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Research article Special Issues

Existence of solutions for Kirchhoff-double phase anisotropic variational problems with variable exponents

  • Received: 20 June 2024 Revised: 17 July 2024 Accepted: 29 July 2024 Published: 05 August 2024
  • MSC : 35J20, 35J60, 35J62

  • This paper is devoted to dealing with a kind of new Kirchhoff-type problem in RN that involves a general double-phase variable exponent elliptic operator ϕ. Specifically, the operator ϕ has behaviors like |τ|q(x)2τ if |τ| is small and like |τ|p(x)2τ if |τ| is large, where 1<p(x)<q(x)<N. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti–Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature.

    Citation: Wei Ma, Qiongfen Zhang. Existence of solutions for Kirchhoff-double phase anisotropic variational problems with variable exponents[J]. AIMS Mathematics, 2024, 9(9): 23384-23409. doi: 10.3934/math.20241137

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  • This paper is devoted to dealing with a kind of new Kirchhoff-type problem in RN that involves a general double-phase variable exponent elliptic operator ϕ. Specifically, the operator ϕ has behaviors like |τ|q(x)2τ if |τ| is small and like |τ|p(x)2τ if |τ| is large, where 1<p(x)<q(x)<N. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti–Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature.



    Let R and R+ denote the sets of all real numbers and nonnegative real numbers respectively, N the set of all positive integers and ¯A the closure and ¯coA the convex hull closure of A. Additionally, Ξ denotes a Banach space, Ω={Λ:Λ, bounded, closed and convex subset of Ξ}, B(Ξ)={Λ:Λ is bounded subset of Ξ}, kerm={ΛB(Ξ):m(Λ)=0} be the kernel of function m:B(Ξ)R+.

    Fixed point theory has been developed in two directions. One deals with contraction mappings on metric spaces, Banach contraction principle being the first important result in this direction. In the second direction, continuous operators are dealt with convex and compact subsets of a Banach space. Brouwer's fixed point theorem and its infinite dimensional form, Schuader's fixed point theorems are the two important theorems in this second direction. In this paper, Fix(Υ) denotes a set of fixed points of a mapping Υ in Λ.

    Theorem 1.1 (Brouwer's Fixed Point Theorem).[2] Every continuous mapping from the unit ball of Rn into itself has a fixed point.

    Theorem 1.2 (Schauder's Fixed Point Theorem).[15] Let Υ:ΛΛ a compact continuous operator, where ΛΩ. Then Fix(Υ).

    In Brouwer's and Schuader's fixed point theorems, compactness of the space under consideration is required as a whole or as a part. However, later the requirement of the compactness was relaxed by making use of the notion of a measure of noncompactness (in short MNC). Using the notion of MNC, the following theorem was proved by Darbo [5].

    Theorem 1.3. [5] Let ΛΩ and Υ:ΛΛ be a continuous function. If there exists k[0,1) such that

    m(Υ(Λ0))km(Λ0),

    where Λ0Λ and m is MNC defined on Ξ. Then Fix(Υ).

    It generalizes the renowned Schuader fixed point result and includes the existence portion of Banach contraction principle. In the sequel many extensions and generalizations of Darbo's theorem came into existence.

    The Banach principle has been improved and extended by several researchers (see [7,13,14,16]). Jleli and Samet [7] introduced the notion of θ-contractions and gave a generalization of the Banach contraction principle in generalized metric spaces, where θ:(0,)(1,) is such that:

    (θ1) θ is non-decreasing;

    (θ2) for every sequence {κj}(0,), we have

    limjθ(κj)=1limjκj=0+;

    (θ3) there exists L(0,) and (0,1) such that

    limκ0+θ(κ)1κ=L.

    Khojasteh et al. [9] introduced the concept of Z-contraction using simulation functions and established fixed point results for such contractions. Isik et al. [6] defined almost Z-contractions and presented fixed point theorems for such contractions. Cho [3] introduced the notion of L-contractions, and proved fixed point results under such contraction in generalized metric spaces. Using specific form of Z and L, we can deduce other known existing contractions. For some results concerning Z-contractions and its generalizations we refer the reader to [3] and the references cited therein. In particular, Chen and Tang [4] generalized Z-contraction with Zm-contraction and established Darbo type fixed point results.

    The aim of the present work is two fold. First we prove fixed point theorems under generalized Zm-contraction and then we prove fixed point results under Darbo type Lm-contraction in Banach spaces. It is interesting to see that several existing results in fixed point theory can be concluded from our main results. Furthermore, as an application of our results, we have proved the existence of solution to the Caputo fractional Volterra–Fredholm integro-differential equation

    cDμ(ϰ)=g(ϰ)+λ1ϰ0J1(ϰ,t)ξ1(t,μ(t))dt+λ2T0J2(ϰ,t)ξ2(t,μ(t))dt,

    under boundary conditions:

    aμ(0)+bμ(T)=1Γ()T0(Tt)1J3(ϰ,t)dt,

    where (0,1], cD is the Caputo fractional derivative, λ1,λ2 are parameters, and a,b>0 are real constants, μ,g:[0,T]R, J1,J2,J3:[0,T]×[0,T]R and ξ1,ξ2:[0,T]×RR are continuous functions. For the validity of existence result we construct an example.

    In this section, we recall some definitions and results which are further considered in the next sections, allowing us to present the results. The concept of MNC was introduced in [1] as follows:

    Definition 2.1. [1] A map m:B(Ξ)R+ is MNC in Ξ if for all Λ1,Λ2B(Ξ) it satisfies the following conditions:

    (i) kerm and relatively compact in Ξ;

    (ii) Λ1Λ2  m(Λ1)m(Λ2);

    (iii) m(¯Λ1)=m(Λ1);

    (iv) m(¯coΛ1)=m(Λ1);

    (v) m(ηΛ1+(1η)Λ2)ηm(Λ1)+(1η)m(Λ2) η[0,1];

    (vi) if {Λn} is a sequence of closed sets in B(Ξ) with Λn+1Λn, nN and limn+m(Λn)=0, then Λ=+n=1Λn.

    The Kuratowski MNC [11] is the function m:B(Ξ)R+ defined by

    m(K)=inf{ε>0:Kni=1Si,SiΞ,diam(Si)<ε},

    where diam(S) is the diameter of S.

    Khojasteh et al. [9] introduced the concept of Z-contraction using simulation functions as follows:

    Definition 2.2. A function Z:R+×R+R is simulation if:

    (Z1) Z(0,0)=0;

    (Z2) Z(κ1,κ2)<κ2κ1, for all κ1,κ2>0;

    (Z3) if {κn} and {κn} are two sequences in (0,) such that limκnn=limκnn>0, then

    limZ(κn,κn)n<0.

    Roldán-López-de-Hierro et al. [12] slightly modified the Definition 2.2 of [9] as follows.

    Definition 2.3. A mapping Z:R+×R+R is simulation if:

    (Z1) Z(0,0)=0;

    (Z2) Z(κ1,κ2)<κ2κ1, for all κ1,κ2>0;

    (Z3) if {κn}, {κn} are sequences in (0,) such that limκnn=limκnn>0 and κn<κn, then

    limZ(κn,κn)n<0.

    Every simulation function in the original Definition 2.2 is also a simulation function in the sense of Definition 2.3, but the converse is not true, see for instance [12]. Note that Z={Z:Z is a simulation function in the sense of Definition 2.3}. The following are some examples of simulation functions.

    Example 2.4. The mapping Z:R+×R+R defined by:

    1. Z(κ1,κ2)=κ2f(κ1)κ1, for all κ1,κ2R+, where f:R+R+ is a lower semi-continuous function such that f1(0)={0},

    2. Z(κ1,κ2)=κ2φ(κ1)κ1, for all κ1,κ2R+, where φ:R+R+ is a continuous function such that φ(κ1)={0}κ1=0,

    3. Z(κ1,κ2)=κ2(κ2)κ1, for all κ1,κ2R+, where :R+R+ is a function such that lim supnr+ (κ1)<1,

    4. Z(κ1,κ2)=ϕ(κ2)κ1, for all κ1,κ2R+, where ϕ:R+R+ is an upper semi-continuous function such that ϕ(κ1)<κ1, for all κ1>0 and ϕ(0)=0,

    5. Z(κ1,κ2)=κ2κ10λ(x)dx, for all κ1,κ2R+, where λ:R+R+ is a function such that ϵ0λ(x)dx exists and ϵ0λ(x)dx>ϵ, for every ϵ>0,

    6. Z(κ1,κ2)=κ2κ2+1κ1, for all κ1,κ2R+,

    are simulation functions.

    Cho [3] introduced the notion of L-simulation function as follows:

    Definition 2.5. An L-simulation function is a function L:[1,)×[1,)R satisfying the following conditions:

    (L1) L(1,1)=1;

    (L2) L(κ1,κ2)<κ2κ1, for all κ1,κ2>1;

    (L3) if {κn} and {κn} are two sequences in (1,) such that limκnn=limκnn>1 and κn<κn, then

    limL(κn,κn)n<1.

    Note that L(1,1)<1, for all t>1.

    Example 2.6. The functions Lb,Lw:[1,)×[1,)R defined by

    1. Lb(κ1,κ2)=κn2κ1, for all κ1,κ21, where n(0,1);

    2. Lw(κ1,κ2)=κ2κ1ϕ(κ2) κ1,κ21, where ϕ:[1,)[1,) is lower semicontinuous and nondecreasing with ϕ1({1})=1,

    are L-simulation functions.

    Definition 2.7. [8] A continuous non-decreasing function ϕ:R+R+ such that φ(t)=0 if and only if t=0 is called an altering distance function.

    In this section, we obtain the results on generalized Zm-contraction. First we give the following definition.

    Definition 3.1. Let ΛΩ. A self-mapping Υ on Λ is called generalized Zm-contraction if there exists ZZ such that

    Z(m(Υ(Λ1)),Δ(Λ1,Λ2))0, (3.1)

    where Λ1 and Λ2 are subsets of Λ, m(Λ1),m(Υ(Λ1)),m(Υ(Λ2))>0, m is MNC defined in Ξ and

    Δ(Λ1,Λ2)=max{m(Λ1),m(Υ(Λ1)),m(Υ(Λ2)),12m(Υ(Λ1)Υ(Λ2))}.

    Using the notion of Zm-contraction, we establish the main result of this section.

    Theorem 3.2. Let Υ:ΛΛ be a continuous function, where ΛΩ. Assume that there exists ZZ such that Z non-decreasing function and Υ is a generalized Zm-contraction. Then Fix(Υ).

    Proof. Define a sequence {Λn}n=0 such that

    Λ0=Λ and  Λn=¯co(ΥΛn1), for all  nN. (3.2)

    We need to prove that Λn+1Λn and ΥΛnΛn, for all nN. For proof of the first inclusion, we use induction. If n=1, then by (3.2), we have Λ0=Λ and Λ1=¯co(ΥΛ0)Λ0. Next, assume that ΛnΛn1, then ¯co(Υ(Λn))¯co(Υ(Λn1)), using (3.2), we get the first inclusion

    Λn+1Λn. (3.3)

    To obtained the second inclusion, using the inclusion (3.3) we have

    ΥΛn¯co(ΥΛn)=Λn+1Λn. (3.4)

    Thus Λn+1Λn and ΥΛnΛn,nN.

    Now, we discuss two cases, depending on the values of m. If we consider m as a non-negative integer with m(Λm)=0, then Λm is a compact set and hence by Theorem 1.2, Υ has a fixed point in ΛmΛ. Instead, assume that m(Λn)>0, nN. Then on setting Λ1=Λn+1 and Λ2=Λn in contraction (3.1), we have

    Z(m(Υ(Λn+1)),Δ(Λn+1,Λn))0, (3.5)

    where

    Δ(Λn,Λn+1)=max{m(Λn),m(Υ(Λn)),m(Υ(Λn+1)),12m(Υ(Λn)Υ(Λn+1))}max{m(Λn),m(Λn),m(Λn+1),12m(ΛnΛn+1)}=max{m(Λn),m(Λn),m(Λn+1),12m(Λn)}=m(Λn),

    that is,

    Δ(Λn,Λn+1)m(Λn). (3.6)

    Using inequality (3.6) and the axiom (Z2) of Z-simulation function, inequality (3.5) becomes,

    0Z(m(Υ(Λn+1)),Δ(Λn+1,Λn))Z(m(Λn+1),m(Λn))m(Λn)m(Λn+1), (3.7)

    that is, m(Λn)m(Λn+1) and hence, {m(Λn)} is a decreasing sequence of positive real numbers. Thus, we can find r0 such that limn m(Λn)=r. Next, we claim that r=0. To support our claim, suppose that r0, that is, r>0. Let un=m(Λn+1) and vn=m(Λn), then since un<vn, so by the axiom (Z3) of Z-simulation function, we have

    lim supn Z(m(Λn+1),m(Λn))=lim supn Z(un,vn)<0,

    which is contradiction to (3.7). Thus r=0 and hence {Λn} is a sequence of closed sets in B(Ξ) with Λn+1Λn, for all nN and limn m(Λn)=0, so the intersection set Λ=+n=1Λn is non-empty, closed and convex subset of Λ. Furthermore, since ΛΛn, for all nN, so by Definition 2.1(ii), m(Λ)m(Λn), for all nN. Thus m(Λ)=0 and hence Λkerm, that is, Λ is bounded. But Λ is closed so that Λ is compact. Therefore by Theorem 1.2, Fix(Υ). From Theorem 3.2 we obtain the following corollaries. We assume that ΛΩ.

    Corollary 3.3. Let Υ:ΛΛ be a continuous function such that

    ψ1(m(Υ(Λ1)))ψ2(max{m(Λ1),m(Υ(Λ1)),m(Υ(Λ2)),12m(Υ(Λ1)Υ(Λ2))}),

    for any non-empty subsets Λ1 and Λ2 of Λ, where m is MNC. Then Fix(Υ).

    Corollary 3.4. Let Υ:ΛΛ be a continuous function such that

    m(Υ(Λ1))Δ(Λ1,Λ2)f(Δ(Λ1,Λ2)),

    for any non-empty subsets Λ1 and Λ2 of Λ, where m is MNC and f:R+R+ is a lower semi-continuous function such that f1(0)={0}. Then Fix(Υ).

    The conclusion of Corollary 3.4 is true if f:R+R+ is continuous function such that f(t)=0   t=0.

    Corollary 3.5. Let Υ:ΛΛ be a continuous function such that

    m(Υ(Λ1))(Δ(Λ1,Λ2))Δ(Λ1,Λ2),

    for any non-empty subsets Λ1 and Λ2 of Λ, where m is MNC and :R+R+ is a function with lim supnr+ (t)<1. Then Fix(Υ).

    Corollary 3.6. Let Υ:ΛΛ be a continuous function such that

    m(Υ(Λ1))ϕ(max{m(Λ1),m(Υ(Λ1)),m(Υ(Λ2)),12m(Υ(Λ1)Υ(Λ2))}),

    for any non-empty subsets Λ1 and Λ2 of Λ, where m is MNC and ϕ:R+R+ is an upper semi-continuous mapping with ϕ(t)<t, t>0 and ϕ(0)=0. Then Fix(Υ).

    Corollary 3.7. Let Υ:ΛΛ be a continuous function such that

    m(Υ(Λ1))0λ(x)dxmax{m(Λ1),m(Υ(Λ1)),m(Υ(Λ2)),12m(Υ(Λ1)Υ(Λ2))},

    for any non-empty subsets Λ1 and Λ2 of Λ, where m is MNC and λ:R+R+ is a mapping such that ϵ0λ(x)dx exists and ϵ0λ(x)dx>ϵ, for every ϵ>0. Then Fix(Υ).

    Corollary 3.8. Let Υ:ΛΛ be a continuous function such that

    m(Υ(Λ1))Δ(Λ1,Λ2)1+Δ(Λ1,Λ2),

    for any non-empty subsets Λ1 and Λ2 of Λ, where m is MNC. Then Fix(Υ).

    In this section, we obtain some results on Lm-contraction. Let us denote by Θ the class of all functions θ:(0,)(1,) that satisfy conditions (θ1) and (θ2). First we introduce the notion of Lm-contraction as:

    Definition 4.1. Let ΛΩ. A self-mapping Υ on Λ is called Lm-contraction with respect to L if there exist θΘ such that, for all ΛΛ with m(Λ)>0,

    L(θ(m(Υ(Λ))),θ(m(Λ)))1, (4.1)

    where m is MNC defined in Ξ.

    Theorem 4.2. Let Υ:ΛΛ be a continuous function. Assume that there exist θΘ and Υ is Lm-contraction with respect to L. Then Fix(Υ).

    Proof. Define a sequence {Λn}n=0 such that

    Λ0=Λ and  Λn=¯co(ΥΛn1), for all  nN. (4.2)

    Then Λn+1Λn and ΥΛnΛn, nN.

    Now, we discuss two cases, depending on the values of m. If we consider m as a non-negative integer with m(Λm)=0, then Λm is a compact set and hence by Theorem 1.2, Υ has a fixed point in ΛmΛ. Instead, assume that m(Λn)>0, for all nN. Then on setting Λ=Λn in contraction (4.1), we have

    1L(θ(m(Υ(Λn))),θ(m(Λn)))θ(m(Λn))θ(m(Υ(Λn))), (4.3)

    that is,

    θ(m(Υ(Λn)))θ(m(Λn)).

    Since θ is nondecreasing, so that

    m(Υ(Λn))m(Λn). (4.4)

    Now, using inequality (4.4), we have

    m(Λn+1)=m(¯co(Υ(Λn)))=m(Υ(Λn))m(Λn),

    that is, m(Λn+1)m(Λn) and hence, {m(Λn)} is a decreasing sequence of positive real numbers. Thus, we can find r0 with limn m(Λn)=r. Next, we claim that r=0. To support our claim, suppose that r0. Then in view of (θ2), we get

    limn θ(m(Λn))1,

    which implies that

    limn θ(m(Λn))>1. (4.5)

    Let un=θ(m(Λn+1)) and vn=θ(m(Λn)), then since unvn, so by the axiom (L3) of L-simulation function, we have

    1lim supn L(m(Λn+1),m(Λn))=lim supn L(un,vn)<1,

    which is contradiction. Thus r=0 and hence {Λn} is a sequence of closed sets from B(Ξ) such that Λn+1Λn, for all nN and limn m(Λn)=0, so the intersection set Λ=+n=1Λn is non-empty, closed and convex subset of Λ. Furthermore, since ΛΛn, for all nN, so by Definition 2.1(ii), m(Λ)m(Λn), for all nN. Thus m(Λ)=0 and hence Λkerm, that is, Λ is bounded. But Λ is closed so that Λ is compact. Therefore by Theorem 1.2, Fix(Υ).

    By taking L=Lb in Theorem 4.2, we obtain the following result.

    Corollary 4.3. Let Υ:ΛΛ be a continuous functions such that, for all ΛΛ with m(Λ)>0,

    θ(m(Υ(Λ)))(θ(m(Λ)))k, (4.6)

    where θΘ and k(0,1). Then Fix(Υ).

    Remark 4.4. Corollary 4.3 is the Darbo type version of Theorem 2.1 in [7].

    By taking L=Lw in Theorem 4.2, we obtain the next result.

    Corollary 4.5. Let Υ:ΛΛ be a continuous functions such that, for all ΛΛ with m(Λ)>0,

    θ(m(Υ(Λ)))θ(m(Λ))ϕ(θ(m(Λ))), (4.7)

    where θΘ and ϕ:[1,)[1,) is lower semi-continuous and nondecreasing with ϕ1({1})=1. Then Fix(Υ).

    By taking θ(t)=et, for all t>0 in Corollary 4.5, we obtain next result.

    Corollary 4.6. Let Υ:ΛΛ be a continuous functions such that, for all ΛΛ with m(Λ)>0,

    m(Υ(Λ))m(Λ)φ(m(Λ)), (4.8)

    where φ:R+R+ is lower semi-continuous and nondecreasing with φ1({0})=0. Then Fix(Υ).

    Remark 4.7. Corollary 4.6 is the Rhoades's Theorem of Darbo type [13].

    Let B(a,r) be the closed ball with center at a and radius r and Br be the ball B(0,r). We check the existence of solution to Caputo fractional Volterra–Fredholm integro differential equation

    cDμ(ϰ)=g(ϰ)+λ1ϰ0J1(ϰ,t)ξ1(t,μ(t))dt+λ2T0J2(ϰ,t)ξ2(t,μ(t))dt, (5.1)

    under boundary conditions:

    aμ(0)+bμ(T)=1Γ()T0(Tt)1J3(ϰ,t)dt, (5.2)

    where (0,1], cD is the Caputo fractional derivative, λ1,λ2 are parameters, and a,b>0 are real constants, μ,g:[0,T]R, J1,J2,J3:[0,T]×[0,T]R and ξ1,ξ2:[0,T]×RR are continuous functions.

    Lemma 5.1. [10] For plR, l=0,1,,r1, we have

    I[cDh(t)]=h(t)+p0+p1t+p2t2+...+pr1tr1.

    Using Lemma 5.1, we can easily establish the following result.

    Lemma 5.2. Problem (5.1) is equivalent to the integral equation

    μ(ϰ)=1Γ()ϰ0(ϰϑ)1g(ϑ)dϑb(a+b)Γ()T0(Tϑ)1g(ϑ)dϑ+1Γ()ϰ0(ϰϑ)1(λ1ϑ0J1(ϑ,t)ξ1(t,μ(t))dt+λ2T0J2(ϑ,t)ξ2(t,μ(t))dt)dϑb(a+b)Γ()T0(Tϑ)1(λ1ϑ0J1(ϑ,t)ξ1(t,μ(t))dt+λ2T0J2(ϑ,t)ξ2(t,μ(t))dt)dϑ+1(a+b)Γ()T0(Tt)1J3(ϰ,t)dt. (5.3)

    Proof. Using Lemma 5.1, we obtain

    μ(ϰ)=c0+I(g(ϰ))+λ1I(ϰ0J1(ϰ,t)ξ1(t,μ(t))dt+λ2T0J2(ϰ,t)ξ2(t,μ(t))dt) (5.4)

    Apply boundary conditions, we deduce that

    c0=b(a+b)Γ()T0(Tϑ)1g(ϑ)dϑ1(a+b)Γ()T0(Tt)1J3(ϰ,t)dt+b(a+b)Γ()T0(Tϑ)1(λ1ϑ0J1(ϑ,t)ξ1(t,μ(t))dt+λ2T0J2(ϑ,t)ξ2(t,μ(t))dt)dϑ.

    Thus by substituting the values of c0 in (5.4), we get integral equation (5.3).

    Notice that the solution of Eq (5.1) is equivalent to Eq (5.3). Now, we are in a position to present the existence result.

    Theorem 5.3. Let μ,νBr,ϑ,τ[0,T], and a,b>0 be real constants. If μ,g:[0,T]R, J1,J2,J3:[0,T]×[0,T]R and ξ1,ξ2:[0,T]×RR are continuous functions satisfying the following axioms:

    1. there exist Ci>0,i=1,2 such that

    |ξi(t,μ(t))ξi(t,ν(t))|Ci (5.5)

    2. there exist real numbers \lambda_{1} and \lambda_{2} with \left|\lambda_{1}\right|C_{1}\mathfrak{K}_{1} +\left|\lambda_{2}\right|C_{2}\mathfrak{K}_{2} < \frac{\Gamma(\wp+1)}{2\mathfrak{T}^{p}} such that

    \begin{equation} \frac{2\left[\left\|\mathfrak{g}\right\|+\left|\lambda_{1}\right|\mathfrak{K}_{1}\mathfrak{F}_{1} +\left|\lambda_{2}\right|\mathfrak{K}_{2}\mathfrak{F}_{2}\right]+\mathfrak{K}_{3}} {\mathfrak{T}^{-\wp}\Gamma(\wp+1)-2\left[\left|\lambda_{1}\right|C_{1}\mathfrak{K}_{1} +\left|\lambda_{2}\right|C_{2}\mathfrak{K}_{2}\right]} \leq r, \end{equation} (5.6)

    where \mathfrak{F}_{1} = {\sup} |\xi_{1}\left(t, 0\right)|, \mathfrak{F}_{2} = {\sup} |\xi_{2}\left(t, 0\right)| and

    \begin{equation} \mathfrak{K}_{1} = \sup\int_{0}^{\vartheta}\left|\mathfrak{J}_{1}\left(\vartheta, \tau\right)\right|d\tau < \infty, \end{equation} (5.7)

    and

    \begin{equation} \mathfrak{K}_{i} = \sup\int_{0}^{\mathfrak{T}}\left|\mathfrak{J}_{i}\left(\vartheta, \tau\right)\right|d\tau < \infty, \ \ i = 2, 3. \end{equation} (5.8)

    Then problem (5.3) has a solution in B_{r} , equivalently problem (5.1) has a solution in B_{r} .

    Proof. Let B_{r} = \left\{\mu\in \mathcal{C}\left(\left[0, \mathfrak{T}\right], \mathbb{R}\right):\left\|\mu\right\|\leq r\right\} . Then, B_{r} is a non-empty, closed, bounded, and convex subset of \mathcal{C}\left(\left[0, \mathfrak{T}\right], \mathbb{R}\right) . Define the operator \Upsilon : B_{r}\rightarrow B_{r} by

    \begin{equation*} \label{App1Z-int-op} \begin{split} \Upsilon \mu(\varkappa) = &\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}\mathfrak{g}(\vartheta)d\vartheta -\frac{b}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}\mathfrak{g}(\vartheta)d\vartheta\\ &+\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}\left(\lambda_{1}\int_{0}^{\vartheta}\mathfrak{J}_{1}\left(\vartheta, t\right) \xi_{1}\left(t, \mu\left(t\right)\right)dt +\lambda_{2}\int_{0}^{\mathfrak{T}}\mathfrak{J}_{2}\left(\vartheta, t\right)\xi_{2}\left(t, \mu\left(t\right)\right)dt\right)d\vartheta\\ &-\frac{b}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}\bigg(\lambda_{1} \int_{0}^{\vartheta}\mathfrak{J}_{1}\left(\vartheta, t\right)\xi_{1}\left(t, \mu\left(t\right)\right)dt\\ &+\lambda_{2}\int_{0}^{\mathfrak{T}}\mathfrak{J}_{2}\left(\vartheta, t\right)\xi_{2}\left(t, \mu\left(t\right)\right)dt\bigg)d\vartheta +\frac{1}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-t)^{\wp-1} \mathfrak{J}_{3}\left(\varkappa, t\right) dt. \end{split} \end{equation*}

    Our first claim is \Upsilon : B_{r}\rightarrow B_{r} is well-defined. Let \mu\in B_{r} , for some r . Then for all \varkappa\in\left[0, \mathfrak{T}\right] , we have

    \begin{align*} \begin{split} \left|\Upsilon \mu(\varkappa)\right| \leq&\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}\left|\mathfrak{g}(\vartheta)\right|d\vartheta +\frac{b}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}\left|\mathfrak{g}(\vartheta)\right|d\vartheta\\ &+\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1} \bigg(\left|\lambda_{1}\right|\int_{0}^{\vartheta}\left|\mathfrak{J}_{1}\left(\vartheta, t\right)\right| \left|\xi_{1}\left(t, \mu\left(t\right)\right)\right|dt\\ &+\left|\lambda_{2}\right|\int_{0}^{\mathfrak{T}}\left|\mathfrak{J}_{2}\left(\vartheta, t\right)\right| \left|\xi_{2}\left(t, \mu\left(t\right)\right)\right|dt\bigg)d\vartheta +\frac{1}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-t)^{\wp-1} \left|\mathfrak{J}_{3}\left(\varkappa, t\right)\right|dt\\ &+\frac{b}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1} \bigg(\left|\lambda_{1}\right|\int_{0}^{\vartheta}\left|\mathfrak{J}_{1}\left(\vartheta, t\right)\right| \left|\xi_{1}\left(t, \mu\left(t\right)\right)\right|dt\\ &+\left|\lambda_{2}\right|\int_{0}^{\mathfrak{T}}\left|\mathfrak{J}_{2}\left(\vartheta, t\right)\right| \left|\xi_{2}\left(t, \mu\left(t\right)\right)\right|dt\bigg)d\vartheta\\ \leq&\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}\left|\mathfrak{g}(\vartheta)\right|d\vartheta +\frac{1}{\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}\left|\mathfrak{g}(\vartheta)\right|d\vartheta\\ &+\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1} \bigg\{\left|\lambda_{1}\right|\int_{0}^{\vartheta}\left|\mathfrak{J}_{1}\left(\vartheta, t\right)\right| \left(\left|\xi_{1}\left(t, \mu\left(t\right)\right)-\xi_{1}\left(t, 0\right)\right| +\left|\xi_{1}\left(t, 0\right)\right|\right)dt\\ &+\left|\lambda_{2}\right|\int_{0}^{\mathfrak{T}}\left|\mathfrak{J}_{2}\left(\vartheta, t\right)\right| \left(\left|\xi_{2}\left(t, \mu\left(t\right)\right)-\xi_{2}\left(t, 0\right)\right| +\left|\xi_{2}\left(t, 0\right)\right|\right)dt\bigg\}d\vartheta\\ &+\frac{1}{\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1} \bigg\{\left|\lambda_{1}\right|\int_{0}^{\vartheta}\left|\mathfrak{J}_{1}\left(\vartheta, t\right)\right| \left(\left|\xi_{1}\left(t, \mu\left(t\right)\right)-\xi_{1}\left(t, 0\right)\right| +\left|\xi_{1}\left(t, 0\right)\right|\right)dt\\ &+\left|\lambda_{2}\right|\int_{0}^{\mathfrak{T}}\left|\mathfrak{J}_{2}\left(\vartheta, t\right)\right| \left(\left|\xi_{2}\left(t, \mu\left(t\right)\right)-\xi_{2}\left(t, 0\right)\right| +\left|\xi_{2}\left(t, 0\right)\right|\right)dt\bigg\}d\vartheta\\ &+\frac{1}{\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-t)^{\wp-1} \left|\mathfrak{J}_{3}\left(\varkappa, t\right)\right|dt \end{split} \end{align*}
    \begin{align*} \begin{split} \leq&\frac{\left\|\mathfrak{g}\right\|}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}d\vartheta +\frac{\left\|\mathfrak{g}\right\|}{\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}d\vartheta +\frac{\mathfrak{K}_{3}}{\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-t)^{\wp-1}dt\\ &+\frac{\left|\lambda_{1}\right|\mathfrak{K}_{1}\left(C_{1}\left\|\mu\right\|+\mathfrak{F}_{1}\right) +\left|\lambda_{2}\right|\mathfrak{K}_{2}\left(C_{2}\left\|\mu\right\|+\mathfrak{F}_{2}\right)} {\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}d\vartheta\\ &+\frac{\left\{\left|\lambda_{1}\right|\mathfrak{K}_{1}\left(C_{1}\left\|\mu\right\|+\mathfrak{F}_{1}\right) +\left|\lambda_{2}\right|\mathfrak{K}_{2}\left(C_{2}\left\|\mu\right\|+\mathfrak{F}_{2}\right)\right\}} {\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}d\vartheta, \end{split} \end{align*}

    with the help of (5.6) , (5.7) and (5.8) , we get

    \begin{equation*} \begin{split} \left|\Upsilon \mu(\varkappa)\right| \leq&\frac{\left\|\mathfrak{g}\right\|}{\Gamma(\wp)}\frac{\varkappa^{\wp}}{\wp} +\frac{\left\|\mathfrak{g}\right\|}{\Gamma(\wp)}\frac{\mathfrak{T}^{\wp}}{\wp} +\frac{\left|\lambda_{1}\right|\mathfrak{K}_{1}\left(C_{1}r+\mathfrak{F}_{1}\right) +\left|\lambda_{2}\right|\mathfrak{K}_{2}\left(C_{2}r+\mathfrak{F}_{2}\right)}{\Gamma(\wp)}\frac{\varkappa^{\wp}}{\wp}\\ &+\frac{\left|\lambda_{1}\right|\mathfrak{K}_{1}\left(C_{1}r+\mathfrak{F}_{1}\right) +\left|\lambda_{2}\right|\mathfrak{K}_{2}\left(C_{2}r+\mathfrak{F}_{2}\right)} {\Gamma(\wp)}\frac{\mathfrak{T}^{\wp}}{\wp} +\frac{r+\mathfrak{K}_{3}}{\Gamma(\wp)}\frac{\mathfrak{T}^{\wp}}{\wp}\\ \leq& \left(2\left\|\mathfrak{g}\right\|+2\left|\lambda_{1}\right|\mathfrak{K}_{1}\left(C_{1}r+\mathfrak{F}_{1}\right) +2\left|\lambda_{2}\right|\mathfrak{K}_{2}\left(C_{2}r+\mathfrak{F}_{2}\right) +\mathfrak{K}_{3}\right)\frac{\mathfrak{T}^{\wp}}{\Gamma(\wp+1)}\\ = & \left\{2\left(\left\|\mathfrak{g}\right\|+\left|\lambda_{1}\right|\mathfrak{K}_{1}\mathfrak{F}_{1} +\left|\lambda_{2}\right|\mathfrak{K}_{2}\mathfrak{F}_{2}\right)+2r\left(\left|C_{1}\lambda_{1}\right|\mathfrak{K}_{1} +\left|\lambda_{2}\right|C_{2}\mathfrak{K}_{2}\right)+\mathfrak{K}_{3}\right\}\frac{\mathfrak{T}^{\wp}}{\Gamma(\wp+1)}\\ \leq& r. \end{split} \end{equation*}

    That is, \left\|\Upsilon\left(\mu\right)\right\|\leq r , for all \mu\in B_{r} , which implies that \Upsilon\left(\mu\right)\in B_{r} and hence \Upsilon : B_{r}\rightarrow B_{r} is well-defined. Now, we have to show that \Upsilon : B_{r}\rightarrow B_{r} is continuous. For this, using (5.7) , (5.5) and (5.8) , we have

    \begin{align*} \begin{split} \bigg|\Upsilon \mu\left(\varkappa\right)&-\Upsilon \nu\left(\varkappa\right)\bigg|\\ = &\bigg|\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}\left(\lambda_{1}\int_{0}^{\vartheta}\mathfrak{J}_{1}\left(\vartheta, t\right) \xi_{1}\left(t, \mu\left(t\right)\right)dt +\lambda_{2}\int_{0}^{\mathfrak{T}}\mathfrak{J}_{2}\left(\vartheta, t\right)\xi_{2}\left(t, \mu\left(t\right)\right)dt\right)d\vartheta\\ &-\frac{b}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}\left(\lambda_{1} \int_{0}^{\vartheta}\mathfrak{J}_{1}\left(\vartheta, t\right)\xi_{1}\left(t, \mu\left(t\right)\right)dt +\lambda_{2}\int_{0}^{\mathfrak{T}}\mathfrak{J}_{2}\left(\vartheta, t\right)\xi_{2}\left(t, \mu\left(t\right)\right)dt\right)d\vartheta\\ &-\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}\left(\lambda_{1}\int_{0}^{\vartheta}\mathfrak{J}_{1}\left(\vartheta, t\right) \xi_{1}\left(t, \nu\left(t\right)\right)dt +\lambda_{2}\int_{0}^{\mathfrak{T}}\mathfrak{J}_{2}\left(\vartheta, t\right)\xi_{2}\left(t, \nu\left(t\right)\right)dt\right)d\vartheta\\ &+\frac{b}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}\left(\lambda_{1} \int_{0}^{\vartheta}\mathfrak{J}_{1}\left(\vartheta, t\right)\xi_{1}\left(t, \nu\left(t\right)\right)dt +\lambda_{2}\int_{0}^{\mathfrak{T}}\mathfrak{J}_{2}\left(\vartheta, t\right)\xi_{2}\left(t, \nu\left(t\right)\right)dt\right)d\vartheta \bigg|\\ \leq&\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1} \left(\left|\lambda_{1}\right|\int_{0}^{\vartheta}\left|\mathfrak{J}_{1}\left(\vartheta, t\right)\right|\left|\xi_{1}\left(t, \mu\left(t\right)\right) -\xi_{1}\left(t, \nu\left(t\right)\right)\right|dt\right)d\vartheta\\ &+\frac{1}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1} \left(\left|\lambda_{2}\right|\int_{0}^{\mathfrak{T}}\left|\mathfrak{J}_{2}\left(\vartheta, t\right)\right|\left|\xi_{2}\left(t, \mu\left(t\right)\right) -\xi_{2}\left(t, \nu\left(t\right)\right)\right|dt\right)d\vartheta\\ &+\frac{b}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1} \left(\left|\lambda_{1}\right|\int_{0}^{\vartheta}\left|\mathfrak{J}_{1}\left(\vartheta, t\right)\right|\left|\xi_{1}\left(t, \mu\left(t\right)\right) -\xi_{1}\left(t, \nu\left(t\right)\right)\right|dt\right)d\vartheta\\ &+\frac{b}{(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1} \left(\left|\lambda_{2}\right|\int_{0}^{\mathfrak{T}}\left|\mathfrak{J}_{2}\left(\vartheta, t\right)\right|\left|\xi_{2}\left(t, \mu\left(t\right)\right) -\xi_{2}\left(t, \nu\left(t\right)\right)\right|dt\right)d\vartheta \end{split} \end{align*}
    \begin{align*} \begin{split} \leq&\frac{C_{1}\mathfrak{K}_{1}\left|\lambda_{1}\right|\left\|\mu-\nu\right\| +C_{2}\mathfrak{K}_{2}\left|\lambda_{2}\right|\left\|\mu-\nu\right\|}{\Gamma(\wp)}\int_{0}^{\varkappa}(\varkappa-\vartheta)^{\wp-1}d\vartheta\\ &+\frac{bC_{1}\mathfrak{K}_{1}\left|\lambda_{1}\right|\left\|\mu-\nu\right\| +bC_{2}\mathfrak{K}_{2}\left|\lambda_{2}\right|\left\|\mu-\nu\right\|} {(a+b)\Gamma(\wp)}\int_{0}^{\mathfrak{T}}(\mathfrak{T}-\vartheta)^{\wp-1}d\vartheta\\ = & \frac{\left(C_{1}\mathfrak{K}_{1}\left|\lambda_{1}\right|\left\|\mu-\nu\right\| +C_{2}\mathfrak{K}_{2}\left|\lambda_{2}\right|\left\|\mu-\nu\right\|\right)\varkappa^{\wp}}{\Gamma(\wp+1)}\\ &+\frac{C_{1}\left(\mathfrak{K}_{1}\left|\lambda_{1}\right|\left\|\mu-\nu\right\| +C_{2}\mathfrak{K}_{2}\left|\lambda_{2}\right|\left\|\mu-\nu\right\|\right)\mathfrak{T}^{\wp}}{\Gamma(\wp+1)}\\ \leq&\frac{2\left(C_{1}\mathfrak{K}_{1}\left|\lambda_{1}\right| +C_{2}\mathfrak{K}_{2}\left|\lambda_{2}\right|\right)\mathfrak{T}^{\wp}}{\Gamma(\wp+1)}\left\|\mu-\nu\right\|. \end{split} \end{align*}

    But \left|\lambda_{1}\right|C_{1}\mathfrak{K}_{1} +\left|\lambda_{2}\right|C_{2}\mathfrak{K}_{2} < \frac{\Gamma(\wp+1)}{2\mathfrak{T}^{p}} , that is \mathbb{h} = \frac{2\left(C_{1}\mathfrak{K}_{1}\left|\lambda_{1}\right| +C_{2}\mathfrak{K}_{2}\left|\lambda_{2}\right|\right)\mathfrak{T}^{\wp}}{\Gamma(\wp+1)}\in\left(0, 1\right) . It follows from above that

    \begin{equation} \left\|\Upsilon \mu-\Upsilon \nu\right\| \leq\mathbb{h}\left\|\mu-\nu\right\|. \end{equation} (5.9)

    That is, \Upsilon : B_{r}\rightarrow B_{r} is contraction and hence continuous. Next, we have to show that \Upsilon is \mathcal{L}_{\rm{m}} -contraction. Let \Lambda be any subset of B_{r} with {\rm{m}} (\Lambda) > 0, and \mu, \nu\in \Lambda . Then from inequality (5.9) , we write

    \begin{equation} \begin{split} \text{diam}\left(\Upsilon\Lambda\right) \leq\mathbb{h} \text{diam}\left(\Lambda\right). \end{split} \end{equation} (5.10)

    Now, let define \theta:(0, \infty)\rightarrow (1, \infty) by \theta(t) = e^{t} , then clearly \theta\in\Theta . Using inequality (5.10) , we have

    \begin{equation*} \begin{split} \theta\left({\rm{m}}\left(\Upsilon\left(\Lambda\right)\right)\right) = &\theta\left(\text{diam}\left(\Upsilon\left(\Lambda\right)\right)\right)\\ = &e^{\text{diam}\left(\Upsilon\left(\Lambda\right)\right)}\\ \leq&e^{\mathbb{h} \text{diam}\left(\Lambda\right)}\\ = &\left(e^{\text{diam}\left(\Lambda\right)}\right)^{\mathbb{h}}\\ = &\left(e^{{\rm{m}}\left(\Lambda\right)}\right)^{\mathbb{h}}\\ = &\left(\theta\left({\rm{m}}\left(\Lambda\right)\right)\right)^{\mathbb{h}}. \end{split} \end{equation*}

    Consequently,

    \begin{equation*} \frac{\left(\theta\left({\rm{m}}\left(\Lambda\right)\right)\right)^{\mathbb{h}}} {\theta\left({\rm{m}}\left(\Upsilon\left(\Lambda\right)\right)\right)}\geq1. \end{equation*}

    Thus for \mathscr{L}(\kappa_{1}, \kappa_{2}) = \frac{\kappa_{2}^{\mathbb{h}}}{\kappa_{1}} , the above inequality becomes

    \begin{equation*} \mathscr{L}(\theta({\rm{m}} (\Upsilon(\Lambda))), \theta({\rm{m}} (\Lambda)))\geq1, \end{equation*}

    That is, \Upsilon : B_{r}\rightarrow B_{r} is \mathcal{L}_{\rm{m}} -contraction and so Theorem 4.2 ensures the existence of a fixed point of \Upsilon in B_{r} , equivalently, the Eq (5.3) has a solution in B_{r} .

    To illustrate the Theorem 5.3 , we present an example.

    Example 5.4. Consider the following Caputo fractional Volterra–Fredholm integro-differential equation

    \begin{equation} \begin{split} ^{c}D^{0.9}\mu(\varkappa) = -\frac{\varkappa^{3}e^{-\varkappa^{4}}}{2} -\frac{1}{7}\int_{0}^{\varkappa}\frac{\varkappa}{3}\sin\left(\frac{\vartheta}{3}\right)\sqrt{2\vartheta+3[\mu(\vartheta)]^{2}}d\vartheta +\frac{1}{22}\int_{0}^{2}\frac{\varkappa^{2}\sqrt{5+2[\mu(\vartheta)]^{2}}}{1+\vartheta\varkappa^{2}}d\vartheta, \end{split} \end{equation} (5.11)

    with boundary condition

    \begin{equation} 5\mu(0)+3\mu(2) = \frac{1}{\Gamma(0.9)}\int_{0}^{2}(2-\vartheta)^{0.9-1}\frac{s\cos\left(\frac{\vartheta}{2}\right)}{\varkappa^{2}+5}d\vartheta, \end{equation} (5.12)

    Compare Eq (5.11) with Eq (5.1) , we get

    \begin{equation*} \begin{split} &\lambda_{1} = \frac{-1}{7}, \lambda_{2} = \frac{1}{22}, a = 5, b = 3, \mathfrak{g}(\varkappa) = -\frac{\varkappa^{3}e^{-\varkappa^{4}}}{2}, \\ &\mathfrak{J}_{1}\left(\varkappa, \vartheta\right) = \frac{\varkappa}{3}\sin\left(\frac{\vartheta}{3}\right), \mathfrak{J}_{2}\left(\varkappa, \vartheta\right) = \frac{\varkappa^{2}}{1+\vartheta\varkappa^{2}}, \mathfrak{J}_{3}\left(\varkappa, \vartheta\right) = \frac{s\cos\left(\frac{\vartheta}{2}\right)}{\varkappa^{2}+5}, \\ &\xi_{1}\left(\vartheta, \mu\left(\vartheta\right)\right) = \sqrt{2\vartheta+3[\mu(\vartheta)]^{2}}, \xi_{2}\left(\vartheta, \mu\left(\vartheta\right)\right) = \sqrt{5+2[\mu(\vartheta)]^{2}}. \end{split} \end{equation*}

    Clearly \mathfrak{g}: [0, 2]\rightarrow \mathbb{R} , \mathfrak{J}_{1}, \mathfrak{J}_{2}, \mathfrak{J}_{3}: [0, 2]\times [0, 2]\rightarrow \mathbb{R} and \xi_{1}, \xi_{2}: [0, 2]\times\mathbb{R}\rightarrow \mathbb{R} are continuous. Now, we have to verify condition (5.5) of Theorem 5.3. Consider

    \begin{equation*} \begin{split} \left|\xi_{1}\left(\vartheta, \mu\left(\vartheta\right)\right)-\xi_{1}\left(\vartheta, \nu\left(\vartheta\right)\right)\right| = &\left|\sqrt{2\vartheta+3[\mu(\vartheta)]^{2}}-\sqrt{2\vartheta+3[\nu(\vartheta)]^{2}}\right|\\ = &\frac{\left|2\vartheta+3[\mu(\vartheta)]^{2}-2\vartheta-3[\nu(\vartheta)]^{2}\right|} {\sqrt{2\vartheta+3[\mu(\vartheta)]^{2}}+\sqrt{2\vartheta+3[\nu(\vartheta)]^{2}}}\\ \leq&\frac{3\left|[\mu(\vartheta)]^{2}-[\nu(\vartheta)]^{2}\right|} {3[\left|\mu(\vartheta)\right|+\left|\nu(\vartheta)\right|]}\\ = &\frac{\left|\mu(\vartheta)-\nu(\vartheta)\right|\left|\mu(\vartheta)+\nu(\vartheta)\right|} {\left|\mu(\vartheta)\right|+\left|\nu(\vartheta)\right|}\\ \leq&\left\|\mu-\nu\right\|. \end{split} \end{equation*}

    Similarly,

    \begin{equation*} \begin{split} \left|\xi_{2}\left(\vartheta, \mu\left(\vartheta\right)\right)-\xi_{2}\left(\vartheta, \nu\left(\vartheta\right)\right)\right| \leq\left\|\mu-\nu\right\|. \end{split} \end{equation*}

    Thus \xi_{1}, \xi_{2}: \mathbb{R}\rightarrow \mathbb{R} are Lipschitz with C_{1} = C_{2} = 1 .

    Next, we have to verify the conditions (5.7) and (5.8) of Theorem 5.3. To do this, we have

    \begin{equation*} \begin{split} \mathfrak{K}_{1} = \sup\int_{0}^{\varkappa}\left|\frac{\varkappa}{3}\sin\left(\frac{\vartheta}{3}\right)\right|d\vartheta = \sup\left(-\left|\varkappa\right|\cos\left(\frac{\varkappa}{3}\right)\right) = 0, \end{split} \end{equation*}
    \begin{equation*} \begin{split} \mathfrak{K}_{2} = \sup\int_{0}^{\varkappa}\left|\frac{\varkappa^{2}}{1+\vartheta\varkappa^{2}}\right|d\vartheta = \sup\left(\ln(1+\varkappa^{3})\right) \approx 2.197, \end{split} \end{equation*}

    and

    \begin{equation*} \begin{split} \mathfrak{K}_{3} & = \sup\left(\frac{1}{\varkappa^{2}+5}\int_{0}^{\varkappa}\vartheta\cos\left(\frac{\vartheta}{2}\right)d\vartheta\right) = \sup\left(\frac{2\varkappa\sin\left(\frac{\varkappa}{2}\right)+4\cos\left(\frac{\varkappa}{2}\right)-4}{\varkappa^{2}+5}\right)\approx 0.17. \end{split} \end{equation*}

    Finally, to verify condition (5.6) of Theorem 5.3 . Let B_{2} = \left\{\mu\in \mathcal{C}\left(\left[0, 2\right], \mathbb{R}\right):\left\|\mu\right\|\leq 2\right\} , then since \left\|\mathfrak{g}\right\| = 0, \mathfrak{K}_{1} = 0, \mathfrak{K}_{2}\approx 2.197, \mathfrak{K}_{3} = 0.17 , \mathfrak{F}_{1} = 2, and \mathfrak{F}_{2} = \sqrt{5} , so we have

    \begin{equation*} \left|\lambda_{1}\right|C_{1}\mathfrak{K}_{1}+\left|\lambda_{2}\right|C_{2}\mathfrak{K}_{2} \approx 0.104619 < 0.2576988\approx \frac{\Gamma(\wp+1)}{2\mathfrak{T}^{p}}, \end{equation*}

    and

    \begin{equation*} \begin{split} \frac{2\left[\left\|\mathfrak{g}\right\|+\left|\lambda_{1}\right|\mathfrak{K}_{1}\mathfrak{F}_{1} +\left|\lambda_{2}\right|\mathfrak{K}_{2}\mathfrak{F}_{2}\right]+\mathfrak{K}_{3}} {\mathfrak{T}^{-\wp}\Gamma(\wp+1)-2\left[\left|\lambda_{1}\right|C_{1}\mathfrak{K}_{1} +\left|\lambda_{2}\right|C_{2}\mathfrak{K}_{2}\right]} \approx & 1.953315 < 2. \end{split} \end{equation*}

    Thus Theorem 5.3 ensures the existence of a solution of (5.11) in B_{2} .

    Darbo type contractions are introduced and fixed point results are established in a Banach space using the concept of measure of non compactness. Various existing results are deduced as corollaries to our main results. Further, our results are applied to prove the existence and uniqueness of solution to the Caputo fractional Volterra–Fredholm integro-differential equation under integral type boundary conditions which is further illustrated by appropriate example. Our study paves the way for further studies on Darbo type contractions and its applications.

    The authors are thankful to the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia, for supporting this research.

    All authors declare no conflicts of interest in this paper.



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