The notion of b-metric space as an extension of metric space was introduced by Bakhtin in [6] and then extensively used by Czerwik in [7, 8, 14]. Since then, a lot of papers have been published on the fixed point theory of various classes of single-valued and multi-valued operators in this type of spaces. We recall here just some basic definitions and notation that we are going to use. ℝ⁺ and ${\mathbb{R}}_0^{+}$ will denote the set of all positive, resp. nonnegative real numbers and ℕ will be the set of positive integers.

A b-metric on a nonempty set ε is a function $\mathcal{E}\times \mathcal{E}\to {\mathbb{R}}_0^{+}$ such that for a constant s ≥ 1 and all x, y, z ∈ ε the following three conditions hold true:

(Ml) d_b(x,y) = 0 ⟺ x = y,

(M2) d_b(x,y) = d_b(y,x),

(M3) d_b(x,y)≤s(d_b(x,z)+ d_b(z,y)).

The triple (𝜀,d_b, s)is called a b-metric space.

Obviously, each metric space is a b-metric space (for s = 1), but the converse need not be true. Standard examples of b-metric spaces that are not metric spaces are the following:

1. 𝜀 = ℝ and d_b : 𝜀 × 𝜀 → ℝ defined by d_b(x, y) = |x – y|²for all x, y ∈ 𝜀, with s = 2.

2. ${\ell }^p(\mathbb{R}):=\{\{x_n\}\subset \mathbb{R}:\sum _{n=1}^{\mathrm{\infty }}|x_n{|}^p<\mathrm{\infty }\},0<p<1,d_b:{\ell }^p(\mathbb{R})\times {\ell }^p(\mathbb{R})\to \mathbb{R}$ given by $$d_b(\{x_n\},\{y_n\})={\left(\sum _{n=1}^{\mathrm{\infty }}|x_n-y_n{|}^p\right)}^{1/p}$$

for all {x_n}, {y_n} ∈ 𝓁^p(ℝ).Here s = 2^1/p. 3. L^p([0,1]) ϶ ƒ : [0,1] → ℝ such that ${\displaystyle {\int }_0^1|f(t){|}^{p}dt<\mathrm{\infty },p>1,d_b:L^p([0,1])\times L^p([0,1])\to \mathbb{R}}$ given by $$d_b(f,g)=\underset{0}{\overset{1}{\int }}|f(t)-g(t){|}^p$$

forali f,g ∈ L^p([0, l]);here s = 2^p-1.

The topology on b-metric spaces and the notions of convergent and Cauchy sequences, as well as the completeness of the space are defined similarly as for standard metric spaces. However, one has to be aware of some differences. For instance, a b-metric need not be a continuous mapping in both variables (see, e.g., [15]).

Now, we give a brief background for multivalued mappings defined in a b-metric space (𝜀,d_b,s).

We denote the class of non-empty and bounded subsets of 𝜀 by 𝒫 _b(𝜀) and the class of non-empty, closed and bounded subsets of 𝜀 by ${\mathcal{P}}_{cb}(\mathcal{E}).\text{For}\mathcal{U},\mathcal{V},\mathcal{W}\in {\mathcal{P}}_b(\mathcal{E})$, we define: $$\begin{array}{lll}{\mathcal{D}}_b(\mathcal{U},\mathcal{V})& =& inf\{d_b(u,v):u\in \mathcal{U},v\in \mathcal{V}\}\text{and}\\ {\delta }_b(\mathcal{U},\mathcal{V})& =& sup\{d_b(u,v):u\in \mathcal{U},v\in \mathcal{V}\}\end{array}$$

with 𝒟_b,(w,𝒲) = 𝒟_b({w},𝒲) = inf{d_b(w,x): x ∈𝒲}.

The following are some easy properties of 𝒟_b and δ_b (see, e.g. [7, 8, 14]):

(i) if 𝒰 = {w}and 𝒱 = {v}then 𝒟_b(𝒰,𝒱) = δ_b(𝒰,𝒱) = d_b(u,v);

(ii) 𝒟_b(𝒰,𝒱)≤δ_b(𝒰,𝒱);

(iii) 𝒟_b(x, 𝒱) ≤ d_b(x, b) for any b ∈ 𝒱;

(iv) δ_b(𝒰,𝒱)≤ s[δ_b(𝒰,𝒱) + δ_b(𝒲,𝒱)];

(v) δ_b(𝒰,𝒱) = 0 iff 𝒰 = 𝒱 = {υ}.

Moreover, we will always suppose that

(vi) the function 𝒟_b is continuous in its variables.

Recall that z ∈ 𝜀 is called a fixed point of a multivalued mapping $\mathcal{J}:\mathcal{E}\to {\mathcal{P}}_b(\mathcal{E})$ if z ∈ 𝒥z.

The concepts of orbit, orbitally complete space and orbitally continuous mapping given in [16-18] for metric spaces can be extended to the case of b-metric spaces, as follows:

Definition 2.1: Let (𝜀,d_b, s) be a b-metric space and $\mathcal{J},{\mathcal{J}}_1,{\mathcal{J}}_2:\mathcal{E}\to {\mathcal{P}}_b(\mathcal{E})$ be three mappings.1. An orbit 𝒪(x₀; 𝒥) of 𝒥 at a point x₀ ∈ 𝜀 is any sequence {x_n} such that $x_n\in \mathcal{J}{x}_{n-1}$ for n = 1,2,....2. If for a point x₀ ∈ 𝜀, there exists a sequence {x_n} in 𝜀 such that $x_{2n+1}\in {\mathcal{J}}_2{x}_{2n},x_{2n+2}\in {\mathcal{J}}_1{x}_{2n+1},n=0,1,2,\dots ,$ then the set 𝒪(x₀; 𝒥₁, 𝒥₂) = {x_n : n = 1,2,...} is called an orbit of (𝒥₁, 𝒥₂) at x₀.3. The space (𝜀, d_b,s) is said to be (𝒥₁, 𝒥₂) -orbitally complete if any Cauchy subsequence $\{x_{n_i}\}$ of 𝒪(x₀; 𝒥₁,𝒥₂) (for some x₀ in 𝜀) converges in 𝜀. In particular, for 𝒥₁ = 𝒥₂ = 𝒥, we say that 𝜀 is 𝒥-orbitally complete.4. The mapping 𝒥 is said to be orbitally continuous at a point x₀ ∈ 𝜀 if for any sequence {x_n }_n≥0 ⊂ 𝒪(x₀; 𝒥) and z ∈ 𝜀, d(x_n, z) → 0 as n → ∞ implies ${\delta }_b(\mathcal{J}{x}_n,\mathcal{J}z)\to 0$ as n → ∞. 𝒥 is called orbitally continuous in 𝜀 if it is orbitally continuous at every point of 𝜀.5. The graph G (𝒥) of 𝒥 is defined as G (𝒥) = {(x,y): x ∈ 𝜀, y ∈ 𝒥x}. The graph G (𝒥) of 𝒥 is called 𝒥-orbitally closed if for any sequence {x_n}, we have (x, x) ∈ G (𝒥) whenever (x_n,x_n+1) ∈ G (𝒥) andlim_n→∞ x_n= x.In his paper [9], Wardowski introduced a new type of contractions which he called F-contractions. Several authors proved various variants of fixed point results using such contractions. In particular, Acar and Altun proved in [10] a fixed point theorem for multivalued mappings under δ-distance.Adapting Wardowski’s approach to b-metric space, Cosentino et al. used in [13] the set of functions 𝔉_s defined as follows

Definition 2.2: Let s ≥ 1 be a real number. We denote by 𝔉_s the family of all functions F : ℝ⁺ → ℝ with thefollowing properties:(F1) F is strictly increasing;(F2) for each sequence {α_n} of positive numbers, lim_n→∞ α_n = 0 if and only if lim_n→∞ F(α_n) = —∞(F3) for each sequence {α_n} of positive numbers with lim_n→∞ α_n = 0, there exists k ∈ (0, 1) such that $\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n^{k}F({\alpha }_n)=0;$(F4) there exists τ ∈ ℝ+ such that for each sequence {α_n} of positive numbers, if τ + F(sα_n) ≤ F (α_n-1) for alln ∈ Ν, then $\tau +F(s^n{\alpha }_n)\le F(s^{n-1}{\alpha }_{n-1})$ for all n ∈ ℕ.

Example 2.3: Let F : ℝ⁺ →ℝ be defined by F (α) = Inα or F (α) = α + In α. It can be easily checked [13, Example 3.2] that F satisfies the properties (F1)-(F4).They proved the following (note that ${\mathcal{H}}_b$ here denotes the b-Hausdorff-Pompeiu metric).

Theorem 2.4: [13, Theorem 3.4] Let (𝜀,d_b, s) be a complete b-metric space and let $\mathcal{J}:\mathcal{E}\to {\mathcal{P}}_{cb}(\mathcal{E}).$ Assume that there exists a continuous from the right function F ∈ 𝔉 and τ ∈ ℝ⁺ such that $$2\tau +F(s{\mathcal{H}}_b(\mathcal{J}x,\mathcal{J}y))\le F(d_b(x,y)),$$for all x, y ∈ 𝜀, 𝒥x ≠ 𝒥y. Then 𝒥 has a fixed point.

We first introduce the notion of multivalued almost (F, δ_b)-contraction in a b-metric space and give relevance to fixed point results.

Definition 3.1: Let (𝜀, d_b, s) be a b-metric space with s > 1. We say that a multivalued mapping $\mathcal{J}:\mathcal{E}\to {\mathcal{P}}_b(\mathcal{E})$ is a multivalued almost (F, δ_b)-contraction if F ∈ 𝔉_s (with parameter τ) and there exists λ ≥ 0 such that $$\tau +F(s{\delta }_b(\mathcal{J}x,\mathcal{J}y))\underset{\_}{<}F({\mathrm{\Theta }}_1(x,y)+\lambda {\mathrm{\Theta }}_2(x,y)),$$(2)for all x, y ∈ 𝜀 with min{δ_b,(𝒥x, 𝒥y), d_b (x, y)} > 0, where $${\displaystyle {\mathrm{\Theta }}_1(x,y)=max\left\{d_b(x,y),{\mathcal{D}}_b(x,\mathcal{J}x),{\mathcal{D}}_b(y,\mathcal{J}y),\frac{{\mathcal{D}}_b(x,\mathcal{J}y)+{\mathcal{D}}_b(y,\mathcal{J}x)}{2s}\right\}}$$(3)and $${\mathrm{\Theta }}_2(x,y)=min\{{\mathcal{D}}_b(x,\mathcal{J}x),{\mathcal{D}}_b(y,\mathcal{J}y),{\mathcal{D}}_b(x,\mathcal{J}y),{\mathcal{D}}_b(y,\mathcal{J}x)\}.$$If (2) is satisfied just for $x,y\in \overline{\mathcal{O}(x_0;\mathcal{J})}$ (for some x₀ ∈ ε), we say that 𝒥 is a multivalued almost orbitally (F, δ_b)-contraction.We are equipped now to state our first main result.

Theorem 3.2: Let (ε, d_b, s) be a b-metric space with s > 1 and let $\mathcal{J}:\mathcal{E}\to {\mathcal{P}}_b(\mathcal{E})$ be a multivalued almost orbitally (F, δ_b)-contraction. Suppose that (ε, d_b, s) is 𝒥-orbitally complete (for the same x₀ ∈ ε). If F is continuous and 𝒥x is closed for all $x,y\in \overline{\mathcal{O}(x_0;\mathcal{J})},$ or 𝒥 has 𝒥-orbitally closed graph, then 𝒥 has a fixed point in ε.

Proof: Starting from the given point x₀, choose a sequence {x_n}in ε such that x_n+1 ∈ 𝒥x_n, for all n ≥ 0. Now, if $x_{n_0}\in \mathcal{J}{x}_{n_0}$for some n₀, then the proof is finished. Therefore, we assume x_n ≠ x_n+1for all n ≥ 0. Sod_b(x_n+1, x_n+2) > 0 andδb(𝒥x_n, 𝒥x_n+1) > 0 for all n ≥ 0.Using the condition (2) for elements x = x_n, y = x_n+1, for arbitrary n ≥ 0 we have $$\tau +F(s{d}_b(x_{n+1},x_{n+2}))\le \tau +F(s{\delta }_b(\mathcal{J}{x}_n,\mathcal{J}{x}_{n+1}))\le F({\mathrm{\Theta }}_1(x_n,x_{n+1})+\lambda {\mathrm{\Theta }}_2(x_n,x_{n+1}))$$where $$\begin{array}{ll}{\mathrm{\Theta }}_1(x_n,x_{n+1})& =max\left\{\begin{array}{l}{d}_b(x_n,x_{n+1}),{\mathcal{D}}_b(x_n,\mathcal{J}{x}_n),{\mathcal{D}}_b(x_{n+1},\mathcal{J}{x}_{n+1}),\\ \frac{1}{2s}[{\mathcal{D}}_b(x_n,\mathcal{J}{x}_{n+1})+{\mathcal{D}}_b(x_{n+1},\mathcal{J}{x}_n)]\end{array}\right\}\\ & \le max\left\{d_b(x_n,x_{n+1}),d_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2}),\frac{1}{2s}{d}_b(x_n,x_{n+2}))\right\}\\ & =max\left\{d_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2}),\frac{1}{2s}{d}_b(x_n,x_{n+2})\right\}\end{array}$$and $${\mathrm{\Theta }}_2(x_n,x_{n+1})=min\{{\mathcal{D}}_b(x_n,\mathcal{J}{x}_n),{\mathcal{D}}_b(x_{n+1},\mathcal{J}{x}_{n+1}),{\mathcal{D}}_b(x_n,\mathcal{J}{x}_{n+1}),{\mathcal{D}}_b(x_{n+1},\mathcal{J}{x}_n)\}=0.$$As ${\displaystyle \frac{1}{2s}{d}_b(x_n,x_{n+2})\le max\{d_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2})\},}$it follows that $${\displaystyle \tau +F(s{d}_b(x_{n+1},x_{n+2}))\le F(max\{d_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2})\}).}$$Suppose that d_b(x_n, x_n+1)≤ d_b(x_n+1, x_n+2),for some positive integer n.Then from (4), we have $$\tau +F(s{d}_b(x_{n+1},x_{n+2}))\le F(d_b(x_{n+1},x_{n+2})),$$a contradiction with (Fl). Hence, $$max\{d_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2})\}=d_b(x_n,x_{n+1}),$$and consequently $$\tau +F(s{d}_b(x_{n+1},x_{n+2}))\le F(d_b(x_n,x_{n+1}))foralln\in \mathbb{N}\cup \{0\}.$$(5)It follows by (5) and the property (F4) that $$\tau +F(s^n{d}_b(x_n,x_{n+1}))\le F(s^{n-1}{d}_b(x_{n-1},x_n))\text{for}\text{all}n\in \mathbb{N}\cup \{0\}.$$(6)Denote ϱ_n = d(x_n, x_n+1) for n = 0,1,2,.... Then, ϱ_n > 0 for all n and, using (6), the following holds: $$F(s^n{\varrho }_n)\le F(s^{n-1}{\varrho }_{n-1})-\tau \le F(s^{n-2}{\varrho }_{n-2})-2\tau \le \cdots \le F({\varrho }_0)-n\tau$$(7)for all n ∈ℕ. From (7), we get F(sⁿ ϱ_n)→ —∞as n →∞. Thus, from (F2), we have $$s^n{\varrho }_n\to 0asn\to \mathrm{\infty }.$$(8)Now, by the property (F3) there exists k ∈(0, 1) such that $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(s^n{\varrho }_n{)}^{k}F(s^n{\varrho }_n)=0.}$$(9)By (7), the following holds for all n ∈ℕ: $$(s^n{\varrho }_n{)}^{k}F(s^n{\varrho }_n)-(s^n{\varrho }_n{)}^{k}F({\varrho }_0)\le (s^n{\varrho }_n{)}^k(-n\tau )\le 0.$$(10)Passing to the limit as n →∞in (10) and using (8) and (9), we obtain $$\underset{n\to \mathrm{\infty }}{lim}n(s^n{\varrho }_n{)}^k=0$$and hence lim_{n →∞} n^1/k sⁿ ϱ_n = 0. Now, the last limit implies that the series $\sum _{n=1}^{\mathrm{\infty }}{s}^n{\varrho }_n$ is convergent and hence {x_n}is a Cauchy sequence in 𝒪(x₀;𝒥). Since 𝜀is 𝒥-orbitally complete, there exists a z ∈ 𝜀 such that $$x_n\to Z\text{as}n\to \mathrm{\infty }.$$Suppose that 𝒥 z is closed.We observe that if there exists an increasing sequence {n_k}⊂ ℕ such that $\{n_k\}\subset \mathbb{N}$for all k ∈ℕ,since 𝒥 z is closed and ${\displaystyle \underset{k\to \mathrm{\infty }}{lim}{x}_{n_k}=z}$, we deduce that z ∈ 𝒥 z and hence the proof is completed. Then we assume that there exists n₀∈ℕ such that x_n ∉𝒥 z for all n ∈ℕ with n ≥ n₀. It follows that δb(𝒥x_n,𝒥z) > 0 for all n ≥ n₀. Using the condition (2) for x = x_n, y = z, we have $$\tau +F(s{\mathcal{D}}_b(x_{n+1},\mathcal{J}z))\le \tau +F(s{\delta }_b(\mathcal{J}{x}_n,\mathcal{J}z))\le F({\mathrm{\Theta }}_1(x_n,z)+\lambda {\mathrm{\Theta }}_2(x_n,z))$$where $$\begin{array}{ll}{\mathrm{\Theta }}_1(x_n,z)& =max\left\{d_b(x_n,z),{\mathcal{D}}_b(x_n,\mathcal{J}{x}_n),{\mathcal{D}}_b(z,\mathcal{J}z),\frac{D_b(x_n,Jz)+D_b(\mathrm{z},J{x}_n)}{2s}\right\}\\ & \le max\left\{d_b(x_n,z),d_b(x_n,x_{n+1}),{\mathcal{D}}_b(z,\mathcal{J}z),\frac{D_b(x_n,Jz)+d_b(z,x_{n+1})}{2s}\right\}\\ & \to {\mathcal{D}}_b(z,\mathcal{J}z),\text{as}n\to \mathrm{\infty },\end{array}$$and $$\begin{array}{ll}{\mathrm{\Theta }}_1(x_n,z)& =min\left\{{\mathcal{D}}_b(x_n,\mathcal{J}{x}_n),{\mathcal{D}}_b(z,\mathcal{J}z),{\mathcal{D}}_b(x_n,\mathcal{J}z),{\mathcal{D}}_b(z,\mathcal{J}{x}_n)\right\}\\ & \le min\{d_b(x_n,x_{n+1}),{\mathcal{D}}_b(z,\mathcal{J}z),{\mathcal{D}}_b(x_n,\mathcal{J}z),d_b(z,x_{n+1})\}\\ & \to 0,\text{as}n\to \mathrm{\infty }.\end{array}$$Since F and D_b are continuous, if D_b(z, 𝒥 z) > 0, passing to the limit as n → ∞ in (11), we obtain $$\tau +F(s{\mathcal{D}}_b(z,\mathcal{J}z))\underset{\_}{<}F({\mathcal{D}}_b(z,\mathcal{J}z)),$$which is impossible since τ > 0, s ≥ 1 and F is strictly increasing. Hence, D_b(z, 𝒥 z) = 0 and, since 𝒥 z is closed, we have z ∈ 𝒥 z. Thus, z is a fixed point of 𝒥.Suppose that G(𝒥) is 𝒥-orbitally closed.Since (x_n,x_n+1) ∈ G(𝒥) for all n ∈ ℕ ∪ {0} and lim_{n → ∞} x_n = z, we have (z, z) ∈ G(𝒥) by the 𝒥-orbitally closedness. Hence, z ∈ 𝒥 z.It is proved that z is a fixed point of 𝒥. DThe following corollaries follow from Theorem 3.2 by taking F(α) = In α, resp. F(α) = α + In α in (2).

In this section, we prove a common fixed point theorem for a pair of multivalued mappings satisfying certain conditions.

First we introduce the notion of multivalued almost (F, δ_b)-contraction pair in b-metric spaces.

Definition 4.1: Let (𝜀, d_b, s) be a b-metric space with s > 1. Two multivalued mappings ${\mathcal{J}}_1,{\mathcal{J}}_2:\mathcal{E}\to {\mathcal{P}}_b(\mathcal{E})$ are said to form a multivalued almost (F, δ_b)-contraction pair, if F ∈ 𝔉_s and there exist τ > 0, λ ≥ 0 such that $$\tau +F(s{\delta }_b({\mathcal{J}}_{1}x,{\mathcal{J}}_{2}y))\le F{\mathrm{\Delta }}_1(x,y)+\lambda {\mathrm{\Delta }}_2(x,y)).$$(13)for all x, y ∈ 𝜀 with min{δ_b (𝒥₁ x, 𝒥₂ y), d_p(x, y)} > 0 where $${\mathrm{\Delta }}_1(x,y)=max\left\{d_b(x,y),{\mathcal{D}}_b(x,{\mathcal{J}}_{1}x),{\mathcal{D}}_b(y,{\mathcal{J}}_{2}y),\frac{1}{2s}[{\mathcal{D}}_b(x,{\mathcal{J}}_{2}y)+{\mathcal{D}}_b(y,{\mathcal{J}}_{1}x)]\right\}$$ $${\mathrm{\Delta }}_1(x,y)=max\{d_b(x,y),{\mathcal{D}}_b(x,{\mathcal{J}}_{1}x),{\mathcal{D}}_b(y,{\mathcal{J}}_{2}y),\frac{1}{2s}[{\mathcal{D}}_b(x,{\mathcal{J}}_{2}y)+{\mathcal{D}}_b(y,{\mathcal{J}}_{1}x)]\}$$and $${\mathrm{\Delta }}_2(x,y)=min\{{\mathcal{D}}_b(x,{\mathcal{J}}_{1^X}),{\mathcal{D}}_b(y,{\mathcal{J}}_{2\mathcal{Y}}),{\mathcal{D}}_b(x,{\mathcal{J}}_{2\mathcal{Y}}),{\mathcal{D}}_b(y,{\mathcal{J}}_{1^X})\}.$$If (13) is satisfied just for $x,y\in \overline{\mathcal{O}(x_0;{\mathcal{J}}_1,{\mathcal{J}}_2)}$ (for some x₀ ∈ 𝜀), we say that (𝒥₁, 𝒥₂) is a multivalued almost orbitally (F, δ_b)-contraction pair.The main result of this section is the following theorem.

Theorem 4.2: Let (𝜀, d_b, s) be a b-metric space with s > 1 and let ${\mathcal{J}}_1,{\mathcal{J}}_2:\mathcal{E}\to {\mathcal{P}}_b(\mathcal{E})$ form a multivalued almost orbitally (F, δ_b)-contraction pair (for some x₀). Assume that 𝜀 is (𝒥₁, 𝒥₂)-orbitally complete at x₀. If F is continuous and 𝒥₁ and 𝒥₂ are orbitally continuous at x₀, then 𝒥₁ and 𝒥₂ have a common fixed point in 𝜀.

Proof: Starting with the given point x₀, choose a sequence {x_n} ⊂ 𝜀 satisfying $$x_{2n+1}\in {\mathcal{J}}_2{x}_{2n},x_{2n+2}\in {\mathcal{J}}_1{x}_{2n+1},forn\in \{0,1,\dots \}$$and let a_n = d_b (x_n, x_{n + 1})· If $x_{n_0}\in {\mathcal{J}}_2{x}_{n_0}or{x}_{n_0}\in {\mathcal{J}}_1{x}_{n_0}$ for some n₀, then the proof is finished. So assume x_n ≠ x_{n + 1} for all n ≥ 0. We claim that $$\underset{n\to \infty }{lim}{s}^n{a}_n=0.$$Suppose that n is an odd number. Substituting x = x_n and y = x_{n+ 1} in (13), we obtain $$\tau +F(s{d}_b(x_n,x_{n+1}))\le \tau +F(s{\delta }_b({\mathcal{J}}_1{x}_n,{\mathcal{J}}_2{x}_{n+1}))\le F{\mathrm{\Delta }}_1(x_n,x_{n+1})+\lambda {\mathrm{\Delta }}_2(x_n,x_{n+1})),$$(14)where $$\begin{array}{ll}\mathrm{\Delta }(x_n,x_{n+1})& =max\left\{\begin{array}{l}{d}_b(x_n,x_{n+1}),{\mathcal{D}}_b(x_n,{\mathcal{J}}_1{x}_n),{\mathcal{D}}_b(x_{n+1},{\mathcal{J}}_2{x}_{n+1}),\\ \frac{1}{2s}{\mathcal{D}}_b(x_n,{\mathcal{J}}_2{x}_{n+1})+{\mathcal{D}}_b(x_{n+1},{\mathcal{J}}_1{x}_n)\end{array}\right\}\\ & \le max\{d_b(x_n,x_{n+1}),d_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2}),\frac{1}{2s}{d}_b(x_n,x_{n+2})\}\\ & \le max\{d_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2})\}\end{array}$$as ${\displaystyle \frac{1}{2s}{\delta }_b(x_n,x_{n+2})\le max\{{\delta }_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2})\}}$ and $$\mathrm{\Delta }(x_n,x_n+1)=min\{{\mathcal{D}}_b(x_n,{\mathcal{J}}_{1^{X}n}),{\mathcal{D}}_b(x_{n+1},{\mathcal{J}}_{2^{X}n+1}),{\mathcal{D}}_b(x_n,{\mathcal{J}}_{2^{X}n+1}),{\mathcal{D}}_b(x_{n+1},{\mathcal{J}}_{1^{X}n})\}=0.$$Therefore it follows from (14) that $${\displaystyle \tau +F(d_b(x_n,x_{n+1}))\le F(max\{d_b(x_{n-1},x_n),d_b(x_n,x_{n+1})\}).}$$(15)Suppose that d_b(x_n-1, x_n) ≤ d_b(x_n, x_n+1). Then from (15), we have $$\tau +F(s{d}_b(x_n,x_{n+1}))\le F(d_b(x_n,x_{n+1})),$$a contradiction, which means that $$max\{d_b(x_n,x_{n+1}),d_b(x_{n+1},x_{n+2})\}=d_b(x_n,x_{n+1}).$$Consequently, τ + F(sd_b(x_n, x_n+1)) ≤ F(d_b(x_n-1, x_n)),that is $$\tau +F(s{a}_n)\le F(a_{n-1}).$$(16)In a similar way, we can establish inequality (16) when n is an even number.It follows by (16) and property (F4) that $$\tau +F(s^n{a}_n)\le F(s^{n-1}{a}_{n-1})\text{for\hspace{0.17em}\hspace{0.17em}all}n\in \mathbb{N}\cup \{0\}.$$Similarly as in Theorem 3.2, we can prove that the sequence {x_n} is a b-Cauchy sequence in $\mathcal{O}(x_0;{\mathcal{J}}_1,{\mathcal{J}}_2)$ Since 𝜀 is (𝒥₁, 𝒥₂)-multivalued orbitally complete at x₀, there exists a z ∈ 𝜀 such that $$x_n\to Zasn\to \mathrm{\infty }.$$If 𝒥₁ and 𝒥₂ are orbitally continuous, then clearly 𝒥₂z = 𝒥₁z = z.Consequences similar to Corollaries 3.3 and 3.4 can be formulated in an obvious way.If in Theorem 4.2, 𝒥₁ and 𝒥₂ are single-valued mappings, we deduce the following result.

Corollary 4.3: Let (𝜀, d_b, s) be α b-metric space with s > 1 and let 𝒥₁, 𝒥₂ :𝜀 → 𝜀 be self-mappings such that 𝜀 is (𝒥₁, 𝒥₂)-orbitally complete (at some x₀). Suppose that F ∈ 𝔉_s and there exist τ > 0, λ ≥ 0 such that $$\tau +F(s{d}_b({\mathcal{J}}_{1}x,{\mathcal{J}}_{2}y))\le F{\mathrm{\Delta }}_1(x,y)+\lambda {\mathrm{\Delta }}_2(x,y)),$$for all $x,y\in \overline{\mathcal{O}(x_0;{\mathcal{J}}_1,{\mathcal{J}}_2)}$ (for the same x₀) with min{d_b (𝒥x, 𝒥y), d_b(x, y)} > 0, where $${\mathrm{\Delta }}_1(x,y)=\text{max}\{d_b(x,y),d_b(x,{\mathcal{J}}_{1}x),d_b(y,{\mathcal{J}}_{2}y),\frac{d_b(x,{\mathcal{J}}_{2}y)+d_b(y,{\mathcal{J}}_{1}x)}{2s}\}$$and $$\mathrm{\Delta }(x,y)=min\{d_b(x,{\mathcal{J}}_{1}x),d_b(y,{\mathcal{J}}_{2}y),d_b(x,{\mathcal{J}}_{2}y),d_b(y,{\mathcal{J}}_{1}x)\}.$$If F is continuous and 𝒥₁ and 𝒥₂ are (𝒥₁, 𝒥₂)-orbitally continuous at x₀, then 𝒥₁ and 𝒥₂ have a common fixed point.We illustrate the preceding result with the following example (inspired by [20, Example 2.10]).

Example 4.4: Let the set 𝜀 = [0, +∞) be equipped with b-metric d_b(x, y) = (x — y)² (s = 2) and define 𝒥₁, 𝒥₂ : 𝜀 → 𝜀 by $${\mathcal{J}}_{1}x=\left\{\begin{array}{ll}\frac{1}{2}x,& 0\underset{\_}{<}x\le \frac{1}{2},\\ x,& x>\frac{1}{2},\end{array}\right.{\mathcal{J}}_{2}x=\left\{\begin{array}{ll}\frac{1}{3}x,& 0\le x\le \frac{1}{3},\\ x,& x>\frac{1}{3}.\end{array}\right.$$Take $x_0={\displaystyle \frac{1}{2}.}$ Then it is easy to show that $$\mathcal{O}(x0,{\mathcal{J}}_{1,}{\mathcal{J}}_2)\subset \{\frac{1}{2^k\cdot 3^l}:k_{,}l\in \mathbb{N}\},\overline{\mathcal{O}(x0,{\mathcal{J}}_{1,}{\mathcal{J}}_2)}=\mathcal{O}(x0,{\mathcal{J}}_{1,}{\mathcal{J}}_2)\cup \{0\}.$$We will check that the contractive condition of Corollary 4.3 is ful filled for x, y ∈ 𝓞(x₀ ; 𝒥₁, 𝒥₂) with ${\displaystyle \tau =\ln \frac{9}{8}}$ and F(α) = In α. Indeed, it takes the form $$2{\left(\frac{x}{2}-\frac{y}{3}\right)}^2\le \frac{8}{9}max\left\{(x-y{)}^2,\frac{1}{4}{x}^2,\frac{4}{9}{y}^2,\frac{1}{2}\left[{\left(x-\frac{y}{3}\right)}^2+{\left(y-\frac{x}{2}\right)}^2\right]\right\},$$which, after the substitution y = tx, t ≥ 0 reduces to $$2{\left(\frac{1}{2}-\frac{1}{3}t\right)}^2\le \frac{8}{9}max\left\{(1-t{)}^2,\frac{1}{4},\frac{4}{9}{t}^2,\frac{1}{2}\left[{\left(1-\frac{t}{3}\right)}^2+{\left(t-\frac{1}{2}\right)}^2\right]\right\}.$$The last inequality can be easily checked by considering possible values of the parameter t > 0.All other conditions are also fulfilled, and hence, by Corollary 4.3, we conclude that 𝒥₁ and 𝒥₂ have a common fixed point (which is z = 0).

In this section we apply the obtained results to achieve the existence of solutions for a certain Fredholm-type integral inclusion. The application is inspired by [3, 21].

Consider the following integral inclusion of Fredholm type. $$x(t){\displaystyle \in f(t)+\underset{a}{\overset{b}{\int }}\mathcal{K}(t_{,}{s}_{,}x(s))ds,t\in [a_{,}b].}$$(17)

Here, ƒ ∈ C[a, b] is a given real function and $\mathcal{K}\mathcal{:}[a_{,}b]\times [a_{,}b]\times \mathbb{R}\to {\mathcal{P}}_{cb}(\mathbb{R})$ a given set-valued operator; x ∈ C [a, b] is the unknown function.

Now, for ρ ≥ 1, consider the b-metric d_b, on C [a, b] defined by $$d_b(x_{,}y)=({\displaystyle \underset{t\in [a,b]}{max}|x(t)-y(t)|{)}^p=\underset{t\in [a,b]}{max}|x(t)-y(t){|}^p}$$(18)

for all x, y ∈ C[a,b]. Then (C[a, b], d_b, 2^ρ–1) is a complete b-metric space. Let 𝓓_b and δ_b, have the respective meanings.

We will assume the following:

(I) For each x ∈ C[a, b], the operator 𝒦_x(t, s) := 𝒦,(t, s, x(s)), (t,s) ∈ [a,b] × [a, b] is continuous.

(II) there exists a continuous function Υ : [a, b]² ∈ [0, +∞) such that $$|k_u(t,s)-k_v(t,s){|}^p$$

for all t, s ∈ [a, b], all u, ύ ∈ C[a, b] and all k_u(t, s) ∈ 𝒦_u(t, s), 𝒦_v(t, s) ∈ 𝒦_v(t, s), where λ > 0, p > 1; (III) there exists τ ∈ [1, +∞) such that $$\underset{t\in [a,b]}{sup}\underset{a}{\overset{b}{\int }}\mathrm{{\rm Y}}(t,\tau )d\tau \le \frac{e^{-\tau }}{2^{p-1}}.$$

Theorem 5.1: Under the conditions (I)-(III), the integral inclusion (17) has a solution in C [a, b].

Proof: Let 𝜀 = C [a, b] (with b-metric d_b as defined in (18)) and consider the set-valued operator 𝒥 : 𝜀 → 𝒫_cb (𝜀) defined by $$\mathcal{J}x=\left\{y\in \mathcal{E}:y(t)\in f(t)+\underset{a}{\overset{b}{\int }}\mathcal{K}(t,s,x(s))ds,t\in [a,b]\right\}.$$(19)It is clear that the set of solutions of the integral inclusion (17) coincides with the set of fixed points of the operator 𝒥. Hence, we have to prove that under the given conditions, 𝒥 has at least one fixed point in 𝜀. For this, we shall check that the conditions of Theorem 3.2 hold true.Let x ∈ 𝜀 be arbitrary. For the set-valued operator 𝒦_x(t, s) : [a, b] × [a, b] → 𝒫_cb (ℝ), it follows from the Michael's selection theorem that there exists a continuous operator k_x : [a, b] x [a, b] → ℝ such that k_x(t, s) ∈ 𝒦_x(t, s) for each (t, s) ∈ [a, b] × [a,b]. It follows that $f(t)+{\displaystyle {\int }_a^b{k}_x(t,s)ds\in \mathcal{J}x.}$ Hence, 𝒥 x ≠ 0. Since ƒ and 𝒦_x are continuous on [a, b], resp. [a, b]², their ranges are bounded and hence 𝒥 x is bounded, i.e., 𝒦 : 𝜀 → 𝒫_Cb (𝜀).We will check that the contractive condition (2) holds for 𝒥 in 𝜀 with some τ > 0, λ ≥ 0 and F ∈ 𝔉_s, i.e., $$\tau +F(s{\delta }_b(\mathcal{J}{x}_1,\mathcal{J}{x}_2))\le F\left(\begin{array}{c}max\left\{d_b(x_1,x_2),{\mathcal{D}}_b(x_1,\mathcal{J}{x}_1),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_2),\frac{D_b(x_1,J{x}_2)+D_b(x_2,J{x}_1)}{2^p}\right\}\\ +\lambda min\{{\mathcal{D}}_b(x_1,\mathcal{J}{x}_1),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_2),{\mathcal{D}}_b(x_1,\mathcal{J}{x}_2),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_1)\}\end{array}\right)$$for elements x₁, x₂ ∈ 𝜀 Let y₁ ∈ 𝒥x₁ be arbitrary, i.e., $$y_1(t)\in f(t)+\underset{a}{\overset{b}{\int }}\mathcal{K}(t,s,x_1(s))ds,t\in [a,b]$$holds true. This means that for all t,s ∈ [a,b] there exists $k_{x_1}(t,s)\in {\mathcal{K}}_{x_1}(t,s)=\mathcal{K}(t,s,x_1(s))$ such that $y_1(t)=f(t)+{\displaystyle {\int }_a^b{k}_{x_1}(t,s)ds}$ for t ∈ [a,b].For all x₁, x₂ ∈ 𝜀, it follows from (II) that

It means that there exists $z(t,s)\in {\mathcal{K}}_{x_2}(t,s)$ such that

for all t, s ∈ [a,b].Denote by 𝒰(t, s) : [a, b] × [a, b] → 𝒫_cb(∞) the operator defined by $$\mathcal{U}(t,s)={\mathcal{K}}_{x_2}(t,s)\cap \{u\in \mathbb{R}:d_b(k_{x_1}(t,s),u))\le R(t,s)\}.$$Since, by (I), 𝒰 is lower semicontinuous, it follows that there exists a continuous operator $k_{x_2}:[a,b]\times [a,b]\to \mathbb{R}$ such that $k_{x_2}(t,s)\in \mathcal{U}(t,s)$ for t ,s ∈ [a, b]. Then$y_2(t):=f(t)+{\displaystyle {\int }_a^b{k}_{x_2}(t,s)ds}$ ds satisfies that $$y_2(t)\in f(t)+\underset{a}{\overset{b}{\int }}\mathcal{K}(t,s,x_2(s))ds,t\in [a,b],$$i.e., y₂ ∈ 𝒥 x₂ and

for all t, s ∈ [a,b].Thus, we obtain that $${\delta }_b(\mathcal{J}{x}_1,\mathcal{J}{x}_2)\le \frac{e^{-\tau }}{2^{p-1}}\left(\begin{array}{c}max\left\{d_b(x_1,x_2),{\mathcal{D}}_b(x_1,\mathcal{J}{x}_1),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_2),\frac{D_b(x_1,J{x}_2)+D_b(x_2,J{x}_1)}{2^p}\right\}\\ +\lambda min\{{\mathcal{D}}_b(x_1,\mathcal{J}{x}_1),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_2),{\mathcal{D}}_b(x_1,\mathcal{J}{x}_2),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_1)\end{array}\right)$$(This shows again that the sets 𝒥 x₁ and 𝒥 x₂ are bounded.) By passing to logarithms, we write $$\ln \left(s{\delta }_b\left(\mathcal{J}{x}_1,\mathcal{J}{x}_2\right)\right)\le \ln e^{-\tau }\left(\begin{array}{c}max\left\{d_b(x_1,x_2),{\mathcal{D}}_b(x_1,\mathcal{J}{x}_1),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_2),\frac{D_b(x_1,J{x}_2)+D_b(x_2,J{x}_1)}{2^p}\right\}\\ +\lambda min\{{\mathcal{D}}_b(x_1,\mathcal{J}{x}_1),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_2),{\mathcal{D}}_b(x_1,\mathcal{J}{x}_2),{\mathcal{D}}_b(x_2,\mathcal{J}{x}_1)\end{array}\right)$$Taking the function F ∈ 𝔉_s defined by F (α) = In α, we obtain that the condition (19) is fulfilled. Using Theorem 3.2, we conclude that the given integral inclusion has a solution.