Fixed points of non-linear multivalued graphic contractions with applications

1.   Introduction and preliminaries

Fixed point theory is one of the main tools in modern functional analysis. Its primary role is in the existence criteria for solutions of different types of equations arising in science and engineering. One of the first most celebrated results in this context is the Banach contraction principle (BCP). The prototypical idea of the BCP has been fine-tuned by many examiners in different domains.

Definition 1.1. A metric space (MS) $(\widetilde{\bigwedge}, \rho)$ is called $\epsilon$-chainable, for some $\epsilon > 0$, if for any $u, v\in \widetilde{\bigwedge}$, we can find $\alpha\in \mathbb{N}$ and a sequence $\{\jmath_i\}_{i = 0}^\alpha$ in $\widetilde{\bigwedge}$ such that ${\jmath}_0 = u$, ${\jmath}_\alpha = v$ and $\rho({\jmath}_{i-1}, {\jmath}_i) < \epsilon$ for $i = \overline{1, \alpha}$.

Definition 1.2. Let $(\widetilde{\bigwedge}, \rho)$ be an MS, $\epsilon > 0, 0\leq l < 1$ and $u, v\in \widetilde{\bigwedge}$. A mapping $g: \widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ is called $(\epsilon, l)$-uniformly locally contractive, if $0 < \rho(u, v) < \epsilon$ implies $\rho(gu, gv) < ld(u, v)$.

As one of the improvements of the BCP, Edelstein ^[6] proved that every $(\epsilon, l)$-uniformly locally contractive mapping on a complete $\epsilon$-chainable MS has a unique fixed point

Let $(\widetilde{\bigwedge}, \rho)$ be an MS. Consistent with Nadler ^[18] and Hu ^[7], denote by $C\widehat{B}(\widetilde{\bigwedge}), \mathcal{K}(\widetilde{\bigwedge})$ and $2^{\widetilde{\bigwedge}}$, the collection of all non-empty closed and bounded, compact and non-empty subsets of $\widetilde{\bigwedge}$, respectively. Let ${\widehat{A}}, {\widehat{B}}\in C{\widehat{B}}(\widetilde{\bigwedge})$. The Pompeiu-Hausdorff distance ${\aleph}$ on $C{\widehat{B}}(\widetilde{\bigwedge})$ induced by the metric $\rho$ is given as:

where

In 1969, Nadler ^[18] brought up a multivalued version of the BCP by availing the Hausdorff distance function. Along this line, Reich ^[26] presented a fixed point theorem for multivalued mappings (MVM) on compact subsets of an MS and noted the puzzle: "can $\mathcal{K}(\widetilde{\bigwedge})$ be replaced with $C\widehat{B}(\widetilde{\bigwedge})$?". Mizoguchi and Takahashi [16,Theorem 5] gave an affirmative response to this puzzle. In similar development, the multivalued fixed point theorem given by Nadler was extended to an $\epsilon$-chainable MS by Hu ^[7]. Azam and Arshad ^[3] improved [18,Theorem 6] by investigating fixed point results of a sequence of locally contractive MVMs in an $\epsilon$-chainable MS. Muhammad et al. ^[17] refined the ideas in ^[3,18] to the setting of an MS with a directed graph. On similar line, Phikul and Suthep ^[24] improved the ideas of Berinde ^[5], Jachysmki ^[8] and Nadler ^[18] by examining common fixed point results of a pair of two MVMs on an MS endowed with a graph. On the other side, Jachysmki ^[8] studied the fixed point notions putforward by Nieto and Rodriguez-Lopez ^[23] and Ran and Reuring ^[25] by launching the concept of a graphic contraction (also named a ${G_r}$-contraction) on an MS.

Following the above trend of investigations (in particular, the ideas in ^{[3,5,8,16,24]} and the references therein), we noticed that fixed point results of set-valued maps connecting the notions of $M$-function, graphic contractions and Berinde-type weak contractions have not been sufficiently examined. Hence, this paper introduces a more general notion of a graphic contraction, viz. a non-linear multivalued ${G_r}_g$-contraction. Sufficient conditions for the existence of coincidence points (CoPs) of the sequence of MVMs and a single-valued map given on an MS endowed with a graph are examined. Comparative examples which dwell on the preeminence of our obtained results are constructed. A significant number of results in an $\epsilon$-chainable MS and cyclic contractions are derived as some special cases of our results. From an application point, one of our findings is employed to investigate new criteria for solution to a more general form of Fredholm integral equation. As a future assignment of the ideas presented here, we note down an open problem regarding a discretized population balance model whose solution may be analyzed using any of the concepts proposed in this work.

We now present some concepts and results that will be needed hereafter. Let $(\widetilde{\bigwedge}, \rho)$ be an MS and $\nabla$ denote the diagonal of the Cartesian product $\widetilde{\bigwedge}\times \widetilde{\bigwedge}$. Given a directed graph ${G_r}$, let $\widetilde{\bigwedge} = {V_{et}}({G_r})$, where ${V_{et}}({G_r})$ depicts the set of vertices of the graph ${G_r}$ and ${E_{ed}}({G_r})$ be the set of all edges of ${G_r}$. Assume that ${G_r}$ has no parallel edges so that ${G_r} = ({V_{et}}({G_r}), {E_{ed}}({G_r}))$.

If ${G_r}$ is a directed graph, then ${G_r}^{-1}$ depicts the graph derived from ${G_r}$ by inverting the direction of edges. And, if we overlook the direction of the edges in ${G_r}$, then, we obtain an undirected graph denoted by $\widetilde{{G_r}}$. The pair $({V_{et}}^\prime, {E_{ed}}^\prime)$ is a subgraph of ${G_r}$, if ${V_{et}}^\prime \subseteq {V_{et}}({G_r})$ and ${E_{ed}}^\prime \subseteq {E_{ed}}({G_r})$, and for each $(p, q)\in {E_{ed}}^\prime$ for all $p, q\in {V_{et}}^\prime$. Moreover, we record the needed concepts of connectivity of graph from ^[9] as follows.

Definition 1.3. A path in a graph ${G_r}$ from the vertex $q$ to $p$ of length $\alpha\in \mathbb{N}\cup\{0\}$, is a sequence $\{\varsigma_i\}_{i = 0}^\alpha$ of $\alpha+1$ vertices such that $\varsigma_0 = q, \varsigma_\alpha = p$ and $(\varsigma_{i-1}, \varsigma_i)\in {E_{ed}}({G_r})$ for $i = \overline{i, \alpha}$.

Definition 1.4. A graph ${G_r}$ is said to be connected if we can find a path between any two of its vertices. ${G_r}$ is weakly connected if $\widetilde{{G_r}}$ is connected.

Definition 1.5. For ${\jmath}, {\ell}, z\in {V_{et}}({G_r})$, $[{\jmath}]_{G_r}$ depicts the equivalence class of the relation $\sim$ given on ${V_{et}}({G_r})$ by the rule ${\ell} \sim z$ if we can find a path in ${G_r}$ from ${\ell}$ to $z$.

For $\eta\in {V_{et}}({G_r})$ and $\alpha\in \mathbb{N}\cup \{0\}$, define the set $[\eta]_{G_r}^\alpha$, as follows:

Jachymski ^[8] brought up the idea of a ${G_r}$-contraction in the following manner.

Definition 1.6. Let $(\widetilde{\bigwedge}, \rho)$ be an MS equipped with a graph ${G_r}$. The mapping $\Upsilon:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ is called a ${G_r}$-contraction if it preserves the edges of ${G_r}$; that is:

and we can find $\beta\in [0, 1)$ such that

Mizoguchi and Takahashi ^[16] introduced an auxiliary function, named an $MT$-function, as follows.

Definition 1.7. ^[16] A function $\psi: \mathbb{R}_+ = [0, \infty) \longrightarrow [0, 1)$ is known as an $MT$-function if it satisfies the Mizoguchi and Takahashi condition, that is, if $\lim_{r\longrightarrow t^+}\sup\psi(t) < 1$ for all $t\in \mathbb{R}_+$.

Now, we present specific results from the theory of MVMs.

Lemma 1.8. ^[2] Let $\{{\widehat{A}}_n\}$ be a sequence in $C{\widehat{B}}(\widetilde{\bigwedge})$ and we can find ${\widehat{A}}\in C{\widehat{B}}(\widetilde{\bigwedge})$ such that $\lim_{n\longrightarrow \infty}{\aleph}({\widehat{A}}_n, {\widehat{A}}) = 0$. If ${\jmath}_n\in {\widehat{A}}_n (n\in \mathbb{N})$ and we can find ${\jmath}\in \widetilde{\bigwedge}$ such that $\lim_{n\longrightarrow \infty}\rho({\jmath}_n, {\jmath}) = 0$, then ${\jmath}\in {\widehat{A}}$.

Lemma 1.9. ^[7] Let $(\widetilde{\bigwedge}, \rho)$ be an MS and ${\widehat{A}}, {\widehat{{\widehat{B}}}}\in C{\widehat{B}}(\widetilde{\bigwedge})$ with ${\aleph}({\widehat{A}}, {\widehat{{\widehat{B}}}}) < \epsilon$ for every $\epsilon > 0$. Then for each ${\jmath}\in {\widehat{A}}$, we can find ${\ell}\in {\widehat{{\widehat{B}}}}$ such that $\rho({\jmath}, {\ell}) < \epsilon$.

2.   Main results

We begin this section by introducing a new type of sequence of MVM in an MS with a directed graph.

Definition 2.1. Let $(\widetilde{\bigwedge}, \rho)$ be an MS, ${G_r} = ({V_{et}}({G_r}), {E_{ed}}({G_r}))$ be a directed graph such that ${V_{et}}({G_r}) = \widetilde{\bigwedge}$ and $g:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ be a surjection. A sequence of MVM $\{\Upsilon_p\}_{p\in \mathbb{N}}$ from $\widetilde{\bigwedge}$ into $CB(\widetilde{\bigwedge})$ is called a non-linear multivalued ${G_r}_g$-contraction, if we can find an $MT$-function $\psi:\mathbb{R}_+\longrightarrow [0, 1)$ and some constant $K\geq0$ such that for $(gu, gv)\in {E_{ed}}({G_r})$,

$\rm (a)$

$\rm (b)$ if ${\jmath}\in \Upsilon_p(u), {\ell}\in \Upsilon_r(v)$ and $\rho(g{\jmath}, g{\ell})\leq \rho(gu, gv)$, then $(g{\jmath}, g{\ell})\in {E_{ed}}({G_r})$.

If $\Upsilon_p:\widetilde{\bigwedge}\longrightarrow CB(\widetilde{\bigwedge})$, then the graph of $\Upsilon_p$ is given by

We now study conditions for the existence of CoPs of a single-valued mapping and a sequence of MVM.

Theorem 2.2. Let $(\widetilde{\bigwedge}, \rho)$ be a complete MS, $\{\Upsilon_p: p\in \mathbb{N}\}$ be a sequence of multivalued ${G_r}_g$-contraction from $\widetilde{\bigwedge}$ into $CB(\widetilde{\bigwedge})$ and $g:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ be a surjection. If we can find $\alpha\in \mathbb{N}$ and $v_0\in \widetilde{\bigwedge}$ such that

(i) $\Upsilon_1(v_0)\cap [g(v_0)]_{G_r}^\alpha \neq\emptyset$;

(ii) for any sequence $\{v_n\}$ in $\widetilde{\bigwedge}$, if $v_n\longrightarrow v$ and $v_n\in \Upsilon_n(v_{n-1})\cap [v_{n-1}]_{G_r}^\alpha$ for all $n\in \mathbb{N}$, then we can find a subsequence $\{v_{n_\gamma}\}$ of $\{v_n\}$ such that $(v_{n_\gamma}, v)\in {E_{ed}}({G_r})$ for all $\gamma\in \mathbb{N}$.

Then we can find $u^*\in \widetilde{\bigwedge}$ such that $gu^*\in \bigcap_{p\in \mathbb{N}}\Upsilon_p(u^*)$.

Proof. Choose $v_1\in \widetilde{\bigwedge}$ such that $gv_1\in \Upsilon_1(v_0)\cap [gv_0]^\alpha_{G_r}$, then, we can find a path from $gv_0$ to $gv_1$, that is,

and $(gu_i^{(1)}, gu_{i+1}^{(1)})\in {E_{ed}}({G_r})$ for all $i = \overline{0, {\alpha}-1}$. Without loss of generality, assume that $gu_{\gamma}^{(1)}\neq gu_{j}^{(1)}$ for each $\gamma, j\in \{0, 1, 2, \cdots, {\alpha}\}$ with $\gamma \neq j$. Rename $gv_1$ as $gu_{0}^{(2)}$. Since $(gu_{0}^{(1)}, gu_{1}^{(1)})\in {E_{ed}}({G_r})$, and as $gu_{0}^{(2)}\in \Upsilon_1(u_0^{(1)})$, then by Lemma 1.9, we can find some $gu_{1}^{(2)}\in \Upsilon_2(u_1^{(1)})$ such that

Again, since $(gu_{1}^{(1)}, gu_{2}^{(1)})\in {E_{ed}}({G_r})$ and $gu_{1}^{(1)}, \cdots, gu_{{\alpha}}^{(1)}\in \Upsilon_2(u_1^{(1)}), {\alpha} = 1, 2$, by Lemma 1.9, we can find some $gu_{2}^{(2)}\in \Upsilon_2(u_1^{(1)})$ such that

Thus, we obtain $\{gu_{0}^{(2)}, gu_{1}^{(2)}, \cdots, gu_{\alpha}^{(2)}\}$ of $\alpha+1$ vertices of $\widetilde{\bigwedge}$ such that $gu_{0}^{(2)}\in \Upsilon_1(u_0^{(1)})$ and $gu_{\rho}^{(2)}\in \Upsilon_2(u_\rho^{(1)})$ for $\rho = \overline{1, \alpha}$ with

for $\rho = \overline{0, \alpha-1}$. Because $(gu_{\rho}^{(1)}, gu_{\rho+1}^{(1)})\in {E_{ed}}({G_r})$ for all $\rho = \overline{0, \alpha-1}$, $(gu_{\rho}^{(2)}, gu_{\rho+1}^{(2)})\in {E_{ed}}({G_r})$ for all $\rho = \overline{0, \alpha-1}$. Let $gu_{\alpha}^{(2)} = gv_2$. Thus, the set of points $gv_1 = gu_{0}^{(2)}, gu_{1}^{(2)}, \cdots, gu_{\alpha}^{(2)} = gv_2\in \Upsilon_2(v_1)$ is a path from $gv_1$ to $gv_2$. Relabel $gv_2$ as $gu_{0}^{(3)}$. Then, by similar steps as above, we obtain a path $gv_2 = gu_{0}^{(3)}, gu_{1}^{(3)}, \cdots, gu_{\alpha}^{(3)} = gv_3\in \Upsilon_3(v_2)$ from $gv_2$ to $gv_3$. Inductively, it follows that $gv_h = gu_{0}^{(h+1)}, gu_{1}^{(h+1)}, \cdots, gu_{\alpha}^{(h+1)} = gv_{h+1}\in \Upsilon_{h+1}(v_h)$ with

thus, $(gu_{t}^{(h+1)}, gu_{t+1}^{(h+1)})\in {E_{ed}}({G_r})$ for $t = \overline{0, \alpha-1}$. Consequently, we construct a sequence $\{gv_h\}_{h = 1}^\infty$ of points of $\widetilde{\bigwedge}$ with

For each $t\in \{0, 1, 2, ..., \alpha-1\}$, and from (2.1), we see that $\{\rho(gu_{t}^{(h)}, gu_{t+1}^{(h)})\}_{h = 1}^\infty$ is a bounded and decreasing sequence of non-negative real numbers, and so it converges. That is, we can find ${\tau^t}\geq0$ such that

Since $\psi$ is an $MT$-function, we can find ${\varrho}^t\in \mathbb{N}$ such that $\psi(\rho(gu_{t}^{(h)}, gu_{t+1}^{(h)})) < \omega({\tau^t})$ for all $h\geq {\varrho}^t$, where $\lim_{t\longrightarrow {\tau^t}^+}\sup\psi(t) < \omega({\tau^t}) < 1$. Now, set

Then, for every $h > {\varrho}^t$, consider

Taking $p = \max\{{\varrho}^t, t = 0, 1, 2, \cdots, \alpha-1\}$, produces

Now, for all $q > h > p$, notice that

Since $\Omega{\tau^t} < 1$ for all $t\in \{0, 1, 2, \cdots, \alpha-1\}$, it follows that $\{gv_h = gu_\alpha^{(h)}\}$ is a Cauchy sequence. By completeness of $\widetilde{\bigwedge}$, we can find $v^*\in \widetilde{\bigwedge}$ such that $gv_h\longrightarrow gv^*$. Now, availing the fact that $gv_n\in \Upsilon(v_{n-1})\cap [gv_{n-1}]_{G_r}^\alpha$ for all $n\in \mathbb{N}$, we can find a subsequence $\{gv_{n_\gamma}\}$ such that $(gv_{n_\gamma}, gv^*)\in {E_{ed}}({G_r})$ for all $\gamma\in \mathbb{N}$. Now, for any $p\in \mathbb{N}$,

Letting $h\longrightarrow \infty$ in (2.2), yields

From (2.3), if $K = 0$, then Lemma 1.8 can be applied to conclude that $gv^*\in \Upsilon_p(v^*)$ for all $p\in \mathbb{N}$. On the other hand, if $K > 0$, assume that $gv^*\notin \Upsilon_p(v^*)$ for some $p\in \mathbb{N}$. So, taking $K = \frac{\rho(gv^*, \Upsilon_p(v^*))}{1+\rho(gv^*, \Upsilon_p(v^*))}$ in (2.3) gives

a contradiction. Consequently, $gv^*\in \bigcap_{p\in \mathbb{N}}\Upsilon_p(v^*)$.

Example 2.3. For $p\in \mathbb{N}$, let $\widetilde{\bigwedge} = \left\{\frac{1}{3^p} \right\}\cup\{0\}\cup\left[\frac{1}{4^p}, 5 \right]$ and $\rho(u, v) = |u-v|$ for all $u, v\in \widetilde{\bigwedge}$. Then, $(\widetilde{\bigwedge}, \rho)$ is a complete MS. Let ${G_r} = ({V_{et}}({G_r}), {E_{ed}}({G_r}))$ be a directed graph such that ${V_{et}}({G_r}) = \widetilde{\bigwedge}$ and ${E_{ed}}({G_r}) = \left\{ (0, 0), \left(\frac{1}{4^p}, 1 \right): p\in \mathbb{N}\right\}$. Let $g:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ be given as $g(u) = 5u$, and $\Upsilon_p:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ be given by

We shall show that $\Upsilon_p$ is a non-linear multivalued ${G_r}_g$-contraction with $\psi(t) = \frac{t}{5}, t\geq0$, and $K = 5$. Now, notice that if $u = v = 0$ and $gu = gv = 0$, then for all $p, r\in \mathbb{N}$, $\Upsilon_p(u) = \Upsilon_r(v) = \{0\}$, thus ${\aleph}(\Upsilon_p(u), \Upsilon_r(v)) = 0$. For $(gu, gv)\in {E_{ed}}({G_r})$ with $u \neq v$, $(u, v) = \left(\frac{1}{4^p}, 1 \right)$ for each $p\in \mathbb{N}$. Thus,

Moreover, let $(gu, gv)\in {E_{ed}}({G_r})$ with $u\neq v$. Then, $(u, v) = \left(\frac{1}{4^p}, 1 \right)$ for all $p\in \mathbb{N}$. Thus, $\Upsilon_p(u) = \Upsilon_p\left(\frac{1}{4^p}\right) = [0, 4]$ and $\Upsilon_r(v) = \Upsilon_r(1) = [0, 4v]$. We observe that if ${\jmath}\in \Upsilon_p(u), {\ell}\in \Upsilon_r(v)$ and $\rho(g{\jmath}, g{\ell})\leq \rho(gu, gv)$, then $({\jmath}, {\ell})$ are $\left(0, \frac{1}{4^{p+2}} \right)$ and $\left(0, \frac{1}{4^{p+3}} \right)$. Thus, $(g{\jmath}, g{\ell})\in {E_{ed}}({G_r})$. Consequently, $\{\Upsilon_p\}_{p\in \mathbb{N}}$ is a non-linear multivalued ${G_r}_g$-contraction. We notice that other conditions of Theorem 2.2 hold obviously. It follows that all conditions of Theorem 1 are obeyed. Thus, $g$ and $\Upsilon_p$ have a CoP $u^* = 0\in \widetilde{\bigwedge}$ such that $g0\in \bigcap_{p\in \mathbb{N}}\Upsilon_p(0)$.

In what follows, we show that our result (Theorem 2.2) cannot be followed from some similar ones in the literature. Now, consider [24,Theorem 3.3], if we take $\psi(\rho(u, v)) = \frac{1}{3}$ with $u = \frac{1}{4}, v = 1$ and $p = 1, r = 2$, then $(u, v) = \left(\frac{1}{4}, 1\right)\in {E_{ed}}({G_r})$, and

for all $K\geq0$. This shows that the mapping $\Upsilon_p$ is not a Berinde graph contractive in the sense of Phikul and Suthep [24,Definition 3.1]. From (2.4), we also observe that $\Upsilon_p$ is not a graph contractive mapping as given by Beg and Butt ^[4]. Thus, the main results in ^[4,24] are not applicable to this illustration.

Theorem 2.4. Let $(\widetilde{\bigwedge}, \rho)$ be a complete MS, $\Upsilon:\widetilde{\bigwedge}\longrightarrow CB(\widetilde{\bigwedge})$ be an MVM and $g:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ a surjection. Assume further that the following conditions are obeyed:

(i) we can find $K\geq0$ such that for all $u, v\in (u\neq v)$, $(gu, gv)\in {E_{ed}}({G_r})$ yields

where $\psi:\mathbb{R}_+\longrightarrow [0, 1)$ is an $MT$-function;

(ii) we can find $\alpha\in \mathbb{N}$ and $v_0\in \widetilde{\bigwedge}$ such that $\Upsilon(v_0)\cap [gv_0]^\alpha_{G_r}\neq \emptyset$;

(iii) for any sequence $\{v_n\}$ in $\widetilde{\bigwedge}$, if $v_n\longrightarrow v$ and $v_n\in \Upsilon(v_{n-1})\cap [v_{n-1}]^\alpha_{G_r}$ for all $n\in \mathbb{N}$, we can find a subsequence $\{v_{n_\gamma}\}$ such that $(v_{n_\gamma}, v)\in {E_{ed}}({G_r})$ for all $\gamma\in \mathbb{N}$.

Then $g$ and $\Upsilon$ have a CoP in $\widetilde{\bigwedge}$; that is, we can find $u^*\in \widetilde{\bigwedge}$ such that $gu^*\in \Upsilon(u^*)$.

Proof. Set $\Upsilon_p = \Upsilon$ for all $q\in \mathbb{N}$ in Theorem 2.2.

The following are further consequences of Theorems 2.2 and 2.4.

Corollary 1. Let $(\widetilde{\bigwedge}, \rho)$ be a complete MS and $\{\Upsilon_p: p\in \mathbb{N}\}$ be a sequence of MVMs from $\widetilde{\bigwedge}$ into $CB(\widetilde{\bigwedge})$. Assume further that:

(i) if for some $K\geq0$ and any $u, v\in \widetilde{\bigwedge} (u \neq v)$ such that $(u, v)\in {E_{ed}}({G_r})$, we have that

for all $p, r\in \mathbb{N}$, where $\psi:\mathbb{R}_+$ is an $MT$-function;

(ii) we can find $\alpha\in \mathbb{N}$ and $v_0\in \widetilde{\bigwedge}$ such that $\Upsilon_1(v_0)\cap [gv_0]^\alpha_{G_r}\neq \emptyset$;

(iii) for any sequence $\{v_n\}$ in $\widetilde{\bigwedge}$, if $v_n\longrightarrow v$ and $v_n\in \Upsilon_n(v_{n-1})\cap[v_{n-1}]_{G_r}^\alpha$ for all $n\in \mathbb{N}$, then we can find a subsequence $\{v_{n_\gamma}\}$ of $\{v_n\}$ such that $(v_{n_\gamma}, v)\in {E_{ed}}({G_r})$ for all $\gamma\in \mathbb{N}$.

Then $\Upsilon_p$ has at least one fixed point in $\widetilde{\bigwedge}$.

Proof. Take $g = I_{\widetilde{\bigwedge}}$, that is, the identity mapping on $\widetilde{\bigwedge}$, in Theorem 2.2.

The following is a consequence of Theorem 2.2 in the case of single-valued mappings.

Corollary 2. Let $(\widetilde{\bigwedge}, \rho)$ be a complete MS, $\Lambda:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ and $g:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ be a surjection. If $u, v\in \widetilde{\bigwedge} (u \neq v)$ such that $(gu, gv)\in {E_{ed}}({G_r})$ implies

for some $K\geq$, where $\psi:\mathbb{R}_+\longrightarrow [0, 1)$ is an $MT$-function. If we can find $\alpha\in \mathbb{N}$ and $v_0\in \widetilde{\bigwedge}$ such that

(i) $\Upsilon(v_0)\cap [gv_0]^\alpha_{G_r} \neq \emptyset$;

(ii) for any sequence $\{v_n\}$ in $\widetilde{\bigwedge}$, if $v_n\longrightarrow v$, and $v_n = \Upsilon(v_{n-1})\cap [v_{n-1}]^\alpha_{G_r}$ for all $n\in \mathbb{N}$, then we can find a subsequence $\{v_{n+\gamma}\}$ such that $(v_{n_\gamma}, v)\in {E_{ed}}({G_r})$ for all $\gamma\in \mathbb{N}$.

Then $\Lambda$ and $g$ have a coincidence in $\widetilde{\bigwedge}$, that is, we can find $u^*\in \widetilde{\bigwedge}$ such that $gu^* = \Lambda(u^*)$.

Proof. Define $\Upsilon:\widetilde{\bigwedge}\longrightarrow CB(\widetilde{\bigwedge})$ by $\Upsilon u = \{\Lambda u\}$ for all $u\in \widetilde{\bigwedge}$, where $\Lambda$ is a single-valued mapping. Then all conditions of Theorem 2.4 and Corollary 2 coincide. In this case, ${\aleph}(\Upsilon(u), \Upsilon(v)) = {\aleph}(\{\Lambda u\}, \{\Lambda v\}) = \rho(\Lambda u, \Lambda v)$. Consequently, we can find $u^*\in \widetilde{\bigwedge}$ such that $gu^*\in \Upsilon u^* = \{\Lambda u^*\}$, which further produces $gu^* = \Lambda u^*$.

3.   Applications in an $\epsilon$-chainable MS and cyclic contractions

Theorem 3.1. Let $(\widetilde{\bigwedge}, \rho)$ be an $\epsilon$-chainable complete MS, $\{\Upsilon_p: p\in \mathbb{N}\}$ be a sequence of MVMs from $\widetilde{\bigwedge}$ into $CB(\widetilde{\bigwedge})$ and $g:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ be a surjection. If we can find an $MT$-function $\psi:\mathbb{R}_+\longrightarrow [0, 1)$ and a constant $K\geq0$ such that $0 < \rho(gu, gv) < \epsilon$ implies

and we can find ${\jmath}_1\in \Upsilon_p(u_0), {\jmath}_2\in \Upsilon_r(v_0)$ such that $0 < \rho(gu_0, gv_0) < \epsilon$.

Then, we can find $u^*\in \widetilde{\bigwedge}$ such that $gu^*\in \bigcap_{p\in \mathbb{N}}\Upsilon_p(u^*)$.

Proof. Let the graph ${G_r}$ be given by ${V_{et}}({G_r}) = \widetilde{\bigwedge}$ and ${E_{ed}}({G_r}) = \nabla\cup\{(gu, gv)\in \widetilde{\bigwedge}\times \widetilde{\bigwedge}: 0 < \rho(gu, gv) < \epsilon\}$. Then connectivity of ${G_r}$ follows from the $\epsilon$-chainable of $(\widetilde{\bigwedge}, \rho)$. If $(gu, gv)\in {E_{ed}}({G_r})$, then

Now, take ${\jmath}\in \Upsilon_p(u), {\ell}\in \Upsilon_r(v)$ and $\rho(g{\jmath}, g{\ell})\leq \rho(gu, gv)$. Since $(gu, gv)\in {E_{ed}}({G_r})$, then $0 < \rho(gu, gv) < \epsilon$. Observe that if $g{\jmath} \neq g{\ell}$, for each ${\jmath}, {\ell}\in \widetilde{\bigwedge}$, then $0 < \rho(g{\jmath}, g{\ell})\leq \rho(gu, gv) < \epsilon$, so that $(g{\jmath}, g{\ell})\in {E_{ed}}({G_r})$. Thus, for each $p\in \mathbb{N}$, $\{\Upsilon_p\}$ is a sequence of non-linear multivalued ${G_r}_g$-contraction. Notice also that if $v_n\longrightarrow v$ and $\rho(v_n, v_{n+1}) < \epsilon$ for all $n\in \mathbb{N}$ with $v_n\in \Upsilon_n(v_{n-1})\cap [v_{n-1}]^\alpha_{G_r}$, then we can find a natural number $\eta(\epsilon)$ such that $\rho(v_n, v) < \epsilon$ for all $n\geq\eta(\epsilon)$. It follows that we can find a subsequence $\{v_{n_\gamma}\}$ of $\{v_n\}$ such that $(v_{n_\gamma}, v)\in {E_{ed}}({G_r})$ for all $\gamma\in \mathbb{N}$. Moreover, since ${\jmath}_1\in \Upsilon_p(u_0)$ and ${\jmath}_2\in \Upsilon_r(v_0)$ such that $0 < \rho(gu_0, gv_0) < \epsilon$, then $(gu_0, gv_0)\in {E_{ed}}({G_r})$. Consequently, Theorem 2.2 can be applied to find $u^*\in \widetilde{\bigwedge}$ such that $gu^*\in \bigcap_{p\in \mathbb{N}}\Upsilon_p$.

The idea of cyclic contractions was introduced by Kirk et. al. ^[15]. Later on, Rus ^[27] brought up the concepts of cyclic representations consistent with ^[15]. Let $\widetilde{\bigwedge}$ be a non-empty set, $\alpha$ be a natural number and $\{{\widehat{A}}_i\}_{i = 1}^\alpha$ be a non-empty closed subset of $\widetilde{\bigwedge}$ with $\xi:\bigcup_{i = 1}^\alpha {\widehat{A}}_i \longrightarrow \bigcup_{i = 1}^\alpha {\widehat{A}}_i$ as an operator. Then $\widetilde{\bigwedge} = \bigcup_{i = 1}^\alpha {\widehat{A}}_i$ is called a cyclic representation of $\widetilde{\bigwedge}$ with respect to $\xi$, if

and the operator $\xi$ is called a cyclic operator (see ^[19]).

In what follows, we initiate the idea of cyclic representations for sequence of MVMs by following ^[24]. Let $\widetilde{\bigwedge}$ be a non-empty set, $\{{\widehat{A}}_i\}_{i = 1}^\alpha$ be a non-empty closed subset of $\widetilde{\bigwedge}$ for each $\alpha\in \mathbb{N}$ and $\{\Upsilon_p: p\in \mathbb{N}\}$ be a sequence of MVMs from $\widetilde{\bigwedge}$ into $2^{\widetilde{\bigwedge}}$. Then $\widetilde{\bigwedge} = \bigcup_{i = 1}^\alpha {\widehat{A}}_i$ is called a cyclic representation of $\widetilde{\bigwedge}$ with respect to $\Upsilon_p, p\in \mathbb{N}$, if

and $\Upsilon_p$ is called a sequence of multivalued operators.

Theorem 3.2. Let $(\widetilde{\bigwedge}, \rho)$ be a complete MS, $\alpha$ be a positive integer, $\{{\widehat{A}}_i\}_{i = 1}^\alpha$ be a non-empty closed subset of $\widetilde{\bigwedge}$, $\Phi = \bigcup_{i = 1}^\alpha {\widehat{A}}_i$, $\{\Upsilon_p: p\in \mathbb{N}\}$ be a sequence of MVMs from $\widetilde{\bigwedge}$ into $2^\Phi$ and $g:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ be a surjection. Suppose that $\bigcup_{i = 1}^\alpha {\widehat{A}}_i$ is a cyclic representation of $\Phi$ with respect to $\{\Upsilon_p\}_{p = 1}^\infty$. If we can find an $MT$-function $\psi:\mathbb{R}_+\longrightarrow [0, 1)$ and a constant $K\geq0$ such that for $g(u) \neq g(v)$,

for $g(u)\in {\widehat{A}}_i, g(v)\in {\widehat{A}}_{i+1}, {\widehat{A}}_{\alpha+1} = {\widehat{A}}_1$.

Then we can find $u^*\in \widetilde{\bigwedge}$ such that $g(u^*)\in \bigcap_{p\in \mathbb{N}}\Upsilon_p(u^*)$.

Proof. Given that ${\widehat{A}}_i, i = \overline{1, \alpha}$ are closed in $\widetilde{\bigwedge}$, it follows that $(\Phi, \rho)$ is a complete MS. Given a graph ${G_r}$ consisting of ${V_{et}}({G_r}) = \Phi$ and ${E_{ed}}({G_r}) = \nabla\cup\{(g(u), g(v))\in \nabla\times \nabla: u\in {\widehat{A}}_{i+1}, i = \overline{1, \alpha}, {\widehat{A}}_{\alpha+1} = {\widehat{A}}_1\}$, let $g(u), g(v)\in \Phi$ be such that $(g(u), g(v))\in {E_{ed}}({G_r}) \Leftrightarrow (u, v)\in {E_{ed}}({G_r})$ with $g(u) \neq g(v)$. Then, $g(u)\in {\widehat{A}}_i, g(v)\in {\widehat{A}}_{i+1}$, for each $i = \overline{1, \alpha}$. It follows that

Let ${\jmath}\in \Upsilon_p(u), {\ell}\in \Upsilon_r(v)$ and $\rho(g{\jmath}, g{\ell})\leq \rho(gu, gv)$. Then ${\jmath}\in \Upsilon_p(u)\subseteq {\widehat{A}}_{i+1}, {\ell}\in \Upsilon_r(v)\subseteq {\widehat{A}}_{i+2}$; thus, $(g{\jmath}, g{\ell})\in {E_{ed}}({G_r})$. Thus, $\{\Upsilon_p\}_{p = 1}^\infty$ is a non-linear multivalued ${G_r}_g$-contraction. Suppose further that $\{v_n\}$ is a sequence in $\Phi$ with $\rho(v_n, v)\longrightarrow 0$ as $n\longrightarrow \infty$, where $v_n\in \Upsilon_n(v_{n-1})\cap [v_{n-1}]_{G_r}^\alpha$ for all $n\in \mathbb{N}$, and $(gv_{n-1}, gv_n)\in {E_{ed}}({G_r})$ for all $n\in \mathbb{N}$. Clearly, infinitely many terms of $\{v_n\}$ are contained in ${\widehat{A}}_i$, thus, we can produce a subsequence $\{v_{n_\gamma}\}$such that $\rho(v_{n_{\gamma}}, v)\longrightarrow 0$ for all $\gamma\in \mathbb{N}$. Since ${\widehat{A}}_i$ is closed for each $i = \overline{1, \alpha}$, then $v\in \bigcap_{i = 1}^\alpha {\widehat{A}}_i$. It follows from the definition of ${E_{ed}}({G_r})$ that $(v_{n_\gamma}, v)\in {E_{ed}}({G_r})$ for all $\gamma\in \mathbb{N}$. Consequently, Theorem 2.2 can be applied to find $u^*\in \widetilde{\bigwedge}$ such that $g(u^*)\in \bigcap_{p\in \mathbb{N}}\Upsilon_p(u^*)$.

4.   Applications to integral equations

Integral equations are found to be of great usefulness in studying dynamical systems and stochastic processes. Some examples are in the areas of oscillation problems, sweeping processes, granular systems, control problems and so on. In a like manner, integral equations arise in several problems in mathematical physics, bio-mathematics, control theory, critical point theory for non-smooth energy functionals, differential variational inequalities, fuzzy set arithmetic, traffic problems, to mention but a few. Usually, the first most concerning problem in the study of differential or integral equations is the conditions for the existence of its solutions. Along this lane, many authors have proposed different fixed point approaches to obtain existence results for differential or integral equations in abstract spaces (see, e.g. ^[1,20,21,22]).

In this section, we examine new conditions for the existence of a unique solution to a more general version of the integral equation analyzed in ^[28], given as

where $f:\varpi\times \mathbb{R}\longrightarrow \mathbb{R}, \Gamma:\varpi^2\longrightarrow \mathbb{R}_+, h:\varpi\longrightarrow \mathbb{R}$ and $g:\varpi\longrightarrow \mathbb{R}$ are given continuous functions. Note that if, in (4.1), $h(t) = 0$ and $g = I_{\widetilde{\bigwedge}}$, which is the identity mapping on $\widetilde{\bigwedge}$, then Problem (4.1) represents an integral reformulation of physical phenomena such as the motion of a spring that is under the influence of a frictional force or a damping force. For some articles modeling real-life problems, via integral/differential equations, see ^{[10,11,12,13,14]} and the references therein.

Let $\widetilde{\bigwedge} = C(\varpi, \mathbb{R})$ be the set of all real-valued continuous functions defined on $\varpi$, and let $\rho(u, v) = \max_{t\in \varpi}|u(t)-v(t)|$. Then $(\widetilde{\bigwedge}, \rho)$ is a complete MS. Define the mapping $\Upsilon:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ by

and $\eta:\widetilde{\bigwedge}\longrightarrow \widetilde{\bigwedge}$ by $\eta u = gu,$ with $(\eta u)(t) = (gu)(t)$, for all $t\in \varpi$. Then, finding a solution of (4.1) is equivalent to showing that $\Upsilon$ and $g$ have a CoP.

Now, we investigate the existence of solution of (4.1) given the following hypotheses.

Theorem 4.1. Given the surjective function $g\in C(\varpi, \mathbb{R})$ and $f:\varpi\times \mathbb{R}\longrightarrow \mathbb{R}$ obeying:

(i) for all $t\in \varpi$,

(ii) we can find a function $\lambda_*:\mathbb{R}_+\longrightarrow [0, 1)$ such that

Then Problem (4.1) has a unique solution in $\widetilde{\bigwedge}$.

Proof. Let $u, v\in \widetilde{\bigwedge}$. Then, we have

for all $K\geq0$. This implies that for each $u, v\in \widetilde{\bigwedge}$, we get $\rho(\Upsilon u, \Upsilon v)\leq \lambda_*(\rho(\eta u, \eta v))\rho(\eta u, \eta v)$. Thus, by applying Corollary 2 with the graph ${G_r} = {G_r}_0$, where ${E_{ed}}({G_r}_0) = \widetilde{\bigwedge}\times \widetilde{\bigwedge}$, we can find $u^*\in \widetilde{\bigwedge}$ such that $\Upsilon u^* = \eta u^*$, where $(\eta u^*)(t) = (gu^*)(t)$ for each $t\in \varpi$. Thus, $u^*$ is the CoP of $\Upsilon$ and $g$, which corresponds to the solution of Problem (4.1).

An open problem

As a future assignment, we suggest the following: a discretized population balance for continuous systems at steady state can be modeled via the following integral equation:

It is not known whether the existence criteria for the solution of (4.3) can be examined using any of the results obtained in this work. The advantage of analyzing this type of problem is that it will allow us to examine the existence criteria of several non-linear physical phenomena.

5.   Conclusions

In this note, a new type of sequence of multivalued contractions under the name non-linear multivalued ${G_r}_g$-contractions on an MS with a graph is introduced (see Definition 2.1). CoP theorems (see Theorem 2.2 and Theorem 2.4) of a single-valued mapping and the new sequence of multivalued mappings were examined via appropriate hypotheses. A comparative illustration (Example 2.3) was constructed to authenticate our assumptions and establish some links between the obtained results herein and their analogues in the literature. Some significant results in an $\epsilon$-chainable MS and cyclic contractions were derived (see Corollaries 1, 2 and Theorems 3.1, 3.2) as some consequences of our findings. From an application view-point, one of the special cases of our theorems was used to investigate novel criteria for solving a more general Fredholm-type integral equation. As a future exercise, an open problem regarding a discretized population balance model whose solution may be discussed using any of the ideas put forward here was unveiled.

Conflict of interest

The authors declare that there are no conflicts of interest.