Hysteroscopic surgery represents a minimally invasive approach to the diagnosis and treatment of intrauterine pathologies. A distension fluid is required to provide visualization and hemostasis of the operative field. The use of a bipolar resectoscope enables the usage of electrolyte solutions, averting dilutional hyponatremia. However, fluid overload that can develop after the absorption of a sufficient amount of the irrigation medium is a complication to be feared.
We report a case of a 23-year-old female patient who developed acute symptomatic fluid overload and pulmonary edema without dilutional hyponatremia (140 mmol/L) secondary to hysteroscopic transcervical endometrial resection (TCER) for a uterine septum, where the distending medium was saline solution 0.9%.
Several precautions could be implemented to reduce the risk of fluid overload induced by the absorption of distention fluid. Namely, reducing the operative time, the flow, the total volume infused, the intrauterine pressure and strictly monitoring the absorbed volume. The instrumentation should support visual and auditory alarms. Moreover, all staff members should be acquainted with the clinical presentation and management, which mainly revolves around early identification. Therefore, regular simulated cases, to sharpen pathology-related knowledge and teamwork, should become standard practice.
Citation: Niccolò Stomeo, Giacomo Simeone, Leonardo Ciavarella, Giulia Lionetti, Arosh S. Perera Molligoda Arachchige, Francesco Cama. Fluid overload during operative hysteroscopy for metroplasty: A case report[J]. AIMS Medical Science, 2023, 10(4): 310-317. doi: 10.3934/medsci.2023024
[1] | Qun Dai, Shidong Liu . Stability of the mixed Caputo fractional integro-differential equation by means of weighted space method. AIMS Mathematics, 2022, 7(2): 2498-2511. doi: 10.3934/math.2022140 |
[2] | Thabet Abdeljawad, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez . On a new structure of multi-term Hilfer fractional impulsive neutral Levin-Nohel integrodifferential system with variable time delay. AIMS Mathematics, 2024, 9(3): 7372-7395. doi: 10.3934/math.2024357 |
[3] | Choukri Derbazi, Zidane Baitiche, Mohammed S. Abdo, Thabet Abdeljawad . Qualitative analysis of fractional relaxation equation and coupled system with Ψ-Caputo fractional derivative in Banach spaces. AIMS Mathematics, 2021, 6(3): 2486-2509. doi: 10.3934/math.2021151 |
[4] | J. Vanterler da C. Sousa, E. Capelas de Oliveira, F. G. Rodrigues . Ulam-Hyers stabilities of fractional functional differential equations. AIMS Mathematics, 2020, 5(2): 1346-1358. doi: 10.3934/math.2020092 |
[5] | Kaihong Zhao, Shuang Ma . Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses. AIMS Mathematics, 2022, 7(2): 3169-3185. doi: 10.3934/math.2022175 |
[6] | Jiqiang Zhang, Siraj Ul Haq, Akbar Zada, Ioan-Lucian Popa . Stieltjes integral boundary value problem involving a nonlinear multi-term Caputo-type sequential fractional integro-differential equation. AIMS Mathematics, 2023, 8(12): 28413-28434. doi: 10.3934/math.20231454 |
[7] | Hasanen A. Hammad, Hassen Aydi, Hüseyin Işık, Manuel De la Sen . Existence and stability results for a coupled system of impulsive fractional differential equations with Hadamard fractional derivatives. AIMS Mathematics, 2023, 8(3): 6913-6941. doi: 10.3934/math.2023350 |
[8] | Dongming Nie, Usman Riaz, Sumbel Begum, Akbar Zada . A coupled system of p-Laplacian implicit fractional differential equations depending on boundary conditions of integral type. AIMS Mathematics, 2023, 8(7): 16417-16445. doi: 10.3934/math.2023839 |
[9] | Thanin Sitthiwirattham, Rozi Gul, Kamal Shah, Ibrahim Mahariq, Jarunee Soontharanon, Khursheed J. Ansari . Study of implicit-impulsive differential equations involving Caputo-Fabrizio fractional derivative. AIMS Mathematics, 2022, 7(3): 4017-4037. doi: 10.3934/math.2022222 |
[10] | Subramanian Muthaiah, Dumitru Baleanu, Nandha Gopal Thangaraj . Existence and Hyers-Ulam type stability results for nonlinear coupled system of Caputo-Hadamard type fractional differential equations. AIMS Mathematics, 2021, 6(1): 168-194. doi: 10.3934/math.2021012 |
Hysteroscopic surgery represents a minimally invasive approach to the diagnosis and treatment of intrauterine pathologies. A distension fluid is required to provide visualization and hemostasis of the operative field. The use of a bipolar resectoscope enables the usage of electrolyte solutions, averting dilutional hyponatremia. However, fluid overload that can develop after the absorption of a sufficient amount of the irrigation medium is a complication to be feared.
We report a case of a 23-year-old female patient who developed acute symptomatic fluid overload and pulmonary edema without dilutional hyponatremia (140 mmol/L) secondary to hysteroscopic transcervical endometrial resection (TCER) for a uterine septum, where the distending medium was saline solution 0.9%.
Several precautions could be implemented to reduce the risk of fluid overload induced by the absorption of distention fluid. Namely, reducing the operative time, the flow, the total volume infused, the intrauterine pressure and strictly monitoring the absorbed volume. The instrumentation should support visual and auditory alarms. Moreover, all staff members should be acquainted with the clinical presentation and management, which mainly revolves around early identification. Therefore, regular simulated cases, to sharpen pathology-related knowledge and teamwork, should become standard practice.
Fractional differential equations have played an important role and have presented valuable tools in the modeling of many phenomena in various fields of science and engineering [6,7,8,9,10,11,12,13,14,15,16]. There has been a significant development in fractional differential equations in recent decades [2,3,4,5,26,23,33,37]. On the other hand, many authors studied the stability of functional equations and established some types of Ulam stability [1,17,18,19,20,21,22,24,27,28,29,30,31,32,33,34,35,36,37] and references there in. Moreover, many authors discussed local and global attractivity [8,9,10,11,34].
Benchohra et al. [13] established some types of Ulam-Hyers stability for an implicit fractional-order differential equation.
A. Baliki et al. [11] have given sufficient conditions for existence and attractivity of mild solutions for second order semi-linear functional evolution equation in Banach spaces using Schauder's fixed point theorem.
Benchohra et al. [15] studied the existence of mild solutions for a class of impulsive semilinear fractional differential equations with infinite delay and non-instantaneous impulses in Banach spaces. This results are obtained using the technique of measures of noncompactness.
Motivated by these works, in this paper, we investigate the following initial value problem for an implicit fractional-order differential equation
{CDα[x(t)−h(t,x(t))]=g1(t,x(t),Iβg2(t,x(t)))t∈J,1<α≤2,α≥β,(x(t)−h(t,x(t)))|t=0=0andddt[x(t)−h(t,x(t))]t=0=0 | (1.1) |
where CDα is the Caputo fractional derivative, h:J×R⟶R,g1:J×R×R⟶R and g2:J×R⟶R are given functions satisfy some conditions and J=[0,T].
we give sufficient conditions for the existence of solutions for a class of initial value problem for an neutral differential equation involving Caputo fractional derivatives. Also, we establish some types of Ulam-Hyers stability for this class of implicit fractional-order differential equation and some applications and particular cases are presented.
Finally, existence of at least one mild solution for this class of implicit fractional-order differential equation on an infinite interval J=[0,+∞), by applying Schauder fixed point theorem and proving the attractivity of these mild solutions.
By a solution of the Eq (1.1) we mean that a function x∈C2(J,R) such that
(i) the function t→[x(t)−h(t,x(t))]∈C2(J,R) and
(ii) x satisfies the equation in (1.1).
Definition 1. [23] The Riemann-Liouville fractional integral of the function f∈L1([a,b]) of order α∈R+ is defined by
Iαaf(t)=∫ta(t−s)α−1Γ(α)f(s)ds. |
and when a=0, we have Iαf(t)=Iα0f(t).
Definition 2. [23] For a function f:[a,b]→R the Caputo fractional-order derivative of f, is defined by
CDαh(t)=1Γ(n−α)∫tah(n)(s)(t−s)n−α−1ds, |
where where n=[α]+1 and [α] denotes the integer part of the real number α.
Lemma 1. [23]. Let α≥0 and n=[α]+1. Then
Iα(CDαf(t))=f(t)−n−1∑k=0fk(0)k!tk |
Lemma 2. Let f∈L1([a,b]) and α∈(0,1], then
(i) CDαIαf(t)=f(t).
(ii) The operator Iα maps L1([a,b]) into itself continuously.
(iii) For γ,β>0, then
IβaIγaf(t)=IγaIβaf(t)=Iγ+βaf(t), |
For further properties of fractional operators (see [23,25,26]).
Consider the initial value problem for the implicit fractional-order differential Eq (1.1) under the following assumptions:
(i) h:J×R⟶R is a continuous function and there exists a positive constant K1 such that:
∣h(t,x)−h(t,y)∣⩽K1∣x−y∣ for each t∈J and x,y∈R. |
(ii) g1:J×R×R⟶R is a continuous function and there exist two positive constants K,H such that:
∣g1(t,x,y)−∣g1(t,˜x,˜y)∣⩽K∣x−˜x∣+H∣y−˜y∣ for each t∈J and x,˜x,y,˜y∈R |
(iii) g2:J×R⟶R is a continuous function and there exists a positive constant K2 such that:
∣g2(t,x)−g2(t,y)∣⩽K2∣x−y∣ for each t∈J andx,y∈R. |
Lemma 3. Let assumptions (i)-(iii) be satisfied. If a function x∈C2(J,R) is a solution of initial value problem for implicit fractional-order differential equation (1.1), then it is a solution of the following nonlinear fractional integral equation
x(t)=h(t,x(t))+1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds | (3.1) |
Proof. Assume first that x is a solution of the initial value problem (1.1). From definition of Caputo derivative, we have
I2−αD2(x(t)−h(t,x(t)))=g1(t,x(t),Iβg2(t,x(t))). |
Operating by Iα−1 on both sides and using Lemma 2, we get
I1D2(x(t)−h(t,x(t)))=Iα−1g1(t,x(t),Iβg2(t,x(t))). |
Then
ddt(x(t)−h(t,x(t)))−ddt(x(t)−h(t,x(t)))|t=0=Iα−1g1(t,x(t),Iβg2(t,x(t))). |
Using initial conditions, we have
ddt(x(t)−h(t,x(t)))=Iα−1g1(t,x(t),Iβg2(t,x(t))). |
Integrating both sides of (1.1), we obtain
(x(t)−h(t,x(t)))−(x(t)−h(t,x(t)))|t=0=Iαg1(t,x(t),Iβg2(t,x(t))). |
Then
x(t)=h(t,x(t))+1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds |
Conversely, assume that x satisfies the nonlinear integral Eq (3.1). Then operating by CDα on both sides of Eq (3.1) and using Lemma 2, we obtain
CDα(x(t)−h(t,x(t)))=CDαIαg1(t,x(t),Iβg2(t,x(t)))=g1(t,x(t),Iβg2(t,x(t))). |
Putting t=0 in (3.1) and since g1 is a continuous function, then we obtain
(x(t)−h(t,x(t)))|t=0=Iαg1(t,x(t),Iβg2(t,x(t)))|t=0=0. |
Also,
ddt(x(t)−h(t,x(t)))=Iα−1g1(t,x(t),Iβg2(t,x(t))). |
Then we have
ddt(x(t)−h(t,x(t)))|t=0=Iα−1g1(t,x(t),Iβg2(t,x(t)))|t=0=0. |
Hence the equivalence between the initial value problem (1.1) and the integral Eq (3.1) is proved. Then the proof is completed.
Definition 3. The Eq (1.1) is Ulam-Hyers stable if there exists a real number cf>0 such that for each ϵ>0 and for each solution z∈C2(J,R) of the inequality
∣CDα[z(t)−h(t,z(t))]−g1(t,z(t),Iβg2(t,z(t)))∣⩽ϵ,t∈J, |
there exists a solution y∈C2(J,R) of Eq (1.1) with
∣z(t)−y(t)|⩽cfϵ,t∈J. |
Definition 4. The Eq (1.1) is generalized Ulam-Hyers stable if there exists ψf∈C(R+,R+),ψf(0)=0, such that for each solution z∈C2(J,R) of the inequality
∣CDα[z(t)−h(t,z(t))]−g1(t,z(t),Iβg2(t,z(t)))∣⩽ϵ,t∈J, |
there exists a solution y∈C2(J,R)of Eq (1.1) with
∣z(t)−y(t)|⩽ψf(ϵ),t∈J. |
Definition 5. The Eq (1.1) is Ulam-Hyers-Rassias stable with respect to φ∈C(J,R+) if there exists a real number cf>0 such that for each ϵ>0 and for each solution z∈C2(J,R) of the inequality
∣CDα[z(t)−h(t,z(t))]−g1(t,z(t),Iβg2(t,z(t)))∣⩽ϵφ(t),t∈J, |
there exists a solution y∈C2(J,R) of Eq (1.1) with
∣z(t)−y(t)|⩽cfϵφ(t),t∈J. |
Definition 6. The Eq (1.1) is generalized Ulam-Hyers-Rassias stable with respect to φ∈C(J,R+) if there exists a real number cf,φ>0 such that for each solution z∈C2(J,R) of the inequality
∣CDα[z(t)−h(t,z(t))]−g1(t,z(t),Iβg2(t,z(t)))∣⩽φ(t),t∈J, |
there exists a solution y∈C2(J,R) of Eq (1.1) with
∣z(t)−y(t)|⩽cf,φφ(t),t∈J. |
Now, our aim is to investigate the existence of unique solution for (1.1). This existence result will be based on the contraction mapping principle.
Theorem 1. Let assumptions (i)-(iii) be satisfied. If K1+KTαΓ(α+1)+K2HTα+βΓ(β+1)Γ(α+1)<1, then there exists a unique solution for the nonlinear neutral differential equation of fractional order.
Proof. Define the operator N by:
Nx(t)=h(t,x(t))+1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds,t∈J. |
In view of assumptions (i)-(iii), then N:C2(J,R)→C2(J,R) is continuous operator.
Now, let x and ,˜x∈C2(J,R), be two solutions of (1.1)then
∣Nx(t)−N˜x(t)∣=|h(t,x(t))+1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds−h(t,˜x(t))−1Γ(α)∫t0(t−s)α−1g1(s,˜x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,˜x(θ))dθ)ds|⩽K1|x(t)−˜x(t)|+1Γ(α)∫t0(t−s)α−1|g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)−g1(s,˜x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,˜x(θ))dθ)|ds⩽K1|x(t)−˜x(t)|+1Γ(α)∫t0(t−s)α−1K∣x(s)−˜x(s)|ds+H1Γ(α)∫t0(t−s)α−11Γ(β)∫s0(s−θ)β−1∣g2(θ,x(θ))−g2(θ,˜x(θ))|dθds⩽K1|x(t)−˜x(t)|+KΓ(α)∫t0(t−s)α−1∣x(s)−˜x(s)|ds+HΓ(α)∫t0(t−s)α−1K2Γ(β)∫s0(s−θ)β−1∣x(θ)−˜x(θ)|dθds. |
Then
||Nx(t)−N˜x(t)||⩽K1||x−˜x||+K||x−˜x||Γ(α)∫t0(t−s)α−1ds+||x−˜x||HΓ(α)∫t0(t−s)α−1K2Γ(β)∫s0(s−θ)β−1dθds⩽K1||x−˜x||+K||x−˜x||TαΓ(α+1)+||x−˜x||K2TβΓ(β+1)HΓ(α)∫t0(t−s)α−1ds⩽K1||x−˜x||+K||x−˜x||TαΓ(α+1)+||x−˜x||K2TβΓ(β+1)HTαΓ(α+1)≤[K1+KTαΓ(α+1)+K2HTα+βΓ(β+1)Γ(α+1)]||x−˜x|| |
Since K1+KTαΓ(α+1)+K2HTα+βΓ(β+1)Γ(α+1)<1. It follows that N has a unique fixed point which is a solution of the initial value problem (1.1) in C2(J,R).
Theorem 2. Let assumptions of Theorem 1 be satisfied. Then the fractional order differential Eq (1.1) is Ulam-Hyers stable.
Proof. Let y∈C2(J,R) be a solution of the inequality
∣CDα[y(t)−h(t,y(t))]−g1(t,y(t),Iβg2(t,y(t)))∣⩽ϵ,ϵ>0,t∈J. | (4.1) |
Let x∈C2(J,R) be the unique solution of the initial value problem for implicit fractional-order differential Eq (1.1). By using Lemma 3, The Cauchy problem (1.1) is equivalent to
x(t)=h(t,x(t))+1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds. |
Operating by Iα−1 on both sides of (4.1) and then integrating, we get
|y(t)−h(t,y(t))−1Γ(α)∫t0(t−s)α−1g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)ds|⩽1Γ(α)∫t0(t−s)α−1ϵds,≤ϵTαΓ(α+1). |
Also, we have
|y(t)−x(t)|=|y(t)−h(t,x(t))−1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds|=|y(t)−h(t,x(t))−1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds+h(t,y(t))+1Γ(α)∫t0(t−s)α−1g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)ds−h(t,y(t))−1Γ(α)∫t0(t−s)α−1g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)ds|≤|y(t)−h(t,y(t))−1Γ(α)∫t0(t−s)α−1g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)ds|+|h(t,y(t))−h(t,x(t))|+1Γ(α)∫t0(t−s)α−1|g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)−g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|ds≤ϵTαΓ(α+1)+K1|y(t)−x(t)|+1Γ(α)∫t0(t−s)α−1[K|y(t)−x(t)|+HΓ(β)∫s0(s−θ)β−1|g2(θ,y(θ))−g2(θ,x(θ))|dθ]ds |
||y−x||≤ϵTαΓ(α+1)+K1||y−x||+1Γ(α)∫t0(t−s)α−1[K||y−x||+HK2||x−y||TβΓ(β+1)]ds≤ϵTαΓ(α+1)+K1||y−x||+KTα||y−x||Γ(α+1)+HK2Tβ+α||x−y||Γ(β+1)Γ(α+1). |
Then
||y−x||≤ϵTαΓ(α+1)[1−(K1+KTαΓ(α+1)+HK2Tβ+αΓ(β+1)Γ(α+1))]−1=cϵ, |
thus the intial value problem (1.1) is Ulam-Heyers stable, and hence the proof is completed. By putting ψ(ε)=cε,ψ(0)=0 yields that the Eq (1.1) is generalized Ulam-Heyers stable.
Theorem 3. Let assumptions of Theorem 1 be satisfied, there exists an increasing function φ∈C(J,R) and there exists λφ>0 such that for any t∈J, we have
Iαφ(t)⩽λφφ(t), |
then the Eq (1.1) is Ulam-Heyers-Rassias stable.
Proof. Let y∈C2(J,R) be a solution of the inequality
∣CDα[y(t)−h(t,y(t))]−g1(t,y(t),Iβg2(t,y(t)))∣⩽ϵφ(t),ϵ>0,t∈J. | (4.2) |
Let x∈C2(J,R) be the unique solution of the initial value problem for implicit fractional-order differential Eq (1.1). By using Lemma 3, The Cauchy problem (1.1) is equivalent to
x(t)=h(t,x(t))+1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds. |
Operating by Iα−1 on both sides of (4.2) and then integrating, we get
|y(t)−h(t,y(t))−1Γ(α)∫t0(t−s)α−1g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)ds|⩽ϵΓ(α)∫t0(t−s)α−1φ(s)ds,≤ϵIαφ(t)≤ϵλφφ(t). |
Also, we have
|y(t)−x(t)|=|y(t)−h(t,x(t))−1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds|=|y(t)−h(t,x(t))−1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds+h(t,y(t))+1Γ(α)∫t0(t−s)α−1g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)ds−h(t,y(t))−1Γ(α)∫t0(t−s)α−1g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)ds|≤|y(t)−h(t,y(t))−1Γ(α)∫t0(t−s)α−1g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)ds|+|h(t,y(t))−h(t,x(t))|+1Γ(α)∫t0(t−s)α−1|g1(s,y(s),1Γ(β)∫s0(s−θ)β−1g2(θ,y(θ))dθ)−g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|ds≤ϵλφφ(t)+K1|y(t)−x(t)|+1Γ(α)∫t0(t−s)α−1[K|y(t)−x(t)|+HΓ(β)∫s0(s−θ)β−1|g2(θ,y(θ))−g2(θ,x(θ))|dθ]ds |
||y−x||≤ϵλφφ(t)+K1||y−x||+1Γ(α)∫t0(t−s)α−1[K||y(t)−x(t)||+HK2||x−y||TβΓ(β+1)]ds≤ϵλφφ(t)+K1||y−x||+KTα||y−x||Γ(α+1)+HK2Tβ+α||x−y||Γ(β+1)Γ(α+1). |
Then
||y−x||≤ϵλφφ(t)[1−(K1+KTαΓ(α+1)+HK2Tβ+αΓ(β+1)Γ(α+1))]−1=cϵφ(t), |
then the initial problem (1.1) is Ulam-Heyers-Rassias stable, and hence the proof is completed.
In this section, we prove some results on the existence of mild solutions and attractivity for the neutral fractional differential equation (1.1) by applying Schauder fixed point theorem. Denote BC=BC(J),J=[0,+∞) and consider the following assumptions:
(I) h:J×R⟶R is a continuous function and there exists a continuous function Kh(t) such that:
∣h(t,x)−h(t,y)∣⩽Kh(t)∣x−y∣ for each t∈J and x,y∈R, |
where K∗h=supt≥0Kh(t)<1,limt→∞Kh(t)=0, and limt→∞h(t,0)=0.
(II) g1:J×R×R⟶R satisfies Carathéodory condition and there exist an integrable function a1:R+⟶R+ and a positive constant b such that:
∣g1(t,x,y)∣≤a1(t)1+|x|+b|y| for eacht∈J and x,y∈R. |
(III) g2:J×R⟶R satisfies Carathéodory condition and there exists an integrable function
a2:R+⟶R+ such that:
∣g2(t,x)∣≤a2(t)1+|x| for eacht∈J and x∈R. |
(IV) Let
limt→∞∫t0(t−s)α−1Γ(α)a1(s)ds=0a∗1=supt∈J∫t0(t−s)α−1Γ(α)a1(s)dslimt→∞∫t0(t−s)α+β−1Γ(α+β)a2(s)ds=0a∗2=supt∈J∫t0(t−s)α+β−1Γ(α+β)a2(s)ds |
By a mild solution of the Eq (1.1) we mean that a function x∈C(J,R) such that x satisfies the equation in (3.1).
Theorem 4. Let assumptions (I)-(IV) be satisfied. Then there exists at least one mild solution for the nonlinear implicit neutral differential equation of fractional order (1.1). Moreover, mild solutions of IVP (1.1) are locally attractive.
Proof. For any x∈BC, define the operator A by
Ax(t)=h(t,x(t))+1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds. |
The operator A is well defined and maps BC into BC. Obviously, the map A(x) is continuous on J for any x∈BC and for each t∈J, we have
|Ax(t)|≤|h(t,x(t))|+1Γ(α)∫t0(t−s)α−1|g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|ds≤|h(t,x(t))−h(t,0)|+|h(t,0)|+1Γ(α)∫t0(t−s)α−1[a1(s)1+|x(s)|+b1Γ(β)∫s0(s−θ)β−1|g2(θ,x(θ))|dθ]ds≤Kh(t)|x(t)|+|h(t,0)|+1Γ(α)∫t0(t−s)α−1[a1(s)+b1Γ(β)∫s0(s−θ)β−1a2(θ)1+|x(θ)|dθ]ds≤K∗h|x(t)|+|h(t,0)|+a∗1+b1Γ(α+β)∫t0(t−s)α+β−1a2(θ)1+|x(θ)|dθ≤K∗hM+|h(t,0)|+a∗1+b1Γ(α+β)∫t0(t−s)α+β−1a2(θ)dθ≤K∗hM+|h(t,0)|+a∗1+ba∗2≤M. |
Then
||Ax(t)||BC≤M,M=(|h(t,0)|+a∗1+ba∗2)(1−K∗h)−1. | (5.1) |
Thus A(x)∈BC. This clarifies that operator A maps BC into itself.
Finding the solutions of IVP (1.1) is reduced to find solutions of the operator equation A(x)=x. Eq (5.1) implies that A maps the ball BM:=B(0,M)={x∈BC:||x(t)||BC≤M} into itself. Now, our proof will be established in the following steps:
Step 1: A is continuous.
Let {xn}n∈N be a sequence such that xn→x in BM. Then, for each t∈J, we have
∣Axn(t)−Ax(t)∣=|h(t,xn(t))+1Γ(α)∫t0(t−s)α−1g1(s,xn(s),1Γ(β)∫s0(s−θ)β−1g2(θ,xn(θ))dθ)ds−h(t,x(t))−1Γ(α)∫t0(t−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds|⩽Kh(t)|xn(t)−x(t)|+1Γ(α)∫t0(t−s)α−1|g1(s,xn(s),1Γ(β)∫s0(s−θ)β−1g2(θ,xn(θ))dθ)−g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|ds⩽K∗h|xn(t)−x(t)|+1Γ(α)∫t0(t−s)α−1|g1(s,xn(s),1Γ(β)∫s0(s−θ)β−1g2(θ,xn(θ))dθ)−g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|ds |
Assumptions (II) and (III) implies that:
g1(t,xn,Iβg2(t,xn))→g1(t,x,Iβg2(t,x))\; as \; n→∞. |
Using Lebesgue dominated convergence theorem, we have
||Axn(t)−Ax(t)||BC→0 asn→∞. |
Step 2: A(BM) is uniformly bounded.
It is obvious since A(BM)⊂BM and BM is bounded.
Step 3: A(BM) is equicontinuous on every compact subset [0,T] of J,T>0 and t1,t2∈[0,T],t2>t1 (without loss of generality), we get
∣Ax(t2)−Ax(t1)∣≤|h(t2,x(t2))+1Γ(α)∫t20(t2−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds−h(t1,x(t1))+1Γ(α)∫t10(t1−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds|≤∣h(t2,x(t2))−h(t1,x(t1))| |
+1Γ(α)|∫t20(t2−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds−∫t10(t1−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds|≤∣h(t2,x(t2))−h(t1,x(t1))+h(t2,x(t1))−h(t2,x(t1))|+1Γ(α)|∫t10(t2−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds+1Γ(α)∫t2t1(t2−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds−∫t10(t1−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds|≤Kh(t)∣x(t2)−x(t1)∣+|h(t2,x(t1))−h(t1,x(t1))|+1Γ(α)|∫t10(t1−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds+1Γ(α)∫t2t1(t2−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds−∫t10(t1−s)α−1g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)ds|≤Kh(t)∣x(t2)−x(t1)∣+|h(t2,x(t1))−h(t1,x(t1))|+1Γ(α)∫t2t1(t2−s)α−1|g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|ds≤Kh(t)∣x(t2)−x(t1)∣+|h(t2,x(t1))−h(t1,x(t1))|+1Γ(α)∫t2t1(t2−s)α−1[a1(s)1+|x(s)|+b1Γ(β)∫s0(s−θ)β−1|g2(θ,x(θ))|dθ]ds≤K∗h∣x(t2)−x(t1)∣+|h(t2,x(t1))−h(t1,x(t1))|+1Γ(α)∫t2t1(t2−s)α−1[a1(s)+b1Γ(β)∫s0(s−θ)β−1a2(θ)1+|x(θ)|dθ]ds≤K∗h∣x(t2)−x(t1)∣+|h(t2,x(t1))−h(t1,x(t1))|+1Γ(α)∫t2t1(t2−s)α−1[a1(s)+b1Γ(β)∫s0(s−θ)β−1a2(θ)dθ]ds. |
Thus, for ai=supt∈[0,T]ai,i=1,2 and from the continuity of the functions ai we obtain
∣Ax(t2)−Ax(t1)∣≤K∗h∣x(t2)−x(t1)∣+|h(t2,x(t1))−h(t1,x(t1))|+1Γ(α)∫t2t1(t2−s)α−1[a1(s)+ba2Γ(β+1)sβ]ds.≤K∗h∣x(t2)−x(t1)∣+|h(t2,x(t1))−h(t1,x(t1))|+a1Γ(α+1)(t2−t1)α+ba2Γ(α+β+1)(t2−t1)α+β. |
Continuity of h implies that
|(Ax)(t2)−(Ax)(t1)|→0ast2→t1. |
Step 4: A(BM) is equiconvergent.
Let t∈J and x∈BM then we have
|Ax(t)|≤|h(t,x(t))−h(t,0)|+|h(t,0)|+1Γ(α)∫t0(t−s)α−1|g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|ds≤Kh(t)|x(t)|+|h(t,0)|+1Γ(α)∫t0(t−s)α−1[a1(s)1+|x(s)|+b1Γ(β)∫s0(s−θ)β−1|g2(θ,x(θ))|dθ]ds≤Kh(t)|x(t)|+|h(t,0)|+1Γ(α)∫t0(t−s)α−1[a1(s)+b1Γ(β)∫s0(s−θ)β−1a2(θ)1+|x(θ)|dθ]ds≤Kh(t)|x(t)|+|h(t,0)|+1Γ(α)∫t0(t−s)α−1a1(s)ds+b1Γ(α+β)∫t0(t−s)α+β−1a2(s)ds. |
In view of assumptions (I) and (IV), we obtain
|Ax(t)|→0 as t→∞. |
Then A has a fixed point x which is a solution of IVP (1.1) on J.
Step 5: Local attactivity of mild solutions. Let x∗ be a mild solution of IVP (1.1). Taking x∈B(x∗,2M), we have
|Ax(t)−x∗(t)|=|Ax(t)−Ax∗(t)|≤∣h(t,x(t))−h(t,x∗(t))∣+1Γ(α)∫t0(t−s)α−1|g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)−g1(s,x∗(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x∗(θ))dθ)|ds≤Kh(t)∣x(t)−x∗(t)∣+1Γ(α)∫t0(t−s)α−1[|g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|+|g1(s,x∗(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x∗(θ))dθ)|]ds≤K∗h∣x(t)−x∗(t)∣+2Γ(α)∫t0(t−s)α−1|a1(s)+bΓ(β)∫s0(s−θ)β−1a2(θ)dθ|ds≤K∗h∣x(t)−x∗(t)∣+2a∗1+2b1Γ(β+α)∫s0(s−θ)α+β−1a2(θ)dθ≤2(K∗h∣x(t)∣+|h(t,0)|+a∗1+ba∗2)≤2(K∗hM+|h(t,0)|+a∗1+ba∗2)≤2M. |
We have
||Ax(t)−x∗(t)||BC≤2M. |
Hence A is a continuous function such that A(B(x∗,2M))⊂B(x∗,2M).
Moreover, if x is a mild solution of IVP (1.1), then
|x(t)−x∗(t)|=|Ax(t)−Ax∗(t)|≤∣h(t,x(t))−h(t,x∗(t))∣+1Γ(α)∫t0(t−s)α−1|g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)−g1(s,x∗(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x∗(θ))dθ)|ds≤Kh(t)∣x(t)−x∗(t)∣+1Γ(α)∫t0(t−s)α−1[|g1(s,x(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x(θ))dθ)|+|g1(s,x∗(s),1Γ(β)∫s0(s−θ)β−1g2(θ,x∗(θ))dθ)|]ds≤K∗h∣x(t)−x∗(t)∣+2Γ(α)∫t0(t−s)α−1a1(s)ds+2bΓ(α+β)∫t0(t−θ)α+β−1a2(θ)dθ. |
Then
|x(t)−x∗(t)|≤(1−K∗h)−1[2Γ(α)∫t0(t−s)α−1a1(s)ds+2bΓ(α+β)∫t0(t−θ)α+β−1a2(θ)dθ]. | (5.2) |
In view of assumption of (IV) and estimation (5.2), we get
limt→∞|x(t)−x∗(t)|=0. |
Then, all mild solutions of IVP (1.1) are locally attractive.
As particular cases of the IVP (1.1), we have
● Taking g1(t,x,y)=g1(t,x), we obtain the initial value problem
{CDα[x(t)−h(t,x(t))]=g1(t,x(t))t∈J,1<α≤2,(x(t)−h(t,x(t)))|t=0=0andddt[x(t)−h(t,x(t))]t=0=0 |
● Letting α→2,β→1, as a particular case of Theorem 1 we can deduce an existence result for the initial value problem for implicit second-order differe-integral equation
{d2dt2(x(t)−h(t,x(t)))=g1(t,x(t),∫t0g2(s,x(s))ds)t∈J,(x(t)−h(t,x(t)))|t=0=0andddt[x(t)−h(t,x(t))]t=0=0 |
As particular cases we can deduce existence results for some initial value problem of second order differential equations (when h=0) and α→2, we get:
● Taking g1(t,x,y)=−λ2x(t),λ∈R+, then we obtain a second order differential equation of simple harmonic oscillator
{d2x(t)dt2=−λ2x(t)t∈J,x(0)=0andx′(0)=0 |
● Taking g1(t,x,y)=(t2−kt2)x+q(x),k∈R where q(x) is continuous function, then we obtain Riccati differential equation of second order
{t2d2x(t)dt2−(t2−k)x(t)=t2q(x(t))t∈J,x(0)=0andx′(0)=0 |
● Taking g1(t,x,y)=−(t2−2lt−k)x+q(x),k∈R where q(x) is continuous function and l is fixed, then we obtain Coulomb wave differential equation of second order
{d2x(t)dt2+(t2−2lt−k)x=q(x(t))t∈J,x(0)=0andx′(0)=0 |
● Taking g1(t,x,y)=(−8π2mℏ2)(Ex−kt22x)+q(x),k∈R where q(x) is continuous function and ℏ is the Planket's constant and E,k are positive real numbers, then we obtain of Schrödinger wave differential equation for simple harmonic oscillator
{d2x(t)dt2=(−8π2mℏ2)(Ex(t)−kt22x(t))+q(x(t))t∈J,x(0)=0andx′(0)=0. |
Sufficient conditions for the existence of solutions for a class of neutral integro-differential equations of fractional order (1.1) are discussed which involved many key functional differential equations that appear in applications of nonlinear analysis. Also, some types of Ulam stability for this class of implicit fractional differential equation are established. Some applications and particular cases are presented. Finally, the existence of at least one mild solution for this class of equations on an infinite interval by applying Schauder fixed point theorem and the local attractivity of solutions are proved.
The authors express their thanks to the anonymous referees for their valuable comments and remarks.
The authors declare that they have no competing interests.
[1] |
Di Spiezio Sardo A, Taylor A, Tsirkas P, et al. (2008) Hysteroscopy: A technique for all? Analysis of 5000 outpatient hysteroscopies. Fertil Steril 89: 438-443. https://doi.org/10.1016/j.fertnstert.2007.02.056 ![]() |
[2] | Jansen FW, Vredevoogd CB, van Ulzen K, et al. (2000) Complications of hysteroscopy: A prospective, multicenter study. Obstet Gynecol 96: 266-270. https://doi.org/10.1016/s0029-7844(00)00865-6 |
[3] |
Zeltser I, Pearle MS, Bagley DH (2009) Saline is our friend. Urology 74: 28-29. https://doi.org/10.1016/j.urology.2009.01.061 ![]() |
[4] |
Mohta M, Bhagchandani T, Tyagi A, et al. (2008) Haemodynamic, electrolyte and metabolic changes during percutaneous nephrolithotomy. Int Urol Nephrol 40: 477-482. https://doi.org/10.1007/s11255-006-9093-6 ![]() |
[5] |
Issa MM (2008) Technological advances in transurethral resection of the prostate: bipolar versus monopolar TURP. J Endourol 22: 1587-1595. https://doi.org/10.1089/end.2008.0192 ![]() |
[6] |
Mamoulakis C, Skolarikos A, Schulze M, et al. (2012) Results from an international multicentre double-blind randomized controlled trial on the perioperative efficacy and safety of bipolar vs monopolar transurethral resection of the prostate. BJU Int 109: 240-248. https://doi.org/10.1111/j.1464-410X.2011.10222.x ![]() |
[7] |
Michielsen DPJ, Coomans D, Braeckman JG, et al. (2010) Bipolar transurethral resection in saline: the solution to avoid hyponatremia and transurethral resection syndrome. Scand J Urol Nephrol 44: 228-235. https://doi.org/10.3109/00365591003720275 ![]() |
[8] |
Aydeniz B, Gruber IV, Schauf B, et al. (2002) A multicenter survey of complications associated with 21676 operative hysteroscopies. Eur J Obstet Gynecol Reprod Biol 104: 160-164. https://doi.org/10.1016/s0301-2115(02)00106-9 ![]() |
[9] |
Shirk GJ, Gimpelson RJ (1994) Control of intrauterine fluid pressure during operative hysteroscopy. J Am Assoc Gynecol Laparosc 1: 229-233. https://doi.org/10.1016/s1074-3804(05)81015-1 ![]() |
[10] |
Berg A, Sandvik L, Langebrekke A, et al. (2009) A randomized trial comparing monopolar electrodes using glycine 1.5% with two different types of bipolar electrodes (TCRis, Versapoint) using saline, in hysteroscopic surgery. Fertil Steril 91: 1273-1278. https://doi.org/10.1016/j.fertnstert.2008.01.083 ![]() |
[11] |
Olsson J, Berglund L, Hahn RG (1996) Irrigating fluid absorption from the intact uterus. Br J Obstet Gynaecol 103: 558-561. https://doi.org/10.1111/j.1471-0528.1996.tb09806.x ![]() |
[12] |
Hahn RG (2006) Fluid absorption in endoscopic surgery. Br J Anaesth 96: 08-20. https://doi.org/10.1093/bja/aei279 ![]() |
[13] | Hahn RG (1988) Hallucination and visual disturbances in transurethral prostatic resection. Intensive Care Med 14: 668-671. https://doi.org/10.1007/BF00256777 |
[14] |
Hahn RG (1990) Fluid and electrolyte dynamics during development of the TURP syndrome. Br J Urol 66: 79-84. https://doi.org/10.1111/j.1464-410x.1990.tb14869.x ![]() |
[15] |
Beal JL, Freysz M, Berthelon G, et al. (1989) Consequences of fluid absorption during transurethral resection of the prostate using distilled water or glycine 1.5 percent. Can J Anaesth 36: 278-282. https://doi.org/10.1007/BF03010765 ![]() |
[16] |
Wilkes NJ, Woolf R, Mutch M, et al. (2001) The effects of balanced versus saline-based hetastarch and crystalloid solutions on acid-base and electrolyte status and gastric mucosal perfusion in elderly surgical patients. Anesth Analg 93: 811-816. https://doi.org/10.1097/00000539-200110000-00003 ![]() |
[17] |
Paschopoulos M, Polyzos NP, Lavasidis LG, et al. (2006) Safety issues of hysteroscopic surgery. Ann N Y Acad Sci 1092: 229-234. https://doi.org/10.1196/annals.1365.019 ![]() |
[18] | Corson SL, Brooks PG, Serden SP, et al. (1994) Effects of vasopressin administration during hysteroscopic surgery. J Reprod Med 39: 419-423. |
[19] | Emmett M, Istre O, Hahn RG (2022) Hyponatremia following transurethral resection, hysteroscopy, or other procedures involving electrolyte-free irrigation. Available from: https://www.uptodate.com/contents/hyponatremia-following-transurethral-resection-hysteroscopy-or-other-procedures-involving-electrolyte-free-irrigation. Accessed on 30 Sep 2020 |
[20] |
Goldenberg M, Zolti M, Bider D, et al. (1996) The effect of intracervical vasopressin on the systemic absorption of glycine during hysteroscopic endometrial ablation. Obstet Gynecol 87: 1025-1029. https://doi.org/10.1016/0029-7844(96)00063-4 ![]() |
1. | Xuming Chen, Jianfa Zhu, Liangxiao Li, Chengwen Long, Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation, 2022, 0, 2444-8656, 10.2478/amns.2022.2.0104 | |
2. | Abha Singh, Abdul Hamid Ganie, Mashael M. Albaidani, Antonio Scarfone, Some New Inequalities Using Nonintegral Notion of Variables, 2021, 2021, 1687-9139, 1, 10.1155/2021/8045406 | |
3. | Liang Song, Shaodong Chen, Guoxin Wang, Oscillation Analysis Algorithm for Nonlinear Second-Order Neutral Differential Equations, 2023, 11, 2227-7390, 3478, 10.3390/math11163478 |