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} (𝒥_{1} *x*, 𝒥_{2} *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 x*_{0} ∈ 𝜀*), we say that* (𝒥_{1}, 𝒥_{2})* 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 x_{0}). Assume that 𝜀* is* (𝒥_{1}, 𝒥_{2})*-orbitally complete at x*_{0}. If F is continuous and 𝒥_{1}* and* 𝒥_{2}* are orbitally continuous at x*_{0}, then 𝒥_{1}* and* 𝒥_{2}* have a common fixed point in 𝜀*.

**Proof:** *Starting with the given point **x*_{0}, choose a sequence {*x*_{n}} ⊂ 𝜀 satisfying
$${x}_{2n+1}\in {\mathcal{J}}_{2}{x}_{2n},{x}_{2n+2}\in {\mathcal{J}}_{1}{x}_{2n+1},for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\in \{0,1,\dots \}$$and let *a*_{n} = *d*_{b} (*x*_{n}, *x*_{n + 1})· If ${x}_{{n}_{0}}\in {\mathcal{J}}_{2}{x}_{{n}_{0}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}or\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{{n}_{0}}\in {\mathcal{J}}_{1}{x}_{{n}_{0}}$ for some n_{0}, 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 $\frac{1}{2s}{\delta}_{b}({x}_{n},{x}_{n+2})\le max\{{\delta}_{b}({x}_{n},{x}_{n+1}),{d}_{b}({x}_{n+1},{x}_{n+2})\}$ 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
$$\tau +F({d}_{b}({x}_{n},{x}_{n+1}))\le F(max\{{d}_{b}({x}_{n-1},{x}_{n}),{d}_{b}({x}_{n},{x}_{n+1})\}).$$(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
$$\tau +F({s}^{n}{a}_{n})\le F({s}^{n-1}{a}_{n-1})\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{for\hspace{0.17em}\hspace{0.17em}all}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\in \mathbb{N}\cup \{0\}.$$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 (𝒥_{1}, 𝒥_{2})-multivalued orbitally complete at *x*_{0}, there exists a z ∈ 𝜀 such that
$${x}_{n}\to Z\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}as\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\to \mathrm{\infty}.$$If 𝒥_{1} and 𝒥_{2} are orbitally continuous, then clearly 𝒥_{2}*z* = 𝒥_{1}*z =* z.Consequences similar to Corollaries 3.3 and 3.4 can be formulated in an obvious way.If in Theorem 4.2, 𝒥_{1} and 𝒥_{2} 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* 𝒥_{1}, 𝒥_{2} :𝜀 → 𝜀 *be self-mappings such that* 𝜀 *is* (𝒥_{1}, 𝒥_{2})*-orbitally complete* (*at some x*_{0})*. 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 x*_{0})* with* min{*d*_{b} (𝒥*x*, 𝒥*y*), *d*_{b}(*x*, *y*)} > 0, *where*
$${\mathrm{\Delta}}_{1}(x,y)=\text{max}\{{d}_{b}(x,y),{d}_{b}(x,{\mathcal{J}}_{1}x),{d}_{b}(y,{\mathcal{J}}_{2}y),\frac{{d}_{b}(x,{\mathcal{J}}_{2}y)\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{d}_{b}(y,{\mathcal{J}}_{1}x)}{2s}\}$$*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* 𝒥_{1}* and* 𝒥_{2}* are* (𝒥_{1}, 𝒥_{2})*-orbitally continuous at x*_{0}*, then* 𝒥_{1}* and* 𝒥_{2}* 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* 𝒥_{1}, 𝒥_{2} : 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(*x*_{0}* ;* 𝒥_{1}, 𝒥_{2})* with $\tau =\mathrm{ln}\frac{9}{8}$* *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* 𝒥_{1}* and* 𝒥_{2}* have a common fixed point (which is z* = *0)*.

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