>, T κ) ≤ *d*(<sup>κ</sup>, *y*) *implies*

$$d(\mathcal{T} \varkappa, \mathcal{T}y) \le P(\varkappa, y),$$

*where*

$$P(\varkappa, y) := (1 - \varepsilon) \max \left\{ d(\varkappa, y), d(\varkappa, \mathcal{T} \varkappa), d(y, \mathcal{T}y), \frac{1}{2} \left[ d(\varkappa, \mathcal{T}y) + d(y, \mathcal{T} \varkappa) \right] \right\},$$

$$+ \Lambda(\varepsilon)^{\mathfrak{a}} \psi(\varepsilon) \left[ 1 + \|\varkappa\| + \|y\| + \|\mathcal{T}\varkappa\| + \|\mathcal{T}y\| \right]^{\mathfrak{b}}.$$

*for all* κ, *y* ∈ *X with* κ *y and for every ε* ∈ [0, 1]*. Suppose also that the following conditions hold:*

*(i) there exists x*0 ∈ *X such that x*0 *Tx*0*;*


*Then T has a fixed point.*

*Moreover, if for all* κ, *y* ∈ *X there exists z* ∈ *X such that* κ, *z and y z, we have uniqueness of the fixed point.*

**Proof.** Set *α* : *X* × *X* → [0, ∞) in a way that

$$\mathfrak{a}(\mathfrak{x}, \mathfrak{y}) = \begin{cases} 1 \text{ if } \varkappa \preceq \mathfrak{y} \text{ or } \varkappa \succeq \mathfrak{y},\\ 0 \text{ otherwise.} \end{cases}$$

It is apparent that *T* is an *α*-Suzuki-Pata contractive mapping, i.e.,

$$
\alpha(\varkappa, y)d(T\varkappa, Ty) \le P(\varkappa, y),
$$

for all κ, *y* ∈ *X*. By assumption, the inequality *<sup>α</sup>*(<sup>κ</sup>0, *<sup>T</sup>*<sup>κ</sup>0) ≥ 1 is observed. In addition, for all κ, *y* ∈ *X*, due to the fact that *T* is nondecreasing, we find

$$a(\mathbf{x}, \mathbf{y}) \ge 1 \implies \mathbf{x} \succeq \mathbf{y} \text{ or } \mathbf{x} \preceq \mathbf{y} \implies \mathbf{T} \mathbf{x} \succeq \mathbf{T} \mathbf{y} \text{ or } \mathbf{T} \mathbf{x} \preceq \mathbf{T} \mathbf{y} \implies a(T\mathbf{x}, T\mathbf{y}) \ge 1.$$

Consequently, we note that *T* is *<sup>α</sup>*−admissible. Now, assume that (*<sup>X</sup>*, , *d*) is regular. Let {<sup>κ</sup>*n*} be a sequence in *X* such that *<sup>α</sup>*(<sup>κ</sup>*n*,κ*n*+<sup>1</sup>) ≥ 1 for all *n* and κ*n* → *x* ∈ *X* as *n* → ∞. From the regularity hypothesis, there exists a subsequence {<sup>κ</sup>*n*(*k*)} of {*xn*} such that <sup>κ</sup>*n*(*k*) *x* for all *k*. On account of *α* we derive that *<sup>α</sup>*(<sup>κ</sup>*n*(*k*),<sup>κ</sup>) ≥ 1 for all *k*. Consequently, the existence and uniqueness of the fixed point is derived by Theorem 3.
