Coupled Fixed Point Theorem for the Generalized Langevin Equation with Four-Point and Strip Conditions

By considering a metric space with partially ordered sets, we employ the coupled ﬁ xed point type to scrutinize the uniqueness theory for the Langevin equation that included two generalized orders. We analyze our problem with four-point and strip conditions. The description of the rigid plate bounded by a Newtonian ﬂ uid is provided as an application of our results. The exact solution of this problem and approximate solutions are compared.


Introduction
The fractional calculus concept is not absolutely intuitive, where it has no clear geometrical interpretation. Several distinct forms have appeared, to the point that the necessity for order has developed in the field [1,2]. The variety of potential implementations is even more difficult. One has to think closely about what the inserting of fractional derivatives in the model can provide. Fractional derivatives are generally inserted for modeling processes of mass transport, optics, diffusion, etc. [3,4]. With the inserting of these derivatives, fractional-order models have described its advantages when modeling supercapacitor capacitances [5] and controllers for temperature [6], DC motors [7], or RC, LC, and RLC electric circuits [8]. Through the present paper, we deem the generalized Langevin differential equation of two generalized different orders: where λ ∈ ℝ, c D β , and c D α are the Caputo generalized derivatives with 0 < α ≤ 1 and 1 < β ≤ 2, and the continu-ously differentiable function f : ½0, 1 × ℝ ⟶ ℝ is given. This equation is supplemented with the strip and fourpoint conditions: where γ > 0 and 0 < η, ξ < 1. The third boundary condition, which appears as a linear combination of nonlocal point and Riemann-Liouville fractional integral condition on an arbitrary segment ð0, ηÞ ⊂ ½0, 1, can be explained as a gathering of the values of the obscure function at local point 1, and nonlocal point ξ ∈ ð0, 1Þ) is proportionate to the strip contribution of the unbeknown function. The investigation of the fractional Langevin equation (1) together with four-point and strip conditions (2) makes our problem new especially when applying coupled immutable point type in the case of an existing mixed monotone mapping.
The main goal of our research is to investigate the solution uniqueness for our case ( (1) and (2)) by virtue of applying coupled immutable point type in a metric space with partially ordered set in the case of an existing mixed monotone mapping. It is worth pointing out, as far as we know, that no contribution till now studied the uniqueness of solution for the generalized Langevin differential equation (1) by using coupled immutable point type in a metric space with partially ordered set in the case of an existing mixed monotone mapping except Fazli and Nieto [9]. In a metric space with partially ordered set, the coupled immutable point type in the case of an existing mixed monotone mapping was established at first by Bhaskar and Lakshmikantham [10] and extended by many authors, for instance, [11][12][13]. The prominence of this process rises from the reality that it is a deductive process that accords convergent sequences to the unique solution of our case ( (1) and (2)).
The Brownian motion exceedingly draws through the Langevin equation when the random fluctuation force is submitted to be white noise. If the random oscillation force is not white noise, the object motion is depicted by the generalized Langevin equation [14]. Overall, the ordinary differential equations cannot precisely characterize experimental data and area measurement; as an alternative approach, fractional-order differential equation models are extremely used today [15][16][17].
The generalized Langevin equation is a substantial differential equation in applied mathematics, physics, and other areas of science and engineering. It has been developed and presented by Mainradi and Pironi [18]. With multipoint and multistrip boundary conditions, [19][20][21] investigated some properties and results to the solution of fractional Langevin equation. The uniqueness of solution and other properties for boundary value problems of the generalized Langevin equation have drawn a plentiful attention from diversified contributors within the previous decades, check for epitome [22][23][24] and the spacious roster of references presented therein. Analytical expressions of the correlation functions have been obtained using the two fluctuationdissipation theorems and fractional calculus approaches.
It is worth mentioning that the Langevin equation is extremely applied to characterize the development of physical phenomena in fluctuating environments. However, for the systems in complex media, the ordinary order to the Langevin equation does not give the true depiction of the dynamics. One of the most important possible generalizations to the Langevin equation is by replacing the derivative of positive integer order by a derivative of fractional order which gives rise to a fractional Langevin equation (see [25] and the references therein).
The Hyers-Ulam-Rassias along with Hyers-Ulam (HU) stability results for fractional Langevin equation has been studied in [26]. An explicit solution to nonhomogeneous fractional delayed Langevin equations has been given in [27]. The stochastic nonlinear fractional Langevin equation with a multiplicative noise has been studied by [28]. The hybrid Sturm-Liouville-Langevin equations with new versions of Caputo fractional derivatives have been investigated in [29].

Preliminaries
The authors in [10] inserted the next basic connotations of coupled immutable point type and mixed monotone mapping.
Let ≤ be a partial order relation on a nonempty set S which is reflexive, antisymmetric, and transitive. We refer the pair ðS, ≤Þ to a partially ordered set. Let s be an element in S, and then s is an upper bound (lower bound) for a subset U ∈ S if u ≤ s (s ≤ u) for each u ∈ U. If there are lower and upper bounds for S, then ðS, ≤Þ is called bounded partially ordered set. Let r and s be two elements in ðS, ≤Þ, and then r and s are said to be comparable if either s ≤ r or r ≤ s (or both, in case of r = s).
Let us recall the next definitions related to coupled immutable point type and mixed monotone mapping: Definition 1. Consider that ðS, ≤Þ is partially ordered and mapping P : S × S ⟶ S. It is called that the mapping P has mixed monotone property if P ðp, qÞ is nonincreasing with respect to q and nondecreasing with respect to p for each p and q in S, and this means that Definition 2. Let ðr, sÞ ∈ S × S, and then ðr, sÞ is called coupled fixed point of P : S × S ⟶ S if the two relations P ðr, sÞ = r and P ðs, rÞ = s are satisfied.
The next coupled immutable point types represent the fundamental outcomes of the contribution [10]. Theorem 3. Consider the partially ordered ðS, ≤Þ. Postulate that ðS, dÞ is a complete metric space. Assume that P : S × S ⟶ S is a continuous mapping that has mixed monotone property on S. Let ρ ∈ ½0, 1Þ such that If r 0 , s 0 ∈ X satisfy the inequalities, Assume either (i) P is a continuous mapping or (ii) The set S satisfies at least one of the following properties:

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(a) If the sequence r n ⟶ r is a nondecreasing, then r n ≤ r for all n (b) If the sequence s n ⟶ s is a nonincreasing, then s ≤ s n for all n Then, there are r, s ∈ S that satisfy the equalities Let us introduce the next partial order on the space Theorem 4 (addendum to the presumptions of Theorem 3).

Theorem 5 (addendum to the presumptions of Theorem 3).
Assume that each pair of elements of S has a lower or an upper bound in S. Then, r * = y * . Furthermore, where Next, let us render sundry famous definitions and identities for fractional calculus. For additional specifics, check [30,31].

Definition 6. A generalized integral for Riemann-Liouville has the integral form
where Γð·Þ is Gamma function and h ∈ Cð½0,∞ÞÞ, provided that the integral exists.
Definition 7. Suppose m ∈ ℕ and ι are positively real with m − 1 < ι ≤ m, the Caputo derivative of h ∈ C m ð½0,∞ÞÞ has the integral form provided that the integral exists. We remark that c D ι c = 0 where c is constant.
If h ∈ C m ð½0,∞ÞÞ, then we have Lemma 9. Let n be positive integer and n − 1 < ι ≤ n. Then, we have Lemma 10. The generalized Langevin equation (1) with the conditions in (2) has a unique representation of the solution xðtÞ if and only if the function xðtÞ is a solution of the integral equation Proof. In view of Lemmas 8 and 9, we can find x Inserting the condition xð0Þ = 0 in (17) gives c 2 = 0, and also inserting the boundary condition c Dxð0Þ = 0 in (16) gives c 0 = 0. Using the third boundary condition in (2) gives

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Substituting these values of c 0 , c 1 , and c 2 in the equation (17), we acquire the desirable results. Conversely, by aiding of the third identity of Lemma 8 and the second identity of Lemma 9, one can see that the solution xðtÞ satisfies the fractional differential equation (16). Also, it is easy to see that the unique solution xðtÞ satisfies all boundary conditions in (17).

Main Results
Presume that ðS, ≤Þ is partially ordered where S = Cð½0, 1, ℝÞ; it is evident that ðS, dÞ is a complete metric space of all continuous functions endowed with the distance Distinctly, if the sequence fr m g m∈ℕ is a nondecreasing in S and converges to r ∈ S and the sequence fs m g m∈ℕ is a nonincreasing in S and converges to s ∈ S, it follows that r m ≤ r and s ≤ s m , for all m ∈ ℕ.
Define the space ðS × S, dÞ, and then, it is a complete metric space endowed with the distance Furthermore, the set ðS × S,≤Þ is partially ordered if we acquaint the next ordered relation in S For any r, s ∈ S, the functions min fr, sg and max fr, sg are also in S and are lower bound and upper bound of r and s, respectively. Therefore, for every ðr, sÞ, ðp, qÞ ∈ S × S, there exists a ðmax fr, pg, min fs, qgÞ ∈ S × S that is comparable to ðr, sÞ and ðp, qÞ.
Previously starting and showing the fundamental outcomes, we insert the next presumptions: Assume that (H 1 ) The function f : ½0, 1 × ℝ ⟶ ℝ is a jointly continuous (H 2 ) The function f satisfies where Ł > 0 is the Lipschitz constant For the sake of computational convenience, we set Our mainly investigation is based on the sign of the value of λ ∈ ℝ, so we introduce the results in two ways when λ ≥ 0 and when λ < 0 as in the following two subsections.
3.1. In the Case of λ ≥ 0. Consider the following two operators: Definition 11. An element ðx 0 , y 0 Þ ∈ X × X is called a coupled lower and upper solution of the boundary value problems (1) and (2) if Theorem 12. Through the accompanying presumptions (H 1 ) and (H 2 ), if the problems (1) and (2) have a coupled upper and lower solutions and Q < 1 where Q = max fQ , 1, Q 2 g and Q 1 and Q 2 are defined as in (23) and (24), respectively, then it has a unique solution in S.

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where F 1 and F 2 are defined as in (27) and (28), respectively. The continuity of the operator F comes according to the assumption (H 1 ), so F ∈ S and it is well defined. Now, let x 1 , x 2 ∈ S such that x 1 ≤ x 2 , and then for each t ∈ ½0, 1 and aiding of the assumption (H 2 ), we have Thus, with fix y ∈ S, we find that which leads to Fðx 1 , yÞ ≤ Fðx 2 , yÞ, and thus, Fðx, yÞ is monotonously nondecreasing in x. Again, let y 1 , y 2 ∈ S such that y 2 ≤ y 1 , and then for each t ∈ ½0, 1 and aiding of the assumption (H 2 ), we have Thus, with fix x ∈ S, we find that which leads to Fðx, y 1 Þ ≤ Fðx, y 2 Þ, and thus, Fðx, yÞ is monotonously nonincreasing in y. Therefore, Fðx, yÞ has the mixed monotone property.
For each x, u ∈ S with u ≤ x and t ∈ ½0, 1, we have For each y, v ∈ S with y ≤ v and t ∈ ½0, 1, we have Therefore, for ðx, yÞ, ðu, vÞ ∈ S × S, we have Now, we have the partially ordered ðS × S,≤Þ and the continuous operator Fðx, yÞ which has mixed monotone property. Thus, we emphasize that there exists a coupled lower and upper solution ðx 0 ðtÞ, y 0 ðtÞÞ for the problems (1) and (2) such that x 0 ðtÞ ≤ Fðx 0 , y 0 Þ and Fðy 0 , x 0 Þ ≤ y 0 ðtÞ. So, for every ðx, yÞ, ðu, vÞ ∈ S × S, there exists a ðx 0 ðtÞ, y 0 ðtÞÞ that is comparable to ðx, yÞ and ðu, vÞ. These mean that all the assumptions of Theorems 3 and 4 are satisfied. Therefore, F has a unique coupled fixed point in S × S; 5 Advances in Mathematical Physics that is, the boundary value problems (1) and (2) have the unique solution ðx * , y * Þ ∈ ½0, 1 × ½0, 1. This coupled solution, according to Theorem 5, can be obtained as This ends the proof.

3.2.
In the Case of λ < 0. Consider the following two operators: Definition 13. The pair ðx 0 , y 0 Þ ∈ S × S is called a coupled lower and upper solution of the boundary value problems (1) and (2) if Theorem 14. Through the accompanying presumptions ðH 1 Þ and ðH 2 Þ, if the problems (1) and (2) have coupled lower and upper solutions and Q < 1 where Q = max fQ 1 , Q 2 g and Q 1 and Q 2 are defined as in (25) and (26), respectively, then it has a unique solution in S.

Proof. Consider the operator
where F 1 and F 2 are defined as in (23) and (24), respectively. The remnant of proof is identical to the proof of the former theorem.

Motion of an Immersed Plate
Consider now the rigid plate of mass m immersed in a Newtonian fluid of infinite extent and connected by a massless   Advances in Mathematical Physics spring of stiffness K to a fixed point. We assume that the small motions of the spring do not disturb the fluid and that the area of the plate, A, is sufficiently large as to produce in the fluid adjacent to the plate the velocity field and stresses developed in the preceding section. For the previous problem, Bagley and Torvik [32] have found the differential equation describing the displacement xðtÞ of the plate to be where D 3/2 = D 1/2 x ′ ðtÞ is the R-L fractional derivative of order 3/2, ρ is the fluid density, and ν is the viscosity. The equation above was then known as the Bagley and Torvik equation. Thus, the fractional derivative is established to become visible in the differential equation which depicts the motion of a simple, physical system depending on familiar mechanical and fluid components. Furthermore, its existence may be anticipated in any system distinguished by localized motion in a viscous fluid. Such is the case for oscillations of a polymeric material. Bagley and Torvik in their work [32] believe this accounts for the success of a fractional derivative in modeling these materials.
Let the rigid plate of mass m = 40 kg and area A = 0:2 m 2 immersed in the Newtonian fluid, e.g., water with the values of viscosity and density ν = 0:6527 mPa:s and ρ = 992:2 kg/ m 3 at 40°C and connected by a massless spring of stiffness K = 10 N/m to a fixed point which implies that λ = 2 A ffiffiffiffiffi ffi νρ p /m~0:254482.
By carrying out Mathematica 11 software, it is easy to compute Q 1~0 :98855 and Q 2~0 :818395 where Q 1 and Q 2 are defined as in (23) and (24), respectively. According to Theorem 12, since λ > 0, we have to choose Q = Q 10 :98855 = <1 which implies that there is a unique solution of our problem (43). Now, we are seeking to compute the exact solution. For this, apply the Laplace transform to the problem (43) to get