1 Introduction

The Banach fixed point theorem [1] served as the inspiration for metric fixed point theory. Because this approach has many applications in several disciplines, many authors have expanded it in many ways [2–5]. Wardowski [6] provides one such astonishing and significant generalization. He introduced the notion of ℑ-contraction as follows:

Definition 0.1. Let (z, m) be a metric space. A mapping ξ: z → z is said to be an ℑ-contraction, if there is ℑ ∈ Δ(ℑ) and λ > 0 such that for all x, y ∈ z (1) where Δ(ℑ) is the family of all mappings obeying the following conditions:

1. (ℑ₁) ℑ(x) < ℑ(y) for all x < y;
2. (ℑ₂) for all sequences {η_p} ⊆ (0, + ∞), lim_p→∞ η_p = 0, if and only if lim_p→∞ ℑ(η_p) = −∞;
3. (ℑ₃) there exists 0 < ℓ < 1 such that η^ℓℑ(η) = 0.

Wardowski’s result is given as follows:

THEOREM 0.1 [6]. Let (z, m) be a complete metric space and ξ: z → z be ℑ-contraction. Then x* ∈ z is a unique fixed point of ξ and for each x₀ ∈ z the sequence is convergent to x*.

In [7], Secelean demonstrated that condition (ℑ₂) may be overtaken by an equivalent and simpler one.

1. () ℑ(t) = −∞.

Lemma 0.1. Let be a function obeying (ℑ₁) and (), then for all sequence {t_p} ⊆ (0, ∞) Following that, Piri and Kumam [8] established Wardowski’s theorem utilising () and the continuity of ℑ rather than (ℑ₂) and (ℑ₃), respectively. Wardowski [9] later proved a fixed point theorem for ℑ-contractions when λ is treated as a function. Recently, other authors demonstrated (in various methods) Wardowski’s original results in the absence of both criterias (ℑ₂) and (ℑ₃) (see, [10, 11]. To more in this direction, consult [12–15]. Very recently, Derouiche and Ramoul [16] used a relaxed version of (ℑ₂) and also dropped (ℑ₃) to establish some new fixed point results in the context of b-metric spaces.

On the other side, the concept of standard metric space is generalized in numerous ways (see [17–21]. Jleli and Samet [22] recapitulated a huge class of topological spaces by introducing the most prevailing generalizations of metric spaces namely JS-generalized metric spaces. More far, in [23], Karapinar et al. achieved fixed point theorems under very general contractive conditions and Altun et al. [24] proved a Feng-Liu’s type fixed point theorem in JS-generalized metric spaces. While, in [25], Dumitrescu and Pitea presented extensions of some classic results regarding the existence and uniqueness of fixed points of operators fulfilling generalized contractive conditions in the setting of JS-generalized metric spaces. Quite recently, Saleem et al. [26] proved some new fixed point theorems, coincidence point theorems and common fixed point theorems for multivalued ℑ-contractions in the framework of JS-generalized metric spaces. Afterwards, Iqbal et al. [27] derived the coincidence point and common fixed point results for ℑ-type mappings with regard to JS-generalized metric spaces.

A binary relation on z is a non-empty subset of the Cartesian product z × z. For ease of use, we designate if . The concepts of antisymmetry, preorder, reflexivity, transitivity, and partial order can be found in [28]. The trivial preorder on z is denoted by , and is given by for each x, y ∈ z. Here after, and demonstrate the set of real numbers and the set of non-negative integers, respectively. Let z be a non-empty set and J_d: z × z → [0, ∞] be a given mapping. Following Jleli and Samet [22], for every x ∈ z, define the set (2)

Definition 0.2 [22]. Let z be a non-empty set and J_d: z × z → [0, ∞] be a function that complies with the following criteria for all x, y ∈ z:

1. (J_d1) J_d(x, y) = 0 implies x = y;
2. (J_d2) J_d(x, y) = J_d(y, x);
3. (J_d3) there exist κ > 0 such that (x, y) ∈ z × z, {x_p} ∈ J_d, z, x) implies (3)

Then J_d is called a JS-generalized metric and the pair (z, J_d) is called a JS-generalized metric space. We renamed it as κ-JS-generalized metric space (in short, a -MS).

Remark 0.1 [22]. If the set c(J_d, z, x) is empty for every x ∈ z, then (z, J_d) is a -MS if and only if (J_d1) and (J_d2) are satisfied.

Many examples of -MS can be found in [22, 23, 26].

Example 0.1 [22].

1. A modular metric space (z, ρ) is a -MS.
2. A standard metric space is a -MS.
3. A 2-metric space is a -MS.

Definition 0.3. [22] Let (z, J_d) be a -MS and x ∈ z.

1. A sequence {x_p} ⊆ z is said to be J_d-convergent and J_d-converges to x if {x_p} ∈ J_d, z, x). In such case, we will write
2. a sequence {x_p} ⊆ z is said to be J_d-Cauchy if (4)
3. A -MS (z, J_d) is said to be complete if every J_d-Cauchy sequence in z is J_d-convergent.

Proposition 0.1 [22]. Let (z, J_d) be a -MS, {x_p} be a sequence in z and (x, y) ∈ z × z. If {x_p} is J_d-convergent and J_d-converges to x and {x_p} J_d-converges to y, then x = y.

Remark 0.2. Jleli and Samet defined J_d-Cauchy sequence as (5) Clearly, (5) implies (4), the opposite, however, need not be true [23]. From here on, we assume that J_d-Cauchy sequences are given by (5).

Definition 0.4 [23]. Let (z, J_d) be a -MS and ξ: z → z. For x₀ ∈ z, denote δ(J_d, ξ, x₀), the J_d-diameter of the orbit of x₀ by ξ, , and is defined as, (6)

Definition 0.5 [23]. Let be a binary relation on -MS (z, J_d). A sequence {x_p} ⊆ z is -non-decreasing if x_p x_p+1 for each .

Definition 0.6 [23]. A -MS (z, J_d) is called -non-decreasing complete if every -non-decreasing and J_d-Cauchy sequence is J_d-convergent in z.

Remark 0.3. Notice that every complete -MS is also -non-decreasing complete. As evidenced by the case below, the contrary is untrue.

Example 0.2. Let z = (0, 1] furnished with the Euclidean metric m(x, y) = |x − y| for each x, y ∈ z. Define a binary relation on z by Then (z, m) is -non-decreasing complete, however, the metric space is not complete.

Definition 0.7 [23]. Let (z, J_d) be a -MS. A mapping ξ: z → z is -non-decreasing-continuous at ν ∈ z if {ξx_p} ∈ J_d, z, sν) for each -non-decreasing sequence {x_p} ∈ J_d, z, ν). A mapping ξ is -non-decreasing-continuous if it is -non-decreasing-continuous at every point of z.

Remark 0.4. [23] Every continuous mapping is also -non-decreasing-continuous, while the reverse is generally false, as seen in Example 4.6 of [23].

By getting inspiration from the work of Karapinar et al. [23], here, we prove fixed point theorems for ℑ-contractions in the context of JS-generalized metric space.

2 Fundamental results

Let (z, J_d) be a -MS and let ξ be a self-mapping on z. Throughout this section, we denote, for all x, y ∈ z, Following [23], define for given , (7) and (8) By the symmetry of J_d, we can alternatively express (9) Notice that if satisfy q ≥ p, then (10)

Lemma 1.1. Let (z, J_d) be a -MS with a preorder and let ξ: z → z be an -non-decreasing mapping. Let x₀ ∈ z be a point such that x₀ and ξx₀ are -comparable. Assume, there is a function such that (11) for x, y ∈ z satisfying and λ > 0. Then (11) holds for each .

Proof. Consider the Picard sequence of ξ based on x₀. Suppose that . As ξ is -non-decreasing, then . Repeating this argument, we get, for every . Since is a preorder, then for all such that p ≤ q. Furthermore, as condition (11) is symmetric on x and y, then (11) holds for each x_p and x_q (being arbitrary), so it holds for each .

Lemma 1.2. Let (z, J_d) be a -MS and let ξ: z → z be a mapping. Let x₀ ∈ z be a point for which there exists such that . Assume, there is a non-decreasing function obeying

1. : ℑ(sup M) = sup ℑ(M) for all M ⊂ (0, ∞) with sup M > 0

and (12) for all and λ > 0. Then In particular,

Proof. Let be such that ℓ ≥ p₀. From (10), we have Let be such that p ≥ q ≥ ℓ + 1. Denote then Hence (13) From (13) and (13), we obtain (14) By the virtue of and (14), we get (15) Continuing this argument and recognizing that ℑ is non-decreasing, we get for all

3 Main results

THEOREM 2.1. Let (z, J_d) be a -MS with a preorder and let ξ: z → z be an -non-decreasing mapping. Let x₀ ∈ z be a point such that and for some . Assume, there is a non-decreasing function obeying , and (11) for all and λ > 0. Then the sequence based on x₀ is -non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is -non-decreasing-complete, then J_d-converges to a point ν ∈ z that obeys Additionally, if ξ is -non-decreasing-continuous, then ν = ξν.

Proof. Consider the Picard sequence of ξ based on x₀. As shown in the proof of Lemma 1.1, is -non-decreasing. If ξ^px₀ = ξ^qx₀, then J_d(x_p, x_q) = 0 for every q, p ≥ p₀. In particular, Consider ξ^px₀ ≠ ξ^qx₀ and for some , then by using Lemma 1.2, we have (16) Letting limit in (16) as p, q → ∞, we have (17) Taking into account of , we have Hence J_d-Cauchy sequence. Since (z, J_d) is -non-decreasing-complete, there exists ν ∈ z such that . By using (J_d3), we get (18) which implies J_d(ν, ν) = 0.

Moreover, as we additionally assume that ξ is -non-decreasing-continuous, then Proposition 0.1 guarantees that ξν = ν, so ν is a fixed point of ξ.

Example 2.1. Consider the function defined as Then ℑ is non-decreasing and continuous but does not satisfies .

So, in next theorem, we replace the condition of by the continuity of ℑ in Theorem 2.1 and we denote by , the collection of all functions that are continuous and non-decreasing.

THEOREM 2.2. Let (z, J_d) be a -MS with a preorder and let ξ: z → z be an -non-decreasing self-mapping. Let x₀ ∈ z be a point such that , for some and the following holds true: If there exists a function and λ > 0 such that inequality (11) holds for all , then the sequence based on x₀ is -non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is -non-decreasing-complete, then J_d-converges to a point ν ∈ z that satisfies Additionally, if ξ is -non-decreasing-continuous, then ν = ξν.

Proof. Consider the Picard sequence of ξ based on x₀. As shown in the proof of Lemma 1.1, is -non-decreasing. If , then J_d(x_p, x_q) = 0 for all q, p ≥ p₀. In particular, Consider and for some . Denote then Hence (19) Assume that for be an arbitrary such that ℓ ≥ p₀. Then, by using inequalities (11) and (19) and continuity of ℑ, we obtain (20) a contradiction because λ > 0. Hence J_d-Cauchy sequence. Since (z, J_d) is -non-decreasing-complete, there is ν ∈ z such that . By using (J_d3), we get (21) which implies J_d(ν, ν) = 0.

Moreover, as we additionally assume that ξ is -non-decreasing-continuous, then

Proposition 0.1 guarantees that ξν = ν.

Example 2.2. Let z = [0, 1] ∪ {2} and let J_d: z × z → [0, ∞] be a function defined by Then (z, J_d) is complete -MS (see [23]. Define a binary relation on z by then is a preorder and (z, J_d) is a preordered space. Let x₀ = 0.25 ∈ z be a point such that 0 < x₀ = 0.25 < 1 = ξ(0.25) = ξx₀, then and Also, and for any such that ℓ ≥ p₀. Now Define ξ: z → z and ℑ: (0, ∞) → (−∞, ∞) by respectively, then ℑ is continuous and non-decreasing function. Let such that ξx ≠ ξy, then there arises two cases:

Case 1: When x = 0.25, y = 0.5, then there exists λ = 0.25 > 0 such that

Case 2: When x = 0.25, y = 1, then there exists λ = 0.25 > 0 such that This show that ℑ satisfies (11) for all . Thus, all hypotheses of Theorem 2.2 hold true and {0, 0.5} is the set of all fixed points of ξ.

4 Consequences

In this section, we find more results involving stronger contractive conditions.

Corollary 3.1. Let (z, J_d) be a -MS with a preorder and let ξ: z → z be an -non-decreasing mapping. Let x₀ ∈ z be a point such that and for some . Assume there is b ∈ (0, 1) such that (22) for all . Then the sequence based on x₀ is -non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is -non-decreasing-complete, then J_d-converges to a point ν ∈ z that meets Additionally, if ξ is -non-decreasing-continuous, then ν = ξν.

Proof. Define by ℑ(s) = ln s for all s ∈ (0, ∞). Put . Inequality (26) implies (11). Hence, all of the requirements of Theorem 2.1 have been met, and the proof is concluded.

Corollary 3.2. Let (z, J_d) be a -MS with a preorder and let ξ: z → z be an -non-decreasing mapping. Let x₀ ∈ z be a point such that and for some . Assume, there is a non-decreasing function ℑ: (0, ∞) → (−∞, ∞) fulfilling , (23) for λ > 0. Then the sequence based on x₀ is -non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is -non-decreasing-complete, then J_d-converges to a point ν ∈ z that meets Additionally, if ξ is -non-decreasing-continuous, then ν = ξν.

Proof. Let the contractivity condition (27) hold for all x, y ∈ z such that , then Lemma 1.1 guarantees that it also holds for . So due to Theorem 2.1, we obtained the result.

Corollary 3.3. Let (z, J_d) be a -MS with a preorder and let ξ: z → z be an -non-decreasing mapping. Let x₀ ∈ z be a point such that , for some and the following holds true: If there exists a function and λ > 0 fulfilling the inequality (27), then the sequence based on x₀ is -non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is -non-decreasing-complete, then J_d-converges to a point ν ∈ z that meets Additionally, if ξ is -non-decreasing-continuous, then ν = ξν.

Proof. By using the same reason as in the proof of Corollary 3.2, Theorem 2.2 gives the result.

Corollary 3.4. Let (z, J_d) be a -MS with a partial order ≪ and let ξ: z → z be an ≪-non-decreasing mapping. Let x₀ ∈ z be a point such that x₀ ≪ ξx₀ and for some . Assume, there is a non-decreasing function ℑ: (0, ∞) → (−∞, ∞) satisfying , and (11) for all and λ > 0. Then, the sequence based on x₀ is ≪-non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is ≪-non-decreasing-complete, then J_d-converges to a point ν ∈ z that meets Additionally, if ξ is ≪-non-decreasing-continuous, then ν = ξν.

Proof. Due to the fact that a partial order ≪ is a preorder , the conclusion is reached by applying Theorem ref 2.1.

Corollary 3.5. Let (z, J_d) be a -MS with a partial order ≪ and let ξ: z → z be an ≪-non-decreasing mapping. Let x₀ ∈ z be a point such that x₀ ≪ ξx₀, for some and the following holds true: If there exists a function and λ > 0 fulfilling the inequality (11) for all , then the sequence based on x₀ is ≪-non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is ≪-non-decreasing-complete, then J_d-converges to a point ν ∈ z that meets Additionally, if ξ is ≪-non-decreasing-continuous, then ν = ξν.

Proof. Due to the fact that a partial order ≪ is a preorder , the conclusion is reached by applying Theorem ref 2.2.

Corollary 3.6. Let (z, J_d) be a -MS with a preorder and let ξ: z → z be an -non-decreasing mapping. Let x₀ ∈ z be a point such that and for some . Assume, there is a non-decreasing function ℑ: (0, ∞) → (−∞, ∞) satisfying , and (24) for all with ξx ≠ ξy and λ > 0. Then the sequence based on x₀ is -non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is -non-decreasing-complete, then J_d-converges to a point ν ∈ z that satisfies Additionally, if ξ is -non-decreasing-continuous, then ν = ξν.

Proof. Since for all s, r ∈ [0, ∞], so inequality (28) implies inequality (11) and Theorem 2.1 leads to the conclusion.

Corollary 3.7. Let (z, J_d) be a -MS with a preorder and let ξ: z → z be an -non-decreasing mapping. Let x₀ ∈ z be a point such that and for some . Assume, there is a continuous non-decreasing function ℑ: (0, ∞) → (−∞, ∞) obeying (25) for all with ξx ≠ ξy, λ > 0 and the following: Then the sequence based on x₀ is -non-decreasing and J_d-Cauchy sequence.

Furthermore, if (z, J_d) is -non-decreasing-complete, then J_d-converges to a point ν ∈ z that obeys Additionally, if ξ is -non-decreasing-continuous, then ν = ξν.

Proof. Since for all s, r ∈ [0, ∞], so inequality (28) implies inequality (11) and the result follows from Theorem 2.2.

Remark 3.1. Theorem 2.1, Theorem 2.2 and Corollaries 3.1-3.7 remain true if we do one or more of the following changes in their statement:

• exchange the -non-decreasing-completeness of (z, J_d) by the completeness of (z, J_d);
• exchange the -non-decreasing-continuity of ξ by continuity;
• exchange the preorder by the trivial preorder given by for all x, y ∈ z;
• exchange, in the contractivity condition, for all by for all x, y ∈ z such that ;
• exchange the -MS by any of the abstract metric spaces that could be considered as a -MS: b-metric spaces, modular spaces, and standard metric spaces.
• exchange the contractivity condition (11) by for every with ξx ≠ ξy.

5 Existence of solution to RLC circuit’s current differential equation

A tuning circuit is a mathematical representation of the electric current in an RLC parallel circuit to present with a rudimentary knowledge of how light is converted into electricity. Consider the following series of electric circuit (Fig 1), which includes a resistor R, a capacitor C, an inductor L, a voltage V, and an electromotive force E. With the aid of Kirchhoff’s Voltage Law, related problems are mathematically modelled as initial value problems for second order ordinary differential equations of the form: (26) where V_ν(t) = V.

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Fig 1. RLC series circuit.

https://doi.org/10.1371/journal.pone.0314493.g001

In this part, we demonstrate the existence of the solution to the RLC differential equation (26). The problem (26) is identical to the following integral equation (see [29, 30]: (27) where is a monotonically non-decreasing function for all g ∈ [0, 1] and is the Green function defined as (28)τ is a constant computed in terms of R and L. Let be the space of all continuous real valued functions on J, where J = [0, 1]. Then z is a complete metric space with respect to the metric and so z is -MS for κ = 1. Hereafter, we assume that (z, J_d) is a -MS with canonical preorder ≤ and (z, J_d) is ≤-non-decreasing-complete. Define the operator ℵ: z → z as follows: (29) A fixed point of operator (29) is the solution of problem (26). We take into account the following hypotheses:

1. (H1) : J² → [0, ∞) is a continuous function;
2. (H2) |ℏ(t, q(t)) − ℏ(t, p(t))| ≤ |q(t) − p(t)| + 1 for all t ∈ [0, 1];
3. (H3) ;
4. (H4) ℵ is ≤-non-decreasing continuous.

THEOREM 4.1. Suppose that hypothesis (H1)-(H4) hold. Then, the initial value problem (26) has a common solution in z.

Proof. Firstly, note that for for all , inequality (11) is equivalent to the following for all x, y ∈ z: (30) Next, for all and t ∈ J, we have This implies that (31) From (H3) and (31), we have Hence (11) is satisfied for . Thus, all hypotheses of Theorem 2.2 are satisfied and therefore differential Eq (26) has a solution in J.

6 Conclusion

In this work, we establish ℑ-contractions and show some fixed point theorems for these contractive conditions in the JS-generalized metric spaces. Finally, we proved fixed point results, an existence result for the solution of the RLC circuit’s current differential equation is also established.

Acknowledgments

Authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.