*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},(𝒥*x*, *𝒥y*)*, d*_{b} (*x, y*)}* >* 0, *where*
$$\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 x*_{0}* ∈* *ε*)*, we say that 𝒥 is a multivalued almost orbitally* (*F, δ*_{b})*-contraction*.

We are equipped now to state our first main result.

Starting from the given point *x*_{0}, 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 *n*_{0}, then the proof is finished. Therefore, we assume *x*_{n} ≠* x*_{n}_{+1}for all *n* ≥ 0. So

*d*_{b}(*x*_{n+}_{1},* x*_{n+}_{2})* >* 0 andδ*b*(𝒥*x*_{n}, 𝒥*x*_{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 $\frac{1}{2s}{d}_{b}({x}_{n},{x}_{n+2})\le max\{{d}_{b}({x}_{n},{x}_{n+1}),{d}_{b}({x}_{n+1},{x}_{n+2})\},$it follows that
$$\tau +F(s{d}_{b}({x}_{n+1},{x}_{n+2}))\le F(max\{{d}_{b}({x}_{n},{x}_{n+1}),{d}_{b}({x}_{n+1},{x}_{n+2})\}).$$

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
$$\tau +F(s{d}_{b}({x}_{n+1},{x}_{n+2}))\le F({d}_{b}({x}_{n},{x}_{n+1}))for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\in \mathbb{N}\cup \{0\}.$$(5)

It follows by (5) and the property (F4) that
$$\tau +F({s}^{n}{d}_{b}({x}_{n},{x}_{n+1}))\le F({s}^{n-1}{d}_{b}({x}_{n-1},{x}_{n}))\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{all}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\in \mathbb{N}\cup \{0\}.$$(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*(*s*^{n} ϱ_{n})→* —*∞as *n* →∞. Thus, from (F2), we have
$${s}^{n}{\varrho}_{n}\to 0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}as\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\to \mathrm{\infty}.$$(8)

Now, by the property (F3) there exists *k* ∈(0, 1) such that
$$\underset{n\to \mathrm{\infty}}{lim}({s}^{n}{\varrho}_{n}{)}^{k}F({s}^{n}{\varrho}_{n})=0.$$(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}* s*^{n} 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(*x*_{0};𝒥). Since 𝜀is 𝒥-orbitally complete, there exists a *z* ∈ 𝜀 such that
$${x}_{n}\to Z\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\to \mathrm{\infty}.$$

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 $\underset{k\to \mathrm{\infty}}{lim}{x}_{{n}_{k}}=z$, we deduce that *z* ∈ 𝒥 *z* and hence the proof is completed. Then we assume that there exists *n*_{0}∈ℕ such that *x*_{n} ∉𝒥 *z* for all *n* ∈ℕ with *n* ≥ *n*_{0}. It follows that δ*b*(𝒥*x*_{n},𝒥*z*)* >* 0 for all *n* ≥ *n*_{0}. 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
$$\begin{array}{ll}{\mathrm{\Theta}}_{1}({x}_{n},z)& =max\left\{{d}_{b}({x}_{n},z),{\mathcal{D}}_{b}({x}_{n},\mathcal{J}{x}_{n}),{\mathcal{D}}_{b}(z,\mathcal{J}z),\frac{{D}_{b}({x}_{n},Jz)+{D}_{b}(\mathrm{z},J{x}_{n})}{2s}\right\}\\ & \le max\left\{{d}_{b}({x}_{n},z),{d}_{b}({x}_{n},{x}_{n+1}),{\mathcal{D}}_{b}(z,\mathcal{J}z),\frac{{D}_{b}({x}_{n},Jz)+{d}_{b}(z,{x}_{n+1})}{2s}\right\}\\ & \to {\mathcal{D}}_{b}(z,\mathcal{J}z),\phantom{\rule{0.056em}{0ex}}\phantom{\rule{0.056em}{0ex}}\text{as}\phantom{\rule{0.056em}{0ex}}\phantom{\rule{0.056em}{0ex}}n\to \mathrm{\infty},\end{array}$$

and
$$\begin{array}{ll}{\mathrm{\Theta}}_{1}({x}_{n},z)& =min\left\{{\mathcal{D}}_{b}({x}_{n},\mathcal{J}{x}_{n}),{\mathcal{D}}_{b}(z,\mathcal{J}z),{\mathcal{D}}_{b}({x}_{n},\mathcal{J}z),{\mathcal{D}}_{b}(z,\mathcal{J}{x}_{n})\right\}\\ & \le min\{{d}_{b}({x}_{n},{x}_{n+1}),{\mathcal{D}}_{b}(z,\mathcal{J}z),{\mathcal{D}}_{b}({x}_{n},\mathcal{J}z),{d}_{b}(z,{x}_{n+1})\}\\ & \to 0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\to \mathrm{\infty}.\end{array}$$

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. D

The following corollaries follow from Theorem 3.2 by taking *F*(α)* =* In α, resp. *F*(α)* =* α* +* In α in (2).

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