Retail supply chains are intended to empower effectiveness, speed, and cost-savings, guaranteeing that items get to the end client brilliantly, giving rise to the new logistic strategy of cross-docking. Cross-docking popularity depends heavily on properly executing operational-level policies like assigning doors to trucks or handling resources to doors. This paper proposes a linear programming model based on door-to-storage assignment. The model aims to optimize the material handling cost within a cross-dock when goods are unloaded and transferred from the dock area to the storage area. A fraction of the products unloaded at the incoming gates is assigned to different storage zones depending on their demand frequency and the loading sequence. Numerical example considering a varying number of inbound cars, doors, products, and storage areas is analyzed, and the result proves that the cost can be minimized or savings can be intensified based on the feasibility of the research problem. The result explains that a variation in the number of inbound trucks, product quantity, and per-pallet handling prices influences the net material handling cost. However, it remains unaffected by the alteration in the number of material handling resources. The result also verifies that applying direct transfer of product through cross-docking is economical as fewer products in storage reduce the handling cost.
Citation: Taniya Mukherjee, Isha Sangal, Biswajit Sarkar, Qais Ahmed Almaamari. Logistic models to minimize the material handling cost within a cross-dock[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 3099-3119. doi: 10.3934/mbe.2023146
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Retail supply chains are intended to empower effectiveness, speed, and cost-savings, guaranteeing that items get to the end client brilliantly, giving rise to the new logistic strategy of cross-docking. Cross-docking popularity depends heavily on properly executing operational-level policies like assigning doors to trucks or handling resources to doors. This paper proposes a linear programming model based on door-to-storage assignment. The model aims to optimize the material handling cost within a cross-dock when goods are unloaded and transferred from the dock area to the storage area. A fraction of the products unloaded at the incoming gates is assigned to different storage zones depending on their demand frequency and the loading sequence. Numerical example considering a varying number of inbound cars, doors, products, and storage areas is analyzed, and the result proves that the cost can be minimized or savings can be intensified based on the feasibility of the research problem. The result explains that a variation in the number of inbound trucks, product quantity, and per-pallet handling prices influences the net material handling cost. However, it remains unaffected by the alteration in the number of material handling resources. The result also verifies that applying direct transfer of product through cross-docking is economical as fewer products in storage reduce the handling cost.
If the image of a point x under two single-valued mappings is x itself, then x is said to be a fixed point of these mappings. Banach [7] proved a meaningful result for contraction mappings. Due to its significance, several authors, like Acar et al. [3], Altun et al. [5], Aslantas et al. [6], Sahin et al. [27], Hussain et al. [17], Hammad et al. [14,15,16] and Ceng et al. [8,9,10,11] presented many related useful applications in fixed point theory. In [23,31], the authors showed a new iterative scheme for the solution of nonlinear mixed Volterra Fredholm type fractional delay integro-differential equations of different orders. Chistyakov [13] introduced the notion of a modular metric space. Mongkolkeha et al. [21] established some results in modular metric spaces for contraction mappings. Chaipunya et al. [12], Abdou et al. [2] and Alfuraidan et al. [4] showed fixed point results for multivalued mappings in modular metric spaces. Abdou et al. [1] proved fixed point theorems of pointwise contractions in modular metric spaces. Hussain et al. [19] discussed some fixed point theorems for generalized F-contractions in fuzzy metric and modular metric spaces. Later, Padcharoen et al. [22] introduced the concept of α-type F-contractions in modular metric spaces and showed fixed point and periodic point results for such a contraction. Recently, Rasham et al. [26] introduced a modular-like metric space and proved results for families of mappings in such spaces. In this research work, we prove existence of fixed point results for a hybrid pair of multivalued maps fulfilling generalized rational type F-contractions, by using a weaker class of strictly increasing mappings F rather than the class of mappings introduced by Wardowski [30].
Let us state the following preliminary concepts.
Definition 1.1. [26] Let B be a non-empty set. A function υ:(0,∞)×B×B→[0,∞) is said to be a modular-like metric on B, if for each e,i,o∈B and υ(a,i,o)=υa(i,o), the following hold:
(i) υa(i,o)=υa(o,i) for all a>0;
(ii) υa(i,o)=0 for all a>0 implies i=o;
(iii) υl+n(i,o)≤υl(i,e)+υn(e,o) for all l,n>0.
The pair (B,υ) is said to be a modular-like metric space. If we change (ii) by "υl(i,o)=0 for each l>0 iff i=o", then (B,υ) becomes a modular metric space. While, by changing (ii) with "υl(i,o)=0 for some l>0, such that i=o", we obtain a regular modular-like metric space. For s∈B and ε>0, ¯Cυl(s,ε)={t∈B:|υl(s,t)−υl(t,t)|≤ε} is a closed ball in (B,υ).
Example 1.2. Let B=[0,∞)×[0,∞). Define υ:(0,∞)×B×B→[0,∞) as
(i)υ(a,(e,p),(i,o))=e+p+i+oa,(ii)υ(a,(e,p),(i,o))=max{e,p,i,o}a. |
The functions given in (i) and (ii) are examples of a modular-like metric on B.
Definition 1.3. [26] Let (B,υ) be a modular-like metric space.
(i) A sequence (an)n∈N in B is said to be υ-convergent to a point a∈B for some l>0 if limn→+∞υl(an,a)=υl(a,a).
(ii) A sequence (an)n∈N in B is said to be an υ-Cauchy sequence for some l>0 if limn,m→∞υl(am,an) exists and is finite.
(iii) B is called υ-complete if each υ-Cauchy sequence (an)n∈N in B is υ-convergent to some a∈B, that is,
limn→+∞υl(an,a)=υl(a,a). |
(iv) If every sequence has a convergent subsequence, then B is called compact.
Definition 1.4. [26] Let (B,υ) be a modular-like metric space and U⊆B. An element p0 in U verifying
υl(s,U)=infp0∈Uυl(s,p0) |
is called a best approximation in U for s∈B. If each s∈B possesses a best approximation in U, then U is called a proximinal set.
From now on, let P(B) represent the set of proximinal compact subsets in B.
Example 1.5. Let B=[0,∞) and υl(s,r)=1w(s+r) with w>0. Take U=[7,8]. Then for any m∈B,
υl(m,U)=υl(m,[7,8])=infn∈[7,8]υl(m,n)=υl(m,7). |
So 7 is a best approximation in U for any m∈B. Moreover, [7,8] is a proximinal set.
Definition 1.6. [26] The mapping Hυl:P(B)×P(B)→[0,∞), given by
Hυl(X,Y)=max{supσ∈Xυl(σ,Y),supς∈Yυl(ς,X)}, |
is known as an υl- Hausdorff metric. Note that (P(B),Hυl) is named as an υl- Hausdorff metric space.
Example 1.7. Let B=[0,∞) and υl(θ,ϑ)=1l(θ+ϑ) with l>0. Taking W=[5,6] and Q=[9,10] we get Hυl(W,Q)=15l.
Definition 1.8. [26] Let (X,υ) be a modular-like metric space. υ is said to satisfy the △M-condition if limn,m→∞υp(xn,xm)=0, where p∈N implies limn,m→∞υl(xn,xm)=0, for some l>0.
Definition 1.9. [28] Let C≠Φ, Y:C→P(C) be a multivalued mapping, E⊆C and α:C×C→[0,+∞) be a function. Then Y is said to be α∗-admissible on E if α∗(Ye,Yz)=inf{α(l,m):l∈Ye,m∈Yz}≥1, whenever α(e,z)≥1 for all e, z ∈E.
Definition 1.10. [29] Let B≠Φ, Y:B→P(B) be a multi-valued mapping, R⊆B and α:B×B→[0,∞) be a function. Then Y is said to be α∗-dominated on R if for all v∈R, α∗(v,Yv)=inf{α(v,j):j∈Yv}≥1.
Definition 1.11. [30] Let (C,d) be a metric space. A self mapping H:C→C is said to be a Q-contraction if for each g,k∈C, there is τ>0 such that d(Ca,Cg)>0 implies
τ+Q(d(Ca,Cg))≤Q(d(a,g)), |
where Q:(0,∞)→R satisfies the following:
(F1) For any k∈(0,1), limσ→0+σkQ(σ)=0;
(F2) For each u,v>0 such that u<v, Q(u)<Q(v);
(F3) limn→+∞σn =0 if and only if limn→+∞Q(σn)=−∞ for every positive sequence {σn}∞n=1.
Let ϝ denote the set of mappings such that (F1)–(F3) hold.
Lemma 1.12. [26] Let (£,υ) be amodular-like metric space. Let (P(£),Hυl) be aHausdorff υl−metric-like space. Then, for all b∈U and foreach U,Y∈P(£), there is ba∈Y such that Hυl(U,Y)≥υl(a,ba).
Example 1.13. [24] Let W=R. Consider α:W×W→[0,∞) as
α(s,r)={1ifs>r14ifs≯r. |
Define L,N:W→P(W) by
Ls=[−4+s,−3+s]andNr=[−2+r,−1+r]. |
The α∗-dominated property for L and N holds. Note that L and N are not α∗-admissible.
Let (£,υ) be a modular-like metric space, δ0∈£, and R,C:£→ P(£) be two multifunctions on £. For δ1∈Rδ0 with υ1(δ0,Rδ0)=υ1(δ0,δ1), take δ2∈Cδ1 such that υ1(δ1,Cδ1)=υ1(δ1,δ2). Choose δ3∈Rδ2 such that υ1(δ2,Rδ2)=υ1(δ2,δ3). In this way, we get a sequence {CR(δn)} in £, where
δ2n+1∈Rδ2n,δ2n+2∈Cδ2n+1, |
for all n∈N∪{0}. Note that υ1(δ2n,Rδ2n)=υ1(δ2n,δ2n+1) and υ1(δ2n+1,Cδ2n+1)=υ1(δ2n+1,δ2n+2). {CR(δn)} is said to be a sequence in £ generated by δ0. If R=C, then we denote {£R(δn)} instead of {CR(δn)}.
Theorem 2.1. Let (£,υ) be a completemodular-like metric space. Suppose that υ is regular and verifiesthe △M-condition. Let δ0∈ £, α:£×£→[0,∞) and R,C:£→ P(£) be α∗-dominatedmultifunctions on £. Assume there are τ>0 and Q∈ϝsuch that
τ+Q(Hυ1(Rt,Cδ))≤Q(max{υ1(t,δ),υ1(t,Rt),υ2(t,Cδ)2,υ1(t,Rt).υ1(δ,Cδ)1+υ1(t,δ)}) | (2.1) |
where t,δ∈{CR(δn)}, α(t,δ)≥1 or α(δ,t)≥1, and Hυ1(Rt,Cδ)>0. Then the sequence {CR(δn)} generated by δ0 convergesto e∈£ and for each n∈N, α(δn,δn+1)≥1. Furthermore, if e satisfies (2.1), α(δn,e)≥1 and α(e,δn)≥1 forall integers n≥0, then R and C have a common fixed point e in £.
Proof. Consider a sequence {CR(δn)}. Obviously, δn∈£ for each integer n≥0. If j is odd, then j=2ˊı+1 for some ˊı∈N. By definition of α∗-dominated mappings, one has α∗(δ2ˊı,Rδ2ˊı)≥1 and α∗(δ2ˊı+1,Cδ2ˊı+1)≥1. Since α∗(δ2ˊı,Rδ2ˊı)≥1, one gets inf{α(δ2ˊı,b):b∈Rδ2ˊı}≥1. Also, δ2ˊı+1∈Rδ2ˊı and so α(δ2ˊı,δ2ˊı+1)≥1. Moreover, δ2ˊı+2∈Cδ2ˊı+1 and so α(δ2ˊı+1,δ2ˊı+2)≥1. In view of Lemma 1.12, we have
τ+Q(υ1(δ2ˊı+1,δ2ˊı+2))≤τ+Q(Hυ1(Rδ2ˊı,Cδ2ˊı+1))≤Q(max{υ1(δ2ˊı,δ2ˊı+1),υ1(δ2ˊı,Rδ2ˊı),υ2(δ2ˊı,Cδ2ˊı+1)2,υ1(δ2ˊı,Rδ2ˊı).υ1(δ2ˊı+1,Cδ2ˊı+1)1+υ1(δ2ˊı,δ2ˊı+1)})≤Q(max{υ1(δ2ˊı,δ2ˊı+1),υ1(δ2ˊı,δ2ˊı+1),υ1(δ2ˊı,δ2ˊı+1)+υ1(δ2ˊı+1,δ2ˊı+2)2,υ1(δ2ˊı,δ2ˊı+1).υ1(δ2ˊı+1,δ2ˊı+2)1+υ1(δ2ˊı,δ2ˊı+1)})≤Q(max{υ1(δ2ˊı,δ2ˊı+1),υ1(δ2ˊı+1,δ2ˊı+2)}). |
This implies
τ+Q(υ1(δ2ˊı+1,δ2ˊı+2))≤Q(max{υ1(δ2ˊı,δ2ˊı+1),υ1(δ2ˊı+1,δ2ˊı+2)}). | (2.2) |
Now, if
max{υ1(δ2ˊı,δ2ˊı+1),υ1(δ2ˊı+1,δ2ˊı+2)}=υ1(δ2ˊı+1,δ2ˊı+2), |
then from (2.2), we have
Q(υ1(δ2ˊı+1,δ2ˊı+2))≤Q(υ1(δ2ˊı+1,δ2ˊı+2))−τ, |
which is a contradiction. Therefore,
max{υ1(δ2ˊı,δ2ˊı+1),υ1(δ2ˊı+1,δ2ˊı+2)}=υ1(δ2ˊı,δ2ˊı+1) |
for all ˊı≥0. Hence, from (2.2), we have
Q(υ1(δ2ˊı+1,δ2ˊı+2))≤Q(υ1(δ2ˊı,δ2ˊı+1))−τ. | (2.3) |
Similarly, we have
Q(υ1(δ2ˊı,δ2ˊı+1))≤Q(υ1(δ2ˊı−1,δ2ˊı))−τ | (2.4) |
for all ˊı≥0. By (2.3) and (2.4), we have
Q(υ1(δ2ˊı+1,δ2ˊı+2))≤Q(υ1(δ2ˊı−1,δ2ˊı))−2τ. |
Repeating these steps, we get
Q(υ1(δ2ˊı+1,δ2ˊı+2))≤Q(υ1(δ0,δ1))−(2ˊı+1)τ. | (2.5) |
Similarly, we have
Q(υ1(δ2ˊı,δ2ˊı+1))≤Q(υ1(δ0,δ1))−2ˊıτ. | (2.6) |
By (2.5) and (2.6), we obtain
Q(υ1(δn,δn+1))≤Q(υ1(δ0,δ1))−nτ. | (2.7) |
Letting n→∞ in (2.7), one obtains
limn→∞Q(υ1(δn,δn+1))=−∞. |
Since Q∈ϝ,
limn→∞υ1(δn,δn+1)=0. | (2.8) |
Due to (F1) of ϝ, there is k∈(0,1) such that
limn→∞(υ1(δn,δn+1))k(Q(υ1(δn,δn+1))=0. | (2.9) |
By (2.7), for all n∈N, we obtain
(υ1(δn,δn+1))k(Q(υ1(δn,δn+1))−Q(υ1(δ0,δ1))≤−(υ1(δn,δn+1))knτ≤0. | (2.10) |
Using (2.8), (2.9) and taking n→∞ in (2.10), we have
limn→∞n(υ1(δn,δn+1))k=0. | (2.11) |
By (2.11), there is n1∈N such that n(υ1(δn,δn+1))k≤1 for all n≥n1, or
υ1(δn,δn+1)≤1n1kforalln≥n1. |
Letting p>0 and m=n+p> n>n1, we get
υp(δn,δm)≤υ1(δn,δn+1)+υ1(δn+1,δn+2)+⋯+υ1(δm,δm+1)≤∞∑j=n1j1k. |
Since k∈(0,1), 1k>1 and so the series ∑∞j=11j1k converges. Thus,
limn,m→∞υp(δn,δm)=0. |
Since υ satisfies the △M-condition, we have
limn,m→∞υ1(δn,δm)=0. | (2.12) |
Hence {CR(δn)} is Cauchy in the regular complete modular-like metric space (£,υ) and so there is e∈£ such that {CR(δn)}→e as n→∞ and thus
limn→∞υ1(δn,e)=0. | (2.13) |
Now, by Lemma 1.12, one obtains
τ+Q(υ1(δ2n+1,Ce)≤τ+Q(Hυ1(Rδ2n,Ce)). | (2.14) |
Now, there exists δ2n+1∈Rδ2n such that υ1(δ2n,Rδ2n)=υ1(δ2n,δ2n+1). From assumption, α(δn,e)≥1. Assume that υ1(e,Ce)>0. Then there is an integer p>0 such that υ1(δ2n+1,Ce)>0 for n≥p. Now, if Hυ1(Rδ2n,Ce)>0, then by (2.1), we have
τ+Q(υ1(δ2n+1,Ce))≤Q(max{υ1(δ2n,e),υ1(δ2n,e),υ1(δ2n,δ2n+1)+υ1(δ2n+1,Ce)2,υ1(δ2n,Rδ2n).υ1(Q,Ce)1+υ1(δ2n,e)}). |
Letting n→∞ and using (2.13), we get
τ+Q(υ1(e,Ce))≤Q(υ1(e,Ce)). |
Since Q is strictly increasing, (2.14) implies
υ1(e,Ce)<υ1(e,Ce). |
This is a contradiction. Hence υ1(e,Ce)=0 and so e∈Ce.
Similarly, we can show that υ1(e,Re)=0, that is, e∈Re. Hence e is a common fixed point of both mappings R and C in £.
Corollary 2.2. Let (£,υ) be a completemodular-like metric space. Suppose that υ is regular and verifiesthe △M-condition. Let α:£×£→[0,∞) and R,C:£→ P(£)be α∗-dominated multifunctions on £. Assume thereare τ>0 and Q∈ϝ such that
τ+Q(Hυ1(Rt,Cδ))≤Q(max{υ1(t,δ),υ1(t,Rt),υ2(t,Cδ)2,υ1(t,Rt).υ1(δ,Cδ)1+υ1(t,δ)}), |
where t,δ∈£, α(t,δ)≥1 or α(δ,t)≥1, and Hυ1(Rt,Cδ)>0. Then there exists a sequence {δn} in £ converging to e∈£ and for each n∈N, α(δn,δn+1)≥1. Also, if α(δn,e)≥1 and α(e,δn)≥1 for all integers n≥0, then R and C have a common fixed point e in £.
Example 2.3. Let £=R+∪{0}. Take υ2(r,m)=r+m and υ1(e,t)=12(e+t) for all e,t∈£. Define R,C:£→ P(£) by
Rv={[v3,2v3]ifv∈{1,13,112,136,1144,⋯}[7v,10v]ifv∉{1,13,112,136,1144,⋯} |
and
Cv={[v4,3v4]ifv∈{1,13,112,136,1144,⋯}[5v,13v]ifv∉{1,13,112,136,1144,⋯}. |
Suppose that v0=1. Then υ1(v0,Rv0)=υ1(1,R1)=υ1(1,13) and so v1=13. Now, υ1(v1,Cv1)=υ1(13,C13)=υ1(13,112) and thus v2=112. Now, υ1(v2,Rv2)=υ1(112,R112)=υ1(112,136) and so v3=136. Continuing in this way, we have {CR(vn)}={1,13,112,136,1144,⋯}. Define α:£×£→[0,∞) as
α(r,t)={1 ifr>t12otherwise. |
Let v,y∈{CR(vn)} with α(v,y)≥1. Then
Hυ1(Rv,Cy)=max{supa∈Rvυ1(a,Cy),supb∈Cyυ1(Rv,b)}=max{υ1(2v3,[y4,3y4]),υ1([v3,2v3],3y4)}=max{υ1(2v3,y4),υ1(v3,3y4)}=max{2v3+y4,v3+3y4}. |
Also,
max{υ1(v,y),υ1(v,Rv),υ2(v,Cy)2,υ1(v,Rv).υ1(y,Cy)1+υ1(v,y)}=max{v+y,v+v3,14(v+y4),(v+v3).(y+y4)1+v+y}. |
If Q(t)=lnt and τ=ln(1.2), then we have
τ+Q(Hυ1(Rv,Cy))≤Q(max{υ1(v,y),υ1(v,Rv),υ2(v,Cy)2,υ1(v,Rv).υ1(y,Cy)1+υ1(v,y)}). |
Hence all the conditions in Theorem 2.1 hold and so R and C possess a common fixed point.
Note that
Rv={[v3,2v3]ifv∈{1,13,112,136,1144,⋯}[7v,10v]ifv∉{1,13,112,136,1144,⋯} |
and
Cv={[v4,3v4]ifv∈{1,13,112,136,1144,⋯}[5v,13v]ifv∉{1,13,112,136,1144,⋯}. |
If v=2 and y=3, then we have
Hυ1(R2,C3)=max{supa∈R2υ1(a,C3),supb∈C3υ1(R2,b)}=max[{supa∈[14,20]υ1(a,[15,39]),supb∈[15,39]υ1([14,20],b)}]=max[{supa∈[14,20]υ1(a,15),supb∈[15,39]υ1(14,b)}]=max{υ1(20,15),υ1(14,39)}=max{20+15,14+39}=53. |
Also
max{υ1(v,y),υ1(v,Rv),υ2(v,Cy)2,υ1(v,Rv).υ1(y,Cy)1+υ1(v,y)}=max{υ1(2,3),υ1(2,[14,20]),υ2(2,[15,39])2,υ1(2,[14,20]).υ1(3,[15,39])1+υ1(2,3)}=max{5,16,174,(16)(18)6}=48. |
Now,
ln(1.2)+ln(53)>ln(48). |
This implies that
τ+F(Hυ1(R2,C3)>F(υ1(2,3)). |
So the condition (2.1) does not hold on the whole space. Hence Corollary 2.2 and the other existing results in modular metric spaces cannot be applied to ensure the existence of a common fixed point. However, Theorem 2.1 is valid here.
Taking R=C in Theorem 2.1, we may state the following corollary.
Corollary 2.4. Let (£,υ) be a completemodular-like metric space. Suppose υ is regular and the △M-condition holds. Let δ0∈£, α:£×£→[0,∞) and R:£→ P(£) be a α∗-dominated set-valuedfunction on £. Assume there are τ>0 and Q∈ϝ suchthat
τ+Q(Hυl(Rt,Rδ))≤Q(max{υ1(t,δ),υ1(t,Rt),υ2(t,Rδ),υ1(t,Rt).υ1(δ,Rδ)1+υ1(t,δ)}), | (2.15) |
where t,δ∈{£R(δn)}, α(t,δ)≥1, and Hυ1(Rt,Rδ)>0. Then, the sequence {£R(δn)} generated by δ0 converges to e∈£and for each integer n≥0, α(δn,δn+1)≥1.Also, if e satisfies (2.15) and either α(δn,e)≥1 or α(e,δn)≥1 for all integers n≥0, then R has afixed point e in £.
Jachymski [20] initiated the graph concept in fixed point theory. Hussain et al. [18] gave new results for graphic contractions. Recently, Younis et al. [32] discussed a significant result on the graphical structure of extended b-metric spaces and Shoaib et al. [29] established some results on graph dominated set-valued mappings in the setting of b-metric like spaces. Further results on graph theory can be seen in [24,25,28].
Definition 3.1. [29] Let A be a non-empty set and Υ=(V(Υ),L(Υ)) be a graph with V(Υ)=A. A mapping P from A into P(A) is said to be multi-graph dominated on A if for each ı∈A, we have (ı,ȷ)∈L(Υ), where ȷ∈Pa.
Theorem 3.2. Let (U,υ) be a complete modular-likemetric space endowed with a graph Υ and δ0∈R satisfying the following:
(i) R and C are multi-graph dominated functions on {CR(δn)};
(ii) There are τ>0 and Q∈ϝ such that
τ+Q(Hυ1(Rw,Ch))≤Q(max{υ1(w,h),υ1(w,Rw),υ2(w,Ch)2,υ1(w,Rw).υ1(h,Ch)1+υ1(w,h)}), | (3.1) |
where w,h∈{CR(δn)}, (w,h)∈L(Υ)or (h,w)∈L(Υ), and Hυ1(Rw,Ch)>0.Suppose that the regularity of R and the △M-conditionare verified. Then (δn,δn+1)∈L(Υ)and {CR(δn)}→δ∗. Also, if δ∗ satisfies (3.1), (δn,δ∗)∈L(Υ) and (δ∗,δn)∈L(Υ)for all integers n≥0, then R and C have a common fixed point in U.
Proof. Define α:U×U→[0,∞) as α(w,h)=1 if w∈U and (w,h)∈L(Υ), and α(w,h)=0, otherwise. The graph domination on U yields that (w,h)∈L(Υ) for all h∈Rw and (w,h)∈L(Υ) for each h∈Cw. So α(w,h)=1 for all h∈Rw and α(w,h)=1 for each h∈Cw. Thus inf{α(w,h):h∈Rw}=1 and inf{α(w,h):h∈Cw}=1. Hence α∗(w,Rw)=1 and α∗(w,Cw)=1 for any w∈R. So R and C are α∗-dominated on U. Furthermore,
τ+Q(Hυ1(Rw,Ch))≤Q(max{υ1(w,h),υ1(w,Rw),υ2(w,Ch)2,υ1(w,Rw).υ1(h,Ch)1+υ1(w,h)}), |
where w,h∈U∩{CR(δn)}, α(w,h)≥1 and Hυ1(Rw,Ch)>0. Also, (ii) is fulfilled. Due to Theorem 2.1, {CR(δn)} is a sequence in U and {CR(δn)}→δ∗∈U. Here, δn,δ∗∈U and either (δn,δ∗)∈L(Υ) or (δ∗,δn)∈L(Υ) yields that either α(δn,δ∗)≥1 or α(δ∗,δn)≥1. So all the hypotheses of Theorem 2.1 hold. Thus δ∗ is a common fixed point of R and C in U and υ1(δ∗,δ∗)=0.
In this section, some corollaries related to single-valued mappings in modular-like metric space are derived. Let (£,υ) be a modular-like metric space, δ0∈£ and R,C:£→£ be a pair of mappings. Let δ1=Rδ0, δ2=Cδ1, δ3=Rδ2. Consider a sequence {δn} in £ such that δ2n+1=Rδ2n and δ2n+2=Cδ2n+1, for integers n≥0. We represent this type of iteration by {CR(δn)}. {CR(δn)} is a sequence in £ generated by δ0. If R=C, then we use {£R(δn)} instead of {CR(δn)}.
Theorem 4.1. Let (£,υ) be a completemodular-like metric space. Suppose that the regularity of υ andthe △M-condition hold. Take r>0, δ0∈£, α:£×£→[0,∞)and let R,C:£→£ be α∗-dominatedmultifunctions on £. Then there are τ>0 and Q∈ϝsuch that
τ+Q(υ1(Rt,Cδ))≤Q(max{υ1(t,δ),υ1(t,Rt),υ2(t,Cδ)2,υ1(t,Rt).υ1(δ,Cδ)1+υ1(t,δ)}), | (4.1) |
where t,δ∈{CR(δn)}, α(t,δ)≥1, or α(δ,t)≥1, and υ1(Rt,Cδ)>0. Then α(δn,δn+1)≥1 for all integers n≥0and {CR(δn)}→h∈£. Also, if h verifies(4.1), α(δn,h)≥1 and α(h,δn)≥1 forall integers n≥0, then R and C admit a common fixed point h in £.
Proof. The proof is similar to the proof of Theorem 2.1.
Letting R=C in Theorem 4.1, we have the following corollary.
Corollary 4.2. Let (£,υ) be a complete modularlike metric space. Suppose that the regularity of υ and the △M-condition hold. Choose δ0∈£, α:£×£→[0,∞) and let R:£→£ be a single-valued function on £.Then there are τ>0 and Q∈ϝ such that
τ+Q(υ1(Rt,Rδ))≤Q(max{υ1(t,δ),υ1(t,Rt),υ2(t,Rδ)2,υ1(t,Rt).υ1(δ,Rδ)1+υ1(t,δ)}), | (4.2) |
where t,δ∈{£R(δn)}, α(t,δ)≥1, or α(δ,t)≥1, and υ1(Rt,Rδ)>0.Then α(δn,δn+1)≥1 for all integers n≥0and {δn}→h∈£. Also, if (4.2) holds for h, α(δn,h)≥1 and α(h,δn)≥1 forall integers n≥0, then R has a fixed point h.
In this section, we apply our work to solve integral equations.
Theorem 5.1. Let (£,υ) be a completemodular-like metric space. Suppose that the regularity of υ andthe △M-condition hold. Take r>0, δ0∈£ and let R,C:£→£ be α∗-dominatedmultifunctions on £. Then there are τ>0and Q∈ϝ such that
τ+Q(υ1(Rt,Cδ))≤Q(max{υ1(t,δ),υ1(t,Rt),υ2(t,Cδ)2,υ1(t,Rt).υ1(δ,Cδ)1+υ1(t,δ)}), | (5.1) |
where t,δ∈{CR(δn)}, and υ1(Rt,Cδ)>0.Then {CR(δn)}→f∈£. Also, if f verifies (5.1), then R and C admit a unique common fixed point f in £.
Let W=C([0,1],R+) be the family of continuous functions defined on [0,1]. The following are two integral equations:
u(e)=e∫0H(e,f,u(f))df, | (5.2) |
c(e)=e∫0G(e,f,c(f))df | (5.3) |
for all e∈[0,1], where H,G:[0,1]×[0,1]×W→R. For δ∈C([0,1],R+), define supremum norm as ‖, and take \tau > 0 arbitrarily. For all c, w\in C([0, 1], \mathbb{R} _{+}), define
\begin{eqnarray*} \upsilon _{1}(\delta , w) = \frac{1}{2}\sup\limits_{s\in \lbrack 0, 1]}\{\left\vert \delta (s)+w(s)\right\vert e^{-\tau s}\} = \frac{1}{2}\Vert \delta +w\Vert _{\tau }. \end{eqnarray*} |
It is clear that (C([0, 1], \mathbb{R} _{+}), d_{\tau }) is a complete modular-like metric space. So we have the following result.
Theorem 5.2. Suppose that
(i) H, G:[0, 1]\times \lbrack 0, 1]\times C([0, 1], \mathbb{R} _{+})\rightarrow \mathbb{R} ;
(ii) Define
\begin{eqnarray*} (Ru)(e) & = &\int\limits_{0}^{e}H(e, f, u(f))df, \\ (C\delta )(e) & = &\int\limits_{0}^{e}G(e, f, \delta (f))df. \end{eqnarray*} |
Assume that there is \tau > 0 such that
\begin{equation*} \left\vert H(e, f, u)+G(e, f, \delta )\right\vert \leq \frac{\tau M(u, \delta )}{ \tau M(u, \delta )+1} \end{equation*} |
for all e, f\in \lbrack 0, 1] and u, \delta \in C([0, 1], \mathbb{R} ^{+}), where
\begin{equation*} M(u, \delta ) = \max \left( \frac{1}{2}\left\{ \begin{array}{c} \left\Vert u+\delta \right\Vert _{\tau }, \left\Vert u+Ru\right\Vert _{\tau }, \\ \frac{\left\Vert u+Ru\right\Vert _{\tau }+\left\Vert \delta +C\delta \right\Vert _{\tau }}{2}, \\ \frac{\left\Vert u+Ru\right\Vert _{\tau }.\left\Vert \delta +C\delta \right\Vert _{\tau }}{1+\left\Vert u+\delta \right\Vert _{\tau }} \end{array} \right\} \right) . \end{equation*} |
Then (5.2) and (5.3) possess a unique solution.
Proof. By (ⅱ),
\begin{eqnarray*} \left\vert Ru+C\delta \right\vert & = &\int\limits_{0}^{e}\left\vert H(e, f, u)+G(e, f, \delta )\right\vert df \leq \int\limits_{0}^{e}\frac{\tau M(u, \delta )}{\tau M(u, \delta )+1} e^{\tau f}df \\ &\leq &\frac{\tau M(u, \delta )}{\tau M(u, \delta )+1}\int\limits_{0}^{e}e^{ \tau f}df \leq \frac{M(u, \delta )}{\tau M(u, \delta )+1}e^{\tau e}. \end{eqnarray*} |
This implies
\begin{equation*} \left\vert Ru+C\delta \right\vert e^{-\tau e}\leq \frac{M(u, \delta )}{\tau M(u, \delta )+1}, \end{equation*} |
\begin{equation*} \Vert Ru+C\delta \Vert _{\tau }\leq \frac{M(u, \delta )}{\tau M(u, \delta )+1}, \end{equation*} |
\begin{equation*} \frac{\tau M(u, \delta )+1}{M(u, \delta )}\leq \frac{1}{\Vert Ru+C\delta \Vert _{\tau }}, \end{equation*} |
\begin{equation*} \tau +\frac{1}{M(u, \delta )}\leq \frac{1}{\Vert Ru+C\delta \Vert _{\tau }}. \end{equation*} |
Thus
\begin{equation*} \tau -\frac{1}{\Vert Ru(e)+C\delta (e)\Vert _{\tau }}\leq \frac{-1}{ M(u, \delta )}. \end{equation*} |
All the conditions of Theorem 5.1 hold for Q(f) = \frac{-1}{f} for f > 0 and \upsilon _{1}(f, \delta) = \frac{1}{2}\Vert f+\delta \Vert _{\tau } . Hence both the integral Eqs (5.2) and \left(5.3\right) admit a unique common solution.
In this article, we have achieved some new results for a pair of set-valued mappings verifying a generalized rational Wardowski type contraction. Dominated mappings are applied to obtain some fixed point theorems. Applications on integral equations and graph theory are given. Moreover, we investigate our results in a more better new framework. New results in ordered spaces, modular metric space, dislocated metric space, partial metric space, b -metric space and metric space can be obtained as corollaries of our results. One can further extend our results to fuzzy mappings, bipolar fuzzy mappings and fuzzy neutrosophic soft mappings.
The authors declare that we have no conflict of interest.
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