Research article

The case for a clean cooking green bank

  • Received: 08 April 2025 Revised: 11 June 2025 Accepted: 11 June 2025 Published: 16 June 2025
  • JEL Codes: F33, F35, O13, O19

  • The socio-economic and climate benefits of a transition to clean fuels and technology for cooking are gaining prominence on the global policy agenda. However, investment volumes fall critically short of those required to achieve Sustainable Development Goal 7's universal clean cooking access target, while available forms of finance often do not match demand. We investigated the value in creating a specialised public bank, modelled on green state investment banks, to address the clean cooking investment challenge. In so doing, we introduced the green bank concept to the academic literature on clean cooking and provided original data and analysis. Expert interviews revealed a desire for public banks to assume greater risk in their financing activities in clean cooking markets, and to adopt a broader array of financial instruments and structures. Interviewees also recommended that public banks act as pathfinders and first movers in these markets, play a more prominent role in market building, and educate and organise the aggregate funding group. These attributes displayed notable similarities with the roles historically undertaken by green banks. Our findings suggested that a dedicated public bank for clean cooking, modelled on green banks, would be additional to the sector and potentially play a catalytic role in leveraging private investment.

    Citation: Olivia Coldrey, Paul Lant, Peta Ashworth. The case for a clean cooking green bank[J]. Green Finance, 2025, 7(2): 381-405. doi: 10.3934/GF.2025014

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  • The socio-economic and climate benefits of a transition to clean fuels and technology for cooking are gaining prominence on the global policy agenda. However, investment volumes fall critically short of those required to achieve Sustainable Development Goal 7's universal clean cooking access target, while available forms of finance often do not match demand. We investigated the value in creating a specialised public bank, modelled on green state investment banks, to address the clean cooking investment challenge. In so doing, we introduced the green bank concept to the academic literature on clean cooking and provided original data and analysis. Expert interviews revealed a desire for public banks to assume greater risk in their financing activities in clean cooking markets, and to adopt a broader array of financial instruments and structures. Interviewees also recommended that public banks act as pathfinders and first movers in these markets, play a more prominent role in market building, and educate and organise the aggregate funding group. These attributes displayed notable similarities with the roles historically undertaken by green banks. Our findings suggested that a dedicated public bank for clean cooking, modelled on green banks, would be additional to the sector and potentially play a catalytic role in leveraging private investment.



    Differential equations with various types of fractional derivatives such as Caputo fractional derivative, Riemann-Liouville fractional derivative, are intensively studied theoretically and applied to varies models in the last decades. For example, they are successfully applied to study various types of neural networks (see, for example, [1,2,3]). Fractional differential equations with delays are rapidly developed. One of the main studied qualitative questions about fractional delay differential equation is the one about stability. In 1961, Dorato [4] introduced a concept of finite time stability (FTS). FTS is different from asymptotic stability. However, it is regarded as one of the core problems in delay systems from practical considerations. Later this type of stability has been applied to different types of differential equations. Recently, it is applied for Caputo delta fractional difference equations [5,6], for Caputo fractional differential equations [7] for -Hilfer fractional differential equation [8].

    The investigations of the properties of the solutions of Riemann-Liouville (RL) fractional differential equations with delays are still at his initial stage. The asymptotic stability of the zero solution of the linear homogeneous differential system with Riemann-Liouville fractional derivative is studied in [9]. Li and Wang introduced the concept of a delayed Mittag-Leffler type matrix function, and then they presented the finite-time stability results by virtue of a delayed Mittag-Leffler type matrix in [10,11,12]. In connection with the presence of the bounded delay the initial condition is given on a whole finite interval called initial interval. In the above mentioned papers ([10,11,12]) the authors study the case when the lower limit of the RL fractional derivative coincides with the left side end of the initial interval. It changes the meaning of the initial condition in differential equations. In connection with this in the paper we set up an initial condition satisfying two main properties: first, it is similar to the initial condition in differential equations with ordinary derivatives and, second, the RL fractional condition is defined at the right side end of the initial interval which is connected with the presence of RL fractional derivative.

    In this paper we study initial value problems for scalar nonlinear RL fractional differential equations with constant delays. Similarly to the case of ordinary derivative, the differential equation is given to the right of the initial time interval. It requires the lower bound of the RL fractional derivative to coincides with the right side end of the initial time interval. We present an integral representation of of the studied initial value problem. By the help of fractional generalization of Gronwall inequality we study the existence, continuous dependence and finite time stability of the scalar nonlinear RL fractional differential equations.

    The main contributions of the current paper include:

    (ⅰ) An appropriate initial value problem for nonlinear RL fractional differential equations is set up based on the idea of the initial time interval for delay differential equations with ordinary derivatives.

    (ⅱ) A mild solution of the considered initial value problem is defined based on an appropriate integral representation of the solution.

    (ⅲ) The existence, continuous dependence and finite time stability of the scalar nonlinear RL fractional differential equations is studied by the help of fractional generalization of Gronwall inequality.

    The rest of this paper is organized as follows. In Section 2, some notations and preliminary lemmas are presented. In Section 3, main results are obtained. In Section 3.1. mild solution of the studied initial value problem is defined and some sufficient conditions by Banach contraction principle are obtained. In Section 3.2. continuous dependence on the initial functions is investigated based on the fractional extension of Gronwall inequality. In Section 3.3. some sufficient conditions for finite time stability are given.

    Let J=[τ,T], I=[0,T] where τ>0 is a constant, T<. Without loss of generality we can assume there exists a natural number N such that T=(N+1)τ. Let Lloc1(I,R) be the linear space of all locally Lebesgue integrable functions m:IR, PC(J)=C([τ,0),R)C((0,T],R).

    Let xPC(J,R). Denote ||x||J=suptJ|x(t)|.

    In this paper we will use the following definitions for fractional derivatives and integrals:

    Riemann - Liouville fractional integral of order q(0,1) ([13,14])

    0Iqtm(t)=1Γ(q)t0m(s)(ts)1qds,     tI,

    where Γ(.) is the Gamma function.

    Riemann - Liouville fractional derivative of order q(0,1) ([13,14])

    RL0Dqtm(t)=ddt( 0I1qtm(t))=1Γ(1q)ddtt0(ts)qm(s)ds,     tI.

    We will give fractional integrals and RL fractional derivatives of some elementary functions which will be used later:

    Proposition 1. The following equalities are true:

    RL0Dqttβ=Γ(1+β)Γ(1+βq)tβq,       0I1qtβ1=Γ(β)Γ(1+βq)tβq,
    0I1qtq1=Γ(q),       RL0Dqttq1=0.

    The definitions of the initial condition for fractional differential equations with RL-derivatives are based on the following result:

    Proposition 2. (Lemma 3.2 [15]). Let q(0,1), and mLloc1([0,T],R).

    (a) If there exists a.e. a limit limt0+[tq1m(t)]=c, then there also exists a limit

    0I1qtm(t)|t=0:=limt0+ 0I1qtm(t)=cΓ(q).

    (b) If there exists a.e. a limit 0I1qtm(t)|t=0=b and if there exists the limit limt0+[t1qm(t)] then

    limt0+[t1qm(t)]=bΓ(q).

    We will use the Mittag - Leffler functions with one and with two parameters, respectively, (see, for example, [14]) given by Ep(z)=j=0zjΓ(jp+1) and Ep,q(z)=j=0zjΓ(jp+q).

    Proposition 3. [16] (Gronwall fractional inequality) Suppose a(t) is a nonnegative function locally integrable on [0,T) (some T) and b(t) is a nonnegative, nondecreasing continuous function defined on [0,T), b(t)M (constant), and suppose u(t) is nonnegative and locally integrable on [0,T) with

    u(t)a(t)+b(t)t0(ts)q1u(s)ds,   t[0,T).

    Then

    u(t)a(t)+t0(n=1(b(t)Γ(q))nΓ(nq)(ts)nq1a(s))ds, t[0,T).

    Recently, in [17] the non-homogeneous scalar linear Riemann-Liouville fractional differential equations with a constant delay :

     RL0Dqtx(t)=Ax(t)+Bx(tτ)+f(t) for t>0. (2.1)

    with the initial conditions

    x(t)=g(t),   t[τ,0], (2.2)
    limt0+(t1qx(t))=g(0)Γ(q) (2.3)

    where fC(R+,R), gC([τ,0],R) was studied. It was proved the solution is given by the function

     Λq(t)={g(t)t(τ,0]g(0)Eq,q(Atq)tq1+t0(ts)q1Eq,q(A(ts)q)(Bg(sτ)+f(s))ds t(0,τ]g(0)Eq,q(Atq)tq1+t0(ts)q1Eq,q(A(ts)q)f(s)ds        +Bn1i=0(i+1)τiτ(ts)q1Eq,q(A(ts)q)Λq(sτ)ds        +Btnτ(ts)q1Eq,q(A(ts)q)Λq(sτ)ds                                             for   t(nτ,(n+1)τ],n=1,2, (2.4)

    where Eq,q(z)=i=0ziΓ(iq+q) and Eq(z)=i=0ziΓ(iq+1) are Mittag-Leffler functions with two and one parameter, respectively.

    Now, we will study the following nonlinear fractional delay differential equations

     RL0Dqtx(t)=Ax(t)+Bx(tτ)+f(t,x(t)) for tI. (2.5)

    with the initial conditions (2.2), (2.3) where A,BR are given constants, f:I×RR.

    Remark 1. Note that in the case of the linear equation (2.1) we have formula (2.4) for the explicit solution since in the case of nonlinear equation (2.5) we are not able to obtain an explicit formula, we could provide only an integral presentation of the solution (see Example 1 and Example 2).

    Example 1. Consider the special case of (2.1):

     RL0D0.5tx(t)=x(t1)+t for t>0x(t)=t,   t[1,0],limt0+(t0.5x(t))=0. (2.6)

    Then applying Eq,q(0)=1Γ(q) we obtain the solution of (2.6):

     x(t)={t,         t(1,0]1πt0(ts)0.5(s1+s)ds=2tπ(43t1),      t(0,1]1πt0(ts)0.5sds+1π10(ts)0.5(s1)ds+1πtτ(ts)0.5x(s1)ds    =43πt1.543π(t1)1.5+43πt(t1.5)+t(t1)+43πt 2F1[0.5,1.5,2.5,1t]                1615πt 2F1[0.5,2.5,3.5,1t],     t(1,2]. (2.7)

    In connection with Remark 1 we will define a mild solution:

    Definition 1. A function xPC(J,R) is called a mild solution of the IVP (2.5), (2.2), (2.3) if it satisfies the following integral equation

     x(t)={g(t)   for   t[τ,0],g(0)Eq,q(Atq)tq1+Bt0(ts)q1Eq,q(A(ts)q)g(sτ)ds        +t0(ts)q1Eq,q(A(ts)q)f(s,x(s))ds   for   t(0,τ],g(0)Eq,q(Atq)tq1+t0(ts)q1Eq,q(A(ts)q)f(s,x(s))ds        +Bn1i=0(i+1)τiτ(ts)q1Eq,q(A(ts)q)x(sτ)ds        +Btnτ(ts)q1Eq,q(A(ts)q)x(sτ)ds                                             for   t(nτ,(n+1)τ],n=1,2,,N

    Example 2. Consider the partial case of (2.5) (compare with (2.6):

     RL0D0.5tx(t)=x(t1)+sin(x(t)) for t>0x(t)=t,   t[1,0],limt0+(t0.5x(t))=0. (3.1)

    Now similarly to Example 1 we are not able to obtain the exact solution of (2.6). But using Definition 1 we can consider the mild solution x(t) of the IVP (3.1) satisfying:

     x(t)={t,         t(1,0]1πt0(ts)0.5(s1)ds+1πt0(ts)0.5sin(x(s))ds      =2tπ(23t1)+1πt0(ts)0.5sin(x(s))ds,      t(0,1]1πt0(ts)0.5sin(x(s))ds+1π10(ts)0.5(s1)ds+1πtτ(ts)0.5x(s1)ds    =1πt0(ts)0.5sin(x(s))ds43π(t1)1.5+43πt(t1.5)        +1πtτ(ts)0.5x(s1)ds,     t(1,2]. (3.2)

    Examples 1 and 2 show the main difference between the linear RL fractional differential equations and the nonlinear RL fractional differential equations with a linear part.

    We will introduce the following conditions:

    (A1). The function fC([0,T]×R,R) and there exists a function wC(I,R+) such that |f(t,x)|w(t) for all tI,xR.

    (A2). The function fC([0,T]×R,R) and there exists a constant L>0 such that |f(t,x)f(t,y)|L|xy| for all tI,x,yR. First, we will consider the case of Lipschitz nonlinear function.

    Theorem 1. Let A0, the condition (A2) be satisfied and

    1. The function gC([τ,0],R and |g(0)|<.

    2. ρ=Lh1+|B|h2|A|<1 where h1=maxt[0,T]|Eq(Atq)1|,

    h2=maxn=0,1,2,,N{maxt(nτ,(n+1)τ](|Eq(A(tnτ)q)1|+n1j=0|Eq(A(tjτ)q)Eq(A(t(j+1)τ)q)|)}.

    Then the the IVP (2.5), (2.2), (2.3) has a unique mild solution xPC(J,R).

    P r o o f: Existence. Define the operator Ξ:PC(J,R)PC(J,R) by the equality

     Ξ(x(t))={g(t)            t[τ,0]g(0)Eq,q(Atq)tq1+Bt0(ts)q1Eq,q(A(ts)q)g(sτ)ds        +t0(ts)q1Eq,q(A(ts)q)f(s,x(s))ds   for   t(0,τ],g(0)Eq,q(Atq)tq1+t0(ts)q1Eq,q(A(ts)q)f(s,x(s))ds        +Bn1i=0(i+1)τiτ(ts)q1Eq,q(A(ts)q)x(sτ)ds        +Btnτ(ts)q1Eq,q(A(ts)q)x(sτ)ds                                             for   t(nτ,(n+1)τ],n=1,2,,N

    Let z,yPC(J,R). We will prove that

    |Ξ(z(t))Ξ(y(t))|(L|Eq(Atq)1||A|+|B|n1j=0|Eq(A(tjτ)q)Eq(A(t(j+1)τ)q)||A|        +|B||Eq(A(tnτ)q)1||A|)||zy||J for   t(nτ,(n+1)τ], n=0,1,2,,N (3.3)

    Let t(0,τ]. Then applying Definition 1 and the equality

    t0(ts)q1Eq,q(A(ts)q)ds=i=0AiΓ((i+1)q)t0(ts)(i+1)q1ds=i=0Ai(1+i)qΓ((i+1)q)t(i+1)q=Eq(Atq)1A,   t(0,τ], (3.4)

    we obtain

    |Ξ(z(t))Ξ(y(t))||t0(ts)q1Eq,q(A(ts)q)|f(s,z(s))f(s,y(s))|ds|L|t0(ts)q1Eq,q(A(ts)q)|z(s)y(s)|ds|L||zy||J|t0(ts)q1Eq,q(A(ts)q)ds|=L|Eq(Atq)1||A|||zy||J (3.5)

    Let t(τ,2τ]. Then according to Definition 1 and the equality

    tτ(ts)q1Eq,q(A(ts)q)ds=i=0AiΓ((i+1)q)tτ(ts)(i+1)q1ds=i=0Ai(1+i)qΓ((i+1)q)(tτ)(i+1)q=Eq(A(tτ)q)1A,    t(τ,2τ], (3.6)

    we have

    |Ξ(z(t))Ξ(y(t))||t0(ts)q1Eq,q(A(ts)q)|f(s,z(s))f(s,t(s))|ds|        +|B| |τ0(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|        +|B| |tτ(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|(L|Eq(Atq)1|A+|B| |tτ(ts)q1Eq,q(A(ts)q)ds|)||zy||J(L|Eq(Atq)1||A|+|B||Eq(A(tτ)q)1||A|)||zy||J. (3.7)

    Let t(2τ,3τ]. Then according to Definition 1 and the equalities

    t2τ(ts)q1Eq,q(A(ts)q)ds=i=0AiΓ((i+1)q)t2τ(ts)(i+1)q1ds=i=0Ai(1+i)qΓ((i+1)q)(tτ)(i+1)q=Eq(A(t2τ)q)1A,    t(2τ,3τ], (3.8)

    and

    2ττ(ts)q1Eq,q(A(ts)q)ds=i=0AiΓ((i+1)q)2ττ(ts)(i+1)q1ds=i=0Ai(1+i)qΓ((i+1)q)(tτ)(i+1)qi=0Ai(1+i)qΓ((i+1)q)(t2τ)(i+1)q=Eq(A(tτ)q)Eq(A(t2τ)q)A, (3.9)

    we have

    |Ξ(z(t))Ξ(y(t))||t0(ts)q1Eq,q(A(ts)q)|f(s,z(s))f(s,t(s))|ds|        +|B| |τ0(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|        +|B| |2ττ(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|        +|B| |t2τ(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|(L|Eq(Atq)1||A|+|B| |2ττ(ts)q1Eq,q(A(ts)q)ds|         +|B| |t2τ(ts)q1Eq,q(A(ts)q)ds|)||zy||J(L|Eq(Atq)1||A|+|B||Eq(A(tτ)q)Eq(A(t2τ)q)||A|       +|B||Eq(A(t2τ)q)1||A|)||zy||J. (3.10)

    Following the induction process and the definition of ρ we obtain that ||Ξ(z(t))Ξ(y(t))||Jρ||zy||J. Therefore, the operator Ξ:PC(J,R)PC(J,R) is a contraction.

    Uniqueness. Let z(t),y(t) be two mild solutions of the IVP (2.5), (2.2), (2.3). Applying induction process w.r.t. the intervals and from condition 2 we obtain that ||zy||(kτ,(k+1)τ]<ρ||zy||(kτ,(k+1)τ] for k=0,1,,N which proves the uniqueness.

    Remark 2. It is obvious that the condition A0 in Theorem 1 is not a restriction because the nonzero term Ax could be added to the nonlinear part without losing the Lipschitz property.

    Example 3. Consider the IVP (3.1) In this case A=0.1,f(t,x)=sin(x)0.1x,B=1. Then the condition (A2) is satisfied with L=1.1 but the condition 2 of Theorem 1 is not satisfied.

    Now, we change the equation in the IVP (3.1) to  RL0D0.5tx(t)=0.1x(t1)+0.01sin(x(t)). In this case A=0.1,f(t,x)=0.01sin(x)0.1x,B=0.1, h1=h2=0.43581 and ρ=(0.11+0.1)0.435810.1<1. According to Theorem the IVP (3.1) has unique mild solution which is satisfying the integral presentation given in Definition 1.

    In the case of a bounded nonlinear function we have the following result:

    Theorem 2. Let the condition (A1) be satisfied and

    1. The function gC([τ,0],R and |g(0)|<.

    2. ρ=2||w||Ih1+|B|h2|A|<1 where h1 and h2 are defined in Theorem 1.

    Then the the IVP (2.5), (2.2), (2.3) has a unique solution xPC(J,R).

    The proof of Theorem 2 is similar to the one of Theorem 1 and we omit it.

    We will study the continuous dependence of mild solutions of the IVP (2.5), (2.2), (2.3) on the initial functions.

    Consider IVP (2.5), (2.2), (2.3) and the RL fractional equation (2.5) with initial conditions

    x(t)=p(t),   t[τ,0], (3.11)
    limt0+(t1qx(t))=p(0)Γ(q) (3.12)

    Theorem 3. Let the following conditions be satisfied:

    1. The functions g,pC([τ,0],R, |g(0)|<, |p(0)|<.

    2. The function fC([0,T]×R,R) and it is Lipschitz with a constant L>0 on [0,T]×R.

    Then for any number δ>0 there exist numbers Kk,Ck>0, k=0,1,2,,N such that the inequality ||gp||[τ,0]<δ implies

    |x(t)y(t)|δ(Kk(tkτ)q1+Ck) for  t(kτ,(k+1)τ],  k=0,1,2,,N (3.13)

    where x(t),y(t) are mild solutions of the IVPs (2.5), (2.2), (2.3) and (2.5), (3.11), (3.12) respectively.

    P r o o f: We will use the induction w.r.t. the intervals to prove the claim.

    Let M=suptJ|Eq,q(Atq)|.

    Let t(0,τ]. Then from Definition 1 and Eq. (3.4) we get

    |x(t)y(t)|δMtq1+|B|Mδt0(ts)q1ds+LMt0(ts)q1|x(s)y(s)|dsδMtq1+δM|B|τqq+LMt0(ts)q1|x(s)y(s)|dsδMtq1+δP0+LMt0(ts)q1|x(s)y(s)|ds (3.14)

    where P0=M|B|τqq.

    According to Proposition 3, the inequality t0(ts)nq1sq1ds=tnq+q1Γ(q)Γ(nq)Γ(nq+q) we obtain

    |x(t)y(t)|δMtq1+δP0+δP0t0n=1(MLΓ(q))n(ts)nq1Γ(nq)ds+δMt0n=1(MLΓ(q))n(ts)nq1Γ(nq)sq1dsδP0n=0(MLΓ(q))ntnqΓ(nq+1)+δMtq1Γ(q)n=0(MLΓ(q))n tnqΓ(nq+q)=δ(K0tq1+C0),   t(0,τ] (3.15)

    where K0=MΓ(q)Eq,q(MLΓ(q)τq), C0=M|B|τqqEq(MLΓ(q)τq).

    Let t(τ,2τ]. Then applying Definition 1, (3.15), the inequalities τ0(τs)q1sq1ds=τ2q1Γ(q)Γ(q)Γ(2q), qΓ2(q)Γ(2q)2 we get

    |x(t)y(t)|δMtq1+LMt0(ts)q1|x(s)y(s)|ds+|B|δMτ0(ts)q1ds+|B|Mtτ(ts)q1(δK0(sτ)q1+δC0)dsδMτq1+LMδK0(tτ)2q1Γ2(q)Γ(2q)+LMδC0(2τ)qq+|B|δMτqq+δ|B|MK0(tτ)2q1Γ2(q)Γ(2q)+δ|B|MC0(tτ)qq+LMtτ(ts)q1|x(s)y(s)|dsδMτq1+2δ(L+|B|)MK0(tτ)q1(τ)qq+δ|B|MC0τqq+|B|δM(τ)qq+δLMK0τ2q1Γ(q)Γ(q)Γ(2q)+LMtτ(ts)q1|x(s)y(s)|ds2δ|B|MK0τqq(tτ)q1+δP1+LMtτ(ts)q1|x(s)y(s)|ds (3.16)

    where P1=Mτq1+|B|MC0τqq+|B|M(τ)qq+LMK0τ2q1Γ(q)Γ(q)Γ(2q).

    According to Proposition 3 we obtain

    |x(t)y(t)|2|B|MK0τqq(tτ)q1+P1++tτ[n=1(MLΓ(q))n(ts)nq1Γ(nq)(2|B|MK0τqq(sτ)q1+P1]ds2|B|MK0τqq(tτ)q1+P1Eq(MLΓ(q)(tτ)q)+2|B|MK0τqqn=1(MLΓ(q))n(tτ)1+q+nqΓ(q)Γ(nq+q)P1Eq(MLΓ(q)(tτ)q)+2|B|MK0Γ(q)τqq(tτ)q1Eq,q(MLΓ(q)(tτ)q)P1Eq(MLΓ(q)τq)+2|B|MK0Γ(q)τqq(tτ)q1Eq,q(MLΓ(q)τq)=δ(K1(tτ)q1+C1),  t(τ,2τ] (3.17)

    where K1=2|B|M2Γ2(q)E2q,q(MLΓ(q)τq)τqq and C1=P1Eq(MLΓ(q)τq).

    Let t(2τ,3τ]. Then applying Definition 1 and (3.6) we get

    |x(t)y(t)|δMtq1+LMδτ0(ts)q1(K0sq1+C0)ds+LMδ2ττ(ts)q1(K1(sτ)q1+C1)ds+LMt2τ(ts)q1|x(s)y(s)|ds+|B|Mδτ0(ts)q1ds+|B|Mδ2ττ(ts)q1(K0(sτ)q1+C0)ds+|B|Mδt2τ(ts)q1(K1(s2τ)q1+C1)dsδM(2τ)q1+δLMK0(3τ)2q1Beta[1/3,q,q]+δLMC0(3τ)qq+δLMK1(2τ)2q1Beta[1/2,q,q]+δLMC1(2q)qq+LMt2τ(ts)q1|x(s)y(s)|ds+|B|Mδ(3τ)qq+|B|MδK0(2τ)2q1Beta[1/2,q,q]+|B|MδC0(2τ)qq+δ|B|MK1τqΓ2(q)Γ(2q)(t2τ)q1+δ|B|MC1τqqδ|B|MK1τqΓ2(q)Γ(2q)(t2τ)q1+δP2+LMt2τ(ts)q1|x(s)y(s)|ds

    where Beta[x,q,q] is the incomplete beta function.

    According to Proposition 3 we obtain

    |x(t)y(t)|δ|B|MK1τqΓ2(q)Γ(2q)(t2τ)q1+δP2+δP2t2τ[n=1(MLΓ(q))n(ts)nq1Γ(nq)]ds+δ|B|MK1τqΓ2(q)Γ(2q)t2τ[n=1(MLΓ(q))n(ts)nq1Γ(nq)(s2τ)q1]dsδP2n=0(MLΓ(q))n(t2τ)nqΓ(nq+1)+δ|B|MK1τqΓ3(q)Γ(2q)(t2τ)q1n=0(MLΓ(q))n(t2τ)nqΓ(q+nq)=δ|B|MK1τqΓ3(q)Γ(2q)Eq,q(MLΓ(q)(t2τ)q)(t2τ)q1+δP2Eq(MLΓ(q)(t2τ)q)=K2(t2τ)q1+C2,   t(2τ,3τ],

    where K2=δ|B|MK1τqΓ3(q)Γ(2q)Eq,q(MLΓ(q)(t2τ)q) and C2=δP2Eq(MLΓ(q)(τ)q).

    Continue the induction process we prove the claim.

    Corollary 1. Let the conditions of Theorem 3 be satisfied and q>0.5.

    Then for any positive numbers δ,ε: ε<τ there exists a number K,C>0 such that the inequality ||gp||[τ,0]<δ implies

    |x(t)y(t)|δK for  t(ε,T], (3.18)

    where x(t),y(t) are mild solutions of the IVPs (2.5), (2.2), (2.3) and (2.5), (3.11), (3.12) respectively.

    P r o o f: The proof is similar to the one of Theorem 3 applying the inequality (tkτ)2q1qτqq for t(kτ,(k+1)τ], k=0,1,,N.

    In this section we will define and study the finite time stability of mild solutions of the IVP for Riemann-Liouville (2.5), (2.2), (2.3).

    Note that because of the singularity of tq1 at 0, we could prove the FTS on an interval which does not contain 0.

    Theorem 4. Let the function gC([τ,0],R, |g(0)|<, q>0.5 and the condition (A1) be satisfied.

    Then for any real positive numbers δ,ε: ε<τ there exists a number K depending on δ and ε such that the inequality ||g||[τ,0]<δ implies |x(t)|<K for t(ε,T] where x(t) is the mild solution of the IVP (2.5), (2.2), (2.3).

    P r o o f: Let ||g||[τ,0]<δ and M=suptJ|Eq,q(Atq)|.

    Let t(0,τ]. Then according to Definition 1 we have

    |x(t)|δEq,q(Atq)tq1+|B|δt0(ts)q1Eq,q(A(ts)q)|ds        +t0(ts)q1Eq,q(A(ts)q)|f(s,x(s))|dsδMtq1+|B|Mδt0(ts)q1ds+M||w||It0(ts)q1dsδMtq1+M(|B|δ+||w||I)τqq,   t(0,τ]. (3.19)

    From (3.19) it follows that

    |x(t)|δMεq1+M(|B|δ+||w||I)τqq,   t(ε,τ]. (3.20)

    Let t(τ,2τ]. Then we have

    |x(t)|δMτq1+Mt0(ts)q1|f(s,x(s))|ds        +|B|Mδτ0(ts)q1ds+|B|Mtτ(ts)q1x(sτ)dsδMτq1+M((||w||I+|B|δ)(2τ)qq+|B|M(δMτ2q12q1+M(||w||I+|B|δ)(τqq)2)=K1.

    Let t(2τ,3τ]. Then we have

    |x(t)|δM(2τ)q1+Mt0(ts)q1|f(s,x(s))|ds+|B|Mδτ0(ts)q1ds+|B|M2ττ(τs)q1(δM(sτ)q1+M(|B|δ+||w||I)τqq)ds+|B|MK1t2τ(ts)q1dsδM(2τ)q1+|M(|w||I+|B|δ)(3τ)qq+|B|M(δMτ2q12q1+M(|B|δ+||w||I)(τqq))2+K1τq)=K2.

    Let t(3τ,4τ]. Then we have

    |x(t)|δM(3τ)q1+|M(|w||I+|B|δ)(4τ)qq+|B|M(δMτ2q12q1+M(|B|δ+||w||I)(τqq)2)+|B|Mτq(K1+K2)=K3.

    Following the induction process we prove the claim with K=δM(Nτ)q1+|M(|w||I+|B|δ)(Nτ)qq+|B|M(δMτ2q12q1+M(|B|δ+||w||I)(τqq)2)+|B|MτqNi=1Ki.

    In the case the nonlinear Lipschitz functions we obtain the following result:

    Theorem 5. Let the function gC([τ,0],R, |g(0)|<, q>0.5 and the condition (A2) be satisfied.

    Then for any real positive numbers δ,ε: ε<τ there exists a number K depending on δ and ε such that the inequality ||g||[τ,0]<δ implies |x(t)|<K for t(ε,T] where x(t) is the mild solution of the IVP (2.5), (2.2), (2.3).

    P r o o f: According to Theorem 1 the the IVP (2.5), (2.2), (2.3) has a unique solution xPC(J,R). Let ||g||[τ,0]<δ and M=suptJ|Eq,q(Atq)|.

    Let t(ε,τ]. Then according to Definition 1 we have

    |x(t)|δEq,q(Atq)tq1+|B|δt0(ts)q1Eq,q(A(ts)q)ds        +Lt0(ts)q1Eq,q(A(ts)q)|x(s)|dsδMtq1+|B|Mδτqq+LMt0(ts)q1|x(s)|ds. (3.21)

    From (3.21) and Proposition 3 it follows that

    |x(t)|δMtq1+|B|Mδτqq+t0(n=1(LMΓ(q))nΓ(nq)(ts)nq1(δMsq1+|B|Mδτqq))dsδMtq1+|B|Mδτqq+δMtq1Γ(q)n=1(LMΓ(q))nΓ(nq+q)tnq+|B|Mδτqqn=1(LMΓ(q))nΓ(nq+1)(t)nqδMtq1+|B|Mδτqq+δMεq1Γ(q)n=1(LMΓ(q))nΓ(nq+q)tnq+|B|Mδτqqn=1(LMΓ(q))nΓ(nq+1)(t)nqδMtq1Γ(q)Eq,q(LMΓ(q))tq)+|B|MδτqqEq(LMΓ(q))tq). (3.22)

    Therefore,

    x(t)δ(Mεq1Γ(q)+|B|Mτqq)Eq(LMΓ(q))τq)=δK0,  t(ε,τ]. (3.23)

    Let t(τ,2τ]. Then from (3.23) we have

    |x(t)|δMtq1+MLt0(ts)q1|x(s)|ds        +|B|Mδτ0(ts)q1ds+|B|Mtτ(ts)q1x(sτ)dsδMtq1+|B|Mδ(1+K0)τqq+MLt0(ts)q1|x(s)|ds.

    From (3.24) and Proposition 3 it follows that

    |x(t)|δMtq1+|B|Mδ(1+K0)τqq+tτ(n=1(MLΓ(q))nΓ(nq)(ts)nq1(δMsq1+|B|Mδ(1+K0)τqq))dsδMtq1Γ(q)Eq,q(LMΓ(q))tq)+|B|Mδ(1+K0)τqq]Eq(LMΓ(q))tq). (3.24)

    Therefore,

    |x(t)|δ(Mεq1Γ(q)+|B|M(1+K0)τqq)Eq(LMΓ(q))τq)=δK1,  t(τ,2τ] (3.25)

    Following the induction process we obtain

    |x(t)|δ(Mεq1Γ(q)+|B|M(1+Kk1)τqq)Eq(LMΓ(q))τq)=δKk,  t(kτ,(k+1)τ],

    where Kk=(Mεq1Γ(q)+|B|M(1+Kk1)τqq)Eq(LMΓ(q))τq), k=1,2,,N.

    Example 4. Consider the IVP (3.1) with RL fractional equation  RL0D0.5tx(t)=0.1x(t1)+0.01sin(x(t)). According to Example 3 it has unique mild solution x(t) which is satisfying the integral presentation given in Definition 1. Also, according to Theorem 5 for δ=1, ε=0.001 the inequality |x(t)|<K holds for t[0.001,3] where M=sup[0,3]E0.5,0.5(0.1t0.5)=0.7772, K0=(0.77720.0010.51Γ(0.5)+0.10.777210.5)Eq(0.110.7772Γ(0.5))=52.321, K1=62.0518 and K=K2=63.8615.

    We study scalar nonlinear RL fractional differential equations with constant delays. An appropriate initial value problem for studd equations is set up based on the idea of the initial time interval for delay differential equations with ordinary derivatives. A mild solution is defined based on an appropriate integral representation of the solution. The existence, continuous dependence and finite time stability of the scalar nonlinear RL fractional differential equations is studied by the help of fractional generalization of Gronwall inequality. The obtained integral representations could be successfully applied to study many qualitative investigation of the properties of the solutions of nonlinear RL fractional differential equations.

    Research was partially supported by Fund Scientific Research MU19-FMI-009, Plovdiv University.

    All authors declare no conflicts of interest in this paper.



    [1] 60 Decibels (2024) Why Off-Grid Energy Matters 2024. Available from: https://60decibels.com/insights/why-off-grid-energy-matters-2024/ (accessed on 20 March 2024).
    [2] African Development Bank Group, African Green Banks Initiative (2024) Available from: https://www.afdb.org/en/topics-and-sectors/initiatives-and-partnerships/african-green-banks-initiative (accessed on 7 July 2024).
    [3] African Heads of State and Government, The African Leaders Nairobi Declaration on Climate Change and Call to Action, 6 September 2023. Available from: https://www.afdb.org/sites/default/files/2023/09/08/the_african_leaders_nairobi_declartion_on_climate_change-rev-eng.pdf (accessed on 10 September 2023).
    [4] Akomea-Frimpong I, Adeabah D, Ofosu D, et al. (2021) A review of studies on green finance of banks, research gaps and future directions. J Sustain Financ Inv 12: 1241–1264. https://doi.org/10.1080/20430795.2020.1870202 doi: 10.1080/20430795.2020.1870202
    [5] Ameli N, Dessens O, Winning M, et al. (2021) Higher cost of finance exacerbates a climate investment trap in developing economies. Nat Commun 12: 4046. https://doi.org/10.1038/s41467-021-24305-3 doi: 10.1038/s41467-021-24305-3
    [6] Aslam W, Jawaid ST (2023) Systematic Review of Green Banking Adoption: Following PRISMA Protocols. IIM Kozhikode Soc Ma 12: 213–233. https://doi.org/10.1177/22779752231168169 doi: 10.1177/22779752231168169
    [7] Bernard Meka'a C, Landry Djamen B, Noufelie R (2024) Foreign direct investment, Green Technological Innovation and Energy Poverty: Empirical evidences from Sub-Saharan African countries. Renew Energ 231: 120831. https://doi.org/10.1016/j.renene.2024.120831 doi: 10.1016/j.renene.2024.120831
    [8] Bhatia M, Angelou N (2015) Beyond Connections: Energy Access Redefined ESMAP Technical Report; 008/15. Washington, DC: World Bank. Available from: https://hdl.handle.net/10986/24368 (accessed on 15 August 2023).
    [9] Bhattacharyya R (2022) Green finance for energy transition, climate action and sustainable development: overview of concepts, applications, implementation and challenges. Green Financ 4: 1–35. https://doi.org/10.3934/GF.2022001 doi: 10.3934/GF.2022001
    [10] Booth WC, Colomb GG, Williams JM, et al. (2016) The Craft of Research, 4 Ed., Chicago: University of Chicago Press. https://doi.org/10.7208/chicago/9780226239873.001.0001
    [11] Brown R (2021) Mission-oriented or mission adrift? A critical examination of mission-oriented innovation policies. Eur Plan Stud 29: 739–761. https://doi.org/10.1080/09654313.2020.1779189 doi: 10.1080/09654313.2020.1779189
    [12] Cabraal A, Ward WA, Bogach VS, et al. (2021) Living in the Light: The Bangladesh Solar Home Systems Story. Washington, DC: World Bank. Available from: https://documents1.worldbank.org/curated/en/153291616567928411/pdf/Living-in-the-Light-The-Bangladesh-Solar-Home-Systems-Story.pdf (accessed on 20 August 2024).
    [13] Clean Cooking Alliance, Clean Cooking Industry Snapshot (2023) Available from: https://cleancooking.org/wp-content/uploads/2023/12/CCA-2023-Clean-Cooking-Industry-Snapshot.pdf (accessed on 15 December 2023).
    [14] Clean Energy Finance Corporation Act 2012 (Cth) (2012) Available from: https://www.legislation.gov.au/C2012A00104/latest/text (accessed on 10 March 2024).
    [15] Clean Energy Finance Corporation, Clean Energy Finance Corporation Annual Report 2022-23, 26 September 2023. Available from: https://www.cefc.com.au/document?file = /media/l4igzbpf/cefc_ar23_web_sml.pdf (accessed on 10 March 2024).
    [16] Clo S, Frigerio M, Vandone D (2022) Financial support to innovation: The role of European development financial institutions. Res Policy 51: 104566. https://doi.org/10.1016/j.respol.2022.104566 doi: 10.1016/j.respol.2022.104566
    [17] Coalition for Green Capital, What is a Green Bank (2024) Available from: https://coalitionforgreencapital.com/what-is-a-green-bank/ (accessed on 5 March 2024).
    [18] Coldrey O, Lant P, Ashworth P, et al. (2024) Reforming Climate and Development Finance for Clean Cooking. Energies 17: 3720. https://doi.org/10.3390/en17153720 doi: 10.3390/en17153720
    [19] Coldrey O, Lant P, Ashworth P (2023) Elucidating Finance Gaps through the Clean Cooking Value Chain. Sustainability 15: 3577. https://doi.org/10.3390/su15043577 doi: 10.3390/su15043577
    [20] D'Orazio P, Valente M (2019) The role of finance in environmental innovation diffusion: An evolutionary modeling approach. J Econ Behav Organ 162: 417–439. https://doi.org/10.1016/j.jebo.2018.12.015 doi: 10.1016/j.jebo.2018.12.015
    [21] Debrah C, Darko A, Chan APC (2023) A bibliometric-qualitative literature review of green finance gap and future research directions. Clim Dev 15: 432–455. https://doi.org/10.1080/17565529.2022.2095331 doi: 10.1080/17565529.2022.2095331
    [22] Development Committee (Joint Ministerial Committee of the Boards of Governors of the Bank and the Fund on the Transfer of Real Resources to Developing Countries), Ending Poverty on a Livable Planet: Report to Governors on World Bank Evolution, DC2023-0004, September 28, 2023. Available from: https://www.devcommittee.org/content/dam/sites/devcommittee/doc/documents/2023/Final%20Updated%20Evolution%20Paper%20DC2023-0003.pdf (accessed on 3 August 2024).
    [23] Donastorg A, Renukappa S, Suresh S (2017) Financing Renewable Energy Projects in Developing Countries: A Critical Review. IOP Conf Ser Earth Environ Sci 83: 012012. https://doi.org/10.1088/1755-1315/83/1/012012 doi: 10.1088/1755-1315/83/1/012012
    [24] Egli F, Polzin F, Sanders M, et al. (2022) Financing the energy transition: four insights and avenues for future research. Environ Res Lett 17: 051003. https://doi.org/10.1088/1748-9326/ac6ada doi: 10.1088/1748-9326/ac6ada
    [25] ENERGIA, World Bank—Energy Sector Management Assistance Program, UN Women (2018) Policy Brief 12 Global Progress of SDG7—Energy and Gender. New York: United Nations. Available from: https://sustainabledevelopment.un.org/content/documents/17489PB12.pdf (accessed on 15 September 2023).
    [26] ESMAP (2020) The State of Access to Modern Energy Cooking Services. Washington, DC: World Bank. Available from: https://www.esmap.org/the-state-of-access-to-modern-energy-cooking-services (accessed on 8 August 2023).
    [27] Fleta-Asín J, Muñoz F (2021) Renewable energy public–private partnerships in developing countries: Determinants of private investment. Sustain Dev 29: 653–670. https://doi.org/10.1002/sd.2165 doi: 10.1002/sd.2165
    [28] French Presidency, Summit for a New Global Financing Pact Multilateral Development Banks Vision Statement (2023) Available from: https://nouveaupactefinancier.org/pdf/multilateral-development-banks-vision-statement.pdf (accessed on 11 August 2023).
    [29] G7 Climate, Energy and Environment Ministers, Meeting Communiqué (2024) Available from: https://www.g7italy.it/wp-content/uploads/G7-Climate-Energy-Environment-Ministerial-Communique_Final.pdf (accessed on 15 May 2024).
    [30] Gabor, D (2021) The Wall Street Consensus. Dev Change 52: 429–459. https://doi.org/10.1111/dech.12645 doi: 10.1111/dech.12645
    [31] Gautam K, Purkayastha D, Widge V (2023) Proposal for a Global Credit Guarantee Facility (GCGF). Available from: https://www.climatepolicyinitiative.org/wp-content/uploads/2023/10/Discussion-Paper-Proposal-for-a-Global-Credit-Guarantee-Facility-GCGF-Oct-2023.pdf (accessed on 22 July 2024).
    [32] Geddes A (2020) The Role of Green State Investment banks in Financing Low-Carbon Projects, In: Böttcher, J (ed.). Green Banking: Realizing Renewable Energy Projects, Berlin, Boston: De Gruyter Oldenbourg, 349–358. https://doi.org/10.1515/9783110607888
    [33] Geddes A, Schmid N, Schmidt TS, et al. (2020) The politics of climate finance: Consensus and partisanship in designing green state investment banks in the United Kingdom and Australia. Energ Res Soc Sci 69: 101583. https://doi.org/10.1016/j.erss.2020.101583 doi: 10.1016/j.erss.2020.101583
    [34] Geddes A, Schmidt TS (2020) Integrating finance into the multi-level perspective: Technology niche-finance regime interactions and financial policy interventions. Res Policy 49: 103985. https://doi/.org/10.1016/j.respol.2020.103985 doi: 10.1016/j.respol.2020.103985
    [35] Geddes A, Schmidt TS, Steffen B (2018) The multiple roles of state investment banks in low-carbon energy finance: An analysis of Australia, the UK and Germany. Energ Policy 115: 158–170. https://doi.org/10.1016/j.enpol.2018.01.009 doi: 10.1016/j.enpol.2018.01.009
    [36] Gill-Wiehl A, Kammen DM (2022) A pro-health cookstove strategy to advance energy, social and ecological justice. Nat Energ 7: 999–1002. https://doi.org/10.1038/s41560-022-01126-2 doi: 10.1038/s41560-022-01126-2
    [37] Global Distributors Collective, Last Mile Distribution Capital Continuum: Trends, Gaps and Opportunities (2022) Available from: https://infohub.practicalaction.org/server/api/core/bitstreams/22dc4d28-cb0c-4d83-9846-d63836971479/content (accessed on 20 March 2025).
    [38] Government of Barbados, The 2022 Bridgetown Initiative for the Reform of the Global Financial Architecture (2022) Available from: https://pmo.gov.bb/wp-content/uploads/2022/10/The-2022-Bridgetown-Initiative.pdf (accessed on 10 September 2023).
    [39] Green Climate Fund, Executive Director unveils "50 by 30" blueprint for reform, targeting USD 50 billion by 2030, 22 September 2023. Available from: https://www.greenclimate.fund/news/executive-director-unveils-50by30-blueprint-reform-targeting-usd-50-billion-2030 (accessed on 22 August 2024).
    [40] Green Climate Fund, Projects & Programmes FP153 Mongolia Green Finance Corporation, 13 November 2020. Available from: https://www.greenclimate.fund/project/fp153#documents (accessed on 27 July 2024).
    [41] Greenhalgh T, Peacock R (2005) Effectiveness and efficiency of search methods in systematic reviews of complex evidence: audit of primary sources. BMJ 331: 1064. https://doi.org/10.1136/bmj.38636.593461.68 doi: 10.1136/bmj.38636.593461.68
    [42] GuarantCo., Enabling sustainable infrastructure in Africa and Asia, 2024. Available from: https://guarantco.com/ (accessed on 5 August 2024).
    [43] Hundt R (2019) Green banks: a critical boost to clean energy transition. Nature 572: 439–439. https://doi.org/10.1038/d41586-019-02494-8 doi: 10.1038/d41586-019-02494-8
    [44] IEA, IRENA, UNSD, World Bank, WHO (2024) Tracking SDG 7: The Energy Progress Report 2024. Washington, DC: World Bank. Available from: https://www.iea.org/reports/tracking-sdg7-the-energy-progress-report-2024 (accessed on 11 July 2024).
    [45] IEA, IRENA, UNSD, World Bank, WHO (2023) Tracking SDG 7: The Energy Progress Report 2023. Washington, DC: World Bank. Available from: https://www.iea.org/reports/tracking-sdg7-the-energy-progress-report-2023 (accessed on 27 September 2023).
    [46] IEA, The Clean Cooking Declaration: Making 2024 the Pivotal Year for Clean Cooking, 2024a. Available from: https://www.iea.org/news/the-clean-cooking-declaration-making-2024-the-pivotal-year-for-clean-cooking (accessed on 8 July 2024).
    [47] IEA (2024b) World Energy Outlook 2024. Paris: IEA. Available from: https://www.iea.org/reports/world-energy-outlook-2024 (accessed on 18 October 2024).
    [48] IEA (2023) A Vision for Clean Cooking Access for All. Paris: IEA. Available from: https://www.iea.org/reports/a-vision-for-clean-cooking-access-for-all (accessed on 7 September 2023).
    [49] Johnson TP (2014) Snowball Sampling: Introduction, Hoboken: Wiley. https://doi.org/10.1002/9781118445112.stat05720.
    [50] Kalirajan K, Chen H (2018) Private Financing in Low-Carbon Energy Transition: Imbalances and Determinants. In: Anbumozhi, V, Kalirajan, K, Kimura, F (eds.) Financing for Low-carbon Energy Transition. Singapore: Springer, 45–61. https://doi.org/10.1007/978-981-10-8582-6_3
    [51] Khan HHA, Ahmad N, Yusof NM, et al. (2024) Green finance and environmental sustainability: a systematic review and future research avenues, Environ Sci Pollut Res Int 31: 9784–9794. https://doi.org/10.1007/s11356-023-31809-6
    [52] Kwakwa PA, Adusah-Poku F, Adjei-Mantey K (2021) Towards the attainment of sustainable development goal 7: what determines clean energy accessibility in sub-Saharan Africa? Green Financ 3: 268–286. https://doi.org/10.3934/GF.2021014 doi: 10.3934/GF.2021014
    [53] Lyons M, White LV (2023) How Green Banks can create multiple types of value in the transition to net zero emissions. Aust J Public Adm, 1–19. https://doi.org/10.1111/1467-8500.12623 doi: 10.1111/1467-8500.12623
    [54] Marbuah G, Te Velde DW, Attridge S, et al. (2022) Understanding The Role of Development Finance Institutions in Promoting Development: An Assessment of Three African Countries. Stockholm: Stockholm Environment Institute. http://doi.org/10.51414/sei2022.006
    [55] Matthew E (2011) The green investment bank, carbon targets and the challenge for financing. Environ Law Manage 23: 202–203
    [56] Mazzucato M, Penna CCR (2016) Beyond market failures: the market creating and shaping roles of state investment banks. J Econ Policy Reform 19: 305–326. https://doi.org/10.1080/17487870.2016.1216416 doi: 10.1080/17487870.2016.1216416
    [57] McInerney C, Bunn DW (2019) Expansion of the investor base for the energy transition. Energ Policy 129: 1240–1244. https://doi.org/10.1016/j.enpol.2019.03.035 doi: 10.1016/j.enpol.2019.03.035
    [58] McVicar E (2014) Financing the transition to a greener economy. Environ Law Manage 26: 70–77
    [59] MECS & Energy4Impact, Modern Energy Cooking: Review of the Funding Landscape (2022) Available from: https://mecs.org.uk/wp-content/uploads/2022/02/MECS-Landscape-report_final-17-02-2022.pdf (accessed on 18 August 2023).
    [60] Molinari A, Patrucchi L (2023) World Bank Group Evolution: Technical fixes or urgently needed reform? Available from: https://www.brettonwoodsproject.org/2023/07/world-bank-group-evolution-technical-fixes-or-urgently-needed-reform/ (accessed on 28 July 2024).
    [61] Mperejekumana P, Shen L, Saad Gaballah M, et al. (2024) Exploring the potential and challenges of energy transition and household cooking sustainability in sub-sahara Africa. Renew Sustain Energy Rev 199: 114534. https://doi.org/10.1016/j.rser.2024.114534 doi: 10.1016/j.rser.2024.114534
    [62] National Audit Office, The Green Investment Bank, 12 December 2017. Available from: https://www.nao.org.uk/wp-content/uploads/2017/12/The-Green-Investment-Bank.pdf (accessed on 11 March 2024).
    [63] New York Public Service Commission, Order Establishing New York Green Bank and Providing Initial Capitalization, Case 13-M-0412, 19 December 2013. Available from: https://documents.dps.ny.gov/public/MatterManagement/MatterFilingItem.aspx?FilingSeq = 106318 & MatterSeq = 43577 (accessed on 13 March 2024).
    [64] Ngum S, Kim L (2023) Powering a Gender-Just Energy Transition. Geneva: Green Growth Knowledge Partnership. Available from: https://gggi.org/report/powering-a-gender-just-energy-transition/ (accessed on 2 September 2023).
    [65] Njenga M, Gitau JK, Mendum R (2021) Women's work is never done: Lifting the gendered burden of firewood collection and household energy use in Kenya. Energ Res Soc Sci 77: 102071. https://doi.org/10.1016/j.erss.2021.102071 doi: 10.1016/j.erss.2021.102071
    [66] OECD, Development Finance Institutions and Private Sector Development (2024) Available from: https://www.oecd.org/development/development-finance-institutions-private-sector-development.htm (accessed on 6 March 2024).
    [67] OECD (2016) Green Investment Banks: Scaling up Private Investment in Low-carbon, Climate-resilient Infrastructure, Green Finance and Investment. Paris: OECD Publishing. https://dx.doi.org/10.1787/9789264245129-en (accessed on 26 February 2024).
    [68] Osiolo HH, Marwah H, Leach M (2023) The Emergence of Large-Scale Bioethanol Utilities: Accelerating Energy Transitions for Cooking. Energies 16: 6242. https://doi.org/10.3390/en16176242 doi: 10.3390/en16176242
    [69] Ozili PK (2022) Green finance research around the world: a review of literature, Int J Green Econ 16: 56–75. https://doi.org/10.1504/IJGE.2022.125554
    [70] Parker C, Scott S, Geddes A (2019) Snowball Sampling. In: Atkinson, P, Delamont, S, Cernat, A Sakshaug, J.W., Williams, R.A (eds.), SAGE Research Methods Foundations. London: SAGE Publications. https://doi.org/10.4135/9781526421036831710
    [71] Peitz L (2022) Multilateral Development Banks: Mission, Business Model, Financial Management.
    [72] Polzin F, Sanders M, Täube F (2017) A diverse and resilient financial system for investments in the energy transition. Curr Opin Env Sust 28: 24–32. https://doi.org/10.1016/j.cosust.2017.07.004 doi: 10.1016/j.cosust.2017.07.004
    [73] Puzzolo E, Fleeman N, Lorenzetti F, et al. (2024) Estimated health effects from domestic use of gaseous fuels for cooking and heating in high-income, middle-income, and low-income countries: A systematic review and meta-analyses. Lancet Respir Med 12: 281–293. https://doi.org/10.1016/S2213-2600(23)00427-7 doi: 10.1016/S2213-2600(23)00427-7
    [74] Queirós A, Faria D, Almeida F (2017) Strengths and limitations of qualitative and quantitative research methods. Eur J Educ Stud 3: 369–387. https://doi.org/10.5281/zenodo.887089 doi: 10.5281/zenodo.887089
    [75] Rahman S, Hossain Moral I, Hassan M, et al (2022) A systematic review of green finance in the banking industry: perspectives from a developing country. Green Financ 4: 347–363. https://doi.org/10.3934/GF.2022017 doi: 10.3934/GF.2022017
    [76] Rainero C, Modarelli G (2019) Patient Investors Taxonomy: A Behavioral Approach, In: De Vincentiis, P, Culasso, F, Cerrato, S (eds) The Future of Risk Management: Volume II. Palgrave MacMillan, 181–202. https://doi.org/10.1007/978-3-030-16526-0_7
    [77] Ramachandran V (2022) Blanket bans on fossil fuels hurt women and lower-income countries. Nature 607: 9. https://doi.org/10.1038/d41586-022-01821-w doi: 10.1038/d41586-022-01821-w
    [78] Schmidt TS (2014) Low-carbon investment risks and de-risking. Nat Clim Change 4: 237–239. https://doi.org/10.1038/nclimate2112 doi: 10.1038/nclimate2112
    [79] Schub J (2015) Green Banks: Growing Clean Energy Markets by Leveraging Private Investment with Public Financing. J Struct Financ 21: 26–35. https://doi.org/10.3905/jsf.2015.21.3.026 doi: 10.3905/jsf.2015.21.3.026
    [80] Sciences Po, ESMAP (2020) The Smart Economics of Clean Cooking: Placing Women at the Center of the Energy Access Development Agenda. Available from: https://www.sciencespo.fr/students/sites/sciencespo.fr.students/files/sciencespo-projet-co-policy-brief-smart-economics-eng.pdf (accessed on 12 September 2023).
    [81] Sennoga E, Balma L (2022) Fiscal Sustainability in Africa: Accelerating the Post-COVID-19 Recovery through Improved Public Finances. Afr Dev Rev 34: S8–33. https://doi.org/10.1111/1467-8268.12648 doi: 10.1111/1467-8268.12648
    [82] Singh S, Ru J (2022) Accessibility, affordability, and efficiency of clean energy: a review and research agenda. Environ Sci Pollut Res 29: 18333–18347. DOI: https://doi.org/10.1007/s11356-022-18565-9
    [83] Steffen B (2021) A comparative analysis of green financial policy output in OECD countries. Environ Res Lett 16: 074031. https://doi.org/10.1088/1748-9326/ac0c43 doi: 10.1088/1748-9326/ac0c43
    [84] Steffen B, Egli F, Schmidt TS (2020) The Role of Public Banks in Catalyzing Private Renewable Energy Finance, In: Donovan, C (ed.). Renewable Energy Finance: Funding the Future of Energy, 2 Eds., London: World Scientific Publishing Europe Ltd., 197–215. https://doi.org/10.1142/9781786348609_0009
    [85] Sustainable Energy for All, Climate Policy Initiative (2021) Energizing Finance: Understanding the Landscape 2021. Vienna: Sustainable Energy for All. Available from: https://www.seforall.org/system/files/2021-10/EF-2021-UL-SEforALL.pdf (accessed on 10 October 2023).
    [86] The Green Guarantee Company, Guarantees for a Greener World (2024) Available from: https://greenguarantee.co/ (accessed on 5 August 2024).
    [87] UNDP, Human Development Report 2023-24: Breaking the gridlock: Reimagining cooperation in a polarized world (2024) Available from: https://hdr.undp.org/system/files/documents/global-report-document/hdr2023-24reporten.pdf (accessed on 23 March 2024).
    [88] United Nations, Our Common Agenda Policy Brief 6: Reforms to the International Financial Architecture, May 2023. Available from: https://www.un.org/sites/un2.un.org/files/our-common-agenda-policy-brief-international-finance-architecture-en.pdf (accessed on 4 September 2023).
    [89] United Nations, Transforming Our World: The 2030 Agenda for Sustainable Development, 19/35, United Nations General Assembly, Res. 70/1 of 25 September 2015. Available from: https://documents.un.org/doc/undoc/gen/n15/291/89/pdf/n1529189.pdf (accessed on 14 September 2023).
    [90] Waidelich P, Steffen B (2024) Renewable energy financing by state investment banks: Evidence from OECD countries. Energ Econ 132: 107455. https://doi.org/10.1016/j.eneco.2024.107455 doi: 10.1016/j.eneco.2024.107455
    [91] Whitney A, Bodnar P (2018) Beyond Direct Access: How National Green Banks Can Build Country Ownership of Climate Finance, Boulder CO: Rocky Mountain Institute. Available from: https://d231jw5ce53gcq.cloudfront.net/wpcontent/uploads/2018/03/Beyond_Direct_Access_Insight_Brief.pdf (accessed on 23 February 2024).
    [92] Whitney A, Grbusic T, Meisel J, et al. (2020) State of Green Banks 2020, Boulder: Rocky Mountain Institute. Available from: https://rmi.org/insight/state-of-green-banks-2020/ (accessed on 23 February 2024).
    [93] World Bank (2024a) A Focused Assessment of the International Development Association's Private Sector Window: An Update to the Independent Evaluation Group's 2021 Early-Stage Assessment. Independent Evaluation Group. Washington, DC: World Bank. Available from: https://documents1.worldbank.org/curated/en/099921401092428546/pdf/SECBOS188761f900d1a394150090aacfd53.pdf (accessed on 4 August 2024).
    [94] World Bank, Remarks by Ajay Banga at the 2024 G20 Finance Ministers - The Role of Economic Policies in Addressing Inequalities: National Experiences and International Cooperation, 28 February 2024b. Available from: https://www.worldbank.org/en/news/speech/2024/02/28/remarks-by-ajay-banga-at-the-2024-g20-finance-ministers-session-1-the-role-of-economic-policies-in-addressing-inequaliti (accessed on 4 August 2024).
    [95] World Health Organization, Household air pollution: Key facts (2023) Available from: https://www.who.int/news-room/fact-sheets/detail/household-air-pollution-and-health (accessed on 10 December 2023).
    [96] World Health Organization, WHO global air quality guidelines. Particulate matter (PM2.5 and PM10), ozone, nitrogen dioxide, sulfur dioxide and carbon monoxide (2021) Geneva: World Health Organization (accessed on 10 December 2023).
    [97] Wüstenhagen R, Menichetti E (2012) Strategic choices for renewable energy investment: Conceptual framework and opportunities for further research. Energ Policy 40: 1–10. https://doi.org/10.1016/j.enpol.2011.06.050 doi: 10.1016/j.enpol.2011.06.050
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