Coincidence and fixed points of multivalued F-contractions in generalized metric space with application

Abstract

The aim of the paper was to prove some new fixed point theorems, coincidence point theorems and common fixed point theorems for multivalued F-contractions involving a binary relation that is not necessarily a partial order, in the context of generalized metric spaces (in the sense of Jleli and Samet). We also prove existence of common solutions to integral inclusions.

1 Introduction

The Banach’s fixed point theorem (in short BFPT) [5] provides a strong foundation on which metric fixed point theory has been developed. In the past few decades, many authors have extended and generalized the BFPT in several ways. Recently, Wardowski in [39] opened a new window for researchers by introducing a concept of ${F}$-contraction and proved a fixed point theorem. Further, Altun et al. [3] broadened this idea for multivalued ${F}$-contraction. To more in this direction, consult [1, 4, 12, 15,16,17,18, 20, 23, 31, 38, 40]. Nadler proved the multivalued version of BFPT in [28]. Nadler’s fixed point theorem has attained the attention of several researchers. For a study to this direction and more references, see [6,7,8,9, 14, 22, 24, 33,34,35]. The fixed point theorem for multivalued mapping without using generalized Hausdorff distance is proved by Feng and Liu in [11].

On the other hand, notion of standard metric space is generalized in several ways (see [10, 13, 25,26,27, 29, 30, 37]). A new generalization of metric spaces was given by Jleli and Samet in [19]; it recapitulates a huge class of topological spaces including b-metric spaces, standard metric spaces, dislocated metric spaces and modular spaces. They extended fixed point results including BFPT, Ciri$\acute{\mathrm{c}}$’s fixed point theorem and a fixed point result due to Ran and Reurings. Further, in [21], Karapinar et al. obtained fixed point theorems under very general contractive conditions in generalized metric spaces (in the sense of Jleli and Samet). Recently, in [2], Altun et al. obtained a Feng-Liu’s type fixed point theorem in the setting of generalized metric spaces (in the sense of Jleli and Samet).

In this paper, we prove fixed point theorems for multivalued ${F}$-contractions in the context of generalized metric space in the manner of Jleli and Samet.

2 Preliminaries

In the sequel, $\mathbb {N}=\{0,1,2,3, ...\}$ indicates the set of all non-negative integers, $\mathbb {R}$, indicates the set of all real numbers. Let ${\psi }$ be a self mapping on nonempty set $U$. A binary relation on $U$ is a nonempty subset $\mathcal {R}$ of the Cartesian product $U\times U$. For simplicity, we denote $u\mathcal {R}v$ if $(u, v)\in \mathcal {R}$. The notions of reflexivity, transitivity, antisymmetry, preorder and partial order can be found in [36]. The trivial preorder on $U$ is denoted by $\mathcal {R}_{U}$ and is given by $u\mathcal {R}_{U}v$ for all $u, v\in U$.

Denote $\Delta ({F}_*)$ by the collection of all mappings ${F}:(0,+\infty )\rightarrow \mathbb {R}$ satisfying (${F}_{1}$)-(${F}_{4}$):

(${F}_{1}$):

${F}(u)< {F}(v)$ for all $u<v$;

(${F}_{2}$):

for all sequences $\{\pi _p\}\subseteq (0,+\infty )$, $\lim _{p\rightarrow +\infty }\pi _p=0$, if and only if $\lim _{p\rightarrow +\infty }{F}(\pi _p)=-\infty ;$

(${F}_{3}$):

there exists $0< \ell <1$ such that $\lim _{\pi \rightarrow 0^{+}}\pi ^\ell {F}(\pi )=0,$

and

(${F}_{4}$):

${F}(\inf M)=\inf {{F}}(M)$ for all $M\subset (0,+\infty )$ with $\inf M>0$.

$\Delta ({F})$ contains mappings ${{F}}$ which satisfies $({F}_{1})$-$({F}_{3})$.

Note that $\Delta ({F}_*)\subset \Delta ({F})$.

Theorem 2.1

[39] Let $(U,{d})$ be a complete metric space and ${\psi }:U\rightarrow U$ be a self-mapping on $U$. If for $u,v\in U$ with ${d}({\psi }u, {\psi }v)>0$ there exists $\lambda >0$ and ${F}\in \Delta ({F})$ such that

$$\begin{aligned} \lambda +{F}\left( {d}\left( {\psi }u, {\psi }v\right) \right) \le {F}\left( {d}\left( u,v\right) \right) , \end{aligned}$$

(2.1)

then ${\psi }$ has a unique fixed point in $U$, say $u^*$, and for every $u_0\in U$ a sequence $\{{\psi }^p u_0\}_{p\in \mathbb {N}}$ is convergent to $u^*$.

A mapping ${\psi }:U\rightarrow U$, which satisfies (2.1) for $\lambda >0$ and ${F}\in \Delta ({F})$ is known as ${F}$-contraction. The extension of ${F}$-contraction to multivalued ${F}$-contraction is given as follows:

Definition 2.2

Let $(U,{d})$ be a metric space and ${\psi }: U\rightarrow CB(U)$ be a mapping. Then ${\psi }$ is multivalued ${F}$-contraction if ${F}\in \Delta ({F})$ and there exist $\lambda >0$ such that for $u,v\in U$,

$$\begin{aligned} H({\psi }u, {\psi }v)>0 \hbox { implies } \lambda +{F}\left( H\left( {\psi }u, {\psi }v\right) \right) \le {F}\left( {d}\left( u,v\right) \right) . \end{aligned}$$

(2.2)

Theorem 2.3

[3, Theorem 2.2] Let $(U, {d})$ be a complete metric space and ${\psi }:U\rightarrow K(U)$ be a multivalued ${F}$-contraction. Then ${\psi }$ has a fixed point in $U$.

Theorem 2.4

[3, Theorem 2.5] Let $(U, {d})$ be a complete metric space and ${\psi }:U\rightarrow CB(U)$ be a multivalued ${F}$-contraction for ${F}\in \Delta ({F}_*)$. Then ${\psi }$ has a fixed point in $U$.

Henceforth, let $\mathrm {J}: U\times U\rightarrow [0,+\infty ]$ be a given mapping. For every $u\in U$, define the set

$$\begin{aligned} {C}(\mathrm {J},U,u)=\left\{ \{u_{p}\}\subseteq U: \lim _{p\rightarrow +\infty }\mathrm {J}(u_p,u)=0\right\} . \end{aligned}$$

(2.3)

Definition 2.5

[19] Let $U$ be a nonempty set and $\mathrm {J}:U\times U\rightarrow [0,+\infty ]$ be a function which satisfies the following conditions for all $u,v\in U$:

($\mathrm {J}_{1}$):

$\mathrm {J}(u,v)=0$ implies $u=v$;

($\mathrm {J}_{2}$):

$\mathrm {J}(u,v)=\mathrm {J}(v,u)$;

($\mathrm {J}_{3}$):

there exist $\kappa > 0$ such that $(u,v) \in U\times U,~\{u_{p}\} \in {C}(\mathrm {J},U,u)$ implies

$$\begin{aligned} \mathrm {J}(u,v)\le \kappa \lim _{p\rightarrow +\infty }\sup \mathrm {J}(u_{p},v). \end{aligned}$$

(2.4)

Then $\mathrm {J}$ is called a generalized metric and the pair $(U, \mathrm {J})$ is called a generalized metric space (for short, a GMS(JS)).

Remark 2.6

[19] If the set ${C}(\mathrm {J},U,u)$ is empty for every $u\in U$, then $(U, \mathrm {J})$ is a GMS(JS) if and only if $(\mathrm {J}_1)$ and $(\mathrm {J}_2)$ are satisfied.

Many examples of GMS(JS) can be found in [19] and [21]. We add one more here

Example 1

Let $U=\{S_n:n\in \mathbb {N}\}$ where $\{S_n\}_{n\in \mathbb {N}}$ is the sequence defined as $S_n=\frac{n(n+1)}{2}$, $n\in \mathbb {N}$. Define the function $\mathrm {J}:U\times U\rightarrow [0,+\infty ]$ by

$$\begin{aligned} \mathrm {J}(S_n, S_m)=\left\{ \begin{array}{ll} S_1 &{} \hbox {if}~~n=m \\ +\infty &{} \hbox {otherwise}. \end{array} \right. \end{aligned}$$

Let us show that $(U,\mathrm {J})$ is a GMS(JS). Properties ($\mathrm {J}_{1}$) and ($\mathrm {J}_{2}$) are apparent. Hence we have

$$\begin{aligned} \mathrm {J}(S_n, S_m)=\left\{ \begin{array}{ll} 1 &{} \hbox {if}~~n=m \\ +\infty &{} \hbox {otherwise}. \end{array} \right. \end{aligned}$$

Therefore, we cannot find any sequence $\{u_r\}\in U$ such that $\lim _{r\rightarrow +\infty }\mathrm {J}(u_r, S_n)=0,$ for some $S_n\in U$. This shows that ${C}(\mathrm {J},U, S_n)$ is empty for all $S_n\in U$. Thus, by Remark 2.6, $(U,\mathrm {J})$ is a GMS(JS).

Definition 2.7

[19] Let $(U,\mathrm {J})$ be a GMS(JS) and $u\in U$.

($\mathrm {1}$):

A sequence $\{u_{p}\} \subseteq U$ is said to be $\mathrm {J}$-convergent and $\mathrm {J}$-converges to $u$ if $\{u_{p}\}\in {C}(\mathrm {J},U,u)$. In such case, we will write $\{u_p\}\xrightarrow {\mathrm {J}} u.$

($\mathrm {2}$):

a sequence $\{u_{p}\} \subseteq U$ is said to be $\mathrm {J}$-Cauchy if

$$\begin{aligned} \lim _{p,q\rightarrow +\infty } \mathrm {J}(u_{p},u_{p+q})=0. \end{aligned}$$

(2.5)

($\mathrm {3}$):

A GMS(JS) $(U,\mathrm {J})$ is said to be complete if every $\mathrm {J}$-Cauchy sequence in $U$ is $\mathrm {J}$-convergent.

Proposition 2.8

[19] Let $(U,\mathrm {J})$ be a GMS(JS), $\{u_{p}\}$ be a sequence in $U$ and $(u,v)\in U\times U$. If $\{u_{p}\}$ is $\mathrm {J}$-convergent and $\mathrm {J}$-converges to $u$ and $\{u_{p}\}$ $\mathrm {J}$-converges to $v$, then $u=v$.

Remark 2.9

Jleli and Samet defined a $\mathrm {J}$-Cauchy sequence by

$$\begin{aligned} \lim _{p,q\rightarrow +\infty } \mathrm {J}(u_{p},u_{q})=0. \end{aligned}$$

(2.6)

Clearly, (2.6) implies (2.5), but the converse need not to be true [21]. Henceforth, we assume that $\mathrm {J}$-Cauchy sequences are given by (2.6).

Definition 2.10

[21] Let $(U,\mathrm {J})$ be a GMS(JS) and ${\psi }:U\rightarrow U$. For $u_{0}\in U$, denote $\delta (\mathrm {J},{{\psi }}, u_{0})$, the $\mathrm {J}$-diameter of the orbit of $u_{0}$ by ${{\psi }}$, $\mathcal {O}_{{\psi }}(u_{0})=\{{\psi }^{p}u_{0} : p\in \mathbb {N}\}$, and is defined as

$$\begin{aligned} \delta (\mathrm {J},{{\psi }},u_{0})= \sup (\{\mathrm {J}({{\psi }}^{p}u_{0},{{\psi }}^{q}u_{0}):p,q\in \mathbb {N}\}). \end{aligned}$$

(2.7)

Proposition 2.11

[19]

• (1) A standard metric space is a GMS(JS).

• (2) A b-metric space is a GMS(JS).

• (3) A dislocated metric space is a GMS(JS).

Definition 2.12

[21] Let $\mathcal {R}$ be a binary relation on GMS(JS) $(U,\mathrm {J})$. A sequence $\{u_p\}\subseteq U$ is $\mathcal {R}$-nondecreasing if $u_p\mathcal {R}u_{p+1}$ for all $p\in \mathbb {N}$.

Definition 2.13

[21] A GMS(JS) $(U,\mathrm {J})$ is $\mathcal {R}$-nondecreasing complete if every $\mathcal {R}$-nondecreasing and $\mathrm {J}$-Cauchy sequence is $\mathrm {J}$-convergent in $U$.

Remark 2.14

Note that every complete GMS(JS) is also $\mathcal {R}$-nondecreasing complete, but converse is false as shown in the example below.

Example 2

Let $U=(0,1]$ equipped with the Euclidean metric ${d}(u,v)=|u-v|$ for all $u,v\in U$. Define a binary relation $\mathcal {R}$ on $U$ by

$$\begin{aligned} u\mathcal {R}v~\quad ~\hbox {if}~~0<u\le v\le 1. \end{aligned}$$

Then $(U,{d})$ is $\mathcal {R}$-nondecreasing complete but not complete metric space.

Definition 2.15

[21] Let $(U,\mathrm {J})$ be a GMS (JS). A mapping ${\psi }:U\rightarrow U$ is $\mathcal {R}$-nondecreasing-continuous at $w\in U$ if $\{{\psi }u_p\}\in {C}(\mathrm {J}, U, {\psi }w)$ for all $\mathcal {R}$-nondecreasing sequence $\{u_p\}\in {C}(\mathrm {J}, U, w)$. A mapping ${\psi }$ is $\mathcal {R}$-nondecreasing-continuous if it is $\mathcal {R}$-nondecreasing-continuous at each point of $U$.

Remark 2.16

[21] Every continuous mapping is also $\mathcal {R}$-nondecreasing-continuous, but converse is false in general as shown in Example 4.6 of [21].

Definition 2.17

[2] Let $(U, \mathrm {J})$ be a GMS(JS) and $V\subseteq U$. Then, V is called sequentially open if for each sequence $\{u_p\}$ of $U$ such that

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(u_p, u)=0 \end{aligned}$$

for some $u\in V$ is eventually in V, that is, there exists $p_0\in \mathbb {N}$ such that $u_p\in V$ for all $p\ge p_0$.

Theorem 2.18

[2] Let $(U,\mathrm {J})$ be a complete GMS(JS) and let ${\psi }:U\rightarrow C(U)$ be a multivalued mapping, where $C(U)$ is the collection of all nonempty closed subsets of $U$. Assume that there exist a constant $\eta >0$ and $\ell \in (0,1)$ such that for any $u\in U$ there is $v\in I_{\ell }^{u}$ satisfying

$$\begin{aligned} \mathrm {J}(v, {\psi }v)\le \eta \mathrm {J}(u,v), \end{aligned}$$

(2.8)

where

$$\begin{aligned} I_{\ell }^{u}=\{v\in {\psi }u: b\mathrm {J}(u,v)\le D(u, {\psi }u)\}. \end{aligned}$$

(2.9)

If there exists $u_0\in U$ such that $\mathrm {J}(u_0, {\psi }u_0)<+\infty$. then there exist a sequence $\{u_p\}$ in $U$ satisfying

$(\mathrm {1})$:

$u_{p+1}\in {\psi }u_{p}$;

$(\mathrm {2})$:

$\mathrm {J}(u_p, u_{p+1})<+\infty$;

$(\mathrm {3})$:

$\eta \mathrm {J}(u_{p+1}, u_{p+2})\le \ell \mathrm {J}(u_p, u_{p+1})$ and $\eta \mathrm {J}(u_{p+1}, {\psi }u_{p+1})\le \ell \mathrm {J}(u_p, {\psi }u_{p})$.

Moreover, If constructed sequence is $\mathrm {J}$-Cauchy and the function $u\longmapsto \mathrm {J}(u, {\psi }u)$ is lower semi-continuous, then ${\psi }$ has a fixed point in $U$.

Following Altun et al. [2], let $\lambda _{JS}$ be the collection of all sequentially open subsets of $U$; then $(U, \lambda _{JS})$ is a topological space. Henceforth, $C(U)$ indicates the collection of all nonempty closed subsets of $(U, \lambda _{JS})$ and $K(U)$ indicates the collection of all nonempty compact subsets of $(U, \lambda _{JS})$. It is obvious that $K(U)\subseteq C(U)$. Further, a sequence $\{u_p\}$ is convergent and converges to $u$ in $(U, \mathrm {J})$ if and only if it is convergent and converges to $u$ in $(U, \lambda _{JS})$.

Proposition 2.19

[2] Let $(U, \mathrm {J})$ be a GMS(JS). If $\Lambda$ is the collection of all nonempty subsets A of $U$, for all $u\in U$ satisfying

$$\begin{aligned} \mathrm {J}(u, A)=0~~\hbox {implies}~u\in A, \end{aligned}$$

(2.10)

where $\mathrm {J}(u, A)=\inf \{\mathrm {J}(u,v):v\in A\}$. Then $C(U)=\Lambda$.

Remark 2.20

Let $(U, {d})$ be a metric space, then $\lambda _{JS}$ coincides with the metric topology $\lambda _{d}$.

Definition 2.21

[9] Let ${\psi }:U\rightarrow P(U)$ be a multivalued mapping on a GMS(JS) $(U, \mathrm {J})$. An orbit $O_{\psi }(u_0)$ for ${\psi }$ at a point $u_0\in U$ is a sequence $\{u_p: u_p\in {\psi }u_{p-1}\}_{p\in \mathbb {N}}$ based on $u_0$.

Note that $O_{\psi }(u_{l+1})\subseteq O_{\psi }(u_{l})$ for all $l\in \mathbb {N}$.

3 Auxiliary results

Denote by $\mathcal {F}$, the collection of all functions ${F}:(0,+\infty )\rightarrow \mathbb {R}$ fulfilling conditions $({F}1)$ and $({F}2)$ and by $\mathcal {F}*$, the collection of all functions ${F}:(0,+\infty )\rightarrow \mathbb {R}$ fulfilling conditions $({F}1)$, $({F}2)$ and $({F}4)$. Let ${\psi }:U\rightarrow P(U)$ be multivalued mapping, for every $u\in U$, define a set

$$\begin{aligned} {C}(\mathrm {J}, U, {\psi }u)=\left\{ u_p\in U:\lim _{p\rightarrow +\infty }\mathrm {J}(u_p, v)=0\quad \hbox {for some}~~~~v\in {\psi }u\right\} . \end{aligned}$$

(3.1)

Also, for every $u_0\in U$ and for an orbit $O_{\psi }(u_0)$ based on $u_0$, define

$$\begin{aligned} \delta (\mathrm {J}, {\psi }, u_0)=\sup \{\mathrm {J}(u_l, u_j)~:~u_l, u_j\in O_{\psi }(u_0)\}. \end{aligned}$$

(3.2)

Note that if $p,q\in \mathbb {N}$ such that $p\le q$, then

$$\begin{aligned} \delta (\mathrm {J}, {\psi }, u_q)\le \delta (\mathrm {J}, {\psi }, u_p). \end{aligned}$$

(3.3)

Definition 3.1

Let $(U, \mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and ${\psi }:U\rightarrow P(U)$ be multivalued mapping. Mapping ${\psi }$ is called $\mathcal {R}$-nondecreasing if for all $u,v\in U$

$$\begin{aligned} u\mathcal {R}v\quad \hbox {implies}\quad s\mathcal {R}t~~~~\hbox {for all}~~~~s\in {\psi }u,~~t\in {\psi }v. \end{aligned}$$

Definition 3.2

Let $(U, \mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$, $g:U\rightarrow U$ and ${\psi }:U\rightarrow P(U)$ be multivalued mapping. Mapping ${\psi }$ is called $(g, \mathcal {R})$-nondecreasing if for all $u,v\in U$

$$\begin{aligned} gu\mathcal {R}gv\quad \hbox {implies}\quad s\mathcal {R}t~~~~\hbox {for all}~~~~s\in {\psi }u,~~t\in {\psi }v. \end{aligned}$$

Remark 3.3

By defining $g=I$, where I is the identity operator, then Definition 3.2 is back to Definition 3.1.

Definition 3.4

Let $(U, \mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and ${\psi }:U\rightarrow P(U)$ be multivalued mapping. Mapping ${\psi }$ is called $\mathcal {R}$-nondecreasing continuous at $w\in U$ if for all $\mathcal {R}$-nondecreasing sequence $\{w_p\}$ in ${C}(\mathrm {J}, U, w)$, we have $\{u_p\}\in {C}(\mathrm {J}, U, {\psi }w)$ for all $u_p\in {\psi }w_p$. The mapping ${\psi }$ is $\mathcal {R}$-nondecreasing continuous if it is $\mathcal {R}$-nondecreasing continuous at each point of $U$.

Definition 3.5

Let $(U, \mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and ${\psi }:U\rightarrow P(U)$ be set-valued mapping. Mapping ${\psi }$ is called $\mathcal {R}$-nondecreasing lower semi-continuous at $w\in U$ if for all $\mathcal {R}$-nondecreasing sequence $\{w_p\}$ in ${C}(\mathrm {J}, U, w)$, we have

$$\begin{aligned} \mathrm {J}(w, {\psi }w)\le \lim \inf _{p\rightarrow +\infty } \mathrm {J}(w_p, {\psi }w_p). \end{aligned}$$

Lemma 3.6

Let $(U, \mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and ${\psi }:U\rightarrow C(U)$ be multivalued mapping. Assume that the function $\vartheta :U\rightarrow \mathbb {R}$, defined by $\vartheta (u)=\mathrm {J}(u, {\psi }u)$, is lower semi-continuous at $w\in U$. Then ${\psi }$ is $\mathcal {R}$-nondecreasing lower semi-continuous at $w\in U$.

Proof

Let $\{w_p\}$ in ${C}(\mathrm {J}, U, w)$ be $\mathcal {R}$-nondecreasing sequence. Then $w_p\mathcal {R}w_{p+1}$ and

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(w_p, w)=0. \end{aligned}$$

Since $\vartheta$ is lower semi-continuous at $w$,

$$\begin{aligned} \lim _{p\rightarrow +\infty }\inf \vartheta (w_p)\ge \vartheta (w), \end{aligned}$$

which implies

$$\begin{aligned} \lim _{p\rightarrow +\infty }\inf \mathrm {J}(w_p, {\psi }w_p)\ge \mathrm {J}(w, {\psi }w). \end{aligned}$$

Hence, ${\psi }$ is $\mathcal {R}$-nondecreasing lower semi-continuous at $w$. $\square$

Remark 3.7

If ${\psi }$ is $\mathcal {R}$-nondecreasing lower semi-continuous mapping at $w\in U$, then the function $\vartheta :U\rightarrow \mathbb {R}$, defined by $\vartheta (u)=\mathrm {J}(u, {\psi }u)$, need not be lower semi-continuous at $w$.

Henceforth, let a multivalued mapping ${\psi }:U\rightarrow C(U)$ and ${F}\in \mathcal {F}$. For $u\in U$ with $0<\mathrm {J}(u, {\psi }u)$ and $\varrho >0$, define ${F}_{\varrho }^{u}\subseteq U$ as

$$\begin{aligned} {F}_{\varrho }^{u}=\{v\in {\psi }u: {F}(\mathrm {J}(u,v))\le {F}(\mathrm {J}(u, {\psi }u))+\varrho \}. \end{aligned}$$

(3.4)

Lemma 3.8

Let $(U,\mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and let ${\psi }:U\rightarrow C(U)$ be an $\mathcal {R}$-nondecreasing set-valued mapping. Suppose there exist $u_0\in U$, $u_1\in {\psi }u_0$ such that $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$ and $u_0\mathcal {R}u_1$. If for $u\in U$ there exist $\lambda >0$ and $v\in {F}_{\varrho }^{u}$ with $u\mathcal {R}v$ such that

$$\begin{aligned} 0<\mathrm {J}(v,{\psi }v)<+\infty ~~\hbox {implies}~~\lambda +{F}(\mathrm {J}(v,{\psi }v))\le {F}(\mathrm {J}(u,v)). \end{aligned}$$

(3.5)

Then (3.5) holds for all $u,v\in O_{\psi }(u_0)$.

Proof

Let $O_{\psi }(u_0)=\{u_p: u_p\in {\psi }u_{p-1}\}$ for $u_0\in U$. By hypothesis, there exists $u_1\in {\psi }u_0$ such that $\mathrm {J}(u_0, u_1)<+\infty$ and $u_0\mathcal {R}u_1$. Since ${\psi }$ is $\mathcal {R}$-nondecreasing set-valued mapping, we have $u_1\mathcal {R}u_2$. Continuing in the same manner, we get $u_p\mathcal {R}u_{p+1}$ for all $p\in \mathbb {N}$. Now, since $\mathcal {R}$ is a preorder, we have $u_p\mathcal {R}u_q$ for all $p,q\in \mathbb {N}$ such that $p\le q$. Hence (3.5) holds for all $u_p$ and $u_q$. As p and q were arbitrary, there for (3.5) holds for all $u,v\in O_{\psi }(u_0)$. $\square$

4 Fixed point theorems for ${F}$-contractions

Theorem 4.1

Let $(U,\mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and let ${\psi }:U\rightarrow K(U)$ be $\mathcal {R}$-nondecreasing multivalued mapping. Assume that there exist $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\mathcal {R}u_1$ and $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If for any $u\in O_{\psi }(u_0)$ there exist $\lambda >0$ and $v\in {F}_{\varrho }^{u}$ with $u\mathcal {R}v$ satisfying (3.5) provided that $\varrho <\lambda$, then there exist a sequence $\{u_p\}\subseteq O_{\psi }(u_0)$ such that

$(\mathrm {1})$:

$\delta (\mathrm {J}, {\psi }, u_p)<+\infty$;

$(\mathrm {2})$:

$\mathrm {J}(u_p, u_{p+1})<+\infty$.

Moreover, If constructed sequence $\{u_p\}$ is $\mathrm {J}$-Cauchy, $(U,\mathrm {J})$ is $\mathcal {R}$-nondecreasing-complete and ${\psi }$ is $\mathcal {R}$-nondecreasing lower semi-continuous, then

$(\mathrm {3})$:

$\{u_p\}$ $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$;

$(\mathrm {4})$:

${\psi }$ has a fixed point in $U$.

Proof

Let $u_0\in U$ be such that $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$ and $O_{\psi }(u_0)$ be the orbit of ${\psi }$ based on $u_0$. Then for all $u_l, u_j\in O_{\psi }(u_0)$, we have

$$\begin{aligned} \mathrm {J}(u_l, u_j)\le \delta (\mathrm {J}, {\psi },u_0)<+\infty \end{aligned}$$

(4.1)

and we have the following cases:

Case 1: Let $\delta (\mathrm {J}, {\psi }, u_0)=0$; then for all $u_l, u_j\in O_{\psi }(u_0)$, we get

$$\begin{aligned} 0\le \mathrm {J}(u_l, u_j)\le \delta (\mathrm {J}, {\psi },u_0)=0, \end{aligned}$$

which further gives

$$\begin{aligned} \mathrm {J}(u_l, u_j)=0. \end{aligned}$$

In particular, $\mathrm {J}(u_0, u_1)=0$ and

$$\begin{aligned} 0\le \mathrm {J}(u_0, {\psi }u_0)\le \mathrm {J}(u_0, u_1)=0. \end{aligned}$$

This implies that $\mathrm {J}(u_0, {\psi }u_0)=0$. Since ${\psi }u_0$ is compact and so closed, therefore, by Proposition 2.19, we get $u_0\in {\psi }u_0$, that is, $u_0$ is fixed point of ${\psi }$ and we are done.

Case 2: Let $\delta (\mathrm {J}, {\psi }, u_0)>0$. Now, we assume that $\mathrm {J}(u_l, {\psi }u_l)>0$ for all $u_l\in O_{\psi }(u_0)$. Otherwise, if $\mathrm {J}(u_l, {\psi }u_l)=0$ for some $u_l\in O_{\psi }(u_0)$, then due to closeness of ${\psi }u_l$, we get $u_l$ is the fixed point of ${\psi }$. We also claim that $\mathrm {J}(u_l, {\psi }u_l)<+\infty$ for all $u_l\in O_{\psi }(u_0)$; if not then

$$\begin{aligned} \mathrm {J}(u_l, u_{l+1})\ge \mathrm {J}(u_l, {\psi }u_l)=+\infty , \end{aligned}$$

a contradiction to (4.1). By hypothesis, there exists $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\mathcal {R}u_1$. Since ${\psi }$ is $\mathcal {R}$-nondecreasing set-valued mapping, then as in proof of Lemma 3.8, we get $u_p\mathcal {R}u_{p+1}$ and hence $\{u_p\}\subseteq O_{\psi }(u_0)$ is $\mathcal {R}$-nondecreasing sequence.

Next for $u_0\in U$, there exists $\lambda >0$, $u_1\in {F}_{\varrho }^{u_0}$ such that $u_0\mathcal {R}u_1$ and

$$\begin{aligned} \lambda +{F}(\mathrm {J}(u_1,{\psi }u_1))\le {F}(\mathrm {J}(u_0,u_1)). \end{aligned}$$

(4.2)

Since $u_0\in {F}_{\varrho }^{u_1}$, so

$$\begin{aligned} {F}(\mathrm {J}(u_0,u_1))\le {F}(\mathrm {J}(u_0, {\psi }u_0))+\varrho . \end{aligned}$$

(4.3)

Continuing in the same manner, we can construct a sequence $\{u_p\}_{p\in \mathbb {N}}\subseteq O_{\psi }(u_0)$ such that, for all $l\in \mathbb {N}$, $p\in \mathbb {N} (p\ge 1)$, $u_p\mathcal {R}u_{p+1}$ and

$$\begin{aligned} \lambda +{F}(\mathrm {J}(u_{p+l},{\psi }u_{p+l}))\le {F}(\mathrm {J}(u_{p-1+l},u_{p+l})). \end{aligned}$$

(4.4)

Again since $u_{p+1}\in {F}_{\varrho }^{u_p}$, so

$$\begin{aligned} {F}(\mathrm {J}(u_{p-1+l},u_{p+l}))\le {F}(\mathrm {J}(u_{p-1+l}, {\psi }u_{p-1+l}))+\varrho \end{aligned}$$

(4.5)

for all $l\in \mathbb {N}$, $p\in \mathbb {N} (p\ge 1)$. Combining (4.4) and (4.5), we obtain

$$\begin{aligned} {F}(\mathrm {J}(u_{p+l},{\psi }u_{p+l}))\le {F}(\mathrm {J}(u_{p-1+l},{\psi }u_{p-1+l}))-(\lambda -\varrho ). \end{aligned}$$

(4.6)

Since ${\psi }u_{p-1+l}$ and ${\psi }u_{p+l}$ is compact, there exist $u_{p+l}\in {\psi }u_{p-1+l}$ and $u_{p+1+l}\in {\psi }u_{p+l}$ such that $\mathrm {J}(u_{p-1+l},u_{p+l})=\mathrm {J}(u_{p-1+l},{\psi }u_{p-1+l})$ and $\mathrm {J}(u_{p+l},u_{p+1+l})=\mathrm {J}(u_{p+l},{\psi }u_{p+l})$, so, (4.6) implies

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{p+l},u_{p+1+l}\right) \right) \le {F}\left( \mathrm {J}\left( u_{p-1+l},u_{p+l}\right) \right) -\left( \lambda -\varrho \right) . \end{aligned}$$

(4.7)

Since $\varrho <\lambda$ and due to $({F}1)$, (4.7) gives

$$\begin{aligned} \mathrm {J}\left( u_{p+l},u_{p+(l+1)}\right) \le \mathrm {J}\left( u_{p-1+l},u_{(p-1)+(l+1)}\right) , \end{aligned}$$

(4.8)

which further implies

$$\begin{aligned} \mathrm {J}\left( u_{p+l},u_{p+(l+1)}\right) \le \sup \left\{ \mathrm {J}(u_r, u_s):u_r, u_s\in O_{\psi }(u_{p-1})\right\} =\delta \left( \mathrm {J}, {\psi }, u_{p-1}\right) . \end{aligned}$$

(4.9)

Therefore,

$$\begin{aligned} \delta \left( \mathrm {J}, {\psi }, u_{p}\right) =\sup \left\{ \mathrm {J}\left( u_r, u_s\right) :u_r, u_s\in O_{\psi }(u_{p})\right\} \le \delta \left( \mathrm {J}, {\psi }, u_{p-1}\right) \end{aligned}$$

(4.10)

and hence

$$\begin{aligned} \delta \left( \mathrm {J}, {\psi }, u_{p}\right) \le \delta \left( \mathrm {J}, {\psi }, u_0\right) <+\infty . \end{aligned}$$

(4.11)

From (4.1), we also get

$$\begin{aligned} \mathrm {J}\left( u_p, u_{p+1}\right) <+\infty . \end{aligned}$$

(4.12)

Hence $(\mathrm {1})$ and $(\mathrm {2})$ hold.

Now, if constructed sequence $\{u_p\}$ is Cauchy, then by $\mathcal {R}$-nondecreasing completeness of $(U, \mathrm {J})$, there exists $w\in U$ such that $\{u_p\}\xrightarrow {\mathrm {J}} w$. By using $(\mathrm {J}_3)$, we get

$$\begin{aligned} 0\le \mathrm {J}(w,w)\le \kappa \lim _{p\rightarrow +\infty }\sup \mathrm {J}(u_p, w)=0, \end{aligned}$$

(4.13)

which implies $\mathrm {J}(w,w)=0$. From (4.7), we get

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_p, u_{p+1}\right) \right) \le {F}\left( \mathrm {J}\left( u_0, u_1\right) \right) -p(\lambda -\varrho ). \end{aligned}$$

(4.14)

Since $\varrho <\lambda$, letting $p\rightarrow +\infty$ in (4.14), we get

$$\begin{aligned} \lim _{p\rightarrow +\infty }{F}\left( \mathrm {J}\left( u_p, u_{p+1}\right) \right) =+\infty . \end{aligned}$$

(4.15)

Hence, by $({F}2)$, we get

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}\left( u_p, u_{p+1}\right) =0, \end{aligned}$$

(4.16)

By using (4.16), we obtain

$$\begin{aligned} 0\le \mathrm {J}\left( u_p, {\psi }u_p\right) \le \mathrm {J}\left( u_p,u_{p+1}\right) =0. \end{aligned}$$

(4.17)

Letting $p\rightarrow +\infty$ in (4.17), we have

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}\left( u_p, {\psi }u_p\right) =0. \end{aligned}$$

(4.18)

Since ${\psi }$ is $\mathcal {R}$-nondecreasing lower semi-continuous and $\{u_p\}\subseteq {C}(\mathrm {J}, U, w)$, we have

$$\begin{aligned} 0\le \mathrm {J}\left( w, {\psi }w\right) \le \lim _p\inf \mathrm {J}\left( u_p, {\psi }u_p\right) =0. \end{aligned}$$

(4.19)

Since ${\psi }w\in K(U)\subseteq C(U)$, we get $w\in {\psi }w$. Consequently, ${\psi }$ has a fixed point in $U$. $\square$

Theorem 4.2

Let $(U,\mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and let ${\psi }:U\rightarrow C(U)$ be $\mathcal {R}$-nondecreasing multivalued mapping. Assume that there exist $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\mathcal {R}u_1$ and $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If for any $u\in O_{\psi }(u_0)$ there exist $\lambda >0$ and $v\in {F}_{\varrho }^{u}$ with $u\mathcal {R}v$ satisfying (3.5) for ${F}\in \mathcal {F}*$ provided that $\varrho <\lambda$, then there exists a sequence $\{u_p\}\subseteq O_{\psi }(u_0)$ such that $(\mathrm {1})$ and $(\mathrm {2})$ hold. Moreover, If constructed sequence $\{u_p\}$ is $\mathrm {J}$-Cauchy, $(U,\mathrm {J})$ is $\mathcal {R}$-nondecreasing-complete and ${\psi }$ is $\mathcal {R}$-nondecreasing lower semi-continuous, then $\{u_p\}$ is $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$ and ${\psi }$ has a fixed point in $U$.

Proof

Let $u_0\in U$ be such that $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$ and $O_{\psi }(u_0)$ be the orbit of ${\psi }$ based on $u_0$. Then as in the proof of Theorem 4.1 for all $u_l, u_j\in O_{\psi }(u_0)$, we have

$$\begin{aligned} \mathrm {J}\left( u_l, u_j\right) <+\infty \end{aligned}$$

(4.20)

and we have the following cases:

Case 1: Let $\delta (\mathrm {J}, {\psi }, u_0)=0$; then for all $u_l, u_j\in O_{\psi }(u_0)$, we get $u_0$ is fixed point of ${\psi }$.

Case 2: Let $\delta (\mathrm {J}, {\psi }, u_0)>0$. Now, we assume that $\mathrm {J}(u_l, {\psi }u_l)>0$ for all $u_l\in O_{\psi }(u_0)$. Otherwise, if $\mathrm {J}(u_l, {\psi }u_l)=0$ for some $u_l\in O_{\psi }(u_0)$, then due to closeness of ${\psi }u_l$, we get $u_l$ is the fixed point of ${\psi }$. We also claim that $\mathrm {J}(u_l, {\psi }u_l)<+\infty$ for all $u_l\in O_{\psi }(u_0)$; if not, then

$$\begin{aligned} \mathrm {J}(u_l, u_{l+1})\ge \mathrm {J}(u_l, {\psi }u_l)=+\infty , \end{aligned}$$

a contradiction to (4.20). By hypothesis, there exist $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\mathcal {R}u_1$. Since ${\psi }$ is $\mathcal {R}$-nondecreasing multivalued mapping, then as in proof of Lemma 3.8, we get $u_p\mathcal {R}u_{p+1}$ and hence $\{u_p\}\subseteq O_{\psi }(u_0)$ is $\mathcal {R}$-nondecreasing sequence.

Next for $u_0\in U$, there exist $\lambda >0$, $u_1\in {F}_{\varrho }^{u_0}$ such that $u_0\mathcal {R}u_1$ and

$$\begin{aligned} \lambda +{F}\left( \mathrm {J}\left( u_1,{\psi }u_1\right) \right) \le {F}\left( \mathrm {J}\left( u_0,u_1\right) \right) . \end{aligned}$$

(4.21)

Since $u_1\in {F}_{\varrho }^{u_0}$, so

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_0,u_1\right) \right) \le \mathrm {J}\left( u_0, {\psi }u_0\right) +\varrho . \end{aligned}$$

(4.22)

Continuing in this manner, we can construct a sequence $\{u_p\}$ in $U$ such that, for all $l\in \mathbb {N}$, $p\in \mathbb {N} (p\ge 1)$

$$\begin{aligned} \lambda +{F}\left( \mathrm {J}\left( u_{p+l},{\psi }u_{p+l}\right) \right) \le {F}\left( \mathrm {J}\left( u_{p-1+l},u_{p+l}\right) \right) . \end{aligned}$$

(4.23)

Again since $u_{p+1}\in {F}_{\varrho }^{u_p}$,

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{p-1+l},u_{p+l}\right) \right) \le {F}\mathrm {J}\left( u_{p-1+l}, {\psi }u_{p-1+l}\right) )+\varrho \end{aligned}$$

(4.24)

for all $l\in \mathbb {N}$, $p\in \mathbb {N} (p\ge 1)$. Combining (4.4) and (4.24), we obtain

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{p+l},{\psi }u_{p+l}\right) \right) \le {F}\left( \mathrm {J}\left( u_{p-1+l},{\psi }u_{p-1+l}\right) \right) +\varrho -\lambda . \end{aligned}$$

(4.25)

Due to $({F}4)$, we obtain

$$\begin{aligned} \inf _{v\in {\psi }u_{p+l}}{F}\left( \mathrm {J}\left( u_{p+l},v\right) \right) \le {F}\left( \mathrm {J}\left( u_{p-1+l},u_{p+l}\right) \right) +\varrho -\lambda . \end{aligned}$$

(4.26)

Thus, there exist $u_{p+1+l}\in {\psi }u_{p+l}$ such that

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{p+l},u_{p+1+l}\right) \right) \le {F}\left( \mathrm {J}\left( u_{p-1+l},u_{p+l}\right) \right) +\varrho -\lambda . \end{aligned}$$

(4.27)

Next, by using the same steps as in the proof of Theorem 4.1, we get the results. $\square$

Remark 4.3

Note that in Theorems 4.1 and 4.2, we have the following advantages:

• $\mathcal {R}$ need not to be a partial order.

• The contractive condition (3.5) need not to satisfy for all $u,v\in U$.

• The GMS(JS) need not be complete.

• The mapping ${\psi }$ need not be $\mathcal {R}$-nondecreasing continuous.

• ${F}:(0,+\infty )\rightarrow \mathbb {R}$ need not to satisfy $({F}3)$.

Definition 4.4

Let $U$ be a nonempty set and ${\psi }:U\rightarrow P(U)$ be a multivalued mapping.Then an element $u_0\in U$ is said to be a periodic point of ${\psi }$ if $u_0\in {\psi }u_{p}$, for some $u_p\in O_{\psi }(u_0)$.

Example 3

Let $U=\{0,1,2\}$ and define ${\psi }:U\rightarrow P(U)$ by

$$\begin{aligned} {\psi }0=\{2\},\quad {\psi }1=\{0,2\},\quad {\psi }2=\{1\}. \end{aligned}$$

Then $O_{\psi }(1)=\{0,1,2\}$. Note that $1\notin {\psi }1$ but $1\in {\psi }2$, where $2\in O_{\psi }(1)$. Hence 1 is a periodic point of ${\psi }$ but 1 is not a fixed point of ${\psi }$.

Theorem 4.5

Let $(U,\mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and let ${\psi }:U\rightarrow C(U)$ be $\mathcal {R}$-nondecreasing multivalued mapping. Assume that there exists $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\mathcal {R}u_1$ and $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If there exist a function ${F}\in \mathcal {F}$ and $\lambda >0$ satisfying the following:

$({F}5)$:

for any ${x},{y}\in O_{\psi }(u_0)$ and $u\in {\psi }{x}$ there exist $v\in {\psi }{y}$ such that $u=v$ or

$$\begin{aligned} \mathrm {J}(u,v)<+\infty ~~\hbox {implies}~~\lambda +{F}(\mathrm {J}(u,v))\le {F}(\mathrm {J}({x},{y})), \end{aligned}$$

then there exist a sequence $\{u_p\}\subseteq O_{\psi }(u_0)$ such that $\{u_p\}$ $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$. Moreover, if $(U,\mathrm {J})$ is $\mathcal {R}$-nondecreasing-complete, then ${\psi }$ has a fixed point or periodic point in $U$.

Proof

Let $u_0\in U$ be such that $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$ and $O_{\psi }(u_0)$ be the orbit of ${\psi }$ based on $u_0$. Then as in the proof of Theorem 4.1 for all $u_l, u_j\in O_{\psi }(u_0)$, we have

$$\begin{aligned} \mathrm {J}(u_l, u_j)<+\infty \end{aligned}$$

(4.28)

and we have the following cases:

Case 1: Let $\delta (\mathrm {J}, {\psi }, u_0)=0$; then for all $u_l, u_j\in O_{\psi }(u_0)$, we get $u_0$ is fixed point of ${\psi }$.

Case 2: Let $\delta (\mathrm {J}, {\psi }, u_0)>0$. We suppose that ${\psi }$ has no periodic point. By hypothesis, there exists $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\mathcal {R}u_1$ and $u_0\ne u_1$. We claim that $u_l=u_{j}$, for all $j>l$, if $u_l=u_{j}$, for some $j>l$, then $u_l$ is periodic point of ${\psi }$. Since $u_1\notin {\psi }u_1$, otherwise $u_1$ is fixed point of ${\psi }$, from $({F}5)$, there exists $u_2\in {\psi }u_1$ such that

$$\begin{aligned} \lambda +{F}(\mathrm {J}(u_1,u_2))\le {F}(\mathrm {J}(u_0,u_1)). \end{aligned}$$

(4.29)

Since ${\psi }$ is $\mathcal {R}$-nondecreasing multivalued mapping and $u_3\notin {\psi }u_3$, therefore, $u_1\mathcal {R}u_2$ and there exist $u_3\in {\psi }u_2$ such that

$$\begin{aligned} \lambda +{F}\left( \mathrm {J}\left( u_2,u_3\right) \right) \le {F}\left( \mathrm {J}\left( u_1,u_2\right) \right) . \end{aligned}$$

(4.30)

Continuing in the same manner, we can choose a sequence $\{u_p\}$ satisfying $u_{p+1}\in {\psi }u_p$, $u_p\mathcal {R}u_{p+1}$ for all $l\in \mathbb {N}$, $p\in \mathbb {N} (p\ge 1)$, we have

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{p+l},u_{p+l+1}\right) \right) \le {F}\left( \mathrm {J}\left( u_{p-1+l},u_{p+l}\right) \right) -\lambda , \end{aligned}$$

(4.31)

which implies

$$\begin{aligned} 0\le \mathrm {J}\left( u_{p+l},u_{p+l+1}\right) < \mathrm {J}\left( u_{p-1+l},u_{p+l}\right) . \end{aligned}$$

(4.32)

Consequently,

$$\begin{aligned} \mathrm {J}\left( u_{p+l},u_{p+l+1}\right) < \delta \left( \mathrm {J}, {\psi }, u_p\right) \le \delta \left( \mathrm {J}, {\psi }, u_0\right) . \end{aligned}$$

(4.33)

Thus, $\{\mathrm {J}(u_{p+l},u_{p+l+1})\}$ is a decreasing bounded sequence and so converges; thus there exists $c\ge 0$ such that

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}\left( u_{p+l},u_{p+l+1}\right) =c. \end{aligned}$$

(4.34)

We claim that $c=0$; if not, then from (4.31), we get

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{p+l},u_{p+l+1}\right) \right) \le {F}\left( \mathrm {J}\left( u_{l},u_{l+1}\right) \right) -p\lambda . \end{aligned}$$

(4.35)

Letting $p\rightarrow +\infty$, we obtain

$$\begin{aligned} \lim _{p\rightarrow +\infty }{F}\mathrm {J}\left( u_{p+l},u_{p+l+1}\right) )=+\infty . \end{aligned}$$

(4.36)

From $({F}2)$ and (4.36), we have

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}\left( u_{p+l},u_{p+l+1}\right) =0, \end{aligned}$$

(4.37)

which is a contradiction. Now, in order to show that $\{u_p\}\subseteq O_{\psi }(u_0)$ is Cauchy, consider $q,p\in \mathbb {N}$ such that $q>p\ge p_1$ for some $p_1\in N$. We claim that $u_p\ne u_q$; otherwise, $u_p\in {\psi }u_{q-1}$ and so $u_p$ is periodic point of ${\psi }$. Since $u_{p-1}\ne u_{q-1}$ and $\mathrm {J}(u_{p-1}, u_{q-1})<+\infty$, so from $({F}5)$, for $u_{p-1}, u_{q-1}\in O_{\psi }(u_0)$ and $u_{p}\in {\psi }u_{p-1}$, there exists $u_{q}\in {\psi }u_{q-1}$ satisfying

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{q},u_{p}\right) \right) \le {F}\left( \mathrm {J}\left( u_{q-1},u_{p-1}\right) \right) -\lambda . \end{aligned}$$

(4.38)

which further implies

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{q},u_{p}\right) \right) \le {F}\left( \mathrm {J}\left( u_{q-p},u_{0}\right) \right) -p\lambda . \end{aligned}$$

(4.39)

Letting $p\rightarrow +\infty$ in (4.39), we obtain

$$\begin{aligned} \lim _{p,q\rightarrow +\infty }{F}\left( \mathrm {J}\left( u_{q},u_{p}\right) \right) =+\infty \end{aligned}$$

(4.40)

and further by using $({F}2)$, we have

$$\begin{aligned} \lim _{p,q\rightarrow +\infty }\mathrm {J}\left( u_{q},u_{p}\right) =0. \end{aligned}$$

(4.41)

Hence $\{u_p\}\subseteq O_{\psi }(u_0)$ is Cauchy sequence; then by $\mathcal {R}$-nondecreasing completeness of $(U, \mathrm {J})$, there exists $w\in U$ such that $\{u_p\}\xrightarrow {\mathrm {J}} w$. By using $(\mathrm {J}_3)$, we get

$$\begin{aligned} 0\le \mathrm {J}(w,w)\le \kappa \lim _{p\rightarrow +\infty }\sup \mathrm {J}\left( u_p, w\right) =0, \end{aligned}$$

(4.42)

which implies $\mathrm {J}(w,w)=0$. For $p\in \mathbb {N}$, there exists $w_p\in {\psi }w$ such that either $w_p=u_{p+1}$ or

$$\begin{aligned} \lambda +{F}\left( \mathrm {J}\left( u_{p+1}, w_p\right) \right) \le {F}\left( \mathrm {J}\left( u_p, w\right) \right) . \end{aligned}$$

(4.43)

holds. We claim that $w_p\ne u_{p+1}$; otherwise, if $w_p=u_{p+1}$, then $w_p\in {\psi }u_p$. Now, since ${\psi }u_p$ is closed and $\{u_p\}\xrightarrow {\mathrm {J}} w$, so, $w\in {\psi }u_p$. Consequently, $w$ is periodic point of ${\psi }$. Hence, from (4.434.44), we get

$$\begin{aligned} {F}\left( \mathrm {J}\left( u_{p+1}, w_p\right) \right) \le {F}\left( \mathrm {J}\left( u_p, w\right) \right) -\lambda . \end{aligned}$$

(4.44)

Since $\{u_p\}\xrightarrow {\mathrm {J}} w$, by letting $p\rightarrow +\infty$ in (4.434.44) and using $({F}2)$, we obtain

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}\left( u_{p+1}, w_p\right) =0. \end{aligned}$$

(4.45)

By using $(\mathrm {J}_3)$, we get

$$\begin{aligned} 0\le \mathrm {J}\left( w_p,w\right) \le \kappa \lim _{p\rightarrow +\infty }\sup \mathrm {J}\left( u_{p+1}, w\right) =0, \end{aligned}$$

(4.46)

which implies

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(w_p,w)=0. \end{aligned}$$

Since ${\psi }w$ is closed, we have $w\in {\psi }w$. Hence ${\psi }$ has a fixed point. $\square$

Example 4

Let $U=[0,1]\cup \{2\}$ and let $\mathrm {J}:U\times U$ be a function defined by

$$\begin{aligned} \mathrm {J}({x},{y})=\left\{ \begin{array}{ll} 10 &{} \hbox {if either}~({x},{y})=(0,2)~\hbox {or}~({x},{y})=(2,0), \\ |{x}-{y}| &{} \hbox {otherwise}. \end{array} \right. \end{aligned}$$

Then $(U,\mathrm {J})$ is complete GMS(JS) (see [21]). Define a binary relation $\mathcal {R}$ on $U$ by

$$\begin{aligned} {x}\mathcal {R}{y}~~~\hbox {if}~~~0<{x}\le {y}\le 1; \end{aligned}$$

then $\mathcal {R}$ is a preorder and $(U,\mathcal {R})$ is a preordered space. Define ${\psi }:U\rightarrow C(U)$ and ${F}:(0,+\infty )\rightarrow \mathbb {R}$ by

$$\begin{aligned} {\psi }{x}=\left\{ \begin{array}{ll} \{0.25, 0.5\} &{} \hbox {if}~{x}=1, \\ \{1\} &{} \hbox {if}~{x}\in \{0.25,0.5\},\\ {[0, 1]} &{} \hbox {otherwise}, \end{array} \right. \end{aligned}$$

and ${F}(r)=\ln (r)$ for all $r\in (0,+\infty )$,, respectively. Then $O_{\psi }(u_0)=\{0.25,1,0.25,1,...\}$, ${F}\in \mathcal {F}$ and ${\psi }$ is multivalued $\mathcal {R}$-nondecreasing mapping. There exist $u_0=0.25\in U$ and $u_1=1\in {\psi }u_0$ such that $u_0\mathcal {R}u_1$ and

$$\begin{aligned} \delta (\mathrm {J}, {\psi }, u_0)=|1-0.25|=0.75<+\infty . \end{aligned}$$

Also, for $x,y\in O_{\psi }(u_0)$, that is, ${x}=1$, ${y}=0.25$ and $u=0.5\in {\psi }1$, there exist $v=1\in {\psi }(0.25)$ and $\tau =0.25>0$ such that

$$\begin{aligned} \mathrm {J}(u,v)=\mathrm {J}(0.5,1)=|0.5-1|=0.5<+\infty \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {F}(\mathrm {J}(u,v))-{F}(\mathrm {J}({x},{y}))&={F}(0.5)-{F}(0.75)\\&=\ln (0.5)-\ln (0.75)\\&=-0.4055\\&<-0.25=-\tau . \end{aligned} \end{aligned}$$

Thus, all hypotheses of Theorem 4.5 hold true and $U-\{0.25,0.5,1,2\}$ is the set of all fixed points of ${\psi }$.

Corollary 4.6

Let $(U,\mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$ and let ${\psi }:U\rightarrow C(U)$ be $\mathcal {R}$-nondecreasing multivalued mapping. Assume that there exists $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\mathcal {R}u_1$ and $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If $(U,\mathrm {J})$ is $\mathcal {R}$-nondecreasing-complete and for any $u\in O_{\psi }(u_0)$ there exist $\eta >0$, $\ell \in (0,1)$ and $v\in I_{b}^{u}$ with $u\mathcal {R}v$ satisfying (2.8) and (2.9). Then there exists a sequence $\{u_p\}\subseteq O_{\psi }(u_0)$ $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$. Moreover, If constructed sequence $\{u_p\}$ is $\mathrm {J}$-Cauchy, ${\psi }$ is $\mathcal {R}$-nondecreasing lower semi-continuous and $c<b$, then $w$ is a fixed point of ${\psi }$.

Proof

Define ${F}:(0,+\infty )\rightarrow \mathbb {R}$ by ${F}(r)=\ln r$, for all $r\in (0,+\infty )$. Put $\ell =\frac{1}{\varrho }$ and $\eta =\frac{1}{\lambda }$, where $\varrho ,\lambda >0$, then $\varrho <\lambda$, (2.9) implies (3.4) and (2.8) implies (3.5). Hence all conditions of Theorem 4.2 are satisfied and the proof is complete.

Remark 4.7

In the light of Remark 2.14, from Corollary 4.6 and Lemma 3.6, we obtain Theorem 2.18.

Corollary 4.8

Let $(U,\mathrm {J})$ be a $\ll$-nondecreasing complete GMS(JS) equipped with a partial order $\ll$ and ${\psi }:U\rightarrow K(U)$ be a $\ll$-nondecreasing multivalued mapping. Assume that there exists $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\ll u_1$ and $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If $(U,\mathrm {J})$ is $\ll$-nondecreasing-complete and for any $u\in O_{\psi }(u_0)$ there exist $\lambda >0$ and $v\in {F}_{\varrho }^{u}$ with $u\ll v$ satisfying (3.5) provided that $\varrho <\lambda$, then there exists a sequence $\{u_p\}\subseteq O_{\psi }(u_0)$ $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$.

Moreover, If ${\psi }$ is $\mathcal {R}$-nondecreasing continuous, then ${\psi }$ has a fixed point in $U$.

Proof

Since a partial order $\ll$ is a preorder $\mathcal {R}$, by using Theorem 4.1 we get the result. $\square$

Corollary 4.9

Let $(U,\mathrm {J})$ be a $\ll$-nondecreasing complete GMS(JS) endowed with partial order $\ll$ and let ${\psi }:U\rightarrow C(U)$ be an $\ll$-nondecreasing multivalued mapping. Assume that there exists $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\ll u_1$ and $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If $(U,\mathrm {J})$ is $\ll$-nondecreasing-complete and for $u\in U$ there exist $\lambda >0$ and $v\in {F}_{\varrho }^{u}$ with $u\mathcal {R}v$ satisfying (3.5) for ${F}\in \mathcal {F}*$, provided that $\varrho <\lambda$, then there exists a sequence $\{u_p\}\subseteq O_{\psi }(u_0)$ $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$.

Moreover, If ${\psi }$ is $\ll$-nondecreasing continuous, then ${\psi }$ has a fixed point in $U$.

Proof

Since a partial order $\ll$ is a preorder $\mathcal {R}$, by using Theorem 4.2 we get the result. $\square$

Corollary 4.10

Let $(U,\mathrm {J})$ be a $\ll$-nondecreasing complete GMS(JS) with respect to partial order $\ll$ and ${\psi }:U\rightarrow C(U)$ be a $\ll$-nondecreasing multivalued mapping. Assume that there exist $u_0\in U$ and $u_1\in {\psi }u_0$ such that $u_0\ll u_1$ and $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If there exists a function ${F}\in \mathcal {F}$ and $\lambda >0$ satisfying the $({F}5)$, Then there exists a sequence $\{u_p\}\subseteq O_{\psi }(u_0)$ such that $\{u_p\}$ $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$ and ${\psi }$ has a fixed point or periodic point in $U$.

Corollary 4.11

Let $(U,\mathrm {J})$ be a complete GMS(JS) and ${\psi }:U\rightarrow K(U)$ be a multivalued mapping. Assume that there exists $u_0\in U$ such that $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If for any $u,v\in U$ satisfying $\frac{1}{2}\mathrm {J}(u, {\psi }u)\le \mathrm {J}(u,v)$, there exist $\lambda >0$ and ${F}\in \mathcal {F}$ such that

$$\begin{aligned} 0\le D\left( v, {\psi }v\right) <+\infty ~~\hbox {implies}~~\lambda +{F}\left( \mathrm {J}\left( v,{\psi }v\right) \right) \le {F}\left( \mathrm {J}\left( u,v\right) ,\right) . \end{aligned}$$

(4.47)

then there exists a sequence, $\{u_p\}\subseteq U$, $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$. Moreover, if function $u\rightarrow \mathrm {J}(u, {\psi }u)$ is lower semi-continuous, then $w$ is the fixed point of ${\psi }$ in $U$.

Proof

Define a preorder $\mathcal {R}$ on $U$ by $\mathcal {R}_U$. Let $u\in U$ with $0<\mathrm {J}(u,{\psi }u)$ and $v\in {F}_{\varrho }^{u}$ with $\varrho <\lambda$ and $u\mathcal {R}v$. Then $v\in {\psi }u$ and

$$\begin{aligned} \frac{1}{2}\mathrm {J}(u, {\psi }u)\le \mathrm {J}(u, {\psi }u)\le \mathrm {J}(u,v). \end{aligned}$$

So, (4.47) together with Lemma 3.8 implies that (3.5) holds. Also, from Lemma 3.6, we have ${\psi }$ is $\mathcal {R}$-nondecreasing continuous mapping. Hence by using Theorem 4.1, we get the result and proof is complete. $\square$

Corollary 4.12

Let $(U,\mathrm {J})$ be a complete GMS(JS) and ${\psi }:U\rightarrow C(U)$ be a multivalued mapping. Assume that there exist $u_0\in U$ such that $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If for any $u,v\in U$ satisfying $\frac{1}{2}\mathrm {J}(u, {\psi }u)\le \mathrm {J}(u,v)$, there exist $\lambda >0$ and ${F}\in \mathcal {F}*$ such that (4.47) holds. Then there exists a sequence, $\{u_p\}\subseteq U$, $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$. Moreover, if function $u\longmapsto \mathrm {J}(u, {\psi }u)$ is lower semi-continuous, then $w$ is the fixed point of ${\psi }$ in $U$.

In the light of Remark 2.14, from Theorem 4.5, we obtain the following:

Corollary 4.13

Let $(U,\mathrm {J})$ be a complete GMS(JS) and ${\psi }:U\rightarrow C(U)$ be a multivalued mapping. Assume that there exists $u_0\in U$ and $u_1\in {\psi }u_0$ such that $\delta (\mathrm {J}, {\psi }, u_0)<+\infty$. If there exists a function ${F}\in \mathcal {F}$ and $\lambda >0$ satisfying the following:

$({F}5)$:

for any ${x},{y}\in U$ and $u\in {\psi }{x}$ there exist $v\in {\psi }{y}$ such that $u=v$ or

$$\begin{aligned} \mathrm {J}(u,v)<+\infty ~~\hbox {implies}~~\lambda +{F}(\mathrm {J}(u,v))\le {F}(\mathrm {J}({x},{y})), \end{aligned}$$

then there exists a sequence $\{u_p\}$ in $U$ such that $\{u_p\}$ $\mathrm {J}$-converges to a point $w\in U$ satisfying $\mathrm {J}(w,w)=0$ and ${\psi }$ has a fixed point or periodic point in $U$.

5 Coincidence point theorems for ${F}$-contractions

In this section, we prove the existence of coincidence point for ${F}$-type contraction.

Theorem 5.1

Let $(U,\mathrm {J})$ be a GMS(JS) equipped with a preorder $\mathcal {R}$, $\xi :U\rightarrow U$ and ${\psi }:U\rightarrow C(U)$. Assume that there exists $u_0, u_1\in U$ such that $\xi u_1\in {\psi }u_0$, $\xi u_0\mathcal {R}\xi u_1$, ${\psi }$ is an $(\xi , \mathcal {R})$-nondecreasing set-valued mapping and $\sup \{\mathrm {J}(\xi u_l, \xi u_j): \xi u_l\in {\psi }u_{l-1},~\xi u_j\in {\psi }u_{j-1}\}<+\infty$. If there exists a function ${F}\in \mathcal {F}$ and $\lambda >0$ satisfying

$$\begin{aligned} 0<\mathrm {J}(u,v)<+\infty ~~\hbox {implies}~~\lambda +{F}(\mathrm {J}(u,v))\le {F}(\mathrm {J}(\xi {x}, \xi {y})) \end{aligned}$$

(5.1)

for all ${x},{y}\in U$ with ${x}\mathcal {R}{y}$ and $u\in {\psi }{x}$, $v\in {\psi }{y}$, then there exists a sequence $\{\xi u_p: \xi u_p\in {\psi }u_{p-1}\}_{P\in \mathbb {N}}$ such that

$(\mathrm {1})$:

$\lim _{p\rightarrow +\infty }\mathrm {J}(\xi u_p, \xi u_{p+1})=0$;

$(\mathrm {2})$:

If $\xi u_l\notin {\psi }u_j$ for all $l>j$, then $\{\xi u_p\}$ is $\mathrm {J}$-Cauchy.

Moreover, if for all $\{w_p\}\subseteq \xi (U)$ we have $\{w_p\}\rightarrow w~~\hbox {implies}~~ w_p\mathcal {R}w$ for all $p\in \mathbb {N}$ and $\xi (U)$ is $\mathcal {R}$-nondecreasing-complete, then $\xi$ and ${\psi }$ has a coincidence point in $U$.

Proof

By hypothesis, there exists $u_0, u_1\in U$ such that $\xi u_1\in {\psi }u_0$ and $\xi u_0\mathcal {R}u_1$. Construct the sequence $\{\xi u_p : \xi u_p\in {\psi }u_{p-1}\}$. Since $\sup \{\mathrm {J}(\xi u_l, \xi u_j): \xi u_l\in {\psi }u_{l-1},~\xi u_j\in {\psi }u_{j-1}\}<+\infty$, we have

$$\begin{aligned} \mathrm {J}(\xi u_l, \xi u_j)<+\infty , \end{aligned}$$

(5.2)

for all $\xi u_l, \xi u_j\subseteq \{\xi u_p\}$. Here arises two cases:

Case 1: If $\sup \{\mathrm {J}(\xi u_l, \xi u_j): \xi u_l\in {\psi }u_{l-1},~\xi u_j\in {\psi }u_{j-1}\}=0$, then for all $\xi u_l, \xi u_j\in \{\xi u_p\}$, we get

$$\begin{aligned} 0\le \mathrm {J}(\xi u_l, \xi u_j)\le \sup \{\mathrm {J}(\xi u_l, \xi u_j): \xi u_l\in {\psi }u_{l-1},~\xi u_j\in {\psi }u_{j-1}\}=0, \end{aligned}$$

which further gives

$$\begin{aligned} \mathrm {J}(\xi u_l, \xi u_j)=0. \end{aligned}$$

In particular,

$$\begin{aligned} 0\le \mathrm {J}(\xi u_0, {\psi }u_0)\le \mathrm {J}(\xi u_0, \xi u_1)=0. \end{aligned}$$

This implies that $\mathrm {J}(\xi u_0, {\psi }u_0)=0$. Since ${\psi }u_0$ is closed, we get $\xi u_0\in {\psi }u_0$, that is, $u_0$ is coincidence point of $\xi$ and ${\psi }$.

Case 2: Let $\sup \{\mathrm {J}(\xi u_l, \xi u_j): \xi u_l\in {\psi }u_{l-1},~\xi u_j\in {\psi }u_{j-1}\}>0$. Assume that $\mathrm {J}(\xi u_1, \xi u_2)>0$, where $\xi u_1, \xi u_2\in \{\xi u_p\}$ otherwise if $\mathrm {J}(\xi u_1, \xi u_2)=0$, then

$$\begin{aligned} 0\le \mathrm {J}(\xi u_1, {\psi }u_1)\le \mathrm {J}(\xi u_1, \xi u_2)=0. \end{aligned}$$

This gives $\mathrm {J}(\xi u_1, {\psi }u_1)=0$; since ${\psi }u_1$ is closed, so, $\xi u_1\in {\psi }u_1$. Since ${\psi }$ is $(\xi , \mathcal {R})$-nondecreasing set-valued mapping, $\xi u_1\mathcal {R}\xi u_2$. Hence from (5.1), we get

$$\begin{aligned} \lambda +{F}(\mathrm {J}(\xi u_1, \xi u_2))\le {F}(\mathrm {J}(\xi u_0, \xi u_1)). \end{aligned}$$

By induction, we have $\{\xi u_p\}_p\in \mathbb {N}$ satisfying $\xi u_p\in {\psi }u_{p-1}$, $\xi u_p\mathcal {R}\xi u_{p+1}$, $\mathrm {J}(\xi u_p, \xi u_{p+1})>0$ and

$$\begin{aligned} \lambda +{F}(\mathrm {J}(\xi u_p, \xi u_{p+1}))\le {F}(\mathrm {J}(\xi u_{p-1}, \xi u_p)), \end{aligned}$$

(5.3)

for all $p\in \mathbb {N}\setminus \{0\}$. This implies

$$\begin{aligned} 0<\mathrm {J}(\xi u_p, \xi u_{p+1})< \mathrm {J}(\xi u_{p-1}, \xi u_p). \end{aligned}$$

(5.4)

Thus, $\{\mathrm {J}(\xi u_{p},\xi u_{p+1})\}$ is a decreasing bounded sequence and so converges; thus there exists $c\ge 0$ such that

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(\xi u_{p},\xi u_{p+1})=c. \end{aligned}$$

(5.5)

We claim that $c=0$; if not, then from (5.3), we get

$$\begin{aligned} {F}(\mathrm {J}(\xi u_{p},\xi u_{p+1}))\le {F}(\mathrm {J}(\xi u_{0},\xi u_{1}))-p\lambda . \end{aligned}$$

(5.6)

Letting $p\rightarrow +\infty$, we obtain

$$\begin{aligned} \lim _{p\rightarrow +\infty }{F}(\mathrm {J}(\xi u_{p},\xi u_{p+1}))=+\infty . \end{aligned}$$

(5.7)

From $({F}2)$, we have

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(\xi u_{p},\xi u_{p+1})=0, \end{aligned}$$

(5.8)

which is a contradiction. This proves $(\mathrm {1})$.

Next, we show that $\{\xi u_p\}$ is $\mathrm {J}$-Cauchy sequence. Consider $q>p>p_1$ for some $p_1\in \mathbb {N}$. We claim that $\mathrm {J}(\xi u_{p},\xi u_{q})>0$. If not, then $\xi u_{q}=\xi u_{p}$. This gives $\xi u_{q}\in {\psi }u_{p-1}$, which is contradiction to the fact that $gu_l\notin Tu_j$ for all $l>j$. Further, since $\mathcal {R}$ is pre-order, by transitivity, we have $\xi u_{q}\mathcal {R}\xi u_{p}$ for all $p,q\in \mathbb {N}$, $p\le q$. Then from (5.1), we obtain

$$\begin{aligned} \lambda +{F}(\mathrm {J}(\xi u_q, \xi u_p))\le {F}(\mathrm {J}(\xi u_{q-1}, \xi u_{p-1})). \end{aligned}$$

(5.9)

which implies

$$\begin{aligned} {F}(\mathrm {J}(\xi u_q, \xi u_p))\le {F}(\mathrm {J}(\xi u_{q-p},\xi u_{0}))-p\lambda . \end{aligned}$$

(5.10)

Letting $p\rightarrow +\infty$ in (5.10), we obtain

$$\begin{aligned} \lim _{p,q\rightarrow +\infty }{F}(\mathrm {J}(\xi u_q, \xi u_p))=+\infty \end{aligned}$$

(5.11)

and further by using $({F}2)$, we have

$$\begin{aligned} \lim _{p,q\rightarrow +\infty }\mathrm {J}(\xi u_q, \xi u_p)=0. \end{aligned}$$

(5.12)

Hence $\{\xi u_p\}$ is $\mathrm {J}$-Cauchy sequence. This proves $(\mathrm {2})$. Since $\xi (U)$ is $\mathcal {R}$-nondecreasing complete, there exists a point $\xi z\in \xi (U)$ such that $\{\xi u_p\}\xrightarrow {\mathrm {J}} \xi z$. Also, by hypothesis, $\xi u_p\mathcal {R}\xi z$; then from (5.1), for $p\in \mathbb {N}$, there exists $\xi z_p\in {\psi }z$ satisfying

$$\begin{aligned} \lambda +{F}(\mathrm {J}(\xi u_{q+1}, \xi z_p))\le {F}(\mathrm {J}(\xi u_{p}, \xi z)). \end{aligned}$$

(5.13)

Since $\{\xi u_p\}\xrightarrow {\mathrm {J}} \xi z$, by letting $p\rightarrow +\infty$ in (5.13) and using $({F}2)$, we obtain

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(\xi u_{p+1}, \xi z_p)=0. \end{aligned}$$

(5.14)

By using $(\mathrm {J}_3)$, we get

$$\begin{aligned} 0\le \mathrm {J}(\xi z_p,\xi z)\le \kappa \lim _{p\rightarrow +\infty }\sup \mathrm {J}(\xi u_{p+1}, \xi z)=0, \end{aligned}$$

(5.15)

which implies

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(\xi z_p,\xi z)=0. \end{aligned}$$

Since ${\psi }z$ is closed, we have $\xi z\in {\psi }z$. Hence $z$ is the coincident point of ${\psi }$ and $\xi$. $\square$

In the light of Remark 2.14, from Theorem 5.1, we obtain the following:

Corollary 5.2

Let $(U,\mathrm {J})$ be a GMS(JS), $\xi :U\rightarrow U$ and ${\psi }:U\rightarrow C(U)$. Assume that there exists $u_0, u_1\in U$ such that $\xi u_1\in {\psi }u_0$ and $\sup \{\mathrm {J}(\xi u_l, \xi u_j): \xi u_l\in {\psi }u_{l-1},~\xi u_j\in {\psi }u_{j-1}\}<+\infty$. If there exists a function ${F}\in \mathcal {F}$ and $\lambda >0$ satisfying (5.1) for all ${x},{y}\in U$ and $u\in {\psi }{x}$, $v\in {\psi }{y}$, then there exists a sequence $\{\xi u_p: \xi u_p\in {\psi }u_{p-1}\}_{P\in \mathbb {N}}$ such that

$(\mathrm {1})$:

$\lim _{p\rightarrow +\infty }\mathrm {J}(\xi u_p, \xi u_{p+1})=0$;

$(\mathrm {2})$:

if $\xi u_l\notin {\psi }u_j$ for all $l>j$, then $\{\xi u_p\}$ is $\mathrm {J}$-Cauchy.

Moreover, if $\xi (U)$ is complete, then $\xi$ and ${\psi }$ have a coincidence point in $U$.

6 Common fixed point theorems for ${F}$-contractions

Theorem 6.1

Let $(U,\mathrm {J})$ be a complete GMS(JS) and ${\psi }, \psi _{1}:U\rightarrow C(U)$. Assume that there exists $u_0, u_1, u_2\in U$ such that $u_1\in {\psi }u_0$, $u_2\in \psi _{1}u_1$, and $\sup \{\mathrm {J}(u_{2l+1}, u_{2j+2}): u_{2l+1}\in {\psi }u_{2l},u_{2j+2}\in \psi _{1}u_{2j+1}\}<+\infty$. If there exists a function ${F}\in \mathcal {F}$ and $\lambda >0$ satisfying

$$\begin{aligned} 0<\mathrm {J}(u,v)<+\infty ~~\hbox {implies}~~\lambda +{F}(\mathrm {J}(u,v))\le {F}(\mathrm {J}({x}, {y})) \end{aligned}$$

(6.1)

for all ${x},{y}\in U$ and $u\in {\psi }{x}$, $v\in \psi _{1}{y}$. Then there exists a sequence $\{u_p: u_{2p+1}\in {\psi }u_{2p}, u_{2p+2}\in \psi _{1}u_{2p+1}\}_{p\in \mathbb {N}}$ such that

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(u_p, u_{p+1})=0. \end{aligned}$$

Then $\{u_p\}$ is $\mathrm {J}$-Cauchy and $\psi _{1}$ and ${\psi }$ has a common fixed point in $U$.

Proof

By hypothesis, there exists $u_0, u_1, u_2\in U$ such that $u_1\in {\psi }u_0$, $u_2\in \psi _{1}u_1$ and $\sup \{\mathrm {J}(u_{2l+1}, u_{2j+2}): u_{2l+1}\in {\psi }u_{2l},u_{2j+2}\in \psi _{1}u_{2j+1}\}<+\infty$; then we have

$$\begin{aligned} \mathrm {J}(u_{2l+1}, u_{2j+2})<+\infty , \end{aligned}$$

(6.2)

for all $u_{2l+1}\in {\psi }u_{2l},u_{2j+2}\in \psi _{1}u_{2j+1}$. Now if $u_1\in \psi _{1}u_1\cap {\psi }u_1$, then $u_1$ is common fixed point of $\psi _{1}$ and ${\psi }$, so let $u_1\notin \psi _{1}u_1$. We claim that $\mathrm {J}(u_1, u_2)>0$; otherwise, if $\mathrm {J}(u_1, u_2)=0$, then $u_1\in \psi _{1}u_1$, which is a contradiction. Hence from (6.1), we obtain

$$\begin{aligned} \lambda +{F}(\mathrm {J}(u_1,u_2))\le {F}(\mathrm {J}(u_2,u_1)). \end{aligned}$$

(6.3)

Next, if $u_2\in \psi _{1}u_2\cap {\psi }u_2$, then $u_2$ is common fixed point of $\psi _{1}$ and ${\psi }$, so let $u_2\notin {\psi }u_2$. We claim that $\mathrm {J}(u_2, u_3)>0$; otherwise if $\mathrm {J}(u_2, u_3)=0$, then $u_2\in {\psi }u_2$, which is a contradiction. Hence from (6.1), we obtain

$$\begin{aligned} \lambda +{F}(\mathrm {J}(u_2,u_3))\le {F}(\mathrm {J}(u_1,u_2)). \end{aligned}$$

(6.4)

Continuing in this manner, we obtain a sequence $\{u_p\}$ such that $u_{2p+1}\in {\psi }u_{2p}$ and $u_{2p+2}\in \psi _{1}u_{2p+1}\}$ with $u_{2p+1}\notin \psi _{1}u_{2p+1}$ and $u_{2p}\notin {\psi }u_{2p}$ satisfying

$$\begin{aligned} \lambda +{F}(\mathrm {J}(u_p,u_{p+1}))\le {F}(\mathrm {J}(u_{p-1},u_p)), \end{aligned}$$

(6.5)

for all $p\in \mathbb {N}\setminus \{0\}$. This implies

$$\begin{aligned} 0<\mathrm {J}(u_p,u_{p+1})< \mathrm {J}(u_{p-1}, u_p). \end{aligned}$$

(6.6)

Thus, $\{\mathrm {J}(u_{p},u_{p+1})\}$ is a decreasing bounded sequence and so converges; thus there exists $c\ge 0$ such that

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(u_{p},u_{p+1})=c. \end{aligned}$$

(6.7)

We claim that $c=0$, if not, then from (6.5), we get

$$\begin{aligned} {F}(\mathrm {J}(u_{p},u_{p+1}))\le {F}(\mathrm {J}(u_{0},u_{1}))-p\lambda . \end{aligned}$$

(6.8)

Letting $p\rightarrow +\infty$, we obtain

$$\begin{aligned} \lim _{p\rightarrow +\infty }{F}(\mathrm {J}(u_{p},u_{p+1}))=+\infty . \end{aligned}$$

(6.9)

From $({F}2)$, we have

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(\xi u_{p},\xi u_{p+1})=0, \end{aligned}$$

(6.10)

which is a contradiction.

Next, we show that $\{u_p\}$ is $\mathrm {J}$-Cauchy sequence. Consider $q>p>p_1$ for some $p_1\in \mathbb {N}$. We claim that $\mathrm {J}(u_{2q+1},u_{2p+2})>0$. If not then $u_{2q+1}=u_{2p+2}$. This gives $u_{2q+1}\in \psi _{1}u_{2p+1}$, which is contradiction to the fact that $u_{2l+1}\notin \psi _{1}u_{2j+1}$ for all $l\ne j$. Then from (6.1), we obtain

$$\begin{aligned} \lambda +{F}(\mathrm {J}(u_{2q+1},u_{2p}))\le {F}(\mathrm {J}(\xi u_{2q}, \xi u_{2p-1})). \end{aligned}$$

(6.11)

Which implies

$$\begin{aligned} {F}(\mathrm {J}(u_{2q+1}, u_{2p}))\le {F}(\mathrm {J}(u_{2q-2p+1},u_{0}))-2p\lambda . \end{aligned}$$

(6.12)

Letting $p\rightarrow +\infty$ in (6.12), we obtain

$$\begin{aligned} \lim _{p,q\rightarrow +\infty }{F}(\mathrm {J}(u_{2q+1}, u_{2p}))=+\infty \end{aligned}$$

(6.13)

and further by using $({F}2)$, we have

$$\begin{aligned} \lim _{p,q\rightarrow +\infty }\mathrm {J}(u_{2q+1}, u_{2p})=0. \end{aligned}$$

(6.14)

Hence, $\{u_p\}$ is $\mathrm {J}$-Cauchy sequence. Since $U$ is complete, so there exist a point $z\in U$ such that $\{u_p\}\xrightarrow {\mathrm {J}} z$. From (6.1), for $p\in \mathbb {N}$ and $u_{2p+1}\in {\psi }u_{2p}$, there exist $z_{2p}\in \psi _{1}z$ satisfying

$$\begin{aligned} \lambda +{F}(\mathrm {J}(u_{2p+1}, z_{2p}))\le {F}(\mathrm {J}(u_{2p}, z)). \end{aligned}$$

(6.15)

Since $\{u_p\}\xrightarrow {\mathrm {J}} z$, by letting $p\rightarrow +\infty$ in (6.15) and using $({F}2)$, we obtain

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(u_{2p+1}, z_{2p})=0. \end{aligned}$$

(6.16)

By using $(\mathrm {J}_3)$, we get

$$\begin{aligned} 0\le \mathrm {J}(z_{2p},z)\le \kappa \lim _{p\rightarrow +\infty }\sup \mathrm {J}(u_{2p+1}, z)=0, \end{aligned}$$

(6.17)

which implies

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(z_{2p},z)=0. \end{aligned}$$

Since $\psi _{1}z$ is closed, we have $z\in \psi _{1}z$.

Similarly, from (6.1), for $p\in \mathbb {N}$ and $u_{2p}\in \psi _{1}u_{2p-1}$, there exist $z_{2p+1}\in {\psi }z$ satisfying

$$\begin{aligned} \lambda +{F}(\mathrm {J}(z_{2p+1}, u_{2p}))\le {F}(\mathrm {J}(z, u_{2p-1})). \end{aligned}$$

(6.18)

Since $\{u_p\}\xrightarrow {\mathrm {J}} z$, by letting $p\rightarrow +\infty$ in (6.18) and using $({F}2)$, we obtain

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(z_{2p+1}, u_{2p})=0. \end{aligned}$$

(6.19)

By using $(\mathrm {J}_3)$, we get

$$\begin{aligned} 0\le \mathrm {J}(z_{2p+1},z)\le \kappa \lim _{p\rightarrow +\infty }\sup \mathrm {J}(u_{2p}, z)=0, \end{aligned}$$

(6.20)

which implies

$$\begin{aligned} \lim _{p\rightarrow +\infty }\mathrm {J}(z_{2p+1},z)=0. \end{aligned}$$

Since ${\psi }z$ is closed, we have $z\in {\psi }z$. Hence $z\in \psi _{1}z\cap {\psi }z$. $\square$

7 Application to integral inclusions

In this section, we give existence theorem for integral inclusion. For this, let $U=C(I,\mathbb {R})$ be the space of all continuous real-valued functions on I, where $I=[0,1]$. Then $U$ is a complete metric space with respect to metric $\mathrm {J}(x,y)=\sup _{t\in I}|x(t)-y(t)|$. Since every metric space is GMS(JS), throughout this section, we assume that $(U,\mathrm {J})$ is complete is GMS(JS). Consider the integral inclusions

$$\begin{aligned} {x}(t)\in \int _{0}^{\sigma (t)}k(t,s)L(s, {x}(s))ds+q(t) \end{aligned}$$

(7.1)

and

$$\begin{aligned} {y}(t)\in \int _{0}^{\sigma (t)}k(t,s)M(s, {y}(s))ds+q(t) \end{aligned}$$

(7.2)

for $t\in I$, where $\sigma :I\rightarrow I$, $q:I\rightarrow U$, $k:I\times I\rightarrow \mathbb {R}$ are continuous and $L,~M:I\times U\rightarrow P(\mathbb {R})$, $P(\mathbb {R})$ denote the collection of all nonempty, compact and convex subsets of $\mathbb {R}$. For each $x\in U$, the operators L(., x) and M(., y) are lower semi-continuous.

Define the multivalued operators ${\psi },\psi _{1}:U\rightarrow {C}(U)$ as follows:

$$\begin{aligned} {\psi }{x}(t)=\left\{ u\in U~:~u\in \int _{0}^{\sigma (t)}k(t,s)L(s, {x}(s))ds+q(t), t\in I\right\} \end{aligned}$$

(7.3)

and

$$\begin{aligned} \psi _{1}{y}(t)=\left\{ v\in U~:~v\in \int _{0}^{\sigma (t)}k(t,s)M(s, {y}(s))ds+q(t), t\in I\right\} \end{aligned}$$

(7.4)

Note that a common fixed point of multivalued operators (7.3) and (7.4) is the common solutions of integral inclusions (7.1) and (7.2). We consider the following set of assumptions in the following:

• (H1) The function k(t, s) is continuous and nonnegative on $I\times I$ with $\Vert k\Vert _{\infty }=\sup \{k(t,s):t,s\in I\}$.

• (H2) $|l_x-m_y|\le \tau |x(s)-y(s)|$ for all $l_x(s)\in L(s, x(s))$ and $m_y(s)\in M(s, y(s))$ for some $\tau >0$.

Theorem 7.1

Assume that hypothesis (H1)-(H2) holds. If $\Vert k\Vert _{\infty }\tau <e^{-\frac{1}{\tau }}$ for some $\tau >0$, then integral inclusions (7.1) and (7.2) have a common solution in $U$.

Proof

Let $x,y\in U$. Denote $L_x=L_x(s, x(s))$ and $M_y=M_y(s, y(s))$. Now for $L_x:I\rightarrow P(\mathbb {R})$ and $M_y:I\rightarrow P(\mathbb {R})$, by Micheal’s selection theorem, there exist continuous operators $l_x, m_y:I\times I \rightarrow \mathbb {R}$ with $l_x(s)\in L_x(s)$ and $m_y(s)\in M_y(s)$ for $s\in I$. So, we have $u=\int _{0}^{\sigma (t)}k(t,s)l_x(s)ds+q(t)\in {\psi }{x}(t)$ and $v=\int _{0}^{\sigma (t)}k(t,s)m_y(s)ds+q(t)\in \psi _{1}{y}(t)$. Thus, the operators ${\psi }{x}$ and $\psi _{1}{y}$ are nonempty and closed (see [32]). By hypothesis (H1)–(H2) and by using Cauchy–Schwartz inequality, we get

$$\begin{aligned} \begin{aligned} \mathrm {J}(u,v)&=\sup _{t\in I}|u(t)-v(t)|\\&=\sup _{t\in I}\left| \int _{0}^{\sigma (t)}k(t,s)\left( l_x(s)-m_y(s)\right) ds\right| \\&\le \sup _{t\in I}\int _{0}^{\sigma (t)}k(t,s)|l_x(s)-m_y(s)|ds\\&\le \sup _{t\in I}\int _{0}^{\sigma (t)}k(t,s)\tau |x(s)-y(s)|ds\\&\le \tau \sup _{t\in I}\left( \int _{0}^{\sigma (t)}k^2(t,s)ds\right) ^{\frac{1}{2}}\left( \int _{0}^{\sigma (t)}|x(s)-y(s)|^2ds\right) ^{\frac{1}{2}}\\&\le \tau \Vert k\Vert _\infty \sup _{t\in I}|x(t)-y(t)|\\&\le e^{-\frac{1}{\tau }}\sup _{t\in I}|x(t)-y(t)|\\&=e^{-\frac{1}{\tau }}\mathrm {J}(x,y). \end{aligned} \end{aligned}$$

Hence (6.1) is satisfied for ${F}(\alpha )=\ln (\alpha )$ and $\lambda =\frac{1}{\tau }>0$. Thus, all hypotheses of Theorem 6.1 are satisfied and, therefore, ${\psi }$ and $\psi _{1}$ have common fixed point. It further implies that integral inclusions (7.1) and (7.2) have a common solution in I. $\square$