Research article

Existence of solutions of fractal fractional partial differential equations through different contractions

  • Received: 28 January 2024 Revised: 08 March 2024 Accepted: 18 March 2024 Published: 29 March 2024
  • MSC : 47H10, 54H25

  • In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the -ψ-contraction -type of the ˜F-contraction, within the context of ˆF-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.

    Citation: Muhammad Sarwar, Aiman Mukheimer, Syed Khayyam Shah, Arshad Khan. Existence of solutions of fractal fractional partial differential equations through different contractions[J]. AIMS Mathematics, 2024, 9(5): 12399-12411. doi: 10.3934/math.2024606

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  • In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the -ψ-contraction -type of the ˜F-contraction, within the context of ˆF-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.



    Fractional order problems are receiving considerable attention in different scientific fields because they can model complicated processes more precisely than conventional integer-order equations. Artificial neural networks (ANNs) have been one of the notable applications used in recent studies to solve fractional higher-order linear integro-differential equations [1]. By utilizing the natural parallel processing skills of ANNs, these equations can be effectively solved, leading to progress in areas such as physics, engineering, and finance. The stability study of pandemics like the COVID-19 outbreak has been improved by using Caputo-Fabrizio fractional differential equations (FDEs) [2,3]. This method allows for a more detailed comprehension of epidemic dynamics, which helps create successful containment tactics. Using spectral methods with fractional basis functions provides a robust framework for solving integral equations in various scientific fields, namely, fractional Fredholm integro-differential equations. Researching stability and the existence of solutions in complex structures, such as the triple problem of fractional hybrid delay differential equations, along with modern mathematical modeling techniques, provides valuable insights into the behavior of intricate dynamic systems [4,5,6].

    Nonlinear differential equations are widely recognized for describing a variety of physical events. Partial differential equations, specifically belonging to the category of the Cauchy problem, have long been recognized as effective mathematical tools for modeling real-world problems in various areas of engineering and science. There are numerous differential operators in use today in the literature, the most popular of which is related to the rate of change[7]. Because of a newly introduced parameter known as fractal dimension, it has now been demonstrated in several exceptional studies that the fractal derivative, a differential operator introduced by Wenfeng Chen, predicts aspects of nature more precisely than ordinary differentiation. The fractal derivative is a term used in applied mathematics to describe a variable scaled according to tα. New avenues of research for science, engineering, and technological progress have been opened because of these new mathematical tools. The fractal derivative was developed to describe physical phenomena that are beyond the scope of classical physical rules. Media having non-integral fractal dimensions do not conform to these supposedly Euclidean geometrical considerations. Fractal features are frequently seen in practical situations such as porous materials, aquifers turbulence, etc. [8,9].

    As a result, utilizing diverse numerical and analytical approaches for solving the nonlinear differential equations is essential for scientific problem identification [10,11]. The majority of researchers (see [12,13]) have looked at theoretical conclusions in different ways that prove the existence results for the FDEs. Afshari and Baleanu[14] have recently investigated the theoretical solution (existence and uniqueness) for some Atangana-Baleanu FDEs in the sense of Caputo. Also, Karapinar et al. [15] demonstrate the existence of solutions to ordinary and fractional boundary value problems (BVPs) with integral type boundary conditions (BC) in the context of some Caputo-type fractional operators.

    Therefore, fixed point theory has attracted much attention in recent decades; it is a beautiful technique for determining the solution of existence to differential/integral equations. For this reason, one of the efficient results proposed by Wardowski[16] guarantees the existence and uniqueness of a fixed point in the context of the usual metric space. Samet et al.[17] introduce the idea of -admissibility of mappings, which was subsequently expanded upon by Karapinar and Samet [18]. Gopal introduced a novel idea of -type ˜F-contraction mapping in their paper [19]. Recently, Jleli with Samet[20] has proposed the concept of F-metric space as an approach to extend the applicability of the Banach contraction principle.

    This manuscript employs various contractions, including the -ψ-contraction and -type of ˜F-contraction, along with outcomes from ˆF-metric and orbital metric spaces. These are utilized to obtain the theoretical solution for a general partial differential equation involving a fractal fractional differential operator

    FF0D(q,)=F(q,,(q,)),0<<1,(q,0)=, (1.1)

    where

    (q,)[0,]×[0,H],V=[0,]×[0,H](q,)C(V,R),

    and F is a function that is continuous and non-linear, i.e., F(0,0,(0,0))=0.

    Definition 1.1. [10] For a non-empty set , define md: ×[0,+), which is termed as a b-metric if these conditions hold:

    (1) If md(w,s)=0, then w=s for all w,s.

    (2) md(w,s)=md(s,w) for all w,s.

    (3) md(w,s)b[md(w,q)+md(q,s)].

    Then, we call (,md) a b-metric space.

    Consider (,md) to represent a complete b-metric space, and P is the set of functions ψ: [0,+)[0,+) possessing the following two properties:

    (1)ψ being increasing and continuous.

    (2)ψ(q)ψ(q)q, provided >0.

    Further, we assume that Q consist of elements in the form as non-decreasing mapping Φ, which are defined by Φ: [0,+)[0,1a) with a1.

    Definition 1.2. [17] Let be a non-empty set, Υ: , and : ×R such that

    (g,s)1(Υg,Υs)1for allg,s.

    Then Υ is termed as -admissible

    Definition 1.3. [11] Let (,md) represent a complete b-metric space, and Υ: , and : ×[0,+) such that,

    (g,s)ψ(a3md(Υg,Υs))Φ(ψ(md(g,s)))ψ(md(g,s)),

    where g,s, ΦQ, a1, and ψP. Then Υ is termed as an -ψ-contraction function.

    The upcoming result illustrates that -ψ-contractive mapping have a fixed point.

    Corollary 1.4. [17] Assume (,md) represents a complete b-metric space, and Υ: is an -ψ-contraction in a way that:

    (1) There exists in a manner that (,ψ)1.

    (2) {n},limn+n=, where and (n,n+1)1 implies (n,)1.

    Then Υ possesses a fixed point.

    For the proof of the following theorem consider

    =C(ς,R)andmd(,ω)=supq,∍∈ς(q)ω()2.

    Theorem 2.1. Let J: R2R in a way that

    ()

    F(q,,(q,))F(q,,ω(q,))ς2221B(,),

    then, Φ(ψ((q,)ω(q,)2))ψ((q,)ω(q,)2) for (q,)ς and (q,),ω(q,)C(ς,R) with J(,ω)0.

    () There exists 1C(ς,R) with J(,Υ1)0, where Υ: CC is defined as

    Υ()=+1ς()t01()1F(q,,(q,))d.

    () (q,)ς and ,ωC,J(,ω)0 emphasize the fact that J(Υ,Υω)0.

    () {n}C,n, where C and J(n,n+1)0, for nN; then there exists at least one solution of the problem (1.1).

    Proof. In problem (1.1) F a is nonlinear mapping, and

    FF0D(q,)=1ς(1)dd0(q,)()d. (2.1)

    Since 0(q,)()d is differentiable, Eq (2.1) can be converted into

    111ς(1)dd0(q,)()d.

    Consequently, Eq (1.1) could be transformed into

    (q,)(q,0)=1()1F(,,)d,1()1F(q,,)d.

    Consequently,

    (q,)=+1ς()01()1F(q,,)=Υ. (2.2)

    Here, we show that Υ has a fixed point

    ΥΥω2=1ς()01()1(F(q,,)F(q,,ω))d2{1ς()t01()1F(q,,)F(q,,ω)d}2{12221B(,)01()1Φ(ψ((q,)ω(q,)2))ψ((q,)ω(q,)2)d}2={12221B(,)Φ(ψ((q,)ω(q,)2))ψ((q,)ω(q,)2)01()1d}2{12221B(,)Φ(ψ(md(ω)))ψ(md(ω))21B(,)}2={122Φ(ψ(md(ω)))ψ(md(ω))}2=18Φ(ψ(md(ω)))ψ(md(ω)),

    for ,ωC(ς,R) with J(,ω)0, we have

    8ΥΥω2Φ(ψ(md(,ω)))ψ(md(,ω)).

    Now defining

    ∝:C(ς,R)×C(ς,R)[0,+)

    by

    (,ω)={1,if J(,ω)0,0,otherwise,

    and

    (,ω)ψ(8md(Υ,Υω))8md(Υ,Υω)=Φ(ψ(md(,ω)))ψ(md(,ω)).

    To justify Υ is -admissible, we have from ()

    (,ω)1J(,ω)0J(Υ,Υω)0⇒∝(Υ,Υω)1,

    ,ωC(ς,R). By (), it is obvious that C(ς,R) in a way that (,Υ)1, from () and the Corollary 1.4 there exist C(ς,R) that ensure =Υ.

    Next, the definition of -type ˜F-contraction is to be presented. For this, we need certain assumptions. Suppose that F represents the mappings of the form, ˜F: R+R in the sense that:

    (k1)˜F needs to be increasing strictly;

    (k2)limg0+gσ˜F(g)=0 for σ(0,1);

    (k3)limn+˜F(gn)= if and only if limn+gn=0 for every {gn}nN.

    Definition 2.2. [19] Let Υ: , : ×{+}(0,+) and ˜Fϝ; and there exists ε>0 such that

    ε+(,ω)˜F(md(Υ,Υω))˜F(md(,ω)),

    for each ,ω, with md(Υ,Υω)>0, and then Υ is called an -type ˜F-contraction on .

    Theorem 2.3. [19] Assume (,md) is a metric space, and Υ: such that

    (λ1) Υ is an -type ˜F-contraction;

    (λ2) There exist with (,Υ)1;

    (λ3) Υ is -admissible;

    (λ4) If {n} with (n,n+1)1 and n, then (n,)1;

    (λ5) ˜F is continuous. Then there exists such that Υ()=, and {Υn}nN converge to .

    For the proof of the next theorem the metric

    md(,ω)=sup(q,)ς|(q,)ω(q,)|=ω

    will be taken under consideration.

    Theorem 2.4. Let J: R2R in a way that

    (p1)

    FFωeες()21B(,)ω

    for (q,)ς and (q,),ω(q,)C(ς,R) with J(,ω)0;

    (p2) There exists 1C(ς,R) with J(,Υ1)0, where Υ:CC, defined by

    Υ()=+1ς()01()1F(q,,(q,))d;

    (p3) (q,)ς and ,ωC,J(,ω)0 imply that J(Υ,Υω)0;

    (p4) {n}C,n, where C and J(n,n+1)0, for nN; Then there exists at least one solution of the problem (1.1).

    Proof. The following integral equation can be formed from Eq (1.1):

    (q,)=+01()1F(q,,)d=Υ.

    To verify the fixed point of Υ, we have

    ΥΥω=1ς()01()1(F(q,,)F(q,,ω))dς()01()1F(q,,F(q,,ω))deε21B(,)ω01()1deε21B(,)ω21B(,).

    Consequently,

    ΥΥωeεω,ε+ln(ΥΥω)ln(ω).

    Or,

    ε+ln(md(ΥΥω))ln(md(ω)).

    Setting F()=ln, then quite smoothly it can be shown that F˜F.

    Next, defining as, : C×C{}[0,+) such that,

    (,ω)={1,if J(,ω)0,,else,

    then we deduce

    ε+(,ω)F(md(Υ,Υω))F(md(,ω))

    for ,ωC, and md(Υ,Υω)>0, and with utilisation of (p3), we have

    (,ω)1J(,ω)0J(Υ,Υω)0⇒∝(Υ,Υω)1

    for ,ωC. Hence, Υ is -admissible by (p2), and we have C such that (,Υ)1. From the condition (p4) and Theorem 2.3, we get =Υ, where C, and then there must be at least one solution for (1.1).

    Let ˆF be the family of functions ϑ: (0,+)R in a way that:

    (ζ1) 0<u<v ϑ(u)ϑ(v);

    (ζ2) n0 if and only if ϑ(n), where {n}(0,+).

    Definition 2.5. [20] Let be a non empty set, md: ×[0,+), ϑF, and ξ[0,+) in a way that, when x,v, the below conditions hold true:

    (υ1) md(x,t)=0x=t;

    (υ2) md(x,t)=md(t,x);

    (υ3) If {xi}ni=1 in the sense that (x1,xn)=(x,t), n2, we have

    md(x,t)>0ϑ(md(x,t))ϑ(n1i=1md(xi,xi+1))+ξ.

    Then (,md) is termed as an ˆF-metric space with ˆF-metric md.

    Convergence, Cauchyness, and sequence completeness are all defined in ˆF-metric space as similar as defined in standard metric space.

    Let η be the mappings ψ: [0,+)[0,+) in a way that:

    (η1) ψ is non-decreasing;

    (η2) +n=1ψn(e)<+, for eR+.

    Definition 2.6. [21] If a mapping Υ: is such that

    (,Υ)1(Υ,Υ2)1,

    for ∍∈ and : ×[0,+), then Υ is called -orbital admissible.

    Corollary 2.7. [21] Let (,md) be a complete ˆF-metric space, and Υ: and ψ: [0,+)[0,+) in the sense that:

    (α1)

    (,ω)md(Υ,Υω)ψ(Md(,ω)),

    where Md(,ω)=maxmd(,ω),md(,Υ),md(ω,Υω),,ω;

    (α2) Υ is -orbital admissible;

    (α3) (s,Υs)1 for s;

    (α4) ΥF is continuous and verify the condition (υ3), and ψ is continuous and satisfying Υ(s)>Υ(ψ(s))+ξ, 0<s<+, where ξ[0,+). Then Υ must have a fixed point.

    Assume =C(ς,R) and md: ×[0,+) defined by

    md(,ω)={eω,if ω,0,if=ω,

    where

    (q,)ω(q,)=sup(q,)ς|(q,)ω(q,)|,

    and then md is an ˆF-metric on . We have ϑF defined by

    ϑ()=1t

    for t>0 as well. So, it is obvious that ϑ(μ)>ϑ(ψ(μ))+ξ,μ>0, such that, ψ possess the properties

    ψ<μ1+μ,eψ(n)ψ(en),

    where n{0,1,2,3,}.

    The problem (1.1) has a solution in ˆF-metric space, which can be observed in the following theorem.

    Theorem 2.8. Suppose there is J: R2R in a way that

    (1)

    F(q,,(q,))F(q,,ω(q,))ς()21B(,)ψ(ω),

    where (q,)ς, and ,ω with J(,ω)0;

    (2) There exists 1 with J(1,Υ1)0 where Υ: is defined by

    Υ()=+1ς()01()1F(q,,(q,))d;

    (3) with J(,Υ)0 implies that J(Υ,Υ2)0. Then, there exist a fixed point of Υ.

    Proof. We can write Eq (1.1) as

    (q,)=+01()1F(q,,)d=Υ.

    To derive the fixed point of Υ, we have

    ΥΥω=1ς()01()1(F(q,,)F(q,,ω))dς()01()1F(q,,)F(q,,ω)d1T21B(,)01()1ψ(ω)d=ψ(ω)T21B(,)01()1dψ(ω).

    Thus, for ,ω having J(,ω)0, we have

    md(Υ,Υω)=eΥΥωeψ(ω)ψ(eω)=ψ(md(,ω))ψ(Md(,ω)).

    Define : ×[0,+) by

    (,ω)={1,if J(,ω),0,otherwise.

    Therefore,

    (,ω)md(Υ,Υω)md(Υ,Υω)ψ(Md(,ω)),

    for ,ω with md(Υ,Υω)0. By (3), we have

    (,Υ)1J(,Υ)0J(Υ,T2)0(Υ,Υ2)1.

    Hence, Υ is an orbital -admissible, and from the condition (2), there exist 1 in a way that (1,Υ1)1. Additionally, by (3) and the Corollary 2.7, we obtain in a way that =Υ. Hence there is a solution of the problem (1.1).

    Now for defining an orbitally complete metric space, suppose (,md) is a metric space and Υ: . If , then has the orbit in the set form as

    O(q)={Υnq:n=0,1,2,3},

    where Υn is the nth iteration of Υ and D(q) is the diameter of O(x). is characterised as Υ-orbitally complete metric space if all the Cauchy sequences from O(x) converge in for some q.

    Theorem 2.9. [22] Let (,md) represent Υ-orbitally complete metric space, Υ: and θ: N. Then Υ has a unique fixed point, if there exists ν>0 and q with 0<D<+ such as

    md(Υθ(q)(),Υθ(q)(ω))eνmd(,ω).

    Let

    =C(ς,R)andmd(,ω)=sup(q,)ς|ω|.

    The below theorem explores the existence and uniqueness of the problem (1.1) in Υ-orbitally complete metric space.

    Theorem 2.10. Let Υ: define by

    Υ()=+1ς()01()1F(q,,(q,))d

    and

    |F(q,,(q,)Fω(q,))|ς()21B(,)eν||||ω||,  
    |F(q,,)|+|F(q,,ω)|ς()21B(,)eν|||+|ω||.

    Then the problem (1.1) must have a unique solution.

    Proof. Equation (1.1) can be written as

    (q,)=+01()1F(q,,)d=Υ.

    To prove a unique solution of Υ, we have

    |ΥΥω|=|1ς()01()1(F(q,,)F(q,,ω))d|1ς()01()1|F(q,,F(q,,ω))|deν21B(,)01()1||||ω||d=eν21B(,)sup||||ω||01(t)1deνsup||||ω||.

    Also,

    |Υ|+|Υω|=|1ς()01()1F(q,,)d|+|1ς()01()1F(q,,ω)d|1ς()01()1(|F(q,,)|+|F(q,,ω)|)deν21B(,)01()1|||+|ω||d=eν21B(,)sup|||+|ω||01()1deνsup|||+|ω||sup|||+|ω||.

    Now,

    md(Υ2,Υ2ω)=sup|Υ2Υ2ω|=sup|ΥΥω|×sup|Υ+Υω|sup|ΥΥω|×sup(|Υ|+|Υω|)eνsup||||ω||×supsup|||+|ω||=eνsup||||ω||eνsup|ω|=eνmd(,ω).

    If we consider θ: N in a way that θ()=2 for every , then all necessities of Theorem 2.9 are true. As a result, the problem (1.1) ensures a unique solution in .

    Exploring the solutions for fractional differential equations has been a central focus of this research. There still needs to be more research regarding solutions for nonlinear differential equations that involve fractal fractional operators. This study has focused on introducing new contraction methods, specifically the -ψ-contraction and -type of ˜F-contraction, in the framework of ˆF-metric and orbitally metric spaces. This paper has developed particular fixed point theorems that provide a novel and direct approach for investigating the existence and uniqueness of solutions for general partial differential equations using fractal fractional operators. This discovery helps further the knowledge and use of fractional calculus in dealing with complicated nonlinear events and opens the door for future development in this field.

    The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

    Authors are thankful to Prince Sultan University for paying the APC and support through TAS research lab.

    The authors declare that they have no competing interests concerning the publication of this article.



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