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Review Special Issues

A rapid review of recent advances in diagnosis, treatment and vaccination for COVID-19

  • COVID-19 is caused by SARS-CoV-2, which originated in Wuhan, Hubei province, Central China, in December 2019 and since then has spread rapidly, resulting in a severe pandemic. The infected patient presents with varying non-specific symptoms requiring an accurate and rapid diagnostic tool to detect SARS-CoV-2. This is followed by effective patient isolation and early treatment initiation ranging from supportive therapy to specific drugs such as corticosteroids, antiviral agents, antibiotics, and the recently introduced convalescent plasma. The development of an efficient vaccine has been an on-going challenge by various nations and research companies. A literature search was conducted in early December 2020 in all the major databases such as Medline/PubMed, Web of Science, Scopus and Google Scholar search engines. The findings are discussed in three main thematic areas namely diagnostic approaches, therapeutic options, and potential vaccines in various phases of development. Therefore, an effective and economical vaccine remains the only retort to combat COVID-19 successfully to save millions of lives during this pandemic. However, there is a great scope for further research in discovering cost-effective and safer therapeutics, vaccines and strategies to ensure equitable access to COVID-19 prevention and treatment services.

    Citation: Srikanth Umakanthan, Vijay Kumar Chattu, Anu V Ranade, Debasmita Das, Abhishekh Basavarajegowda, Maryann Bukelo. A rapid review of recent advances in diagnosis, treatment and vaccination for COVID-19[J]. AIMS Public Health, 2021, 8(1): 137-153. doi: 10.3934/publichealth.2021011

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  • COVID-19 is caused by SARS-CoV-2, which originated in Wuhan, Hubei province, Central China, in December 2019 and since then has spread rapidly, resulting in a severe pandemic. The infected patient presents with varying non-specific symptoms requiring an accurate and rapid diagnostic tool to detect SARS-CoV-2. This is followed by effective patient isolation and early treatment initiation ranging from supportive therapy to specific drugs such as corticosteroids, antiviral agents, antibiotics, and the recently introduced convalescent plasma. The development of an efficient vaccine has been an on-going challenge by various nations and research companies. A literature search was conducted in early December 2020 in all the major databases such as Medline/PubMed, Web of Science, Scopus and Google Scholar search engines. The findings are discussed in three main thematic areas namely diagnostic approaches, therapeutic options, and potential vaccines in various phases of development. Therefore, an effective and economical vaccine remains the only retort to combat COVID-19 successfully to save millions of lives during this pandemic. However, there is a great scope for further research in discovering cost-effective and safer therapeutics, vaccines and strategies to ensure equitable access to COVID-19 prevention and treatment services.



    We investigate a new class of boundary value problems for generalized Caputo-type fractional differential equations and inclusions supplemented with Katugampola type generalized fractional integral boundary conditions. Precisely, we study the following problems:

    {ρcDα0+y(t)=f(t,y(t)),  tJ:=[0,T],y(T)=mi=1σiρIβ0+y(ηi)+κ,  δy(0)=0,  ηi(0,T), (1.1)

    and

    {ρcDα0+y(t)F(t,y(t)),  tJ:=[0,T],y(T)=mi=1σiρIβ0+y(ηi)+κ,  δy(0)=0,  ηi(0,T), (1.2)

    where ρcDα0+ denotes the generalized Caputo-type fractional derivative of order 1<α2,ρ>0, ρIβ0+ is the Katugampola type fractional integral of order β>0,ρ>0, f:J×RR is a continuous function, σiR,i=1,2,,m,κR, δ=t1ρddt, and F:J×RP(R) is a multivalued function (P(R) is the family of all nonempty subjects of R).

    Here we emphasize that the problem considered in the present paper is motivated by Laskin's work [1] on the generalization of the Feynman and Wiener path integrals in the context of fractional quantum mechanics and fractional statistical mechanics. One can find more details in the articles [2, 3]. It is expected that the results obtained in this paper will provide more leverage in dealing with Feynman and Wiener path type integrals involving an index like ρ>0 in (1.1), instead of a fixed choice ρ=1 (Caputo fractional derivative case). Moreover, chaos for a fractional order differential equation involving two parameters (α and ρ>0) becomes more complicated than the one containing Caputo fractional derivative of order α (generalized Caputo-type fractional derivative with ρ=1); one can find more details in [4, 5]. It is worthwhile to notice that Katugampola fractional integral unifies the Riemann-Liouville and Hadamard integrals into a single integral [6]. Thus, our results are more general in the context of integral boundary conditions.

    The topic of fractional-order differential equations and inclusions attracted significant attention in recent years and several results on fractional differential equations involving Riemann-Liouville, Caputo, Hadamard type derivatives, supplemented with a variety of boundary conditions, can be found in the related literature [7, 8, 9]. The interest in the subject owes to its extensive applications in various disciplines of science and engineering, for instance, see the papers [10, 11, 12, 13, 14, 15, 16, 17, 18], and the references cited therein. In a recent paper [19], the authors studied fractional differential equations involving Caputo-Katugampola derivative. In a more recent work [20], the authors studied a fractional order boundary value problem involving Katugampola-type generalized fractional derivative and generalized fractional integral.

    We organize the rest of the paper as follows. Section 2 contains preliminary material related to our work. The existence and uniqueness results for the problem (1.1), obtained with the aid of the standard fixed point theorem, are presented in Section 3. The existence results for the inclusions problem (1.2) are derived in Section 4. Examples are provided to demonstrate the application of the main theorems.

    For cR,1p, let Xpc(a,b) denote the space of all complex-valued Lebesgue measurable functions ϕ on (a,b) endowed with the norm:

    ϕXpc=(ba|xcϕ(x)|pdxx)1/p<.

    We denote by L1(a,b) the space of all Lebesgue measurable functions φ on (a,b) equipped with the norm:

    φL1=ba|φ(x)|dx<.

    Let G=C(J,R) denote the Banach space of all continuous functions from [0,T] to R endowed with the norm defined by y=supt[0,T]|y(t)|.

    Recall that

    ACn(J,R)={h:JR:h,h,,h(n1)C(J,R) and h(n1)is absolutely continuous}.

    For 0ϵ<1, we define Cϵ,ρ(J,R)={f:JR:(tρaρ)ϵf(t)C(J,R)} equipped with the norm fCϵ,ρ=(tρaρ)ϵf(t)C. Moreover, let us introduce ACnδ(J), which consists of the functions f that have absolutely continuous δn1-derivative, where δ=t1ρddt. Thus we define spaces ACnδ(J,R)={f:JR:δn1fAC(J,R), δ=t1ρddt}, and Cnδ,ϵ(J,R)={f:JR:δn1fC(J,R),δnfCϵ,ρ(J,R),δ=t1ρddt} endowed with the norms fCnδ=n1k=0δkfC and fCnδ,ϵ=n1k=0δkfC+δnfCϵ,ρ respectively. Here we use the convention Cnδ,0=Cnδ.

    Definition 2.1. [6] The generalized fractional integral of order α>0 and ρ>0 of fXpc(a,b) for <a<t<b<, is defined by

    (ρIαa+f)(t)=ρ1αΓ(α)tasρ1(tρsρ)1αf(s)ds. (2.1)

    Note that the integral in (2.1) is called the left-sided fractional integral. Similarly we can define right-sided fractional integral ρIαbf as

    (ρIαbf)(t)=ρ1αΓ(α)btsρ1(sρtρ)1αf(s)ds. (2.2)

    Definition 2.2. [21] The generalized fractional derivative, associated with the generalized fractional integrals (2.1) and (2.2) for 0a<t<b<, are defined by

    (ρDαa+f)(t)=(t1ρddt)n(ρInαa+f)(t)=ραn+1Γ(nα)(t1ρddt)ntasρ1(tρsρ)αn+1f(s)ds (2.3)

    and

    (ρDαbf)(t)=(t1ρddt)n(ρInαbf)(t)=ραn+1Γ(nα)(t1ρddt)nbtsρ1(sρtρ)αn+1f(s)ds, (2.4)

    if the integrals in (2.3) and (2.4) exist.

    Definition 2.3. [22] For α0 and fACnδ[a,b], the generalized Caputo-type fractional derivatives ρcDαa and ρcDαb are defined via the above generalized fractional derivatives as follows

    ρcDαa+f(x)=ρDαa+[f(t)n1k=0δkf(a)k!(tρaρρ)k](x),  δ=x1ρddx, (2.5)
    ρcDαbf(x)=ρDαb[f(t)n1k=0(1)kδkf(b)k!(bρtρρ)k](x),  δ=x1ρddx, (2.6)

    where n=[α]+1.

    Lemma 2.1. [22] Let α0,n=[α]+1 and fACnδ[a,b], where 0<a<b<. Then,

    (1) for αN,

    ρcDαa+f(t)=1Γ(nα)ta(tρsρρ)nα1(δnf)(s)dss1ρ=ρInαa+(δnf)(t), (2.7)
    ρcDαbf(t)=1Γ(nα)bt(sρtρρ)nα1(1)n(δnf)(s)dss1ρ=ρInαb(δnf)(t); (2.8)

    (2) for αN,

    ρcDαa+f=δnf,   ρcDαbf=(1)nδnf. (2.9)

    Lemma 2.2. [22] Let fACnδ[a,b] or Cnδ[a,b] and αR. Then

    ρIαa+ρcDαa+f(x)=f(x)n1k=0(δkf)(a)k!(xρaρρ)k,
    ρIαbρcDαbf(x)=f(x)n1k=0(1)k(δkf)(a)k!(bρxρρ)k.

    In particular, for 0<α1, we have

    ρIαa+ρcDαa+f(x)=f(x)f(a),
    ρIαbρcDαbf(x)=f(x)f(b).

    Next we define a solution for the problem (1.1).

    Definition 2.4. A function yAC2δ([0,T],R) is said to be a solution of (1.1) if y satisfies the equation ρcDαy(t)=f(t,y(t)) on [0,T], and the conditions y(T)=mi=1σi ρIβy(ηi)+κ, δy(0)=0.

    Relative to the problem (1.1), we consider the following lemma.

    Lemma 2.3. Let hC(0,T)L1(0,T), yAC2δ(J) and

    Ω=1mi=1σiηρβiρβΓ(β+1)0. (2.10)

    Then the integral solution of the linear boundary value problem (BVP):

    {ρcDα0+y(t)=h(t),  tJ:=[0,T],y(T)=mi=1σiρIβ0+y(ηi)+κ,  δy(0)=0,  ηi(0,T), (2.11)

    is given by

    y(t)=ρIα0+h(t)+1Ω{ρIα0+h(T)+mi=1σiρIα+β0+h(ηi)+κ}. (2.12)

    Proof. Applying ρIα0+ on both sides of the fractional differential equation in (2.11) and using Lemma 2.2, we get

    y(t)=ρIα0+h(t)+c1+c2tρρ=ρ1αΓ(α)t0sρ1(tρsρ)α1h(s)ds+c1+c2tρρ, (2.13)

    for some c1,c2R. Taking δ-derivative of (2.13), we get

    δy(t)=ρIα10+h(t)+c2=ρ2αΓ(α1)t0sρ1(tρsρ)α2h(s)ds+c2. (2.14)

    Using the boundary condition δy(0)=0 in (2.14), we get c2=0. Applying the generalized integral ρIβ0+ to (2.13) after inserting the value of c2 in it, we get

    ρIβ0+y(t)=ρIα+β0+h(t)+c1tρβρβΓ(β+1). (2.15)

    Making use of the first boundary condition y(T)=mi=1σi ρIβy(ηi)+κ in (2.15), we get

    ρIα0+h(T)+c1=mi=1σi ρIα+β0+h(ηi)+mi=1σic1ηρβiρβΓ(β+1)+κ,

    which, on solving for c1 together with (2.10), yields

    c1=1Ω{ρIα0+h(T)+mi=1σi ρIα+β0+h(ηi)+κ}.

    Substituting the values of c1 and c2 in (2.13), we obtain the solution (2.12). The converse follows by direct computation. The proof is completed.

    Using Lemma 2.3, we define an operator N:GG by

    Ny(t)=ρIα0+f(t,y(t))+1Ω{ρIα0+f(T,y(T))+mi=1σiρIα+β0+f(ηi,y(ηi))+κ}. (3.1)

    In the following, for brevity, we set the notation:

    Λ=TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+mi=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}. (3.2)

    Our first existence result for the problem (1.1) relies on Leray-Schauder nonlinear alternative [23].

    Theorem 3.1. Assume that

    (A1) |f(t,y)|p(t)ψ(y),(t,y)[0,T]×R, where pL1([0,T],R+) and ψ:R+ R+ is a nondecreasing function;

    (A2) we can find a positive constant W satisfying the inequality:

    Wψ(W)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+|κ|})>1.

    Then there exists at least one solution for the problem (1.1) on [0,T].

    Proof. Consider the operator N:GG defined by (3.1) and show that it is continuous and completely continuous. We establish in four steps.

    (ⅰ) N is continuous. Let {yn} be a sequence such that yny in G. Then

    |N(yn)(t)N(y)(t)|ρIα0+|f(t,yn(t))f(t,y(t))|+1Ω{ρIα0+|f(T,yn(T))f(T,y(T))|+mi=1|σi| ρIα+β0+|f(ηi,yn(ηi))f(ηi,y(ηi))|}Λf(,yn)f(,y).

    Since f is a continuous function, therefore, we have

    N(yn)N(y)Λf(,yn)f(,y)0,asn.

    (ⅱ) The operator N maps bounded sets into bounded sets in G.

    For any ˉr>0, it is indeed enough to show that there exists a positive constant such that N(y) for yBˉr={yG:yˉr}. By the assumption (A1), for each tJ, we have

    |N(y)(t)|ρIα0+|f(t,y(t))|+1|Ω|{ρIα0+|f(T,y(T))|+mi=1|σi| ρIα+β0+|f(ηi,y(ηi))|+|κ|}ρIα0+p(T)Ω(y)+1|Ω|{ρIα0+p(T)ψ(y)+mi=1|σi| ρIα+β0+p(ηi)ψ(y)+|κ|}ψ(y)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+|κ|}).

    Thus

    N(y)ψ(ˉr)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+|κ|}):=.

    (ⅲ) N maps bounded sets into equicontinuous sets of G.

    Let t1,t2(0,T],t1<t2, Bˉr be a bounded set of G as in (ii) and let yBˉr. Then

    |N(y)(t2)N(y)(t1)||ρIα0+f(t2,y(t2))ρIα0+f(t1,y(t1))|ρ1αψ(ˉr)Γ(α)|t10[sρ1(tρ2sρ)1αsρ1(tρ1sρ)1α]p(s)ds+t2t1sρ1(tρ2sρ)1αp(s)ds|0 as t1t2, independent ~of y.

    From the steps (i)(iii), we deduce by the Arzelá-Ascoli theorem that N:GG is completely continuous.

    (ⅳ) There exists an open set VG with yνN(y) for ν(0,1) and yV.

    Let yG be a solution of yνNy=0 for ν[0,1]. Then, for t[0,T], we obtain

    |y(t)|=|ν(Ny)(t)|ρIα0+|f(t,y(t))|+1|Ω|{ρIα0+|f(T,y(T))|+mi=1|σi| ρIα+β0+|f(ηi,y(ηi))|+|κ|}ρIα0+p(T)ψ(y)+1|Ω|{ρIα0+p(T)ψ(y)+mi=1|σi| ρIα+β0+p(ηi)ψ(y)+|κ|}ψ(y)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+|κ|}),

    which, on taking the norm for t[0,T], implies that

    yψ(y)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+|κ|})1.

    By the assumption (A2), there exists a positive constant W such that yW. Next we define V={yG:y<W} and note that the operator N:¯VG is continuous and completely continuous. By the choice of V, there does not exist any yV satisfying y=νN(y) for some ν(0,1). In consequence, by the nonlinear alternative of Leray-Schauder type [23], we deduce that there exists a fixed point y¯V for the operator N, which is a solution of the problem (1.1).

    In our next result, we make use of Banach contraction mapping principle to establish the uniqueness of solutions for the problem (1.1).

    Theorem 3.2. Suppose that

    (A3) there exists a nonnegative constant L such that

    |f(t,u)f(t,v)|Luv,fort[0,T]and everyu,vR.

    Then the problem (1.1) has a unique solution on [0,T] if

    LΛ<1, (3.3)

    where Λ is defined by (3.2).

    Proof. Consider the operator N:GG associated with the problem (1.1) defined by (3.1). With Λ given by (3.2), we fix

    rΛf0+|κ|/|Ω|1LΛ,f0=supt[0,T]|f(t,0)|,

    and show that FBrBr, where Br={yG:yr}. For yBr, using (A3), we get

    |N(y)(t)|ρIα0+[|f(t,y(t))f(t,0)|+|f(t,0)|]+1|Ω|{ρIα0+[|f(T,y(T))f(T,0)|+|f(T,0)|]+mi=1|σi| ρIα+β0+[|f(ηi,y(ηi))f(ηi,0)|+|f(ηi,0)|]+|κ|}(Ly+f0)[TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+mi=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}]+|κ||Ω|(Lr+f0)Λ+|κ||Ω|r,

    which, on taking the norm for t[0,T], yields N(y)r. This shows that N maps Br into itself. Now we show that the operator N is a contraction. Let y,uG. Then we get

    |N(y)(t)N(u)(t)|ρIα0+|f(t,y(t))f(t,u(t))|+1|Ω|{ρIα0+|f(T,y(T))f(T,u(T))|+mi=1|σi| ρIα+β0+|f(ηi,y(ηi))f(ηi,u(ηi))|}LΛyu.

    Consequently we obtain N(y)N(u)LΛyu, which shows that N is a contraction by means of (3.3). Thus the contraction mapping principle applies and the operator N has a unique fixed point. This shows that there exists a unique solution for the problem (1.1) on [0,T].

    Now we prove the uniqueness of solutions for the problem (1.1) by applying a fixed point theorem for nonlinear contractions due to Boyd and Wong [24].

    Definition 3.1. A mapping H:EE is called a nonlinear contraction if we can find a continuous nondecreasing function ϕ:R+R+ such that ϕ(0)=0, ϕ(ξ)<ξ for all ξ>0 and HyHuϕ(yu),y,uE (E is a Banach space).

    Lemma 3.1. (Boyd and Wong) [24] Let E be a Banach space and let N:EE be a nonlinear contraction. Then N has a unique fixed point in E.

    Theorem 3.3. Assume that

    (A4) |f(t,y)f(t,u)|g(t)|yu|G+|yu|,t[0,T],y,u0, where g:[0,T]R+ is continuous and

    G=ρIα0+g(T)+1|Ω|{ρIα0+g(T)+mi=1|σi|ρIα+β0+g(ηi)}. (3.4)

    Then the problem (1.1) has a unique solution on [0,T].

    Proof. Let Ψ:R+R+ be a continuous nondecreasing function such that Ψ(0)=0 and Ψ(ξ)<ξ for all ξ>0, defined by

    Ψ(ξ)=GξG+ξ,ξ0.

    Let y,uG. Then

    |f(s,y(s))f(s,u(s))|g(s)GΨ(yu),

    so that

    |N(y)(t)N(u)(t)|ρIα0+(g(t)|y(t)u(t)|G+|y(t)u(t)|)+1|Ω|{ρIα0+(g(T)|y(T)u(T)|G+|y(T)u(T)|)+mi=1|σi|ρIα+β0+(g(ηi)|y(ηi)u(ηi)|G+|y(ηi)u(ηi)|)}|y(t)u(t)|G+|y(t)u(t)|{ρIα0+g(T)+1|Ω|{ρIα0+g(T)+mi=1|σi|ρIα+β0+g(ηi)}},

    for t[0,T]. By the condition (3.4), we deduce that N(y)N(u)Ψ(yu) and hence N is a nonlinear contraction. Thus it follows from the fixed point theorem due to Boyd and Wong [24] that the operator N has a unique fixed point in G, which is indeed a unique solution of problem (1.1).

    Example 3.1. Let us consider the following boundary value problem

    {1/3  cD5/40+y=f(t,y), t[0,2],y(2)=21/3I3/4y(1/2)+1/2 1/3I3/4y(3/2)+1/4,   δy(0)=0, (3.5)

    where ρ=1/3,α=5/4,σ1=2,σ2=1/2,β=3/4,η1=1/2,η2=3/2,κ=1/4, T=2 and f(t,y(t)) will be fixed later.

    Using the given data, we find that |Ω|=4.543695998 and Λ=7.572001575, where Ω and Λ are given by (2.10) and (3.2) respectively.

    For illustrating Theorem 3.1, we take

    f(t,y)=(1+t)30(|y||y|+1+y+18), (3.6)

    and find that p(t)=(1+t)30 and ψ(y)=||y||+98. By condition (A2), we have W>0.7066246467. Thus, the hypothesis of Theorem 3.1 holds true, which implies that the problem (3.5) has at least one solution.

    Furthermore, for the uniqueness results, Theorem 3.2 can be illustrated by choosing

    f(t,y)=tan1y+et281+sint. (3.7)

    Clearly the condition (A3) is satisfied with L=1/18. Also

    LΛ0.4206667542<1.

    Obviously all the conditions of Theorem 3.2 hold and consequently the problem (3.5) with f(t,y) given by (3.7) has a unique solution on [0,2] by the conclusion of Theorem 3.2.

    Finally, for illustrating Theorem 3.3, we take

    f(t,y)=t(|y||y|+11+18). (3.8)

    Here we choose g(t)=(1+t) and find that

    G=ρIα0+g(T)+1|Ω|{ρIα0+g(T)+mi=1|σi|ρIα+β0+g(ηi)}9.923097014,

    and

    |f(t,y)f(t,u)|=t(|y||u|11+|y|+|u|+|y||u|11)(1+t)|yu|9.923097014+|yu|.

    So, the conclusion of Theorem 3.3 applies to the problem (3.5) with f(t,y) given by (3.8).

    In this section, we present existence results for the problem (1.2).

    Definition 4.1. A function yAC2δ([0,T],R) is called a solution of the problem (1.2) if y(T)=mi=1σi ρIβ0+y(ηi)+κ,δy(0)=0 and there exists function vL1([0,T],R) such that v(t)F(t,y(t)) a.e. on [0,T] and

    y(t)=ρIα0+v(t)+1Ω{ρIα0+v(T)+mi=1σi ρIα+β0+v(ηi)+κ}.

    Here we prove an existence result for the problem (1.2) by applying nonlinear alternative for Kakutani maps [23] when F has convex values and is of Carathéodory type.

    Theorem 4.1. Assume that

    (B1) F:[0,T]×RPcp,c(R) is L1-Carathéodory, where Pcp,c(R) ={YP(R): Y  is compact and convex};

    (B2) there exist a continuous nondecreasing function φ:[0,)(0,) and a function pL1([0,T],R+) such that

    F(t,y)P:=sup{|x|:xF(t,y)}p(t)φ(y)  for each  (t,y)[0,T]×R;

    (B3) there exists a constant ˆW>0 satisfying

    ˆWφ(ˆW)(ρIα0+p(T)+1Ω( ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+κ))>1.

    Then there exists at least one solution for the problem (1.2) on [0,T].

    Proof. Define an operator M:C([0,T],R)P(C([0,T],R)) by

    M(y)={hC([0,T],R):h(t)=F(y)(t)}, (4.1)

    where

    F(y)(t)=ρIα0+v(t)+1Ω{ρIα0+v(T)+mi=1σi ρIα+β0+v(ηi)+κ},

    for vSF,y. Here SF,y denotes the set of selections of F and is defined by

    SF,y:={vL1([0,T],R):v(t)F(t,y(t)) a.e. on [0,T]},

    for each yC([0,T],R). Notice that the fixed points of the operator M are solutions of the problem (1.2).

    To show that M satisfies the assumptions of Leray-Schauder nonlinear alternative [23], we split the proof in several steps.

    Step 1. M(y) is convex for each yC([0,T],R) as SF,y is convex (F has convex values).

    Step 2. Let Br={yC([0,T],R):yr} be a bounded ball in C([0,T],R), where r is a positive number. Then, for each hM(y),yBr, there exists vSF,y such that

    h(t)=ρIα0+v(t)+1Ω{ρIα0+v(T)+mi=1σi ρIα+β0+v(ηi)+κ}

    with

    hφ(r)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+|κ|}):=1.

    This shows that M maps bounded sets (balls) into bounded sets in C([0,T],R).

    Step 3. In order to show that M maps bounded sets into equicontinuous sets of C([0,T],R), we take t1,t2(0,T],t1<t2, and yBr. Then we find that

    |h(t2)h(t1)|ρ1αφ(r)Γ(α)|t10[sρ1(tρ2sρ)1αsρ1(tρ1sρ)1α]p(s)ds+t2t1sρ1(tρsρ)1αp(s)ds|,

    which tends to zero independently of yBr as t2t10. In view of the foregoing steps, it follows by the Arzelá-Ascoli theorem that M:C([0,T],R)P(C([0,T],R)) is completely continuous.

    Step 4. In our next step, we show that M is upper semi-continuous (u.s.c.). Since M is completely continuous, it is enough to establish that it has a closed graph (see [25, Proposition 1.2]). For that, let yny,hnM(yn) and hnh. Then we have to show that hM(y). Associated with hnM(yn), we can find vnSF,yn such that for each t[0,T],

    hn(t)=ρIα0+vn(t)+1Ω{ρIα0+vn(s)vn(T)+mi=1σi ρIα+β0+vn(ηi)+κ}.

    Next, for each t[0,T], we establish that there exists vSF,y satisfying

    h(t)=ρIα0+v(t)+1Ω{ρIα0+v(T)+mi=1σi ρIα+β0+v(ηi)+κ}.

    Consider the linear operator Θ:L1([0,T],R)C([0,T],R) given by

    vΘv(t)=ρIα0+v(t)+1Ω{ρIα0+v(T)+mi=1σi ρIα+β0+v(ηi)+κ}.

    Notice that hn(t)h(t)0 as n. Thus we deduce by the closed graph theorem [26] that ΘSF is a closed graph operator. Furthermore, we have hn(t)Θ(SF,yn). As yny, we have

    h(t)=ρIα0+v(t)+1Ω{ρIα0+v(T)+mi=1σi ρIα+β0+v(ξ)+κ},for ~some vSF,y.

    Step 5. Finally, we show the existence of an open set UC([0,T],R) such that yλM(y) for any λ(0,1) and all yU. For that we take λ(0,1) and yλM(y). Then there exists vL1([0,T],R) with vSF,y such that, for t[0,T], we have

    y(t)=λρIα0+v(t)+λΩ{ρIα0+v(T)+mi=1σi ρIα+β0+v(ηi)+κ}.

    As in the second step, one can obtain

    |y(t)|ρIα0+|v(T)|+1|Ω|{ρIα0+|v(T)|+mi=1|σi| ρIα+β0+|v(ηi)|+|κ|}φ(y)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+|κ|}),

    which implies that

    yφ(y)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+mi=1|σi| ρIα+β0+p(ηi)+|κ|})1.

    By the hypothesis (B3), we can find ˆW such that yˆW. Setting Y={yC(J,R):y<ˆW}, we notice that the operator M:¯YP(C(J,R)) is compact multi-valued, u.s.c. with convex closed values. From the choice of Y, there does not exist any yY satisfying yλM(y) for some λ(0,1). In consequence, we deduce by the nonlinear alternative of Leray-Schauder type [23] that M has a fixed point y¯Y which is a solution of the problem (1.2). This completes the proof.

    Consider a mapping Hd:P(X)×P(X)R{} defined by

    Hd(S,V)=max{supsSd(a,V),supvVd(S,v)},

    where d(S,v)=infsSd(s;v), d(s,V)=infvVd(s;v) and (X,d) is a metric space induced from the normed space (X;). Note that (Pcl,b(X),Hd) is a metric space (see [27]), where Pcl,b(X)={YP(X):Y is closed and bounded}.

    The following result, dealing with the existence of solutions for the problem (1.2) with nonconvex valued right hand side of the inclusion, relies on Covitz and Nadler's fixed point theorem for multivalued maps [28].

    Theorem 4.2. Assume that

    (C1) F:[0,T]×RPcp(R) is such that F(,y):[0,T]Pcp(R) is measurable for each yR, where Pcp(R)={YP(R):Y is compact};

    (C2) Hd(F(t,y),F(t,ˉy))μ(t)|yˉy| for almost all t[0,T] and y,ˉyR with μC([0,T],R+) and d(0,F(t,0))μ(t) for almost all t[0,T].

    Then the problem (1.2) has at least one solution on [0,T] provided that

    ϑ=μΛ<1, (4.2)

    where Λ is given by (3.2).

    Proof. By the assumption (C1), the set SF,y is nonempty for each yC([0,T],R) and F has a measurable selection by Theorem Ⅲ.6 in [29]. Now we proceed to show that the operator M:(C[0,T],R)Pcl(C([0,T],R)) (Pcl(C([0,T],R))={YP(C([0,T],R)):Y  is closed}) is a contraction so that Covitz and Nadler's Theorem [28] is applicable.

    In the first step, we show that M(y)Pcl((C[0,T],R)) for each yC([0,T],R). Let {un}n0M(y) with unu (n) in C([0,T],R). Then uC([0,T],R) and there exists vnSF,yn satisfying

    un(t)=ρIα0+vn(t)+1Ω{ρIα0+vn(T)+mi=1σi ρIα+β0+vn(ηi)+κ}for each t[0,T].

    In view of the compact values of F, we pass onto a subsequence (if necessary) to find that vn converges to v in L1([0,T],R). For vSF,y and for each t[0,T], we have

    un(t)u(t)=ρIα0+v(t)+1Ω{ρIα0+v(T)+mi=1σi ρIα+β0+v(ηi)+κ}.

    Thus uM(y).

    Now, for each y,ˉyC([0,T],R), we establish that there exists ϑ<1 (defined by (4.2)) satisfying

    Hd(M(y),M(ˉy))ϑyˉy.

    Let y,ˉyC([0,T],R) and h1M(y). Then there exists v1(t)F(t,y(t)) satisfying

    h1(t)=ρIα0+v1(t)+1Ω{ρIα0+v1(T)+mi=1σi ρIα+β0+v1(ηi)+κ},

    for each t[0,T]. By (C2), we have

    Hd(F(t,y),F(t,ˉy))μ(t)|y(t)ˉy(t)|.

    Therefore, we can find wF(t,ˉy(t)) satisfying

    |v1(t)w|μ(t)|y(t)ˉy(t)|,  t[0,T].

    Introduce U:[0,T]P(R) by

    U(t)={wR:|v1(t)w|μ(t)|y(t)ˉy(t)|}.

    As U(t)F(t,ˉy(t)) is measurable (Proposition Ⅲ.4 [29]), we can find a measurable selection v2(t) for U such that v2(t)F(t,ˉy(t)) satisfying |v1(t)v2(t)|μ(t)|y(t)ˉy(t)| for each t[0,T].

    Define

    h2(t)=ρIα0+v2(t)+1Ω{ρIα0+v2(T)+mi=1σi ρIα+β0+v2(ηi)+κ},

    for each t[0,T]. Then

    |h1(t)h2(t)|ρIα0+|v1(t)v2(t)|+1|Ω|{ρIα0+|v1(T)v2(T)|+mi=1|σi| ρIα+β0+|v1(ηi)v2(ηi)|}μ[TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+mi=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}]yˉy.

    Hence

    h1h2μ[TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+mi=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}]yˉy.

    Analogously, switching the roles of y and ¯y, we can obtain

    Hd(M(y),M(ˉy))μ[TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+mi=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}]yˉy.

    So M is a contraction. Thus, by Covitz and Nadler's fixed point theorem [28], the operator M has a fixed point y, which corresponds to a solution of (1.2).

    Example 4.1. Consider the following boundary value problem

    {1/3  cD5/40+yF(t,y), t[0,2],y(2)=21/3I3/4y(1/2)+1/2 1/3I3/4y(3/2)+1/4,   δy(0)=0, (4.3)

    where F(t,y) will be fixed later.

    For illustrating Theorem 4.1, we take

    F(t,y)=[et900+t(siny+12) ,(1+t)30(|y||y|+1+y+18)]. (4.4)

    Using the given data, we find p(t)=(1+t)30,φ(y)=||y||+98, and by condition (B3), we have ˆW>0.7066246467. Thus all conditions of Theorem 4.1 are satisfied and consequently, there exists at least one solution for the problem (4.3) with F(t,y) given by (4.4) on [0,2].

    In order to demonstrate the application of Theorem 4.2, let us choose

    F(t,y)=[et900+t(tan1y+12) ,(1+t)30(|y||y|+1+18)]. (4.5)

    Clearly

    Hd(F(t,y),F(t,ˉy))(t+1)30yˉy.

    Letting μ(t)=(t+1)30, it is easy to check that d(0,F(t,0))μ(t) holds for almost all t[0,2] and ϑ0.7572001575<1 (ϑ is given by 4.2). As the hypotheses of Theorem 4.2 are satisfied, we conclude that the problem (4.3) with F(t,y) given by (4.5) has at least one solution on [0,2].

    We have developed the existence theory for fractional differential equations and inclusions involving Caputo-type generalized fractional derivative equipped with generalized fractional integral boundary conditions (in the sense of Katugampola). Standard fixed point theorems for single-valued and multi-valued maps are employed to obtain the desired results, which are well illustrated with the aid of examples. Our results are new in the given configuration and contribute significantly to the existing literature on the topic.

    The authors gratefully acknowledge the referees for their useful comments on their paper.

    The authors declare that they have no conflict of interests.


    Acknowledgments



    The authors thank Mr. Shajan K Varghese (Graphic designer) for providing the image.

    Conflict of interest



    The authors declare no conflict of interest.

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