Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-018-0625-x

On Ulam’s Stability for a Coupled Systems of Nonlinear Implicit Fractional Differential Equations Zeeshan Ali1 · Akbar Zada1 · Kamal Shah2

Received: 29 November 2017 / Revised: 31 March 2018 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Abstract In this manuscript, we study the existence, uniqueness and various kinds of Ulam stability including Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability and generalized Ulam–Hyers–Rassias stability of the solutions to a nonlinear coupled systems of implicit fractional differential equations involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach contraction principle and Leray–Schauder of cone type. For stability, we utilize classical functional analysis. Also, an example is given to demonstrate our main theoretical results. Keywords Caputo derivative · Fractional-order differential equation · Coupled system · Green function · Boundary conditions · Ulam stability Mathematics Subject Classification 34A08 · 34B15 · 34B27

Communicated by Norhashidah Hj. Mohd. Ali.

B

Kamal Shah [email protected] Zeeshan Ali [email protected] Akbar Zada [email protected]

1

Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan

2

Department of Mathematics, University of Malakand, Lower Dir, Khyber Pakhtunkhwa, Pakistan

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1 Introduction In previous few decades, fractional differential equations turned out to be the best region of interest for the researchers due to high accuracy and usability in numerous subject of science and technology. A lot of physical and natural phenomena can be modeled through fractional differential equations, which give best results than integerorder differential equations. Due to this, fractional differential equations are counted as a special tool for molding. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, electrochemistry, aerodynamics, viscoelasticity, polymer rheology, economics, biology, electrodynamics of complex medium, etc. For details, see [5,14,21,22,26–29] and references therein. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, the subject of aforesaid equations is gaining great importance and attention from the researchers. Also, the enriched material on theoretical aspects and analytic/numerical methods for solving fractional-order models attracts the modelers. As compared to ordinary differential equations, fractional-order differential equations can more accurately describe various process and phenomenons of applied sciences. In the most recent years, many researchers have focused on the existence of solution to the aforesaid differential equations, for instance see [2,4,12,33] and references therein. On the other hand, the study of coupled systems involving fractional differential equations is also important. Because, such systems occur in various problems of applied nature like biology, chemistry and physics can be modeled in the form of systems of aforesaid equations, for instance see [4,34–36] and references cited therein. Another aspect of research, which has been exclusively studied for integer-order differential equations and got much attraction from the researchers is Ulam stability and their various types. The above-mentioned stability was first introduced by Ulam [39], in 1940 and then explained by Hyers [15] in the following year. The Hyers results are extended and generalized by many researchers for integer-order differential equations. Plenty of best manuscripts can be found in the literature, and few of them are [16,18,19,24,25,31,32,38,43–45] and references cited therein. The foregoing stabilities [9] for fractional differential equations are quite significant in realistic problems, numerical analysis, biology and economics. For details, see [1,6,8,10,11,17,23,40– 42] and references therein. Recently, to the best of our knowledge only few of researchers devoted their research work to the study of various kind of Ulam stabilities for coupled system of fractional differential equations. Because, the above-mentioned stability has not properly studied for aforesaid systems and very few papers can be found in the literature. For details, we refer the reader to see [7,20,37]. Ali et al. [7], studied the Ulam–Hyers stability of the following

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⎧c p ⎪ ⎪ D y(t) − f (t, z(t)) = 0; t ∈ [0, 1], ⎪ ⎪ c q ⎪ D z(t) − g(t, y(t)) = 0; t ∈ [0, 1], ⎪ ⎪ ⎪  T ⎨   1 γ   y(t) = 0, y(t) = I h(y) = (T − ς )γ −1 h(y(ς ))dς, 0 T t=0 t=1 ⎪ (γ ) ⎪ 0 ⎪ ⎪  T ⎪ ⎪   1 ⎪ δ ⎪ ⎩ z(t)t=0 = 0, z(t)t=1 = 0 IT j (z) = (T − ς )δ−1 j (z(ς ))dς, (δ) 0 where p, q, γ , δ ∈ (1, 2], h, j ∈ L[0, 1] are boundary functions and f, g : [0, 1] × R → R. Khan et al. [20], studied the Ulam–Hyers stability of the following ⎧ p D y(t) − f (t, y(t), z(t)) = 0; t ∈ (0, 1), ⎪ ⎪ ⎪ q ⎪ ⎨ D z(t) − g(t, y(t), z(t)) = 0; t ∈ (0, 1),      ⎪ y(t)t=0 = y  (t)t=0 = y  (t)t=0 = · · · = y n−2 (t)t=0 = y(t)t=1 = 0, ⎪ ⎪ ⎪      ⎩ z(t)t=0 = z  (t)t=0 = z  (t)t=0 = · · · = z n−2 (t)t=0 = z(t)t=1 = 0, where f, g : [0, 1] × R × R → R are continues and n − 1 < p, q ≤ n, n ≥ 2. Shah and Tunc [37] studied the Ulam–Hyers stability of the following ⎧ p D y(t) − f (t, y(t), z(t)) = 0; t ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ ⎨ Dq z(t) − g(t, y(t), z(t)) = 0; t ∈ [0, 1],    ⎪ I 3− p y(t)t=0 = D p−2 y(t)t=0 = 0, ; y(t)t=1 = 0, ⎪ ⎪ ⎪    ⎩ 3−q I z(t)t=0 = Dq−2 z(t)t=0 = 0, ; z(t)t=1 = 0, where f, g : [0, 1] × R × R → R are continues and p, q ∈ (2, 3]. In this manuscript, we investigate the existence, uniqueness as well as different kinds of Ulam stability for the considered coupled system involving Caputo derivative as given by ⎧c p D y(t) − f (t, z(t),c D p y(t)) = 0; p ∈ (2, 3]; t ∈ J, ⎪ ⎪ ⎪ ⎪ c ⎨ Dq z(t) − g(t, y(t),c Dq z(t)) = 0; q ∈ (2, 3]; t ∈ J,    ⎪ y  (t)t=0 = y  (t)t=0 = 0, y(t)t=1 = λy(η), λ, η ∈ (0, 1), ⎪ ⎪   ⎪ ⎩   z (t) t=0 = z  (t)t=0 = 0, z(t)t=1 = λz(η), λ, η ∈ (0, 1),

(1)

where J = [0, 1] and f, g : J × R × R → R are continuous functions. The manuscript is structured as follows. In Sect. 2, we give some definitions, theorems, lemmas and remarks. In Sect. 3, we built up some appropriate conditions for the existence and uniqueness of solutions to the considered problem (1) using fixed point theorems of Leray–Schauder and Banach contraction type. In Sect. 4, we set up applicable results under which the solution of the considered boundary value problem (1) fulfills the conditions of different kinds of Ulam stability. The established results are illustrate by an example in Sect. 5.

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2 Background Materials and Auxiliary Results This section is devoted to some basic definitions, theorems, lemmas and remarks which are useful in existence and stability results.  Definition 2.1 [21] For a function y ∈ C (0, ∞), R , the Caputo derivative of fractional-order p ∈ R+ is defined as  t 1 c p D y(t) = (t − ς )n− p−1 y (n) (ς )dς, n = [ p] + 1, (n − p) 0 where [ p] denotes the integer part of p. Definition 2.2 [21] The fractional integral of order p ∈ R+ of function y ∈ L 1 (J, R+ ) is given by  t 1 I p y(t) = (t − ς ) p−1 y(ς )dς, ( p) 0 where



∞

( p) =

t p−1 e−t dt, p > 0.

0

Lemma 2.3 [21] Let p > 0, then the differential equations c

D p y(t) = 0

has solution given by y(t) = C0 + C1 t + C2 t 2 + · · · + Cn−1 t n−1 , Ci ∈ R, i = 0, 1, 2, . . . , n − 1, where n = [ p] + 1. Lemma 2.4 [21] Let p > 0, then the solution of the differential equation c

D p y(t) = h¯ (t)

will be in the following form

 I p D p y(t) = I p h¯ (t) + C0 + C1 t + C2 t 2 + · · · + Cn−1 t n−1 , Ci ∈ R, i = 0, 1, 2, . . . , n − 1, where n = [ p] + 1. Theorem 2.5 [13](Leray–Schauder of cone type) Consider a Banach space B together with cone C ⊂ B and if S ⊂ C is relatively open set with 0 ∈ S. Let T : S → S be a completely continuous operator. Then, one of the given condition exists: (1) The operator T has a fixed point in S; (2) There exist y ∈ ∂S and μ ∈ (0, 1) such that y = μTy.

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Lemma 2.6 The space B  =  y(t)| y ∈ C(J) is a Banach space under the defined norm y B = maxt∈J  y(t). Similarly, the norm on product space is defined by (y, z) B ×B = y B + z B . Obviously B × B, (y, z) B ×B is a Banach space. Also, we define the cone C ⊂ B × B by

C = (y, z) ∈ B × B : y(t) ≥ 0, z(t) ≥ 0 . Ulam Stability We adopt the following definitions from [30]. Definition 2.7 Our considered system (1) is said to be Ulam–Hyers stable, if there exist K p,q = (K p , Kq ) > 0 such that for some = ( p , q ) > 0 and for every solution (y, z) ∈ B × B of the inequality  c p   D y(t) − f (t, z(t),c D p y(t)) ≤ p , t ∈ J, (2) c q   D z(t) − g(t, y(t),c Dq z(t)) ≤ q , t ∈ J. There exists a unique solution (ϑ, χ ) ∈ B × B with   (y, z)(t) − (ϑ, χ )(t) ≤ K p,q , t ∈ J.

(3)

Definition 2.8 Our considered system (1) is said to be generalized Ulam–Hyers stable, if there exist ∈ C(R+ , R+ ) with (0) = 0, such that for every solution (y, z) ∈ B × B of the inequality (2), there exist a unique solution (χ , ϑ) ∈ B × B of (1) which satisfies   (y, z)(t) − (ϑ, χ )(t) ≤ ( ), t ∈ J. (4) Definition 2.9 The proposed coupled system (1) is said to be Ulam–Hyers–Rassias stable with respect to p,q = ( p , q ) ∈ C(J, R), if there exist constants K p , q = (K p , K q ) > 0 such that for some = ( p , q ) > 0 and for every solution (y, z) ∈ B × B of the inequality  c p   D y(t) − f (t, z(t),c D p y(t)) ≤ p (t) p , t ∈ J, (5) c q   D z(t) − g(t, y(t),c Dq z(t)) ≤ q (t) q , t ∈ J. There exists a unique solution (ϑ, χ ) ∈ B × B with   (y, z)(t) − (ϑ, χ )(t) ≤ K , p,q , t ∈ J. p q

(6)

Definition 2.10 The considered system (1) is said to be generalized Ulam–Hyers– Rassias stable with respect to p,q = ( p , q ) ∈ C(J, R), there exist constants K p , q = (K p , K q ) > 0, such that for every solution (y, z) ∈ B × B of the inequality (5), there exist a unique solution (χ , ϑ) ∈ B × B of (1) which satisfies   (y, z)(t) − (ϑ, χ )(t) ≤ K , p,q (t), t ∈ J. (7) p q

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Remark 2.11 We say that (y, z) ∈ B × B is a solution of the system of inequality (2) if there exist functions ϕ, ψ ∈ C(J, R) which depend upon y, z, respectively, such that    (h 1 ) ϕ(t) ≤ p , big|ψ(t) ≤ q , t ∈ J; (h 2 ) and  c p D y(t) = f (t, z(t),c D p y(t)) + ϕ(t), t ∈ J, c q D z(t) = g(t, y(t),c Dq z(t)) + ψ(t), t ∈ J.

3 Main Results Theorem 3.1 For h¯ ∈ C(J, R), the following linear boundary value problem c

D p y(t) − h¯ (t) = 0, t ∈ J, 2 < p ≤ 3,    y  (t)t=0 = y  (t)t=0 = 0, y(t)t=1 = λy(η), λ, η ∈ (0, 1),

has a solution of the form



1

y(t) = 0

G p (t, ς )h¯ (ς )dς,

where G p (t, ς ) is the Green’s function given by G p (t, ς)

=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

1 ( p) ⎪ ⎪ ⎪ ⎪ ⎩

−1 p−1 + (t − ς) p−1 ,  (1 − ς) −1 p−1 ,  (1 − ς) λ p−1 − 1 (1 − ς) p−1 + (t  (η − ς) λ p−1 − 1 (1 − ς) p−1 ,  (η − ς) 

0 ≤ η ≤ ς ≤ t ≤ 1, 0 ≤ η ≤ t ≤ ς ≤ 1, − ς) p−1 , 0 ≤ ς ≤ t ≤ η ≤ 1,

(8)

0≤t ≤ς ≤η≤1

Proof By Lemma 2.4, we have y(t) = C0 + C1 t + C2 t 2 + I p h¯ (t).

(9)

Using condition y  (0) = y  (0) = 0, we get C1 = C2 = 0. Further by using boundary condition y(1) = λy(η), we get C0 =

 1 p λI h¯ (η) − I p h¯ (1) . 

Plugging the values of C0 , C1 and C2 in (9), we have  1 p λI h¯ (η) − I p h¯ (1) + I p h¯ (t)     η  1 1 λ 1 (η − ς ) p−1 h¯ (ς )dς − (1 − ς ) p−1 h¯ (ς )dς =  ( p) 0 ( p) 0  t 1 (10) (t − ς ) p−1 h¯ (ς )dς. + ( p) 0

y(t) =

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Therefore, we can write (10) as 

1

y(t) = 0

G p (t, ς )h¯ (ς )dς,

(11)

where G p (t, ς ) is the Green’s function given in (8).



In view of Theorem 3.1, an equivalent system of Fredholm integral equations to the proposed system (1) is given by ⎧  ⎪ ⎪ ⎪ ⎨ y(t) = ⎪ ⎪ ⎪ ⎩ z(t) =

1

0  1

G p (t, ς ) f (t, z(t),c D p y(t))(ς )dς, (12) Gq (t, ς )g(t, y(t), D z(t))(ς )dς. c

q

0

We use the following notions for convince u(t) = f (t, z(t),c D p y(t)) = f (t, z(t), u(t)) v(t) = g(t, y(t),c Dq z(t)) = g(t, y(t), v(t)). Hence,(12) can be written as ⎧  1 ⎪ ⎪ ⎪ y(t) = G p (t, ς )u(ς )dς, ⎨ 0  1 ⎪ ⎪ ⎪ Gq (t, ς )v(ς )dς, ⎩ z(t) =

(13)

0

where u, v ∈ B satisfies the implicit functional equations and Gq (t, ς ) is the Green’s function given by ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

1 Gq (t, ς) = (q) ⎪ ⎪ ⎪ ⎪ ⎩

−1 q−1 + (t − ς)q−1 ,  (1 − ς) −1 q−1 ,  (1 − ς) λ q−1 − 1 (1 − ς)q−1 + (t  (η − ς) λ q−1 − 1 (1 − ς)q−1 ,  (η − ς)

0 ≤ η ≤ ς ≤ t ≤ 1, 0 ≤ η ≤ t ≤ ς ≤ 1, − ς)q−1 ,

0 ≤ ς ≤ t ≤ η ≤ 1,

(14)

0 ≤ t ≤ ς ≤ η ≤ 1,

where  = 1 − λ.

 Lemma 3.2 We use G(t, ς ) = G p (t, ς ), Gq (t, ς ) for the Green’s function of our considered problem (1), which satisfy the following properties: (i) G(t, ς ) is continuous over J × J for all t, ς ∈ J;     p−1 q−1 = G p (1, ς ), maxt∈J Gq (t, ς ) = (1−ς) (ii) maxt∈J G p (t, ς ) = (1−ς) ( p) (q) = Gq (1, ς ), var sigma ∈ J;   1 1 1 (iii) maxt∈J 0 G p (t, ς )dς ≤ (1p+1) , maxt∈J 0 Gq (t, ς )dς ≤ (q+1) , ς ∈ J.

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Proof (i) and (ii) are easy to prove, here we leave it. (iii) : We omit the proof, because it is similar to the proof of Lemma 3.1 in [8]. Further, the proofs of  t∈J

1 . ( p + 1)

 Gq (t, ς )dς ≤

1 . (q + 1)

0

and



1

max t∈J

 G p (t, ς )dς ≤

1

max

0



are obvious.

Now, to transform the problem (1) into a fixed point problem, let define the operator T : B × B → B × B as ⎛ ⎞ 1  ⎜ G p (t, ς ) f (t, z(ς ), u(ς ))dς ⎟  T p (z, u)(t) ⎜0 ⎟ T(y, z)(t) = ⎜ 1 . (15) ⎟= Tq (y, v)(t) ⎝ ⎠ Gq (t, ς )g(t, y(ς ), v(ς ))dς 0

Then, the fixed point of the operator T coincides with the solution of coupled system (1). Assume that the following assumptions are hold: (A1 ) 

1

A1 =

G p (1, ς )α(ς )dς, B1 =

Kf + Lf <1 ( p + 1)

Gq (1, ς )a(ς )dς, B2 =

Kg + Lg < 1. (q + 1)

0

and  A2 = 0

1

(A2 ) For all z, u, h, m ∈ R and for each t ∈ J there exist constants K f > 0, 0 < L f < 1, such that        f (t, z, u) − f (t, h, m) ≤ K f z − h  + L f u − m . ˜ m˜ ∈ R and for each t ∈ J, there exist constants Kg > 0, Similarly, for all y, v, h, 0 < Lg < 1, such that       g(t, y, v) − g(t, h, ˜ m) ˜  ≤ Kg  y − h˜  + Lg v − m˜ . Theorem 3.3 If f, g : J × R × R → R and (A2 ) hold. Then, the operator T : C → C defined in (15) is completely continuous.

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Proof Since f, g, G p (t, ς ) and Gq (t, ς ) are continuous, so the operator T is also continuous for all (y, z) ∈ C. Let us define a bounded set  B of C. Suppose α =   f (t, 0, 0) and a = g(t, 0, 0) with α ∗ = maxt∈[0,1]  f (t, 0, 0) and a ∗ = maxt∈[0,1] g(t, 0, 0). Then, for every y ∈ B and |t| ≤ 1, we have     T p y(t) =  

1

0 1



  G p (t, ς )u(ς )dς 

  G p (t, ς )u(ς )dς.

≤

(16)

0

Now by (A2 ) with z ≤ ξ p , we have     u(t) =  f (t, z(t), u(t))     ≤  f (t, z(t), u(t)) − f (t, 0, 0) +  f (t, 0, 0)       ≤ K f z − 0 + L f u − 0 +  f (t, 0, 0) ≤ α ∗ + K f ξ p + L f u B . Therefore, we get u B ≤

α∗ + K f ξ p = Np. 1 − Lf

(17)

Now by using (iii) of Lemma 3.2 and equation (17) in equation (16), we obtain the result given by Np . (18) T p (y) B ≤ ( p + 1)   Using a ∗ = maxt∈[0,1] g(t, 0, 0) and repeating the same fashion, we obtain the result given by Mq Tq (z) B ≤ , (19) (q + 1) where Mq =

a ∗ + Kg ξq 1 − Lg

with y ≤ ξq . Thus from the relations (18) and (19), we get p (y) B

+ Tq (z) B ≤

Mq Np + = M, ( p + 1) (q + 1)

which further yields the following result T(y, z) B ×B ≤ M .

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Hence, T is uniformly bounded. Next, we prove that T is equicontinuous. Let t < τ ∈ J and suppose   T p (y)(τ ) − T p (y)(t) ≤



  G p (t, ς ) − G p (τ, ς )u(ς )dς

1 0

 1 Np p−1 p−1 ≤ (t −τ ) (1 − ς ) p−1 dς ( p) 0  τ

 Np + (τ − ς ) p−1 − (t − ς ) p−1 dς ( p) 0   τ p−1 (τ − ς ) dς + t  Np ≤ (t p−1 − τ p−1 ) + t p − τ p ( p + 1)  p p (20) − (t − τ ) + (τ − t) . In same way, we can show that

 Mq (t q−1 − τ q−1 ) + t q − τ q (q + 1)  q q − (t − τ ) + (τ − t) .

  Tq (z)(τ ) − Tq (z)(t) ≤

(21)

The right-hand sides of (20) and (21) tend to zero, when t → τ. So by Arzela–Ascoli theorem, we conclude that T is equicontinuous and hence T is uniformly equicontinuous. Also, it is easy to show that T(B) ⊂ B. Thus, the operator T is completely continuous. 

Theorem 3.4 Suppose the assumption (A2 ) hold. Then, the considered system (1) has a unique solution if Kg Kf + < 1. (1 − L f )( p + 1) (1 − Lg )(q + 1)

(22)

Proof Let y, y ∈ C and consider

    1 

  T p (y)(t) − T p (y)(t) =  G p (t, ς ) u(ς ) − u(ς ) dς   0  1    G p (t, ς )u(ς ) − u(ς )dς, ≤ 0

where u(t) = f (t, z(t), u(t)), u(t) = f (t, z(t), u(t)).

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Now by using assumption (A2 ), we have     u(t) − u(t) =  f (t, z(t), u(t)) − f (t, z(t), u(t))     ≤ K f z(t) − z(t) + L f u(t) − u(t), which further gives

  u(t) − u(t) ≤

 K f  z(t) − z(t). 1 − Lf

(24)

Put (24) in (23) and taking maximum of both side over J, we obtain  Kf z − z B . (1 − L f )( p + 1)

(25)

 Kg y − y B . (1 − Lg )(q + 1)

(26)

 T p (y) − T p (y) |B ≤ On the similar way, we can obtain  Tq (z) − Tq (z) |B ≤

Therefore from (25) and (26), we get the result given as T(y, z) − T(y, z) B ×B   Kg Kf (y, z) − (y, z) B ×B . ≤ + (1 − L f )( p + 1) (1 − Lg )(q + 1)

(27)

Thus from (27), we conclude that the operator T is contraction. Thus, it has unique fixed point, which is the unique solution of the considered system (1). 

Theorem 3.5 Under the continuity of f, g : J × R × R → R and let assumptions (A1 ) and (A2 ) hold. Then, the coupled system (1) has at least one solution. Proof Let us define a set

S = (y, z) ∈ B × B : (y, z) B ×B < r , 

 where max

2A1 2A2 1−2B1 , 1−2B2

< r. Further, the operator T : S → C as defined in (15)

is completely continuous. If (y, z) ∈ S, then by definition (y, z) B ×B < r.  T p (z, u) B ≤ max t∈J



  G p (t, ς ) f (ς, z(ς ), u(ς ))dς

1 0

  G p (t, ς ) f (ς, 0, 0)dς

1

≤ max t∈J

0



+ max t∈J

  G p (t, ς ) f (ς, z(ς ), u(ς )) − f (ς, 0, 0)dς

1

0

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1

≤

G p (1, ς )α(ς )dς + r

0

= A1 + rB1 ≤

Kf + Lf ( p + 1)

(28)

r . 2

Similarly,

r . 2

(29)

T(y, z) B ×B ≤ r.

(30)

Tq (y, v) B ≤ Hence from (28) and (29), we get

Thus from (30), we can say that T(y, z) ∈ S. Therefore, by Theorem 3.3, the operator T : S → S is completely continuous. Now, let us recall the eigenvalue problem as (y, z) = μT(y, z), μ ∈ (0, 1).

(31)

Then, in view of the solution (y, z) of (31) for μ ∈ (0, 1), we get y B = μT(z, u) B  1    G p (t, ς ) f (ς, z(ς ), u(ς ))dς ≤ max t∈J

0



  G p (t, ς ) f (ς, 0, 0)dς

1

≤ max t∈J

0



  G p (t, ς ) f (ς, z(ς ), u(ς )) − f (ς, 0, 0)dς

1

+ max  ≤

t∈J 1

0

G p (1, ς )α(ς )dς + r

0

Kf + Lf r = A1 + rB1 ≤ . ( p + 1) 2

Similarly, z B ≤

r . 2

(32)

(33)

Hence from (32) and (33), we get (y, z) B ×B ≤ r.

(34)

From the relation (34), we get (y, z) ∈ / ∂S. Therefore, in view of Theorem 2.5, T has at least one fixed point which lies in S. Consequently, the coupled system (1) has at least one solution. 

4 Ulam Stability Analysis The current section is devoted to the stability analysis of the considered system (1).

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Lemma 4.1 Suppose (y, z) ∈ B × B be the solution of the inequality (2), then we have the system of inequalities given as ⎧   1   ⎪ c p ⎪  y(t) −  ≤ K p p , t ∈ J, ⎪ G (t, ς ) f (ς, z(ς ), D y(ς ))dς p ⎨  0    1 ⎪   ⎪ ⎪ Gq (t, ς )g(ς, y(ς ),c Dq z(ς ))dς  ≤ Kq q , t ∈ J. ⎩ z(t) − 0

Proof By using (h 2 ) of Remark 2.11, we have ⎧c p c p ⎪ ⎪ D y(t) = f (t, z(t), D y(t)) + ϕ(t), t ∈ J, ⎪ ⎪ c q c ⎨ D z(t) = g(t, y(t), Dq z(t)) + ψ(t), t ∈ J,    ⎪ y  (t)t=0 = y  (t)t=0 = 0, y(t)t=1 = λy(η), λ, η ∈ (0, 1), ⎪ ⎪   ⎪ ⎩ z  (t) = z  (t) = 0, z(t) = λz(η), λ, η ∈ (0, 1). t=0

t=0

(35)

t=1

Now in view of Lemma 2.4, the solution of (35) is given by ⎧  1  1 ⎪ c p ⎪ ⎪ y(t) = G (t, ς ) f (ς, z(ς ), D y(ς ))dς + G p (t, ς )ϕ(ς )dς, t ∈ J, p ⎨ 0 0  1  1 ⎪ ⎪ c q ⎪ Gq (t, ς )g(ς, y(ς ), D z(ς ))dς + Gq (t, ς )ψ(ς )dς, t ∈ J. ⎩ z(t) = 0

0

(36)

From first equation of the system (36), we have     y(t) + 

1

0

    G p (t, ς ) f (ς, z(ς ), D y(ς ))dς  =  c

  G p (t, ς )ϕ(ς )dς  0  1    G p (t, ς )ϕ(ς )dς. ≤ max

p

t∈J

1

0

Using (h 1 ) of Remark 2.11 and by (iii) of Lemma 3.2, we get     y(t) + 

0

1

  G p (t, ς ) f (ς, z(ς ),c D p y(ς ))dς  ≤

p = Kp p, ( p + 1)

(37)

where K p = (1p+1) . Repeating the same procedure for second equation of the system (36), we have    z(t) + 

0

where Kq =

1 (q+1) .

1

  Gq (t, ς )g(ς, y(ς ), D z(ς ))dς  ≤ Kq q , c

q

(38) 

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Theorem 4.2 Under the assumption (A2 ) and if =1−

K p K f Kq Kg >0 (1 − L f )(1 − Lg )

(39)

holds, then the coupled system (1) is Ulam–Hyers stable and consequently generalized Ulam–Hyers stable. Proof Let (y, z) ∈ B × B be the solution of the inequalities given in (2) and (ϑ, χ ) ∈ B × B be the unique solution to the following considered system ⎧c p D ϑ(t) − f (t, χ (t),c D p ϑ(t)) = 0; p ∈ (2, 3]; t ∈ J, ⎪ ⎪ ⎪ ⎪ ⎨ c Dq χ (t) − g(t, ϑ(t),c Dq χ (t)) = 0; q ∈ (2, 3]; t ∈ J,    ⎪ ϑ  (t)t=0 = ϑ  (t)t=0 = 0, ϑ(t)t=1 = λϑ(η), λ, η ∈ (0, 1), ⎪ ⎪   ⎪ ⎩ χ  (t) = χ  (t) = 0, χ (t) = λχ (η), λ, η ∈ (0, 1). t=0

t=0

(40)

t=1

Then, in view of Lemma 2.4, the solution of (40) is provided by ⎧  1 ⎪ ⎪ ⎪ G p (t, ς ) f (ς, χ (ς ),c D p ϑ(ς ))dς, t ∈ J, ⎨ ϑ(t) = 0  1 ⎪ ⎪ ⎪ Gq (t, ς )g(ς, ϑ(ς ),c Dq χ (ς ))dς, t ∈ J. ⎩ χ (t) =

(41)

0

Consider    1     c p  y(t) − ϑ(t) =  y(t) − G p (t, ς) f (ς, χ(ς), D ϑ(ς))dς   0    1   ≤  y(t) − G p (t, ς) f (ς, z(ς),c D p y(ς))dς  0  1  +  G p (t, ς) f (ς, z(ς),c D p y(ς))dς 0   1  G p (t, ς) f (ς, χ(ς),c D p ϑ(ς))dς  − 0    1   ≤  y(t)− G p (t, ς) f (ς, z(ς),c D p y(ς))dς  + 0   1  − G p (t, ς)u ϑ (ς)dς  0  1     G p (t, ς) u(ς) − u ϑ (ς) dς, ≤ Kp p + 0

where u, u ϑ ∈ B. We use u(t) = f (t, z(t), u(t)),

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1

G p (t, ς)u(ς)dς

0

(42)

On Ulam’s Stability for a Coupled Systems of Nonlinear…

and u ϑ (t) = f (t, χ (t), u ϑ (t)). Now by (A2 ), one has     u(t) − u ϑ (t) =  f (t, z(t), u(t)) − f (t, χ (t), u ϑ (t))     ≤ K f z(t) − χ (t) + L f u(t) − u ϑ (t). Thus, we get

  u(t) − u ϑ (t) ≤

 K f  z(t) − χ (t). 1 − Lf

(43)

KpK f z − χ B 1 − Lf

(44)

Kq Kg y − ϑ B , 1 − Lg

(45)

Put (43) in (42), we obtain y − ϑ B ≤ K p p + and similarly, we have z − χ B ≤ Kq q + where v, vχ ∈ B. Also we use v(t) = g(t, y(t), v(t)), and vχ (t) = g(t, ϑ(t), vχ (t)). From (44) and (45), we write as KpK f z − χ B ≤ K p p 1 − Lf Kq Kg z − χ B − y − ϑ B ≤ Kq q 1 − Lg ⎤ ⎡ ⎤ ⎡  K p K f  y − ϑ Kp p B 1 − 1−L f ⎦≤⎣ ⎦. ⎣ Kq Kg − 1−L 1 z − χ B Kq q g y − ϑ B −

Solving the above inequality, we get ⎡ ⎣

y − ϑ B z − χ B

⎤ ⎦≤



KpK f 1  (1−L f ) Kq Kg 1 (1−Lg ) 

⎡K ⎤ p p ⎦, ⎣ Kq q

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where =1−

K p K f Kq Kg > 0. (1 − L f )(1 − Lg )

Further simplification of above system gives Kq K p K f q Kp p +  (1 − L f ) Kq K p Kg p Kq q + z − χ B ≤  (1 − Lg ) y − ϑ B ≤

from which we have y − ϑ B + z − χ B ≤

Kp p 

+

Kq q 

+

Kq K p K f q (1−L f )

+

Kq K p Kg p (1−Lg ) .

Let max p , q = , then from (4) we have (y, z) − (ϑ, χ ) B ×B ≤ K p,q ,

(46)

where  K p,q =

 Kq Kq K p K f Kq K p Kg Kp + + + .   (1 − L f ) (1 − Lg )

Further, if we can write (y, z) − (ϑ, χ ) B ×B ≤ K p,q ( ), where (0) = 0,

(47)

then the solutions of the proposed coupled system (1) are generalized Ulam–Hyers stable. 

For the next result, the given assumption is provided as: (A3 ) If there exist two increasing function p , q ∈ C(J, R+ ), then 

1

G p (t, ς )ψ f (ς )dς ≤

p

p (t), where t ∈ J ( p + 1)

Gq (t, ς )ψg (ς )dς ≤

q

q (t), where t ∈ J (q + 1)

0

and 

1 0

are satisfied.

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On Ulam’s Stability for a Coupled Systems of Nonlinear…

Corollary 4.3 Under the assumption (A3 ) and (39) and by using Definition 2.9 and 2.10, by repeating the same procedure of Lemma 4.1 and Theorem 4.2, we can prove that the proposed system (1) is Ulam–Hyers–Rassias and generalized Ulam–Hyers– Rassias stable.

5 Example Example 5.1 ⎧ c 5    z(t) ⎪ cos  D 2 y(t) ⎪ c 25 ⎪  −   D y(t) − = 0, t ∈ J, ⎪ ⎪ ⎪ 35 + t 2 35(t + 2) 1 + z(t) ⎪ ⎪ ⎪ c 5  ⎪ ⎪  D 2 z(t) ⎪ 5 1 ⎪ c ⎪ ⎨ D 2 z(t) − t cos y(t) − y(t) sin(t) − c 5  = 0, t ∈ J, 50 25 +  D 2 z(t) ⎪   ⎪ ⎪ 2 1 ⎪   ⎪ y y , (0) = y (0) = 0, y(1) = ⎪ ⎪ ⎪ 3 5 ⎪ ⎪   ⎪ ⎪ 2 1 ⎪ ⎪ ⎩ z  (0) = z  (0) = 0, z(1) = z . 3 5

(48)

From coupled systems (48), we have p = q = 25 , λ = 23 , η = 15 and  = 13 . Further, 1 1 and Kg = Lg = 25 . Therefore one computation we have K f = L f = 35 Kg Kf + ≈ 0.06415 < 1. (1 − L f )( p + 1) (1 − Lg )(q + 1) Hence, the coupled system (48) has unique solution. Furthermore, condition (39) in Theorem 4.2 also satisfied. So, the coupled system (48) is Ulam–Hyers stable, generalized Ulam–Hyers stable. Also, it is easy to show that the system (48) is Ulam– Hyers–Rassias and generalized Ulam–Hyers–Rassias stable.

6 Conclusion In this manuscript, we have successfully established an existence theory for a coupled systems (1) of implicit fractional differential equation. In addition, we have developed conditions for various kinds of Ulam stability by the help of classical nonlinear functional analysis. As a conclusion we have obtained the required results for different kinds of Ulam stability including Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias and generalized Ulam–Hyers–Rassias stability of the solutions to the proposed problem (1). Also, we have provided an example for verification of the main results. Acknowledgements We are very thankful to the anonymous referees for their careful reading and suggestions which improved the quality of this paper.

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Z. Ali et al. Author contributions All authors have equal contribution in this manuscript. Compliance with Ethical Standards Conflict of interest All authors declare that they have no Conflict of interest.

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