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Research article

Maximum likelihood DOA estimation based on improved invasive weed optimization algorithm and application of MEMS vector hydrophone array

  • Received: 26 January 2022 Revised: 10 April 2022 Accepted: 17 April 2022 Published: 25 April 2022
  • MSC : 65K05, 65K10

  • Direction of arrival (DOA) estimation based on Maximum Likelihood is a common method in array signal processing, with many practical applications, but the huge amount of calculation limits the practical application. To deal with such an Maximum Likelihood (ML) DOA estimation problem, firstly, the DOA estimation model with ML for acoustic vector sensor array is developed, where the optimization standard in various cases can be unified by converting the maximum of objective function to the minimum. Secondly, based on the Invasive Weed Optimization (IWO) method which is a novel biological evolutionary algorithm, a new Improved IWO (IIWO) algorithm for DOA estimation of the acoustic vector sensor array is proposed by using ML estimation. This algorithm simulates weed invasion process for DOA estimation by adjusting the non-linear harmonic exponent of IWO algorithm adaptively. The DOA estimation accuracy has been improved, and the computation of multidimensional nonlinear optimization for the ML method has been greatly reduced in the IIWO algorithm. Finally, compared with Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE) method and Tuna Swarm Optimization(TSO) algorithm, numerical simulations show that the proposed algorithm has faster convergence rate, improved accuracy in terms of Root Mean Square Error (RMSE), lower computational complexity and more robust estimation performance for ML DOA estimation. The experiment with tracking the orientation of the motorboat by Microelectronic mechanical systems (MEMS) vector hydrophone array shows the superior performance of proposed IIWO algorithm in engineering application. Therefore, the proposed ML-DOA estimation with IIWO algorithm can take into account both resolution and computation. which can meet the requirements of real-time calculation and estimation accuracy in the actual environment.

    Citation: Peng Wang, Jiajun Huang, Weijia He, Jingqi Zhang, Fan Guo. Maximum likelihood DOA estimation based on improved invasive weed optimization algorithm and application of MEMS vector hydrophone array[J]. AIMS Mathematics, 2022, 7(7): 12342-12363. doi: 10.3934/math.2022685

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  • Direction of arrival (DOA) estimation based on Maximum Likelihood is a common method in array signal processing, with many practical applications, but the huge amount of calculation limits the practical application. To deal with such an Maximum Likelihood (ML) DOA estimation problem, firstly, the DOA estimation model with ML for acoustic vector sensor array is developed, where the optimization standard in various cases can be unified by converting the maximum of objective function to the minimum. Secondly, based on the Invasive Weed Optimization (IWO) method which is a novel biological evolutionary algorithm, a new Improved IWO (IIWO) algorithm for DOA estimation of the acoustic vector sensor array is proposed by using ML estimation. This algorithm simulates weed invasion process for DOA estimation by adjusting the non-linear harmonic exponent of IWO algorithm adaptively. The DOA estimation accuracy has been improved, and the computation of multidimensional nonlinear optimization for the ML method has been greatly reduced in the IIWO algorithm. Finally, compared with Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE) method and Tuna Swarm Optimization(TSO) algorithm, numerical simulations show that the proposed algorithm has faster convergence rate, improved accuracy in terms of Root Mean Square Error (RMSE), lower computational complexity and more robust estimation performance for ML DOA estimation. The experiment with tracking the orientation of the motorboat by Microelectronic mechanical systems (MEMS) vector hydrophone array shows the superior performance of proposed IIWO algorithm in engineering application. Therefore, the proposed ML-DOA estimation with IIWO algorithm can take into account both resolution and computation. which can meet the requirements of real-time calculation and estimation accuracy in the actual environment.



    We are interested to discuss about the existence of positive solutions of the following infinite coupled system of (n1,n)-type semipositone boundary value problems (BVPs) of nonlinear fractional differential equations (IBVP for short) in the sequence space of weighted means c0(W1,W2,Δ)

    {Dα0+ui(ρ)+ηfi(ρ,v(ρ))=0,ρ(0,1),Dα0+vi(ρ)+ηgi(ρ,u(ρ))=0,ρ(0,1),u(j)i(0)=v(j)i(0)=0,0jn2,ui(1)=ζ10ui(ϑ)dϑ, vi(1)=ζ10vi(ϑ)dϑ,iN, (1.1)

    where n3, α(n1,n], η,ζ are real numbers, 0<η<α, Dα0+ is the Riemann-Liouville's (R-L's) fractional derivative, and fi,gi, i=1,2,, are continuous and sign-changing. This kind of problems that the nonlinearity in (1.1) may change signs is mentioned as semipositone problems in the literature.

    Fractional differential equations (FDEs) occur in the various fields of biology [16], economy [20,38], engineering [24,32], physical phenomena [5,7,8,16,25], applied science, and many other fields [3,9,14,21]. Hristova and Tersian [18] solved an FDE with a different strategy, and Harjani, Lˊopez, and Sadarangani [17] solved an FDE using a fixed point approach. Now, we intend to solve an FDE by using the technique of measure of noncompactness. On the other hand, we encounter many real world problems, which can be modeled and described using infinite systems of FDEs (see [4,27,34,36,37]). In the theory of infinite system of FDEs, the measure of noncompactness (MNC) plays a significant role, which was introduced by Kuratowski [23] (see recent works [27,35,36]). The MNC has been utilized in sequence spaces for various classes of differential equations, see [2,6,11,12,13,26,29,30,35,36].

    The difference sequence spaces of weighted means λ(u,v,Δ) (λ=c0,c, and l) first have been introduced in [33]. Thereafter, Mursaleen et al. [28] constructed some estimations for the Hausdorff MNC of some matrix operators on these spaces. They also determined several classes of compact operators in such spaces. Motivated by the mentioned papers, in this work, we first discuss the existence of solutions of IBVP (1.1) in the difference sequence space of weighted means c0(W1,W2,Δ). Then, we find an interval of η such that for any η belongs to this interval, IBVP (1.1) has a positive solution. Eventually, we demonstrate an example illustrating the obtained results. Here, we preliminarily collect some definitions and auxiliary facts applied throughout this paper.

    Suppose that (Λ,) is a real Banach space containing zero element. We mean by D(z,r) the closed ball centered at z with radius r. For a nonempty subset U of Λ, the symbol ¯U denotes the closure of U and the symbol ConvU denotes the closed convex hull of U. We denote by MΛ the family of all nonempty, bounded subsets of Λ and by NΛ the family consisting of nonempty relatively compact subsets of Λ.

    Definition 1.1. [1] The function ˜μ:MΛ[0,+) is called an MNC in Λ if for any U,V1,V2MΛ, the properties (i)(v) hold:

    (i) ker˜μ={UMΛ:˜μ(U)=0} and ker˜μNΛ.

    (ii) If V1V2, then ˜μ(V1)˜μ(V2).

    (iii) ˜μ(¯U)=˜μ(ConvU)=˜μ(U).

    (iv) For each ρ[0,1], ˜μ(ρU+(1ρ)V)ρ˜μ(U)+(1ρ)˜μ(V).

    (v) If for each natural number n, Un is a closed set in MΛ, Un+1Un, and lim, then \mathcal{U}_{\infty} = {\bigcap_{n = 1}^{\infty}}\mathcal{U}_n is nonempty.

    In what follows, we mean by \mathfrak{M}_Y , the family of bounded subsets of the metric space (Y, d) .

    Definition 1.2. [10] Suppose that (Y, d) is a metric space. Also, suppose that \mathcal{P}\in\mathfrak{M}_Y . The Kuratowski MNC of \mathcal{P} , which is denoted by \alpha(\mathcal{P}) , is the infimum of the set of positive real numbers \varepsilon such that \mathcal{P} can be covered by a finite number of sets of diameter less than to \varepsilon . Indeed,

    \alpha(\mathcal{P}) = \inf\Big\{\varepsilon > 0: \mathcal{P}\subset\bigcup\limits_{j = 1}^n K_j, K_j\subset Y, \text{diam}(K_j) < \varepsilon\ (j = 1,\ldots,n);\ n\in\mathbb{N}\Big\},

    when \text{diam}(K_j) = \sup\{d(\varsigma, \nu): \varsigma, \nu\in K_j\} .

    The Hausdorff MNC (ball MNC) of the bounded set \mathcal{P} , which is denoted by \chi(\mathcal{P}) , is defined by

    \chi(\mathcal{P}) = \inf\Big\{\varepsilon > 0:\mathcal{P}\subset\bigcup\limits_{j = 1}^n D(y_j,r_j), y_j\in Y, r_j < \varepsilon\ (j = 1,\ldots,n);\ n\in\mathbb{N}\Big\}.

    Here, we quote a result contained in [10].

    Lemma 1.3. Let (Y, d) be a metric space and let \mathcal{P}, \mathcal{P}_1, \mathcal{P}_2\in\mathfrak{M}_Y . Then

    (i) \beta(\mathcal{P}) = 0 if and only if \mathcal{P} is totally bounded,

    (ii) \mathcal{P}_1\subset \mathcal{P}_2\Rightarrow \beta(\mathcal{P}_1)\leq \beta(\mathcal{P}_2) ,

    (iii) \beta(\overline{\mathcal{P}}) = \beta(\mathcal{P}) ,

    (iv) \beta(\mathcal{P}_1\cup \mathcal{P}_2) = \max\{\beta(\mathcal{P}_1), \beta(\mathcal{P}_2)\} .

    Besides, if Y is a normed space, then

    (v) \beta(\mathcal{P}_1+\mathcal{P}_2)\leq \beta(\mathcal{P}_1)+\beta(\mathcal{P}_2) ,

    (vi) for each complex number \rho , \beta(\rho\mathcal{P}) = |\rho|\beta(\mathcal{P}) .

    Now, we state a version of Darbo's theorem [10], which is fundamental in our work.

    Theorem 1.4. [10] Suppose that \widetilde{\mu} is an MNC in a Banach space \Lambda . Also, suppose that \emptyset\neq\mathfrak{D}\subseteq \Lambda is a bounded, closed, and convex set and that S:\mathfrak{D}\rightarrow \mathfrak{D} is a continuous mapping. If a constant \kappa\in [0, 1) exists such that

    \begin{equation*} \label{6} \widetilde{\mu}(S(\mathcal{X})) \leq \kappa \widetilde{\mu}(\mathcal{X}) \end{equation*}

    for any nonempty subset \mathcal{X} of \mathfrak{D} , then S has a fixed point in the set \mathfrak{D} .

    Suppose that J = [0, s] and that \Lambda is a Banach space. Consider the Banach space C(J, \Lambda) with the norm

    \|z\|_{C(J,\Lambda)}: = \sup\{\|z(\rho)\|:\rho\in J\},\qquad z\in C(J,\Lambda).

    Proposition 1.5. [10] Suppose that \Omega\subseteq C(J, \Lambda) is equicontinuous and bounded. Then \tilde{\mu}(\Omega(\cdot)) is continuous on J and

    \tilde{\mu}(\Omega) = \sup\limits_{\rho\in J}\tilde{\mu}(\Omega(\rho)),\qquad \tilde{\mu}\big(\int_{0}^{\rho}\Omega(\varrho)d\varrho\big)\leq\int_{0}^{\rho}\tilde{\mu}(\Omega(\varrho))d\varrho.

    Definition 1.6. (see [22,31]) Suppose that f:(0, \infty)\rightarrow \mathbb{R} is a continuous function. The R-L's fractional derivative of order \ell (\ell > 0) is defined as

    D_{0_+}^{\ell}f(\jmath) = \frac{1}{\Gamma(n-\ell)}(\frac{d}{d\jmath})^n\int_{0}^{\jmath}\frac{f(\varsigma)} {(\jmath-\varsigma)^{1-n+\ell}}d\varsigma,

    when n = [\ell]+1 and the right-hand side is pointwise defined on (0, +\infty).

    We terminate this section by describing the unique solution of a nonlinear FDE, which will be needed later.

    Lemma 1.7. [39] Let h\in C[0, 1] . Then the BVP

    \begin{equation} \begin{cases} D^{\ell}_{0_+}u(\rho)+h(\rho) = 0,& \rho\in(0,1),\ 2\leq n-1 < \ell\leq n, \\ u^{(j)}(0) = 0,& j\in [0, n-2], \\u(1) = \zeta\int_0^1 u(\varrho)d\varrho, \end{cases} \end{equation} (1.2)

    has a unique solution

    u(\rho) = \int_0^1H(\rho,\varrho)h(\varrho)d\varrho,

    when H(\rho, \varrho) is the Green's function of BVP (1.2) defined as

    \begin{equation*} H(\rho,\varrho) = \begin{cases}\frac{\rho^{\ell-1}(1-\varrho)^{\ell-1}(\ell-\zeta+\zeta \varrho)-(\ell-\zeta)(\rho-\varrho)^{\ell-1}}{(\ell-\zeta)\Gamma(\ell)},& 0\leq \varrho\leq\rho \leq1, \\ \frac{\rho^{\ell-1}(1-\varrho)^{\ell-1}(\ell-\zeta+\zeta\varrho)}{(\ell-\zeta)\Gamma(\ell)}, & 0\leq \rho\leq \varrho\leq1. \end{cases} \end{equation*}

    The function H(\rho, \varrho) has the following properties:

    \zeta\rho^{\ell-1}q(\varrho)\leq H(\rho,\varrho)\leq\frac{M_0\rho^{\ell-1}}{(\ell-\zeta)\Gamma(\ell)},\quad H(\rho,\varrho)\leq M_0q(\varrho), \quad \mathit{\text{for}}\ \rho,\varrho\in[0,1],

    where M_0 = (\ell-\zeta)(\ell-1)+\ell+\zeta and q(\varrho) = \frac{\varrho(1-\varrho)^{\ell-1}}{(\ell-\zeta)\Gamma(\ell)}.

    Suppose that S is the space of complex or real sequences. Any vector subspace of S is said to be a sequence space. We denote by c the space of convergent sequences and by c_0 the space of null sequences.

    A complete linear metric sequence space is called an FK space if it has the property that convergence implies coordinatewise convergence. Moreover, a normed FK space is called a BK space. It is known the spaces c_0 and c are BK spaces with the norm \|z\|_\infty = {\sup_{k\in\mathbb{N}}}|z_k| (see [12]).

    Suppose that \mathcal{X} and \mathcal{Y} are sequence spaces. We denote by (\mathcal{X}, \mathcal{Y}) the class of infinite matrices \mathcal{B} that map \mathcal{X} into \mathcal{Y} . We denote by \mathcal{B} = (b_{mk})_{m, k = 0}^\infty an infinite complex matrix and by \mathcal{B}_m its m th row. Then we can write

    \mathcal{B}_m(x) = {\sum\limits_{k = 0}^{\infty}}b_{mk}x_k\ \text{and}\ \mathcal{B}(x) = (\mathcal{B}_m(x))_{m = 0}^\infty.

    Thus \mathcal{B}\in(\mathcal{X}, \mathcal{Y}) if and only if \mathcal{B}_m(x) converges for all m and all x\in \mathcal{X} and \mathcal{B}(x)\in \mathcal{Y} .

    The set

    \begin{equation} \mathcal{X}_\mathcal{B} = \{x\in S :\mathcal{B}(x)\in \mathcal{X}\} \end{equation} (2.1)

    is called the matrix domain of \mathcal{B} in \mathcal{X} ; see [19]. An infinite matrix Y = (y_{nl}) is said to be a triangle if y_{nn}\neq0 and y_{nl} = 0 for each l > n . The matrix domain of a triangle Y , \mathcal{X}_Y , shares many properties with the sequence space \mathcal{X} . For instance, if \mathcal{X} is a BK space, then \mathcal{X}_Y is a BK space with the norm \|Z\|_{\mathcal{X}_Y} = \|YZ\|_{\mathcal{X}} for each Z\in\mathcal{X}_Y ; see [15].

    Now, let W = (w_k) be a sequence. The difference sequence of W is denoted by \Delta W = (w_k - w_{k-1}) . Suppose that W_1 = (w_k^1) and W_2 = (w_k^2) are the sequences of real numbers such that w_k^1\neq 0 and w_k^2\neq 0 for all k . Also, consider the triangle Y = (y_{nl}) defined by

    \begin{equation*} (y_{nl}) = \begin{cases}w_n^1(w_l^2-w_{l+1}^2),&l\leq n,\\w_n^1w_n^2,&l = n,\\0, & l > n.\end{cases} \end{equation*}

    The difference sequence space of weighted means c_0(W_1, W_2, \Delta) is defined as the matrix domain of the triangle Y in the space c_0 . Evidently, c_0(W_1, W_2, \Delta) is a BK space with the norm defined by

    \|x\| = \|Y(x)\|_\infty = \sup\limits_m|Y_m(x)|,\qquad x\in c_0(W_1,W_2,\Delta).

    Now, we describe the Hausdorff MNC \chi in the space c_0(W_1, W_2, \Delta). For this purpose, we quote the following two theorems.

    Theorem 2.1. [26] Suppose that \mathcal{P}\in \mathfrak{M}_{c_0} . Also, suppose that P_m:c_0\rightarrow c_0 is the operator defined by P_m(z) = (z_0, z_1, \ldots, z_m, 0, 0, \ldots) . Then

    \chi(\mathcal{P}) = \lim\limits_{m\rightarrow \infty}\sup\limits_{z\in \mathcal{P}}\|(\mathcal{I}-P_m)(z)\|_\infty,

    when \mathcal{I} is the identity operator.

    Theorem 2.2. [19] Let \mathcal{X} be a normed sequence space. Also, let \chi_Y and \chi denote the Hausdorff MNC on \mathfrak{M}_{\mathcal{X}_Y} and \mathfrak{M}_{\mathcal{X}} , the family of bounded sets in \mathcal{X}_Y and \mathcal{X}, respectively. Then

    \chi_Y (\mathcal{P}) = \chi(Y(\mathcal{P})),

    where \mathcal{P}\in \mathfrak{M}_{\mathcal{X}_Y}.

    Combining these two facts gives us the following theorem.

    Theorem 2.3. Let \mathcal{P}\in\mathfrak{M}_{c_0(W_1, W_2, \Delta)} . Then the Hausdorff MNC \chi on the space c_0(W_1, W_2, \Delta) can be defined as the following form:

    \chi_Y(\mathcal{P}) = \chi(Y(\mathcal{P})) = \lim\limits_{m\rightarrow \infty}\sup\limits_{x\in \mathcal{P}}\|(\mathcal{I}-P_m)(Y(x))\|_\infty.

    In this section, we first make some sufficient conditions to discuss the existence of solutions of IBVP (1.1) in the space c_0(W_1, W_2, \Delta) . Then, we give an interval of \eta such that any \eta belongs to this interval and the infinite system (1.1) has a positive solution. Eventually, we demonstrate an example to present the effectiveness of the obtained result.

    Here, we consider some assumptions.

    (A1) Let J_1 = [0, 1] , let f_i, g_i\in C(J_1\times\mathbb{R}^\infty, \mathbb{R}) , and let the function K:J_1\times C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta))\rightarrow c_0(W_1, W_2, \Delta)\times c_0(W_1, W_2, \Delta) be defined by

    (\varrho,U,V)\rightarrow K(U,V)(\varrho) = \big((f_i(\varrho,V(\varrho)))_{i = 1}^\infty, (g_i(\varrho,U(\varrho)))_{i = 1}^\infty\big),

    such that the family of functions (K(U, V)(\varrho)) is equicontinuous at each point of the space C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)).

    (A2) For each k\in\mathbb{N} and U = (u_i)\in C(J_1, c_0(W_1, W_2, \Delta)) , the following inequalities hold:

    f_k(\varrho,U(\varrho))\leq p_k(\varrho) u_{k}(\varrho),
    g_k(\varrho,U(\varrho))\leq q_k(\varrho) u_{k}(\varrho),

    where p_k, q_k:J_1\rightarrow \mathbb{R_+} = [0, +\infty) are mappings and the families \{p_k\} and \{q_k\} are equibounded.

    (A3) Let f_i , g_i\in C(J_1\times \mathbb{R_{+}^\infty}, \mathbb{R}) and let a function \theta\in L^1(J_1, (0, +\infty)) exist such that f_i(\rho, \mathfrak{Z}(\rho))\geq-\theta(\rho) and g_i(\rho, \mathfrak{Z}(\rho))\geq-\theta(\rho) , for each i\in \mathbb{N} , \rho\in J_1, and nonnegative sequence (\mathfrak{Z}(\rho)) in c_0(W_1, W_2, \Delta) .

    (A4) For any i\in\mathbb{N} and \rho\in J_1 , let f_i(\rho, U^0(\rho)) > 0 , where U^0(\rho) = (u_i^0(\rho)) and u_i^0(\rho) = 0 for all i and all \rho . Also, the sequence (f_i(\rho, U^0(\rho))) is equibounded.

    (A5) There exists \sigma > 0 such that g_i(\rho, \mathfrak{Z}(\rho)) > 0 , where i\in\mathbb{N} and (\rho, \mathfrak{Z}(\rho))\in J_1\times([0, \sigma])^\infty. Put

    P: = \sup\limits_{k\in \mathbb{N}}\sup\limits_{\varrho\in J_1}|p_k(\varrho)|,

    and

    Q: = \sup\limits_{k\in \mathbb{N}}\sup\limits_{\varrho\in J_1}|q_k(\varrho)|.

    Theorem 3.1. Assume that IBVP (1.1) fulfills the hypotheses (A1) , (A2) and \frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}(Q+P) < 1 , then it has at least one solution.

    Proof. Let (U, V) = ((u_i), (v_i)) be in C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) and satisfy the initial conditions of IBVP (1.1) and let each u_i and v_i be continuous on J_1 . We define the operator T:C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta))\rightarrow C(I, c_0(W_1, W_2, \Delta)\times c_0(W_1, W_2, \Delta)) by

    T(U,V)(\rho) = \bigg(\big(\eta\int_0^1H(\rho,\varrho)f_i(\varrho,V(\varrho))d\varrho\big)_{i = 1}^\infty, \big(\eta\int_0^1H(\rho,\varrho)g_i(\varrho,U(\varrho))d\varrho\big)_{i = 1}^\infty\bigg).

    Note that the product space C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) is equipped with the norm

    \|(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))} = \|U\|_{C(J_1,c_0(W_1,W_2,\Delta))}+\|V\|_{C(J_1,c_0(W_1,W_2,\Delta))}

    for each (U, V)\in C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)). Then, using our assumptions for any \rho\in J_1 , we can write

    \begin{align*} \|T(U,V)&(\rho)\|_{c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta)}\\ = &\|\big(\eta\int_0^1H(\rho,\varrho)f_i(\varrho,V(\varrho))d\varrho\big)\|_{c_0(W_1,W_2,\Delta)}\\&+ \|\big(\eta\int_0^1H(\rho,\varrho)g_i(\varrho,U(\varrho))d\varrho\big)\|_{c_0(W_1,W_2,\Delta)} \\ = &|\eta|\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk}\int_0^1H(\rho,\varrho)f_k(\varrho,V(\varrho))d\varrho| \\&+|\eta|\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk}\int_0^1H(\rho,\varrho)g_k(\varrho,U(\varrho))d\varrho| \\\leq&\frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}\big(\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk} \int_0^1p_k(\varrho)v_k(\varrho)d\varrho|+\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk} \int_0^1q_k(\varrho)u_k(\varrho)d\varrho|\big) \\\leq&\frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}(P+Q)\big(\|U\|_{C(J_1,c_0(W_1,W_2,\Delta))}+\|V\|_{C(J_1,c_0(W_1,W_2,\Delta))}\big) \\ = &\frac{M_0|\eta|(P+Q)}{(\alpha-\zeta)\Gamma(\alpha)}\|(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}. \end{align*}

    Accordingly, we obtain

    \begin{align*} &\|T(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta))}\\&\qquad\leq\frac{M_0|\eta|(P+Q)}{(\alpha-\zeta)\Gamma(\alpha)}\|(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}. \end{align*}

    It implies that

    \begin{equation} r\leq \big(\frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}(P+Q)r. \end{equation} (3.1)

    Let r_0 denote the optimal solution of inequality (3.1). Take

    \begin{align*} D& = D((U^0,U^0),r_0)\\& = \Big\{(U,V)\in C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta)):\\ &\qquad \|(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}\leq r_0,\ u_{i}^{(j)}(0) = v_{i}^{(j)}(0) = 0,\\&\qquad j\in [0, n-2],\ u_{i}(1) = \zeta \int_{0}^{1}u_{i}(\varrho)d\varrho,\quad v_{i}(1) = \zeta \int_{0}^{1}v_{i}(\varrho)d\varrho \Big\}. \end{align*}

    Clearly, D is bounded, closed, and convex and T is bounded on D . Now, we prove that T is continuous. Let (U_1, V_1) be a point in D and let \varepsilon be an arbitrary positive number. Employing assumption (A1) , there exists \delta > 0 such that if (U_2, V_2)\in D and \|(U_1, V_1)-(U_2, V_2)\|_{C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta))}\leq\delta, then

    \|K((U_1,V_1))-K((U_2,V_2))\|_{C(J_1,c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta))}\leq\frac{(\alpha-\zeta)\Gamma(\alpha)\varepsilon}{M_0|\eta|}.

    Therefore, for any \rho in I , we get

    \begin{align*} \|T&(U_1,V_1)(\rho)-T(U_2,V_2)(\rho)\|_{c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta)}\\ = &\|\Big(\big(\eta\int_0^1H(\rho,\varrho)f_i(\varrho,V_1(\varrho))d\varrho\big), \big(\eta\int_0^1H(\rho,\varrho)g_i(\varrho,U_1(\varrho))d\varrho\big)\Big) \\&-\Big(\big(\eta\int_0^1H(\rho,\varrho)f_i(\varrho,V_2(\varrho))d\varrho\big), \big(\eta\int_0^1H(\rho,\varrho)g_i(\varrho,U_2(\varrho))d\varrho\big)\Big)\|_{c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta)} \\ = &\|\big(\eta\int_0^1H(\rho,\varrho)(f_i(\varrho,V_1(\varrho))-f_i(\varrho,V_2(\varrho))d\varrho\big)\|_{c_0(W_1,W_2,\Delta)} \\&+\|\big(\eta\int_0^1H(\rho,\varrho)\big(g_i(\varrho,U_1(\varrho))-g_i(\varrho,U_2(\varrho))d\varrho\big)\|_{c_0(W_1,W_2,\Delta)} \\ = &|\eta|\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk}\int_0^1H(\rho,\varrho)\big(f_k(\varrho,V_1(\varrho)) -f_k(\varrho,V_2(\varrho))\big)d\varrho| \\&+|\eta|\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk}\int_0^1H(\rho,\varrho)\big(g_k(\varrho,U_1(\varrho)) -g_k(\varrho,U_2(\varrho))\big)d\varrho| \\\leq&\frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}\bigg(\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk} \sup\limits_{\rho\in[0,1]}\big(f_k(\varrho,V_1(\varrho)) -f_k(\varrho,V_2(\varrho))\big)| \\&+\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk} \sup\limits_{\rho\in[0,1]}\big(g_k(\varrho,U_1(\varrho)) -g_k(\varrho,U_2(\varrho))\big)|\bigg)\\ = &\frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}\|K(U_1,V_1)-K(U_2,V_2)\|_{C(I,c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta))} \\\leq&\varepsilon. \end{align*}

    Accordingly, we get

    \begin{equation*} \|T(U_1,V_1)-T(U_2,V_2)\|_{C(J_1,c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta))}\leq \varepsilon. \end{equation*}

    Thus, F is continuous.

    Next, we show that T(U, V) is continuous on the open interval (0, 1). To this aim, let \rho_1\in(0, 1) and \varepsilon > 0 be arbitrary. By applying the continuity of H(\rho, \varrho) with respect to \rho , we are able to find \delta = \delta(\rho_1, \varepsilon) > 0 such that if |\rho-\rho_1| < \delta, then

    |H(\rho,\varrho)-H(\rho_1,\varrho)| < \frac{\varepsilon}{|\eta|(P+Q)\|(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}}.

    We can write

    \begin{align*} \|T&(U,V)(\rho)-T(U,V)(\rho_1)\|_{c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta)}\\ = &\|\big(\eta\int_0^1(H(\rho,\varrho)-H(\rho_1,\varrho))f_i(\varrho,V(\varrho))d\varrho\big)\|_{c_0(W_1,W_2,\Delta)} \\&+\|\big(\eta\int_0^1(H(\rho,\varrho)-H(\rho_1,\varrho))g_i(\varrho,U(\varrho))d\varrho\big)\|_{c_0(W_1,W_2,\Delta)} \\ = &|\eta|\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk}\int_0^1(H(\rho,\varrho)-H(\rho_1,\varrho))f_k(\varrho,V(\varrho)) d\varrho| \\&+|\eta|\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk}\int_0^1(H(\rho,\varrho)-H(\rho_1,\varrho))g_k(\varrho,U(\varrho)) d\varrho| \\\leq&\frac{|\eta|P\varepsilon}{(P+Q)|\eta|\|(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}} \big(\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk} \sup\limits_{\rho\in[0,1]}V_k(\rho)|\big) \\&+\frac{|\eta|Q\varepsilon}{(P+Q)|\eta|\|(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}} \big(\sup\limits_{n}|\sum\limits_{k = 1}^{\infty}Y_{nk} \sup\limits_{\rho\in[0,1]}U_k(\rho)|\big) \\\leq&\frac{(P+Q)\varepsilon}{(P+Q)\|(U,V)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}}\\&\times\big(\|U\|_{C(J_1,c_0(W_1,W_2,\Delta))}+\|V\|_{C(J_1,c_0(W_1,W_2,\Delta))}\big) \\ = &\varepsilon. \end{align*}

    Eventually, we are going to show that T:D\rightarrow D fulfills the conditions of Theorem 1.4. Due to Proposition 1.5 and Theorem 2.3, for any nonempty subset \mathcal{X}_1\times \mathcal{X}_2\subset D , we obtain

    \begin{align*} \widetilde{\mu}(T(&\mathcal{X}_1\times \mathcal{X}_2))\\ = &\sup\limits_{\rho\in J_1}\sup\limits_{(U,V)\in \mathcal{X}_1\times \mathcal{X}_2}\widetilde{\mu}(T(U,V)(\rho))\\ = &\sup\limits_{\rho\in [0,1]}\sup\limits_{(U,V)\in \mathcal{X}_1\times \mathcal{X}_2}\widetilde{\mu}\big((\eta\int_0^1H(\rho,\varrho)f_i(\varrho,V(\varrho))d\varrho), (\eta\int_0^1H(\rho,\varrho)g_i(\varrho,U(\varrho))d\varrho)\big) \\ = &|\eta|\sup\limits_{\rho\in [0,1]}\lim\limits_{r\rightarrow \infty}\sup\limits_{V\in\mathcal{X}_2} \sup\limits_{n > r}|\sum\limits_{k = 1}^{\infty}Y_{nk}\int_0^1H(\rho,\varrho)f_k(\varrho,V(\varrho))d\varrho| \\&+|\eta|\sup\limits_{\rho\in [0,1]}\lim\limits_{r\rightarrow \infty}\sup\limits_{U\in\mathcal{X}_1} \sup\limits_{n > r}|\sum\limits_{k = 1}^{\infty}Y_{nk}\int_0^1H(\rho,\varrho)g_k(\varrho,U(\varrho))d\varrho| \\\leq&\frac{M_0|\eta|P}{(\alpha-\zeta)\Gamma(\alpha)}\sup\limits_{\rho\in [0,1]}\lim\limits_{r\rightarrow \infty}\sup\limits_{V\in\mathcal{X}_2} \sup\limits_{n > r}|\sum\limits_{k = 1}^{\infty}Y_{nk}v_k(\rho)| \\&+\frac{M_0|\eta|Q}{(\alpha-\zeta)\Gamma(\alpha)}\sup\limits_{\rho\in [0,1]}\lim\limits_{r\rightarrow \infty}\sup\limits_{U\in\mathcal{X}_1} \sup\limits_{n > r}|\sum\limits_{k = 1}^{\infty}Y_{nk}u_k(\rho)| \\ = &\frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}(P+Q)\widetilde{\mu}(\mathcal{X}_1\times \mathcal{X}_2). \end{align*}

    Using Theorem 1.4, we conclude that T has a fixed point in D , and hence IBVP (1.1) admits at least one solution in C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) .

    We are now in a position to discuss about the existence of positive solutions of IBVP (1.1) in the space c_0(W_1, W_2, \Delta) . To this end, consider the following IBVP

    \begin{equation} \begin{cases} -D^{\alpha}_{0_+}x_i(\rho) = \eta(f_i(\rho,(y_i(\rho)-K(\rho))^*)+\theta(\rho)),&\rho\in(0,1),\\ -D^{\alpha}_{0_+}y_i(\rho) = \eta (g_i(\rho,(x_i(\rho)-K(\rho))^*)+\theta(\rho)),& \rho\in(0,1),\\x_i^{(j)}(0) = y_{i}^{(j)}(0) = 0,& j\in[0, n-2], \\ x_{i}(1) = \zeta\int_0^1 x_i(\vartheta)d\vartheta, \ y_{i}(1) = \zeta\int_0^1 y_i(\vartheta)d\vartheta,& i\in\mathbb{N},\\ \end{cases} \end{equation} (3.2)

    where

    \begin{equation*} \mathcal{Z}(\rho)^* = \begin{cases}\mathcal{Z}(\rho),& \mathcal{Z}(\rho)\geq0,\\ 0,& \mathcal{Z}(\rho) < 0, \quad\end{cases} \end{equation*}

    and K(\rho) = \eta\int_{0}^{1}H(\rho, \vartheta)\theta(\vartheta)d\vartheta, which is the solution of the BVP

    \begin{equation*} \begin{cases} -D^{\alpha}_{0_+}K(\rho) = \eta \theta(\rho),& \rho\in(0,1), \\ K^{(j)}(0) = 0, & j\in[0,n-2], \\K(1) = \zeta\int_{0}^{1}K(\vartheta)d\vartheta. \end{cases} \end{equation*}

    We are going to show that there exists a solution (x, y) = ((x_i), (y_i)) for IBVP (1.1) with x_i(\rho)\geq K(\rho) and y_i(\rho)\geq K(\rho) for each i\in\mathbb{N} and for each \rho\in[0, 1].

    Accordingly, (U, V) is a nonnegative solution of IBVP (1.1), where U(\rho) = (x_i(\rho)-K(\rho)) and V(\rho) = (y_i(\rho)-K(\rho)) . Indeed, for any i\in\mathbb{N} and each \rho\in(0, 1) , we have

    \begin{equation*} \begin{cases} -D^{\alpha}_{0_+}x_i(\rho) = -D^{\alpha}_{0_+}u_i(\rho)+(-D^{\alpha}_{0_+}K(\rho)) = \eta(f_i(\rho, v(\rho))+\theta(\rho)), &\ \\ -D^{\alpha}_{0_+}y_i(\rho) = -D^{\alpha}_{0_+}v_i(\rho)+(-D^{\alpha}_{0_+}K(\rho)) = \eta(g_i(\rho, u(\rho))+\theta(\rho)). \end{cases} \end{equation*}

    It implies that

    \begin{equation*} \begin{cases} -D^{\alpha}_{0_+}u_i(\rho) = \eta(f_i(\rho, v(\rho)), &\ \\ -D^{\alpha}_{0_+}v_i(\rho) = \eta(g_i(\rho, v(\rho)). \end{cases} \end{equation*}

    Therefore, we concentrate our attention to the study of IBVP (3.2). We know that (3.2) is equal to

    x_i(\rho) = \eta\int_{0}^{1} H(\rho,\vartheta)(f_i(\vartheta, (y_i(\vartheta)-K(\vartheta))^*)+\theta(\vartheta))d\vartheta,
    \begin{equation} y_i(\rho) = \eta\int_{0}^{1} H(\rho,\vartheta)(g_i(\vartheta, (x_i(\vartheta)-K(\vartheta))^*)+\theta(\vartheta))d\vartheta. \end{equation} (3.3)

    In view of (3.3), we get

    \begin{equation} x_i(\rho) = \eta\int_{0}^{1} H(\rho,\vartheta)(f_i(\vartheta,(\eta\int_{0}^{1} H(\vartheta,\varsigma)g_i(\varsigma, (x_i(\varsigma)-K(\varsigma))^*)d\varsigma)^*)+\theta(\vartheta))d\vartheta. \end{equation} (3.4)

    In what follows, we demonstrate our main result.

    Theorem 3.2. Assume that IBVP (1.1) fulfills the hypotheses (A1) (A5) and \frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}(Q+P) < 1 . Then there exists a positive real constant \widetilde{\eta} such that for each 0 < \eta\leq \widetilde{\eta} , IBVP (1.1) has at least one positive solution.

    Proof. Take any \delta\in(0, 1) . Regarding assumptions (A4) and (A5) , we are able to find 0 < \varepsilon < \min\{1, \sigma\} such that for each i\in\mathbb{N} , \rho\in J_1 and the nonnegative sequence \mathfrak{Z} in C(J_1, c_0(W_1, W_2, \Delta)) with \|\mathfrak{Z}\|_{C(J_1, c_0(W_1, W_2, \Delta))} < \varepsilon , we have

    f_i(\rho,\mathfrak{Z}(\rho))\geq \delta f_i(\rho,U^0(\rho)),\qquad g_i(\rho,\mathfrak{Z}(\rho)) > 0.

    Suppose that

    0 < \eta < \widetilde{\eta}: = \min\Big\{\frac{\varepsilon}{2\Upsilon\widetilde{f}(\varepsilon)},\frac{1}{Q\Upsilon}\Big\},

    where \widetilde{f}(\varepsilon) = \max\{f_i(\rho, \mathfrak{Z}(\rho))+\theta(\rho), \quad i\in\mathbb{N}, \ 0\leq \rho\leq1, \ 0\leq\|\mathfrak{Z}\|_{C(J_1, c_0(W_1, W_2, \Delta))}\leq\varepsilon\} and \Upsilon = \int_{0}^{1}M_0q(\vartheta)d\vartheta . Since {\lim_{\varsigma\rightarrow0}}\frac{\widetilde{f}(\varsigma)}{\varsigma} = +\infty and \frac{\widetilde{f}(\varepsilon)}{\varepsilon} < \frac{1}{2\Upsilon\eta}, then there exists R_0\in(0, \varepsilon) such that \frac{\widetilde{f}(R_0)}{R_0} = \frac{1}{2\Upsilon\eta}. Let

    D_0 = \{x = (x_i)\in C(J_1,c_0(W_1,W_2,\Delta)):\ \|x-K\|_{C(J_1,c_0(W_1,W_2,\Delta))} < R_0,\ x_{i}^{(j)}(0) = 0,
    0\leq j\leq n-2,\ x_i(1) = \zeta\int_{0}^{1}x_i(\vartheta)d\vartheta,\ \text{ for all } i\in\mathbb{N}\}

    Now, for any x\in D_0 and \rho\in J_1 , we have

    \begin{align*} \|(\eta&\int_{0}^{1} H(\rho,\vartheta)(g_i(\rho,(x_i(\rho)-K(\rho))^*))d\vartheta)\|_{c_0(W_1,W_2,\Delta)}\\ = &\sup\limits_{n\in\mathbb{N}}|{\sum\limits_{k = 1}^\infty} Y_{nk}\eta\int_{0}^{1} H(\rho,\vartheta)(g_k(\vartheta,(x_k(\vartheta)-K(\vartheta))^*))d\vartheta|\\ \leq& \eta \int_{0}^{1} M_0q(\vartheta) Q|{\sum\limits_{k = 1}^\infty} Y_{nk}(x_k(\vartheta)-K(\vartheta))^*d\vartheta| \\ = &\eta \int_{0}^{1} M_0q(\vartheta) Q\|x-K\|_{C(J_1,c_0(W_1,W_2,\Delta))}\\ \leq&\eta\int_{0}^{1} M_0q(\vartheta) QR_0d\vartheta \\ < &R_0 < \varepsilon. \end{align*}

    Thus, using (3.4), we deduce that

    \begin{align*} x_i(\rho) = &\eta \int_{0}^{1} H(\rho,\vartheta)(f_i(\vartheta,(\eta\int_{0}^{1} H(\rho,\varsigma)g_i(\varsigma, (x_i(\varsigma)-K(\varsigma)^*)d\varsigma)^*)+\theta(\vartheta)))d\vartheta \\ \geq& \eta \int_{0}^{1} H(\rho,\vartheta)(\delta f_i(\vartheta,U^0(\vartheta))+\theta(\vartheta))d\vartheta \\ = &\eta(\delta\int_{0}^{1} H(\rho,\vartheta)f_i(\vartheta,U^0(\vartheta))d\vartheta+\int_{0}^{1} H(\rho,\vartheta)\theta(\vartheta) d\vartheta) \\ > &\eta\int_{0}^{1} H(\rho,\vartheta)\theta(\vartheta) d\vartheta = K(\rho), \end{align*}

    for any \rho\in J_1 , and any i\in\mathbb{N}.

    Thanks to relation (3.3), we get

    \begin{eqnarray*} y_i(\rho)& = &\eta\int_{0}^{1} H(\rho,\vartheta)(g_i(\vartheta,(x(\vartheta)-K(\vartheta))^*)+\theta(\vartheta))d\vartheta\\ & = & \eta\int_{0}^{1} H(\rho,\vartheta)(g_i(\vartheta,x(\vartheta)-K(\vartheta))+\theta(\vartheta))d\vartheta \\& > &\eta\int_{0}^{1} H(\rho,\vartheta)\theta(\vartheta)d\vartheta = K(\rho), \end{eqnarray*}

    for any \rho\in J_1 .

    Thus, if 0 < \eta\leq\widetilde{\eta} , then (x, y) is a positive solution of IBVP (3.2) with x_i(\rho)\geq K(\rho) and y_i(\rho)\geq K(\rho) for each i\in\mathbb{N} and for each \rho\in J_1.

    Let U(\rho) = (u_i(\rho)) = (x_i(\rho)-K(\rho)) and let V(\rho) = (v_i(\rho)) = (y_i(\rho)-K(\rho)) . Then (U, V) is a nonnegative solution of IBVP (1.1).

    Example 3.3. Consider the following IBVP of FDEs

    \begin{equation} \begin{cases} D^{\frac{39}{2}}_{0_+}u_i(\rho)+\frac{1}{40} {\sum_{j = i}^{+\infty}}\frac{e^{-2\rho}\big(\arctan^2(v_j(\rho)+1)+\frac{\pi}{2}\sin^2(v_j(\rho)-1)\big)\cos(\rho)}{j(j+1)(\rho+1)} = 0,& 0 < \rho < 1, \\ D^{\frac{39}{2}}_{0_+}v_i(\rho)+\frac{1}{40}{\sum_{j = i}^{+\infty}} \frac{e^{-5\rho}\big(1+u_j(\rho)+\sin^2(u_j(\rho)-1)\big)}{j^2\cosh(\rho) (2\rho+3)} = 0,& 0 < \rho < 1, \\u_i^{(j)}(0) = v_{i}^{(j)}(0) = 0,& 0\leq j\leq 18, \\ u_{i}(1) = 19.4\int_0^1 u_i(\vartheta)d\vartheta, \ v_{i}(1) = 19.4\int_0^1 v_i(\vartheta)d\vartheta, & i\in\mathbb{N},\\ \end{cases} \end{equation} (3.5)

    in the space C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) . By taking \alpha = \frac{39}{2} , \eta = \frac{1}{40}, \zeta = 19.4 ,

    f_i(\rho,V(\rho)) = {\sum\limits_{j = i}^{+\infty}}\frac{e^{-2\rho}\big(\arctan^2(v_j(\rho)+1)+ \frac{\pi}{2}\sin^2(v_j(\rho)-1)\big)\cos(\rho)}{j(j+1)(\rho+1)},

    and

    g_i(\rho,U(\rho)) = {\sum\limits_{j = i}^{+\infty}} \frac{e^{-5\rho}\big(1+u_j(\rho)+\sin^2(u_j(\rho)-1)\big)}{j^2\cosh(\rho) (2\rho+3)},

    system (3.5) is a special case of IBVP (1.1). Clearly, f_i, g_i\in C(J_1\times \mathbb{R}_+^\infty, \mathbb{R}) for each i\in\mathbb{N} . It can be easily verified that condition (A1) holds. Indeed, suppose that (U, V), (U^1, V^1)\in C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) and that \varepsilon > 0 is arbitrary. Now if \|(U, V)-(U^1, V^1)\|_{C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta))}\leq \frac{6\varepsilon}{\pi^2+12\pi} , then for each \rho\in [0, 1] , we obtain

    \begin{align*} \|K&(U,V)(\rho)-K(U^1,V^1)(\rho)\|_{c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta)}\\ = &\|\big((f_i(\rho,V(\rho))-f_i(\rho,V^1(\rho))), (g_i(\rho,U(\rho))-g_i(\rho,U^1(\rho)))\big)\|_{c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta)}\\ = &\|(f_i(\rho,V(\rho))-f_i(\rho,V^1(\rho)))\|_{c_0(W_1,W_2,\Delta)}+\|(g_i(\rho,U(\rho))-g_i(\rho,U^1(\rho)))\|_{c_0(W_1,W_2,\Delta)}\\ = &\sup\limits_{n}|\sum\limits_{i = 1}^{\infty}Y_{ni} {\sum\limits_{j = i}^{+\infty}}\frac{e^{-2\rho}\cos(\rho)}{j(j+1)(\rho+1)}\big((\arctan^2(v_j(\rho)+1)-\arctan^2(v_j^1(\rho)+1))\\ &+\frac{\pi}{2}(\sin^2(v_j(\rho)-1)-\sin^2(v_j^1(\rho)-1))\big)|\\ &+\sup\limits_{n}|\sum\limits_{i = 1}^{\infty}Y_{ni} {\sum\limits_{j = i}^{+\infty}}\frac{e^{-5\rho}}{j^2\cosh(\rho) (2\rho+3)}\\&\qquad\qquad\big((1+u_j(\rho)-1-u_j^1(\rho))+ (\sin^2(u_j(\rho)-1)-\sin^2(u_j^1(\rho)-1))\big)|\\ \leq&\sup\limits_{n}|\sum\limits_{i = 1}^{\infty}2\pi Y_{ni}(v_j(\rho)-v_j^1(\rho))|+\sup\limits_{n}|\sum\limits_{i = 1}^{\infty}\frac{\pi^2}{6}Y_{ni}(u_j(\rho)-u_j^1(\rho))|\\ \leq&(\frac{\pi^2+12\pi}{6})(\|U(\rho)-U^1(\rho)\|_{c_0(W_1,W_2,\Delta)}+\|V(\rho)-V^1(\rho)\|_{c_0(W_1,W_2,\Delta)})\\ = &(\frac{\pi^2+12\pi}{6})\|(U,V)-(U^1,V^1)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}\\ \leq&\varepsilon. \end{align*}

    Also, we get

    f_i(\rho,V(\rho))\leq \pi v_i(\rho),\qquad g_i(\rho,U(\rho))\leq \frac{\pi^2}{9} u_i(\rho).

    For each natural number i and \rho\in[0, 1] , we put p_i(\rho) = \pi and q_i(\rho) = \frac{\pi^2}{9}. Thus (p_i(\rho)) and (q_i(\rho)) are equibounded on the interval I . Moreover, P = \pi and Q = \frac{\pi^2}{9}. Note that

    f_i(\rho,V(\rho))+\theta(\rho) > 0,\ \mbox{and}\ g_i(\rho,U(\rho))+\theta(\rho) > 0,

    where \theta(\rho) = \tan(\rho) for each \rho\in I . Evidently, f_i(\rho, U^0(\rho)) > 0 , the sequence (f_i(\rho, U^0(\rho))) is equibounded, and g_i(\rho, U(\rho)) > 0. Moreover, \frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}(P+Q) = \frac{101.875\times\sqrt{\pi}}{18.5\times17.5\times\cdots\times1.5\times9} < 1 . Therefore, we conclude from Theorem 3.2 that (3.5) has a positive solution (U, V) in the space C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) .

    Mursaleen et al. [28] constructed a measure of noncompactness in the difference sequence space of weighted means \lambda(u, v, \Delta) . Also, a fractional differential equation was studied by Yuan [39]. Now, in this work, we discuss the existence of solutions of the infinite coupled system of (n-1, n) -type semipositone boundary value problem of nonlinear fractional differential Eq (1.1) in the difference sequence space of weighted means c_0(W_1, W_2, \Delta) .

    We would like to thank the referees for their useful comments and suggestions which have significantly improved the paper.

    The authors declare that they have no conflict of interest.



    [1] M. Hawkes, A. Nehorai, Wideband source localization using a distributed acoustic vector-sensor array, IEEE T. Signal Proces., 51 (2003), 1479–1491. https://doi.org/10.1109/TSP.2003.811225 doi: 10.1109/TSP.2003.811225
    [2] Y. W. Zhang, D. J. Sun, D. L. Zhang, Robust adaptive acoustic vector sensor beamforming using automated diagonal loading, Appl. Acoust., 70 (2009), 1029–1033. https://doi.org/10.1016/j.apacoust.2009.03.004 doi: 10.1016/j.apacoust.2009.03.004
    [3] A. J. Song, A. Abdi, M. Badiey, P. Hursky, Experimental demonstration of underwater acoustic communication by vector sensors, IEEE J. Oceanic Eng., 36 (2011), 454–461. https://doi.org/10.1109/Joe.2011.2133050 doi: 10.1109/Joe.2011.2133050
    [4] D. R. Dall'Osto, J. W. Choi, P. H. Dahl, Measurement of acoustic particle motion in shallow water and its application to geoacoustic inversion, J. Acoust. Soc. Am., 139 (2016), 311. https://doi.org/10.1121/1.4939492 doi: 10.1121/1.4939492
    [5] P. Wang, G. J. Zhang, C. Y. Xue, W. D. Zhang, J. J. Xiong, Self-adapting root-music algorithm and its real-valued formulation for acoustic vector sensor array, EURASIP J. Adv. Sig. Process., 2012 (2012), 228. https://doi.org/10.1186/1687-6180-2012-228 doi: 10.1186/1687-6180-2012-228
    [6] H. P. Hu, L. M. Zhang, H. C. Yan, Y. P. Bai, P. Wang, Denoising and baseline drift removal method of MEMS hydrophone signal based on vmd and wavelet threshold processing, IEEE Access, 7 (2019), 59913–59922. https://doi.org/10.1109/Access.2019.2915612 doi: 10.1109/Access.2019.2915612
    [7] A. Nehorai, E. Paldi, Acoustic vector-sensor array processing, IEEE Trans. Signal Process., 42 (1994), 2481–2491. https://doi.org/10.1109/78.317869 doi: 10.1109/78.317869
    [8] B. C. Ng, C. M. S. See, Sensor-array calibration using a maximum-likelihood approach, IEEE T. Antenn. Propag., 44 (1996), 827–835. https://doi.org/10.1109/8.509886 doi: 10.1109/8.509886
    [9] P. Stoica, A. Nehorai, Music, maximum likelihood, and Cramer-Rao bound, IEEE T. Acoust. Speech Sig. Process., 37 (1989), 720–741. https://doi.org/10.1109/29.17564
    [10] N. Wu, Z. Y. Qu, W. J. Si, S. H. Jiao, DOA and polarization estimation using an electromagnetic vector sensor uniform circular array based on the ESPRIT algorithm, Sensors, 16 (2016), 2109. https://doi.org/10.3390/s16122109 doi: 10.3390/s16122109
    [11] H. W. Chen, J. W. Zhao, Coherent signal-subspace processing of acoustic vector sensor array for DOA estimation of wideband sources, Signal Process., 85 (2005), 837–847. https://doi.org/10.1016/j.sigpro.2004.07.030 doi: 10.1016/j.sigpro.2004.07.030
    [12] P. Palanisamy, N. Kalyanasundaram, P. M. Swetha, Two-dimensional DOA estimation of coherent signals using acoustic vector sensor array, Signal Process., 92 (2012), 19–28. https://doi.org/10.1016/j.sigpro.2011.05.021 doi: 10.1016/j.sigpro.2011.05.021
    [13] S. G. Shi, Y. Li, Z. R. Zhu, J. Shi, Real-valued robust DOA estimation method for uniform circular acoustic vector sensor arrays based on worst-case performance optimization, Appl. Acoust., 148 (2019), 495–502. https://doi.org/10.1016/j.apacoust.2018.12.014 doi: 10.1016/j.apacoust.2018.12.014
    [14] H. L. Van Trees, Optimum array processing: Part IV of detection, estimation, and modulation theory, John Wiley & Sons, 2004.
    [15] P. Stoica, K. C. Sharman, Novel eigenanalysis method for direction estimation, IEE Proc. F (Radar and Signal Process.), 137 (1990), 19–26. https://doi.org/10.1049/ip-f-2.1990.0004 doi: 10.1049/ip-f-2.1990.0004
    [16] A. Lopes, I. S. Bonatti, P. L. D. Peres, C. A. Alves, , Improving the MODEX algorithm for direction estimation, Signal Process., 83 (2003), 2047–2051. https://doi.org/10.1016/S0165-1684(03)00146-4 doi: 10.1016/S0165-1684(03)00146-4
    [17] I. Ziskind, M. Wax, Maximum likelihood localization of multiple sources by alternating projection, IEEE T. Acoust. Speech Sig. Process., 36 (1988), 1553–1560. https://doi.org/10.1109/29.7543 doi: 10.1109/29.7543
    [18] M. Feder, E. Weinstein, Parameter estimation of superimposed signals using the EM algorithm, IEEE T. Acoust. Speech Sig. Process., 36 (1988), 477–489. https://doi.org/10.1109/29.1552 doi: 10.1109/29.1552
    [19] M. I. Miller, D. R. Fuhrmann, Maximum-likelihood narrow-band direction finding and the EM algorithm, IEEE T. Acoust. Speech Sig. Process., 38 (1990), 1560–1577. https://doi.org/10.1109/29.60075 doi: 10.1109/29.60075
    [20] J. A. Fessler, A. O. Hero, Space-alternating generalized expectation-maximization algorithm, IEEE T. Signal Process., 42 (1994), 2664–2677. https://doi.org/10.1109/78.324732 doi: 10.1109/78.324732
    [21] Y. M. Liu, S. Q. Xing, Y. C. Liu, Y. Z. Li, X. S. Wang, Maximum likelihood angle estimation of target in the presence of chaff centroid jamming, IEEE Access, 6 (2018), 74416–74428. https://doi.org/10.1109/Access.2018.2882579 doi: 10.1109/Access.2018.2882579
    [22] W. H. Fang, Y. C. Lee, Y. T. Chen, Maximum likelihood 2-D DOA estimation via signal separation and importance sampling, IEEE Antenn. Wirel. Pr., 15 (2016), 746–749. https://doi.org/10.1109/Lawp.2015.2471800 doi: 10.1109/Lawp.2015.2471800
    [23] W. L. Liu, Y. J. Gong, W. N. Chen, Z. Q. Liu, H. Wang, J. Zhang, Coordinated charging scheduling of electric vehicles: A mixed-variable differential evolution approach, IEEE T. Intell. Transp., 21 (2019), 5094-5109. https://doi.org/10.1109/TITS.2019.2948596 doi: 10.1109/TITS.2019.2948596
    [24] F. Q. Zhao, X. He, L. Wang, A two-stage cooperative evolutionary algorithm with problem-specific knowledge for energy-efficient scheduling of no-wait flow-shop problem, IEEE T. Cybernetics, 51 (2020), 5291–5303. https://doi.org/10.1109/TCYB.2020.3025662 doi: 10.1109/TCYB.2020.3025662
    [25] S. C. Zhou, L. N. Xing, X. Zheng, N. Du, L. Wang, Q. F. Zhang, A self-adaptive differential evolution algorithm for scheduling a single batch-processing machine with arbitrary job sizes and release times, IEEE T. Cybernetics, 51 (2019), 1430–1442. https://doi.org/10.1109/TCYB.2019.2939219 doi: 10.1109/TCYB.2019.2939219
    [26] F. Q. Zhao, L. X. Zhao, L. Wang, H. B. Song, An ensemble discrete differential evolution for the distributed blocking flowshop scheduling with minimizing makespan criterion, Expert Syst. Appl., 160 (2020), 113678. https://doi.org/10.1016/j.eswa.2020.113678 doi: 10.1016/j.eswa.2020.113678
    [27] F. Q. Zhao, R. Ma, L. Wang, A self-learning discrete jaya algorithm for multiobjective energy-efficient distributed no-idle flow-shop scheduling problem in heterogeneous factory system, IEEE T. Cybernetics, 2021, 1–12. https://doi.org/10.1109/TCYB.2021.3086181
    [28] M. Li, Y. Lu, Genetic algorithm based maximum likelihood DOA estimation, RADAR 2002, 2002,502–506. https://doi.org/10.1109/RADAR.2002.1174766
    [29] A. Sharma, S. Mathur, Comparative analysis of ML-PSO DOA estimation with conventional techniques in varied multipath channel environment, Wireless Pers. Commun., 100 (2018), 803–817. https://doi.org/10.1007/s11277-018-5350-0 doi: 10.1007/s11277-018-5350-0
    [30] Y. A. Sheikh, F. Zaman, I. M. Qureshi, M. A. ur Rehman, Amplitude and direction of arrival estimation using differential evolution, 2012 International Conference on Emerging Technologies, 2012. https://doi.org/10.1109/ICET.2012.6375456
    [31] L. Xie, T. Han, H. Zhou, Z. R. Zhang, B. Han, A. Di. Tang, Tuna swarm optimization: A novel swarm-based metaheuristic algorithm for global optimization, Comput. Intel. Neurosc., 2021 (2021), 9210050. https://doi.org/10.1155/2021/9210050. doi: 10.1155/2021/9210050
    [32] L. Boccato, R. Krummenauer, R. Attux, A. Lopes, Application of natural computing algorithms to maximum likelihood estimation of direction of arrival, Signal Process,, 92 (2012), 1338–1352. https://doi.org/10.1016/j.sigpro.2011.12.004
    [33] W. T. Shi, J. G. Huang, Y. S. Hou, Fast DOA estimation algorithm for MIMO sonar based on ant colony optimization, J. Syst. Eng. Electron., 23 (2012), 173–178. https://doi.org/10.1109/Jsee.2012.00022 doi: 10.1109/Jsee.2012.00022
    [34] Z. C. Zhang, J. Lin, Y. W. Shi, Application of artificial bee colony algorithm to maximum likelihood DOA estimation, J. Bionic Eng., 10 (2013), 100–109. https://doi.org/10.1016/S1672-6529(13)60204-8 doi: 10.1016/S1672-6529(13)60204-8
    [35] J. W. Shin, Y. J. Lee, H. N. Kim, Reduced-complexity maximum likelihood direction-of-arrival estimation based on spatial aliasing, IEEE T. Signal Process., 62 (2014), 6568–6581. https://doi.org/10.1109/Tsp.2014.2367454 doi: 10.1109/Tsp.2014.2367454
    [36] H. H. Chen, S. B. Li, J. H. Liu, Y. Q. Zhou, M. Suzukii, Efficient AM algorithms for stochastic ML estimation of DOA, Int. J. Antenn. Propag., 2016 (2016), 4926496. https://doi.org/10.1155/2016/4926496. doi: 10.1155/2016/4926496
    [37] P. Wang, Y. J. Kong, X. F. He, M. X. Zhang, X. H. Tan, An improved squirrel search algorithm for maximum likelihood DOA estimation and application for MEMS vector hydrophone array, IEEE Access, 7 (2019), 118343–118358. https://doi.org/10.1109/Access.2019.2936823 doi: 10.1109/Access.2019.2936823
    [38] A. R. Mehrabian, C. Lucas, A novel numerical optimization algorithm inspired from weed colonization. Ecol. Inform., 1 (2006), 355–366. https://doi.org/10.1016/j.ecoinf.2006.07.003
    [39] J. Yan, W. X. He, X. L. Jiang, Z. L. Zhang, A novel phase performance evaluation method for particle swarm optimization algorithms using velocity-based state estimation, Appl. Soft Comput., 57 (2017), 517–525. https://doi.org/10.1016/j.asoc.2017.04.035. doi: 10.1016/j.asoc.2017.04.035
    [40] R. Mallipeddi, P. N. Suganthan, Q. K. Pan, M. F. Tasgetiren, Differential evolution algorithm with ensemble of parameters and mutation strategies, Appl. Soft Comput., 11 (2011), 1679–1696. https://doi.org/10.1016/j.asoc.2010.04.024 doi: 10.1016/j.asoc.2010.04.024
    [41] B. Bai, Z. M. Ren, J. W. Ding, W. Xu, G. J. Zhang, J. Liu, et al., Cross-supported planar MEMS vector hydrophone for high impact resistance, Sensors Actuat. A-Phys., 263 (2017), 563–570. https://doi.org/10.1016/j.sna.2017.06.010 doi: 10.1016/j.sna.2017.06.010
    [42] G. J. Zhang, J. W. Ding, W. Xu, Y. Liu, R. X. Wang, J. J. Han, et al., Design and optimization of stress centralized MEMS vector hydrophone with high sensitivity at low frequency, Mech. Syst. Signal Pr., 104 (2018), 607–618. https://doi.org/10.1016/j.ymssp.2017.11.027 doi: 10.1016/j.ymssp.2017.11.027
    [43] M. R. Liu, L. Nie, G. J. Zhang, W. D. Zhang, J. Zou, Realization of a composite MEMS hydrophone without left-right ambiguity, Sensors Actuat. A-Phys., 272 (2018), 231–241. https://doi.org/10.1016/j.sna.2018.01.061 doi: 10.1016/j.sna.2018.01.061
    [44] Q. D. Xu, G. J. Zhang, J. W. Ding, R. X. Wang, Y. Pei, Z. M. Ren, et al., Design and implementation of two-component cilia cylinder MEMS vector hydrophone, Sensors Actuat. A-Phys., 277 (2018), 142–149. https://doi.org/10.1016/j.sna.2018.05.005 doi: 10.1016/j.sna.2018.05.005
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