Generalized Riccati Wick differential equation and applications

São Paulo Journal of Mathematical Sciences https://doi.org/10.1007/s40863-020-00184-2 ORIGINAL ARTICLE Generalized Riccati Wick differential equation and applications Marwa Missaoui1 · Hafedh Rguigui2,3 · Slaheddine Wannes4 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020 Abstract Using the Wick derivation operator and the Wick product of elements in a distribution space F∗𝜃 (S�ℂ ), we introduce the generalized Riccati Wick differential equation as a distribution analogue of the classical Riccati differential equation. The solution of this new equation is given. Finally, we finish this paper by building some applications. Keywords Wick product · Wick derivation · Generalized Riccati Wick differential equation · Space of entire functions with 𝜃-exponential growth condition of minimal type Communicated by Carlos Tomei. * Hafedh Rguigui [email protected] Marwa Missaoui [email protected] Slaheddine Wannes [email protected] 1 Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Road of Soukra, 1171 3000 Sfax, Tunisia 2 Department of Mathematics, AL-Qunfudhah University College, Umm Al-Qura University, KSA, Mecca, Saudi Arabia 3 High School of Sciences and Technology of Hammam Sousse, University of Sousse, Rue Lamine Abassi, 4011 Hammam Sousse, Tunisia 4 Department of Mathematics, Faculty of Sciences of Gabes, University of Gabes, 6072 Gabes, Tunisia 13 Vol.:(0123456789) São Paulo Journal of Mathematical Sciences 1 Introduction The generalized Riccati equation is named after Jacopo Francesco Riccati, Italian mathematician (1676−1754) [21]. As soon as differential equations were invented, the Riccati differential equation was the first one to be investigated extensively since the end of the 17th century [22]. This equation appears widely in large variety of applications in engineering and applied science such as random processes, diffusion problems, optimal control etc., [1, 11, 12, 15]. It has the form y� = a(x)y + b(x)y2 + c(x). (1) where a(x), b(x) and c(x) are continuous functions of x. In order to find the general solution for the equation (1), a particular solution is needed even though the equation is nonlinear. Since the usual product in (1) is not defined in the distribution case, we need to use a suitable product to introduce a distribution analogue of Eq. (1). The Wick product, denoted by ⋄, was introduced by Hida and Ikeda [8], and it has been used extensively in the study of generalized Bernoulli Wick differential equations and white noise integral equations, see [2, 9, 10, 19] and references cited therein. This product permits us to extend the equation (1) in infinite dimensional distribution case, which will be called the generalized Riccati Wick differential equation, as follows: D(𝛷) = A ⋄ 𝛷 + B ⋄ 𝛷⋄2 + H (2) L(F𝜃 (S�ℂ ), F∗𝜃 (S�ℂ )) and D is a Wick derivation operator. where A, B, H ∈ The paper is organized as follows: In Sect. 2, we briefly recall some basic notations in quantum white noise calculus. Namely, we give definitions and properties of the test functions space of entire functions with 𝜃-exponential growth condition of minimal type and the associated generalized functions space. In Sect. 3, we give the solution of the generalized Riccati Wick differential equation and we illustrate the method with examples. Section 4 is devoted to build some applications of the generalized Riccati Wick differential equation. 2 Preliminaries In this section we shall briefly recall some of the concepts, notations and known results on nuclear algebras of entire functions and Wick calculus which can be found also in Refs. [2–7, 14, 16–20]. 2.1 Nuclear algebras of entire functions Let E = S(ℝ) be the Schwartz space consisting of rapidly decreasing C∞-functions, E� = S� (ℝ) the space of tempered distributions and H = L2 (ℝ, dt) the 13 São Paulo Journal of Mathematical Sciences Hilbert space with norm |.|0 . The space S(ℝ) can be reconstructed in a standard manner (see Ref. [13]) by the harmonic oscillator A = 1 + t2 − d2 ∕dt2 and H. In fact, S(ℝ) is a nuclear space equipped with the Hilbertian norms |𝜂|p = |Ap 𝜂|0 , 𝜂 ∈ S(ℝ), p∈ℝ and we know that: E = proj lim Sp , p→∞ E� = ind lim S−p p→∞ where, for p ≥ 0, Sp is the completion of S(ℝ) with respect to the norm | ⋅ |p and S−p is the topological dual space of Sp and S� (ℝ) is the topological dual of S(ℝ). The canonical ℂ-bilinear from S� (ℝ) × S(ℝ) which is compatible with the inner product of H is marked by ⟨., .⟩. and S� (ℝ) the space of tempered distributions. We denote by Sℂ its the complexification, i.e., Sℂ = S(ℝ) + iS(ℝ) and Sℂ,p = Sp (ℝ) + iSp (ℝ). Throughout the paper, we fix a Young function 𝜃 , i.e. a continuous, convex and increasing function defined on ℝ+ and satisfies the two conditions: 𝜃(0) = 0 and = +∞. The polar function 𝜃 ∗ of 𝜃 , defined by limx→∞ 𝜃(x) x 𝜃 ∗ (x) = sup(tx − 𝜃(t)), x ≥ 0, t≥0 is also a Young function and (𝜃 ∗ )∗ = 𝜃. For a complex Banach space (B, ‖ ⋅ ‖), let H(B) denotes the space of all entire functions on B i.e. of all continuous ℂ-valued functions on B whose restrictions to all affine lines of B are entire on ℂ. For each m > 0 we denote by Exp(B, 𝜃, m) the space of all entire functions on B with 𝜃-exponential growth of finite type m, i.e., � � Exp(B, 𝜃, m) = f ∈ H(B); ‖f ‖𝜃,m ∶= sup �f (z)�e−𝜃(m‖z‖) < ∞ . z∈B The projective system {Exp(Sℂ,−p , 𝜃, m); p ∈ ℕ, m > 0} and the inductive system {Exp(Sℂ,p , 𝜃, m); p ∈ ℕ , m > 0} gives the two spaces F𝜃 (S�ℂ ) = proj G𝜃 (Sℂ ) = ind lim Exp(Sℂ,−p , 𝜃, m), p→∞;m↓0 lim Exp(Sℂ,p , 𝜃, m). p→∞;m↓∞ The space F𝜃 (S�ℂ ) is named the space of test functions on S′ℂ. Its topological dual space F∗𝜃 (S�ℂ ), equipped with the strong topology, is called the space of distributions on S′ℂ. It is known that every 𝜙 ∈ F𝜃 (Sℂ ) admits a Taylor expansion of the form 𝜙(x) = ∞ ∑ < x⊗n , 𝜙n >, � x ∈ S�ℂ , 𝜙n ∈ S⊗n . ℂ (3) n=0 Let F𝜃 (Sℂ ) be the space of all Taylor coefficients 𝜙n obtained from (3). It is given that 13 São Paulo Journal of Mathematical Sciences F𝜃 (Sℂ ) = proj lim p→∞;m↓0 F𝜃,m (Sℂ,p ), where { � �⃗ = (𝜙n )n≥0 ; 𝜙n ∈ S⊗n �⃗ 𝜃,p,m = F𝜃,m (Sℂ,p ) = 𝜙 , ||𝜙|| ℂ,p ∞ ∑ n=0 } 𝜃n−2 m−n |𝜙n |2p < ∞ and e𝜃(r) , r>0 r n n = 0, 1, 2, .... 𝜃n = inf Moreover, equipped with the projective limit topology, F𝜃 (Sℂ ) is a nuclear Fréchet space and is isomorphic to F𝜃 (S�ℂ ) via the Taylor map determined by )∞ ( 1 (n) . 𝜑 (0) T∶ 𝜑⟼ n! n=0 Let G𝜃 (S�ℂ ) = ind lim p→∞;m→∞ G𝜃,m (Sℂ,−p ), where { G𝜃,m (Sℂ,−p ) = � ��⃗ = (Fn )n≥0 ; Fn ∈ S⊗n , 𝛷 ℂ,−p ∞ ∑ } 2 n (n!𝜃n ) m |Fn |2−p <∞ . n=0 The space G𝜃 (S�ℂ ) carries the dual topology of F𝜃 (Sℂ ) with respect to the ℂ-bilinear pairing given by ⟨⟨𝛷, 𝛗 ⟩⟩ = ∞ � n=0 n!⟨𝛷n , 𝜑n ⟩, (4) where 𝛷 = (𝛷n )∞ ∈ G𝜃 (S�ℂ ) and 𝛗 = (𝜑n )∞ ∈ F𝜃 (Sℂ ). n=0 n=0 The Taylor map T is also a topological isomorphism from G𝜃∗ (Sℂ ) onto G𝜃 (S�ℂ ). The action of a distribution 𝛷 ∈ F∗𝜃 (S�ℂ ) on a test function 𝜑 ∈ F𝜃 (S�ℂ ) can be shown in terms of the Taylor map as follows: ⟨⟨𝛷, 𝜑⟩⟩ = ⟨⟨𝛷, 𝛗 ⟩⟩, where 𝛷 = (T ∗ )−1 𝛷 and 𝛗 = T𝜑. One can easily see that for each 𝜉 ∈ Sℂ, the exponential function e𝜉 (z) = e⟨z,𝜉⟩ , z ∈ S�ℂ , is a test function in the space F𝜃 (S�ℂ ) for any Young function 𝜃 . Thus we can define the Laplace transform of a distribution 𝛷 ∈ F∗𝜃 (S�ℂ ) by 13 São Paulo Journal of Mathematical Sciences L𝛷(𝜉) = ⟨⟨ 𝛷, e𝜉 ⟩⟩ , 𝜉 ∈ Sℂ . (5) The following theorem is useful. Theorem 1 The Laplace transform L brings a topological isomorphism from F∗𝜃 (S�ℂ ) onto G𝜃∗ (Sℂ ). 2.2 Wick calculus Recall that the Wick product of 𝛷1 , 𝛷2 ∈ F∗𝜃 (S�ℂ ) denoted by 𝛷1 ⋄ 𝛷2 is the unique element of F∗𝜃 (S�ℂ ) satisfying (see [14, 20]) L(𝛷1 ⋄ 𝛷2 )(𝜉) = L(𝛷1 )(𝜉)L(𝛷2 )(𝜉), 𝜉 ∈ Sℂ this is verified by Theorem 1. Using this definition, one can easily show that the Wick product is associative and commutative. Moreover, for 𝛷 ∈ F∗𝜃 (S�ℂ ), we have 𝛿0 ⋄ 𝛷 = 𝛷 ⋄ 𝛿0 = 𝛷 where 𝛿0 denoting the Dirac distribution at 0 (which is also the unique distribution satisfying L(𝛿0 )(𝜉) = 1). For n ∈ ℕ, by recurrence one can easily show that the Wick product 𝛷 ⋄ 𝛷 ⋄ ⋯ ⋄ 𝛷 n-times (denoted by 𝛷⋄n ) is given via L(𝛷⋄n ) = (L(𝛷))n . By convention we take 𝛷⋄0 = 𝛿0. Now, for r ∈ ℝ∗ + = (0, ∞), we need to introduce the Wick power in this case. The element 𝛷⋄r (which will be called the Wick r power of 𝛷) is defined via L(𝛷⋄r ) = (L(𝛷))r . Let G0𝜃∗ be the space of generalized functions g in G𝜃∗ (Sℂ ) such that g(𝜉) has no zero, i.e., { } G0𝜃∗ ∶= g ∈ G𝜃∗ (Sℂ ) | g(𝜉) ≠ 0 ∀𝜉 ∈ Sℂ . Definition 1 [2] Let 𝛷 ∈ F∗𝜃 (S�ℂ ). If there exists 𝜓 ∈ F∗𝜃 (S�ℂ ) such that 𝜓 ⋄ 𝛷 = 𝛿0 then we say that 𝛷 is Wick invertible and its Wick inverse is equal to 𝜓 which will be denoted by 𝛷⋄(−1). Let E0𝜃 be the set of all Wick invertible elements on F∗𝜃 (S�ℂ ). Proposition 1 [2] The Laplace transform realizes a topological isomorphism from the space E0𝜃 onto the space G0𝜃∗. 13 São Paulo Journal of Mathematical Sciences Let 𝛷 ∈ F∗𝜃 (S�ℂ ) and r ∈ ℝ+. If there exists 𝜓 ∈ F∗𝜃 (S�ℂ ) such that 𝜓 ⋄ 𝛷⋄r = 𝛿0 then 𝜓 will be denoted by 𝛷⋄(−r). Lemma 1 [2] Let r ∈ ℝ+. If 𝛷 ∈ E0𝜃 , then 𝛷⋄(−r) exists. More precisely, 𝛷⋄(−r) is given by ( )⋄r 𝛷⋄(−r) = 𝛷⋄(−1) and belongs to E0𝜃. Recall that, a continuous linear map D ∶ F∗𝜃 (S�ℂ ) → F∗𝜃 (S�ℂ ) is called a Wick derivation if it is a derivation with respect to the Wick product, i.e., D(𝛷 ⋄ 𝛹 ) = D(𝛷) ⋄ 𝛹 + 𝛷 ⋄ D(𝛹 ), for 𝛷, 𝛹 ∈ F∗𝜃 (S�ℂ ). D is given by D ∶= L−1 dL where d is a derivation on G𝜃∗ (Sℂ ), i.e., d(f .g) = f .d(g) + g.d(f ) (6) for f , g ∈ G𝜃∗ (Sℂ ) ( ) Lemma 2 [2] Let D a Wick derivation. Then, for r ≥ 1 and 𝛷 ∈ F∗𝜃 (S�ℂ ) or for ( ) r < 1 and 𝛷 ∈ E0𝜃 , we have ( ) D 𝛷⋄r = rD(𝛷) ⋄ 𝛷⋄(r−1) . As a standard example, we take the operator D𝜉 , (𝜉 ∈ Sℂ ), given as follows: D𝜉 ∶= L−1 d𝜉 L where (d𝜉 f )(𝜂) ∶= lim t→0 f (𝜂 + t𝜉) − f (𝜉) t for f ∈ G𝜃∗ (Sℂ ). 2.3 Generalized Bernoulli Wick differential equation In this subsection, we summarize some important results about generalized Bernoulli Wick differential equation (see [2]). Recall that (see [3]), for 𝛷 ∈ F∗𝜃 (S�ℂ ) the Wick exponential is defined by 13 São Paulo Journal of Mathematical Sciences e⋄𝛷 = ∞ ∞ ∑ ∑ 1 ⋄n 1 ⋄n 𝛷 = 𝛿0 + 𝛷 n! n! n=1 n=0 which is an element of F∗(e𝜃∗ )∗ (S�ℂ ). Using Lemma 2, one can prove easily that D(e⋄𝛷 ) = D(𝛷) ⋄ e⋄𝛷 . Proposition 2 [2] Let D be a Wick derivation and A, B ∈ F∗𝜃 (S�ℂ ). Assume that Y, Z ∈ F∗𝜃 (S�ℂ ) such that 1. D(Y) = A 2. D(Z) = B ⋄ e⋄(−Y) . Then, 𝛷 is a solution of D𝛷 = A ⋄ 𝛷 + B (7) 𝛷 = (Z + F) ⋄ e⋄Y , (8) if and only if 𝛷 is of the form where D(F) = 0. One can prove that Proposition 2 remains true if we take A, B ∈ E0𝜃 . The generalized Bernoulli Wick differential equation is in the form A ⋄ D(𝛷) + B ⋄ 𝛷 = G ⋄ 𝛷⋄r , r ∈ ℝ+ |{0, 1}, (9) where A, B, G ∈ E0𝜃. Theorem 2 [2] The equation (9) has a unique solution given by ( )⋄( 1 ) 1 1 1−r = (Z + F)⋄( 1−r ) ⋄ e⋄( 1−r Y) , 𝛷 = (Z + F) ⋄ e⋄Y where F, Y and Z satisfy D(F) = 0, D(Y) = (r − 1)B ⋄ A⋄(−1) and D(Z) = (1 − r)A⋄(−1) ⋄ G ⋄ e⋄(−Y) . 3 Generalized Riccati Wick differential equation We introduce the generalized Riccati wick differential equation as follows: 13 São Paulo Journal of Mathematical Sciences D(𝛷) + 𝛼 ⋄ 𝛷 + 𝛽 ⋄ 𝛷⋄2 = 𝛾 (10) where 𝛼, 𝛽 and 𝛾 in E0𝜃. Theorem 3 Let 𝛷p a particular solution of (10), then the solution 𝛷 of (10) is given by 𝛷 = 𝛷p + (Z + F)⋄(−1) ⋄ e⋄(−Y) where F, Y and Z satisfy – D(F) = 0 – D(Y) = 𝛼 + 2𝛽 ⋄ 𝛷p – D(Z) = 𝛽 ⋄ e⋄(−Y). Proof Let 𝛷p a particular solution of (10). Taking 𝜓 = 𝛷 − 𝛷p. Then, we get D(𝜓) = D(𝛷) − D(𝛷p ) ) ( ) ( = 𝛾 − 𝛼 ⋄ 𝛷 − 𝛽 ⋄ 𝛷⋄2 − 𝛾 − 𝛼 ⋄ 𝛷p − 𝛽 ⋄ 𝛷p⋄2 = −𝛼 ⋄ 𝛷 − 𝛽 ⋄ 𝛷 ⋄2 + 𝛼 ⋄ 𝛷p + 𝛽 ⋄ 𝛷p⋄2 replacing 𝛷 by 𝜓 + 𝛷p in (11), we get D(𝜓) = −𝛼 ⋄ (𝜓 + 𝛷p ) − 𝛽 ⋄ (𝜓 + 𝛷p )⋄2 + 𝛼 ⋄ 𝛷p + 𝛽 ⋄ 𝛷p⋄2 = −𝛼 ⋄ 𝜓 − 𝛼 ⋄ 𝛷p − 𝛽 ⋄ 𝜓 ⋄2 − 𝛽 ⋄ 𝛷p⋄2 − 2𝛽 ⋄ 𝜓 ⋄ 𝛷p + 𝛼 ⋄ 𝛷p + 𝛽 ⋄ 𝛷p⋄2 ) ( = −𝛼 − 2𝛽 ⋄ 𝛷p ⋄ 𝜓 − 𝛽 ⋄ 𝜓 ⋄2 . This gives D(𝜓) + (𝛼 + 2𝛽 ⋄ 𝛷p ) ⋄ 𝜓 = −𝛽 ⋄ 𝜓 ⋄2 . Hence, by Theorem 2, we get 𝜓 = (Z + F)⋄−1 ⋄ e⋄(−Y) where F, Y and Z satisfy D(F) = 0 D(Y) = (2 − 1)(𝛼 + 2𝛽 ⋄ 𝜙p ) ⋄ 𝛿0⋄−1 = 𝛼 + 2𝛽 ⋄ 𝛷p and 13 (11) São Paulo Journal of Mathematical Sciences D(Z) = (1 − 2)𝛿0⋄−1 ⋄ (−𝛽) ⋄ e⋄(−Y) = 𝛽 ⋄ e⋄(−Y) . This completes the proof. ◻ Example 1 Let us study the following generalized Riccati Wick differential equation D(𝛷) − 3𝛷 + 𝛷⋄2 = −2𝛿0 (12) for which ( )⋄−1 𝛷p = (Z + F) ⋄ e⋄Y + 𝛿0 is a particular solution of Riccati equation (12), where D(F) = 0, D(Z) = e−Y and D(Y) = −3 + 2𝛿0 . In fact, ( )⋄−2 ( ) D(𝛷p ) = − (Z + F) ⋄ e⋄Y ⋄ D(Z + F) ⋄ e⋄Y + De⋄Y ⋄ (Z + F) ( )⋄2 ( ) = − 𝛷 p − 𝛿0 ⋄ 𝛿0 + (−3 + 2𝛿0 ) ⋄ (𝛷p − 𝛿0 )⋄−1 ( ( )⋄2 )⋄2 ( )⋄−1 ⋄ 𝛿0 − (−3 + 2𝛿0 ) ⋄ 𝛷p − 𝛿0 ⋄ 𝛷p − 𝛿0 = − 𝛷 p − 𝛿0 )⋄2 )⋄1 ( ( = − 𝛷 p − 𝛿0 − (−3 + 2𝛿0 ) ⋄ 𝛷p − 𝛿0 ( ) = − 𝛷p⋄2 − 2𝛷p + 𝛿0⋄2 − (−𝛷p + 3𝛿0 − 2𝛿02 ) = −𝛷p⋄2 + 2𝛷p − 𝛿0 + 𝛷p − 3𝛿0 + 2𝛿0 = −𝛷p⋄2 + 3𝛷p − 2𝛿0 and then we get D(𝛷p ) − 3𝛷p + 𝛷p⋄2 = −2𝛿0 which gives that 𝛷p is a particular solution of (12). Example 2 Let G in E0𝜃 such that D(G) = 0. Consider the following generalized Riccati Wick differential equation D(𝛷) − (2G + 𝛿0 ) ⋄ 𝛷 + 𝛷⋄2 = −G⋄2 − G (13) for which ( )⋄−1 𝛷p = (Z + F) ⋄ e⋄Y +G 13 São Paulo Journal of Mathematical Sciences is a particular solution of the generalized Riccati Wick differential equation (13), where D(F) = 0, D(Z) = e−Y and D(Y) = −𝛿0 . In fact, ( ) ( )⋄−2 D(𝛷p ) = − D(Z) ⋄ e⋄Y + (Z + F) ⋄ D(Y) ⋄ e⋄Y ⋄ (Z + F) ⋄ e⋄Y + D(G) ( ) ( )⋄−2 = − D(Z) ⋄ e⋄Y + 𝛿0 ⋄ (Z + F) ⋄ e⋄Y ⋄ (Z + F) ⋄ e⋄Y ( ) ( )⋄−2 = − e⋄−Y ⋄ e⋄Y + (Z + F) ⋄ e⋄Y ⋄ (Z + F) ⋄ e⋄Y ( ) = − 𝛿0 + (𝛷p − G)⋄−1 ⋄ (𝛷p − G)⋄2 )⋄2 )⋄−1 ( ( )⋄2 ( ⋄ 𝛷p − G + 𝛷p − G = −𝛿0 ⋄ 𝛷p − G )⋄1 ( )⋄2 ( + 𝛷p − G = − 𝛷p − G = −𝛷p⋄2 + 2𝛷p ⋄ G − G⋄2 + 𝛷p − G and then we get D(𝛷p ) − (2G + 𝛿0 ) ⋄ 𝛷p + 𝛷p⋄2 = −G⋄2 − G from which we deduce that 𝛷p is a particular solution of equation (13). 4 Application of the generalized Riccati Wick differential equation 4.1 Generalized Schrodinger Wick differential equation Let us introduce the following equation: −D2 (𝜓) + (V − E) ⋄ 𝜓 = 0 (14) where D is a Wick derivation and V, E ∈ E◦𝜃 . Equation (14) is called generalized Schrodinger Wick differential equation. In the next theorem, we will show that equation (14) leads to a generalized Riccati Wick differential equation. Theorem 4 The generalized Schrodinger Wick differential equation (14) is equivalent to the following generalized Riccati Wick differential equation D(𝛷) + 𝛷⋄2 = (V − E) where 𝛷 is given by 13 (15) São Paulo Journal of Mathematical Sciences 𝛷 = D(𝜓) ⋄ 𝜓 ⋄(−1) . Proof “⇒” Let 𝛷 given by 𝛷 = D(𝜓) ⋄ 𝜓 ⋄(−1) then, we get D(𝛷) = D2 (𝜓) ⋄ 𝜓 ⋄(−1) + D(𝜓) ⋄ (−1)D(𝜓) ⋄ 𝜓 ⋄(−2) (16) = D2 (𝜓) ⋄ 𝜓 ⋄(−1) − D(𝜓) ⋄ D(𝜓) ⋄ 𝜓 ⋄−2 . (17) But, from (14), we have D2 (𝜓) = (V − E) ⋄ 𝜓 then, we obtain D(𝛷) = (V − E) ⋄ 𝜓 ⋄ 𝜓 ⋄(−1) − D(𝜓) ⋄ D𝜓 ⋄ 𝜓 ⋄(−1) ⋄ 𝜓 ⋄(−1) = (V − E) ⋄ 𝛿0 − D(𝜓) ⋄ 𝜓 ⋄(−1) ⋄ D(𝜓) ⋄ 𝜓 ⋄(−1) = (V − E) − 𝛷 ⋄ 𝛷 = (V − F) − 𝛷⋄2 . Hence, we obtain D(𝛷) + 𝛷⋄2 = (V − E). “⇐” Suppose that there exists 𝜓 ∈ E0𝜃 such that 𝛷 = D(𝜓) ⋄ 𝜓 ⋄(−1) . But, we have D(𝛷) + 𝛷⋄2 = (V − E) then, we get D2 (𝜓) ⋄ 𝜓 ⋄(−1) + D(𝜓) ⋄ D(𝜓) ⋄ (−𝜓 ⋄(−2) ) + D(𝜓) ⋄ D(𝜓) ⋄ 𝜓 ⋄(−2) = (V − E). This gives D2 (𝜓) ⋄ 𝜓 ⋄(−1) = (V − E). From which we deduce that D2 (𝜓) = (V − E) ⋄ 𝜓. This completes the proof. ◻ 13 São Paulo Journal of Mathematical Sciences As example of the generalized Schrodinger Wick differential equation (14), we take −D2 (𝜓) + (V − E) ⋄ 𝜓 = 0 where V −E = 1 𝛿. 4 0 Then, by Theorem 4, this equation is equivalent to D(𝛷) + 𝛷⋄2 = 1 𝛿 4 0 where 𝛷 = D(𝜓) ⋄ 𝜓 ⋄(−1) . By Theorem 3, we get ( )⋄−1 1 𝛷p = (Z + F) ⋄ e⋄Y + 𝛿0 2 is a particular solution of Riccati equation (18), where D(F) = 0, D(Z) = e−Y and D(Y) = 𝛿0 . In fact; ( )⋄−2 ( ) D(𝛷p ) = − (Z + F) ⋄ e⋄Y ⋄ D(Z + F) ⋄ e⋄Y + De⋄Y ⋄ (Z + F) ( ) )⋄2 ( 1 = − 𝛷 p − 𝛿0 ⋄ 𝛿0 + DY ⋄ e⋄Y ⋄ (Z + F) 2 ( ( )⋄2 ( )⋄−1 ) 1 1 ⋄ 𝛿 0 + 𝛷p − 𝛿0 = − 𝛷 p − 𝛿0 2 2 ( )⋄2 ( )⋄2 ( )⋄−1 1 1 1 − 𝛷p − 𝛿0 ⋄ 𝛷p − 𝛿0 = − 𝛷 p − 𝛿0 2 2 2 ) ) ( ( 1 1 ⋄2 ⋄2 = − 𝛷p − 𝛷p + 𝛿0 − 𝛷p − 𝛿0 4 2 1 ⋄2 = −𝛷p + 𝛿0 4 then, we get D(𝛷) + 𝛷⋄2 = 1 𝛿 4 0 which gives that 𝛷p is a particular solution of (18). 13 (18) São Paulo Journal of Mathematical Sciences 4.2 Generalized logistic Wick differential equation The generalized logistic Wick differential equation introduced in [2] as follows: D(𝛷) = 𝛷 ⋄ (𝛿0 − 𝛷) (19) for 𝛷 ∈ E0𝜃 . Corollary 1 The solution of equation (19) is given by 𝛷 = 𝛷p + (Z + F)⋄(−1) ⋄ e⋄(−Y) , where F, Y and Z satisfy D(F) = 0, D(Y) = −𝛿0 + 2𝛿0 ⋄ 𝛷p = −𝛿0 + 2𝛷p and D(Z) = 𝛿0 ⋄ e⋄(−Y) = e⋄(−Y) . Proof Equation (19) is equivalent to D(𝛷) = 𝛷 ⋄ 𝛿0 − 𝛷⋄2 = 𝛷 − 𝛷⋄2 . This gives D(𝛷) − 𝛷 + 𝛷⋄2 = 0 which is a generalized Riccati Wick differential equation. Then, by Theorem 3 (for 𝛼 = −𝛿0, 𝛽 = 𝛿0 and 𝛾 = 0), we complete the proof. ◻ 4.3 Generalized Gross–Pitaevskii Wick differential equation Riccati differential equations are known to have many applications in nonlinear physics. The generalized Gross–Pitaevskii equation can be written as follows: i𝜕t u + 𝛽(t) 𝛥u + 𝜒(t) ∣ u ∣2 +𝛼(t)r2 u = i𝛾(t)u. 2 Which can be transformed easily into a Riccati ODE form as follows: du + 2𝛽(t)a2 − 𝛼(t) = 0. dt In the distribution case, as analogue of the above Riccati differential equation, we introduce the Wick generalized Gross–Pitaevskii equation as follows: D(𝛷) + 2𝛽 ⋄ 𝛷⋄2 − 𝛾 = 0, 𝛽, 𝛾 ∈ E0𝜃 (20) 13 São Paulo Journal of Mathematical Sciences Corollary 2 The solution of Eq. (20) is given by 𝛷 = 𝛷p + (Z + F)⋄(−1) ⋄ e⋄(−Y) where F, Y and Z satisfy – D(F) = 0 – D(Y) = 4𝛽 ⋄ 𝛷p – D(Z) = 2𝛽 ⋄ e⋄(−Y). Proof Using Theorem 3 (for 𝛼 = 0 and replacing 𝛽 by 2𝛽 ), we get the result. ◻ 4.4 Generalized cosmological Riccati Wick differential equation An equation of state for a cosmological fluid is of the form p(𝜌) = (𝛾 − 1)𝜌, such as 𝛾 = constant, 𝜌 = the pressure and p = the density, the Friedmann equations give for the scale factor the evolution equation in the conformal time 𝜂 which is written as ( � )2 a�� a + ck = 0 + (c − 1) (21) a a 3𝛾 − 1, and k = ±1, and a prime denotes the derivative with respect to 𝜂 . 2 � By introducing the transformation u = aa , the equation (21) reduces to a Riccati equation as follows: where c = � u + cu2 + kc = 0 where u is the Hubble parameter, c is related to the adiabatic index of the cosmological fluid and the values of k are the curvature indices of the Friedmann–Robertson–Walker universes. In the distribution case, as analogue of the above cosmological Riccati equation, we introduce the generalized cosmological Riccati Wick differential equation as follows: D(𝛷) + c ⋄ 𝛷⋄2 + k ⋄ c = 0, c, k ∈ E0𝜃 Corollary 3 The equation (22) has solution given by 𝛷 = 𝛷p + (Z + F)⋄(−1) ⋄ e⋄(−Y) where F, Y and Z satisfy – D(F) = 0 – D(Y) = 2c ⋄ 𝛷p 13 (22) São Paulo Journal of Mathematical Sciences – D(Z) = c ⋄ e⋄(−Y). Proof Using Theorem 3 (for 𝛼 = 0, 𝛽 = c and 𝛾 = −k ⋄ c), we get the result. ◻ Remark 1 In this paper, we studied the generalized Riccati Wick differential equation using the topological dual space F∗𝜃 (S�ℂ ) of the space of holomorphic functions with 𝜃-exponential growth of finite type. Note that one can use the Hida space (E)∗ of generalized functions or Malliavin calculus to find similar result which can be used in financial mathematics. Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest. References 1. 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