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Coupled system of a fractional order differential equations with weighted initial conditions

  • Ahmed M. A. El-Sayed and Sheren A. Abd El-Salam EMAIL logo
Published/Copyright: December 31, 2019
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Open Mathematics
From the journal Open Mathematics Volume 17 Issue 1

Abstract

Here, a coupled system of nonlinear weighted Cauchy-type problem of a diffre-integral equations of fractional order will be considered. We study the existence of at least one integrable solution of this system by using Schauder fixed point Theorem. The continuous dependence of the uniqueness of the solution is proved.

Keywords: Riemann-Liouville differential operator; coupled system; weighted Cauchy type problem; continuous dependence
MSC 2010: 34A08; 34A12; 45D05

1 Preliminaries and introduction

The coupled system was studied by many authors (see [1] and [11]). Also, the weighted Cauchy-type problem (see [2], [3], [4], [5], [6], [7]). In [5] the author studied the problem:

Dα u(t) = f(t,u) + ∫0tg(t,s,u(s)) ds,   t > 0,t1−α u(t)|t=0 = b,  0 < α < 1, b ∈ R, (1)

such that the functions f and g satisfies the following assumptions

  1. t1−αf(t, u) is continuous on R+ × C1−α0 (R+) and

    |f(t,u)|≤tμφ(t)|u|m1,μ≥0,m1>1,

  2. s1−αg(t, s, u(s)) is continuous on DR+ × C1−α0 (R+) where

    DR+={(t,s)∈R+×R+,0≤s≤t},

    and

    |g(t,s,u(s))|≤(t−s)β−1sσψ(s)|u|m2,0<β<1,σ≥0,m2>1,

    where φ(t) and ψ(s) are such that

  3. φ(t) is continuous and tμ−(1−α)m1 φ(t) is continuous in case

    μ−(1−α)m1<0,

  4. ψ(t) is continuous and tσ−(1−α)m2 ψ(t) is continuous in case

    σ−(1−α)m2<0.

Under these assumptions the author proved the existence of at least one local solution in the space C1−αα ([0, h]), where for h > 0

C1−αα([0,h])={v∈C1−α0([0,h]):∃ c∈R and v∗∈C1−α0([0,h]) such that v(t)=ctα−1+Iαv∗(t)},

and

Cr0([0,h])={v∈C0((0,h]):limt→0+trv(t) exists and finite},

(the space C0((0, h]) is the usual space of continuous functions on [0, h]).

In comparison with earlier results, we study the coupled system of weighted Cauchy-type problems of diffre-integral equations of fractional order

Dα u(t) = f1(t,v(t),∫0tg1(t,s,v(s))ds),Dβ v(t) = f2(t,u(t),∫0tg2(t,s,u(s))ds), (2)

where t ∈ I = [0, 1] and α, β ∈ (0, 1) with the initial conditions

t1−αu(t)|t=0=k1  and  t1−βv(t)|t=0=k2,

such that the functions fi and gi, i = 1, 2 satisfy the following assumptions:

  1. fi : I × R × R → R be a function with the following properties:

    1. for each t ∈ I, fi(t, ⋅, ⋅) is continuous,

    2. for each (u, v) ∈ R × R, fi(⋅, u, v) is measurable,

    3. there exist a real function t → a(t), a ∈ L1(I) and a positive constants b1 and b2 such that

      |fi(t,u,v)| ≤ a(t) + b1 |u| + b2 |v|,  for each  t∈I, (u,v) ∈ R×R;

  2. gi : I × I × R → R be a function with the following properties:

    1. for each (t, s) ∈ I × I, gi(t, s, ⋅) is continuous,

    2. for each u ∈ R, gi(⋅, ⋅, u) is measurable,

    3. there exist a real function (t, s) → k(t, s), k ∈ L1(I) and a positive constant b3 such that

      |gi(t,s,u)| ≤ k(t,s) + b3 |u|,  for each  (t,s)∈I×I,  u∈R;

  3. b1 + b2 b3 < Γ(1 + α) and b1 + b2 b3 < Γ(1 + β).

Note that if α = β, f1 = f2, g1 = g2, k1 = k2 = b and u(t) = v(t), then problem (2) will take the form

Dα u(t) = f(t,u(t),∫0tg(t,s,u(s))ds),t1−αu(t)|t=0=b,

which is the generalization of problem (1).

2 Main results

2.1 Integral representation

Lemma 2.1

Let the assumptions (i-iii) be satisfied. If the solution of the coupled system (2) exists, then it can be represented by the coupled system of nonlinear integral equations of fractional order

u(t)=k1 tα−1 + ∫0t (t − s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ)) dθ) ds,v(t)=k2 tβ−1 + ∫0t (t − s)β−1Γ(β) f2(s,u(s),∫0sg2(s,θ,u(θ)) dθ) ds. (3)

Proof

Let u(t) be a solution of

Dα u(t) = ddtI1−αu(t) = f1(t,v(t),∫0tg1(t,s,v(s)) ds).

Integrate both sides, we get

I1−αu(t) − I1−αu(t)|t=0 = I f1(t,v(t),∫0tg1(t,s,v(s)) ds).

Operating by Iα on both sides of the last equation, we get

Iu(t) − Iα C = I1+α f1(t,v(t),∫0tg1(t,s,v(s)) ds).

Differentiate both sides, we get

u(t) − C1 tα−1  = Iα f1(t,v(t),∫0tg1(t,s,v(s)) ds).

From the initial condition, we find that C1 = k1, then

u(t)=k1 tα−1 + ∫0t (t − s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ)) dθ) ds.

Similarly, we can get

v(t)=k2 tβ−1 + ∫0t (t − s)β−1Γ(β) f2(s,u(s),∫0sg2(s,θ,u(θ)) dθ) ds.

Therefore, the solution (u, v) of system (2) can be represented by system (3).

2.2 Existence of solution

Let L1(I) be a class of Lebesgue integrable functions on the interval I, with the norm ∥x∥ = ∫I |x(t)|dt.

Define the operator T by

T(u,v)(t) = (T1v(t),T2u(t)),

where

T1v(t)=k1 tα−1+∫0t (t − s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ))dθ) ds,T2u(t)=k2 tβ−1+∫0t (t − s)β−1Γ(β) f2(s,u(s),∫0sg2(s,θ,u(θ))dθ) ds.

It is clear that the fixed point of the operator T is the solution of system (3).

Theorem 2.1

Assume that fi and gi satisfy the assumptions (i-iii). Then the coupled system of weighted Cauchy-type problems (2) has at least one solution (u, v) ∈ L1 × L1.

Proof

Define

X={(u(t),v(t))|(u(t),v(t))∈L1×L1 and ||(u,v)||L1×L1=||u||L1+||v||L1≤r}.

For (u, v) ∈ X, we have

||T1v||=∫01 |T1v(t)| dt≤∫01 |k1 tα−1| dt + ∫01 |∫0t (t − s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ))dθ) ds| dt≤k1 tαα01 + ∫01 ∫s1 (t − s)α−1Γ(α) dt |f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)| ds=k1α + ∫01 (t − s)αΓ(1+α)|s1 |f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)| ds=k1α + ∫01 (1 − s)αΓ(1+α) |f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)| ds≤k1α + 1Γ(1+α) ∫01 |f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)| ds≤k1α + 1Γ(1+α) ∫01 [a(s) + b1 |v(s)| + b2 ∫0s|g1(s,θ,v(θ))| dθ] ds≤k1α + 1Γ(1+α) ||a|| + b1Γ(1+α) ||v||L1 + b2Γ(1+α)∫01∫0s[k(s,θ) + b3|v(θ)|] dθ ds≤k1α + 1Γ(1+α) ||a|| + b1Γ(1+α) ||v||L1 + b2 k∗Γ(1+α) + b2 b3Γ(1+α) ||v||L1≤k1α + 1Γ(1+α) ||a|| + b1+b2 b3Γ(1+α) r1 + b2 k∗Γ(1+α) ≤ r1,

where k∗=∫01∫0sk(s,θ) dθ ds and

r1 = k1α + ||a||Γ(1+α) + b2 k∗Γ(1+α)1 − b1+b2 b3Γ(1+α).

Similarly, we get

||T2u||≤k2β + 1Γ(1+β) ||a|| + b1+b2 b3Γ(1+β) r2 + b2 k∗Γ(1+β) ≤ r2,

where

r2 = k2β + ||a||Γ(1+β) + b2 k∗Γ(1+β)1 − b1+b2 b3Γ(1+β).

Let

r = max {r1 + r2}.

Then,

||T(u,v)(t)||=||T1v(t),T2u(t)|| = ||T1v(t)|| + ||T2u(t)||≤r1 + r2 ≤ r.

Therefore, for (u, v) ∈ X, we get T(u, v) ∈ X and hence TX ∈ X. Now, from the assumptions (i-a) - (ii-a), we deduce that T maps X into L1 × L1 continuously. Moreover, we have

||fi||=∫01 |fi(t,w(t),∫0tgi(t,s,w(s))ds)| dt≤∫01 (a(t) + b1 |w(t)| + b2 ∫0t|gi(t,s,w(s))| ds) dt≤||a|| + b1 ||w|| + b2 ∫01 ∫0t[k(t,s) + b3 |w(s)|] ds dt≤||a|| + b1 ||w|| + b2 k∗ + b2 b3 ||w||.

This estimation shows that fi in L1(I).

Now, we will use Kolmogorov compactness criterion (see [8]) to show that T is compact. So, let ℵ be a bounded subset of L1. Then T1(ℵ) is bounded in L1(I). Now we show that (T1v)h → T1 v in L1(I) as h → 0, uniformly with respect to T1 v ∈ T ℵ. Indeed:

||(T1v)h − T1v||=∫01 |(T1v)h(t) − (T1v)(t)| dt=∫01 |1h ∫tt+h (T1v)(s) ds − (T1v)(t)| dt≤∫01 (1h ∫tt+h | (T1v)(s) −(T1v)(t) | ds) dt≤∫01 1h ∫tt+h |k1 sα−1 − k1 tα−1| ds dt+∫01 1h ∫tt+h |Iα f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)−Iα f1(t,v(t),∫0tg1(t,s,v(s))ds)| ds dt,

since f1 ∈ L1(I) we get that Iα f1(.) ∈ L1(I). Moreover tα−1 ∈ L1(I). So, we have (see [10])

1h ∫tt+h | k1 sα−1 − k1 tα−1 | ds → 0

and

1h ∫tt+h |Iα f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)−Iα f1(t,v(t),∫0tg1(t,s,v(s))ds)| ds → 0

for a.e. t ∈ I. Therefore, T1(ℵ) is relatively compact, that is, T1 is a compact operator, similarly T2 is a compact operator. Hence T is a compact operator

Therefore, Schauder fixed point Theorem (see [9]) implies that T has a fixed point (u, v) which is a solution of the coupled system (3).

To complete the proof, let (u(t), v(t)) be a solution of

u(t)=k1 tα−1 + Iα f1(t,v(t),∫0tg1(t,s,v(s))ds),v(t)=k2 tβ−1 + Iβ f2(t,u(t),∫0tg2(t,s,u(s))ds), (4)

which gives

t1−αu(t)|t=0 = k1, t1−βv(t)|t=0 = k2.

Operating on both sides of the first and second equations in (4) by I1−α and I1−β respectively, we get

I1−αu(t)=k1 + I1−α Iα f1(t,v(t),∫0tg1(t,s,v(s))ds),I1−βv(t)=k2 + I1−β Iβ f2(t,u(t),∫0tg2(t,s,u(s))ds).

Differentiate both sides, we obtain

Dα u(t) = f1(t,v(t),∫0tg1(t,s,v(s))ds),Dβ v(t) = f2(t,u(t),∫0tg2(t,s,u(s))ds).

2.3 Uniqueness of the solution

For the uniqueness of the solution we have the following theorem:

Theorem 2.2

Suppose that the functions fi and gi satisfy conditions (i-b), (ii-b) and (iii) of Theorem 2.1 in addition to the following assumptions:

|fi(t,u1,v1) − fi(t,u2,v2)| ≤ b1 |u1 − u2| + b2 |v1 − v2|, i = 1, 2 (5)

and

|gi(t,s,v1) − gi(t,s,v2)| ≤ b3 |v1 − v2|, i = 1, 2. (6)

Then the coupled system of weighted Cauchy-type problems (2) has a unique solution.

Proof

From assumption (5), we get

|fi(t,u,v) − fi(t,0,0)| ≤ b1 |u| + b2 |v|.

But since

|fi(t,u,v)| − |fi(t,0,0)| ≤ |fi(t,u,v) − fi(t,0,0)| ≤ b1 |u| + b2 |v|,

therefore

|fi(t,u,v)| ≤ |fi(t,0,0)| + b1 |u| + b2 |v|,

i.e. assumptions (i − a) and (i − c) of theorem 2.1 are satisfied, similarly assumptions (ii − a) and (ii − c) of Theorem 2.1 are satisfied.

Then from Theorem 2.1 the solution exists. Now we prove the uniqueness of this solution:

Let (u1, v1) and (u2, v2) be two solutions of (3). Then

u2(t) − u1(t)=∫0t (t−s)α−1Γ(α) [f1(s,v2(s),∫0sg1(s,θ,v2(θ))dθ)−f1(s,v1(s),∫0sg1(s,θ,v1(θ))dθ)] ds,≤∫0t(t−s)α−1Γ(α) {b1 |v2(s)−v1(s)|+b2 ∫0s|g1(s,θ,v2(θ))−g1(s,θ,v1(θ))|dθ} ds.

Therefore

∫01|u2(t) − u1(t)| dt≤∫01∫0t(t−s)α−1Γ(α) {b1 |v2(s) − v1(s)|+b2 ∫0s|g1(s,θ,v2(θ)) − g1(s,θ,v1(θ))| dθ} ds dt,
||u2 − u1||L1≤∫01∫s1(t − s)α−1Γ(α) dt {b1 |v2(s) − v1(s)|+b2 ∫0s|g1(s,θ,v2(θ))−g1(s,θ,v1(θ))| dθ} ds=∫01(1−s)αΓ(1+α) {b1 |v2(s) − v1(s)|+b2 ∫0s|g1(s,θ,v2(θ))−g1(s,θ,v1(θ))| dθ} ds≤1Γ(1+α)∫01 {b1|v2(s) − v1(s)| + b2 b3∫0s|v2(θ) − v1(θ)| dθ} ds≤b1 + b2 b3Γ(1+α) ||v2 − v1||L1.

Similarly

||v2 − v1||L1 ≤ b1 + b2 b3Γ(1+β) ||u2 − u1||L1.

Therefore

||(u2,v2) − (u1,v1)||=||u2 − u1||L1 + ||v2 − v1||L1≤b1 + b2 b3Γ(1+α) ||v2−v1||L1 + b1 + b2 b3Γ(1+β) ||u2− u1||L1≤||u2 − u1||L1 + ||v2 − v1||L1=||(u2,v2) − (u1,v1)||,

which implies

||(u2,v2) − (u1,v1)|| = 0  ⇒  (u2,v2) = (u1,v1).

This completes the proof.

2.4 Continuous dependence on initial data

Now we show that the solution of the coupled system (2) is depending continuously on initial data.

Theorem 2.3

Let the assumptions of Theorem 2.2 be satisfied. Then the solution of the weighted Cauchy-type problem (2) is depending continuously on initial data,

Proof

Let (u(t), v(t)) be a solution of the couple

u(t)=k1 tα−1+∫0t (t − s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ)) dθ) ds,v(t)=k2 tβ−1+∫0t (t − s)β−1Γ(β) f2(s,u(s),∫0sg2(s,θ,u(θ)) dθ) ds

and let (u͠(t), v͠(t)) be a solution of the above coupled system such that t1−α u~(t)|t=0=k1~  and  t1−β v~(t)|t=0=k2~. Then

u(t) − u~(t)=(k1−k1~) tα−1 + ∫0t (t−s)α−1Γ(α) [f1(s,v(s),∫0sg1(s,θ,v(θ)) dθ)−f1(s,v~(s),∫0sg1(s,θ,v~(θ)) dθ)] ds,|u(t) − u~(t)|≤|k1−k1~| tα−1+∫0t (t − s)α−1Γ(α) {b1 |v(s)−v~(s)|+b2 ∫0s|g1(s,θ,v(θ))−g1(s,θ,v~(θ))| dθ} ds.

Therefore

∫01 |u(t) − u~(t)| dt≤1α |k1−k1~|+∫01 ∫0t (t − s)α−1Γ(α) {b1 |v(s)−v~(s)|+b2 ∫0s|g1(s,θ,v(θ))−g1(s,θ,v~(θ))| dθ} ds dt,
||u − u~||L1≤1α |k1−k1~|+∫01∫s1(t − s)α−1Γ(α) dt {b1 |v(s) − v~(s)|+b2 ∫0s|g1(s,θ,v(θ))−g1(s,θ,v~(θ))| dθ} ds=1α |k1−k1~|+∫01(1−s)αΓ(1+α) {b1 |v(s) − v~(s)|+b2 ∫0s|g1(s,θ,v(θ))−g1(s,θ,v~(θ))| dθ} ds≤1α |k1−k1~|+1Γ(1+α)∫01 {b1|v(s) − v~(s)| + b2 b3∫0s|v(θ) − v~(θ)| dθ} ds≤1α |k1−k1~|+b1 + b2 b3Γ(1+α) ||v − v~||L1.

Similarly

||v − v~||L1≤1β |k2−k2~| + b1 + b2 b3Γ(1+β) ||u − u~||L1.

Therefore

||(u,v) − (u~,v~)||L1=||u − u~||L1 + ||v − v~||L1≤1α |k1 − k1~| + b1 + b2 b3Γ(1+α) ||v − v~||L1+1β |k2 − k2~| + b1 + b2 b3Γ(1+β) ||u − u~||L1≤max{1α, 1β} (|k1 − k1~| + |k2 − k2~|)+(b1 + b2 b3) max{1Γ(1+α), 1Γ(1+β)} (||v − v~||L1+||u − u~||L1)≤M∗ (|k1 − k1~| + |k2 − k2~|)+N∗ (b1 + b2 b3) (||v − v~||L1+||u − u~||L1)=M∗ (|k1 − k1~| + |k2 − k2~|)+N∗ (b1 + b2 b3) ||(u,v) − (u~,v~)||L1,

where M∗=max{1α, 1β}  and  N∗=max{1Γ(1+α), 1Γ(1+β)}.

(1−N∗ (b1 + b2 b3)) ||(u,v)−(u~,v~)||L1≤M∗ (|k1 − k1~| + |k2 − k2~|)⇒ ||(u,v)−(u~,v~)||L1≤(M∗1−N∗ (b1+b2 b3)) (|k1−k1~|+|k2−k2~|).

Therefore, if |k1−k1~| < δ(ε)2  and  |k2−k2~| < δ(ε)2, then ∥(u, v) − (u͠, v͠)∥L1 < ε. Now from the equivalence we get that the solution of the weighted Cauchy-type problem (2) is depending continuously on initial data.

3 Solution in C1−α([0, T])

Now, define the space C1−α([0, T]) by

C1−α([0,T]) = u:t1−αu(t) ∈C([0,T]),

with norm

||u||C1−α =  ||t1−α u||C

and C([0, T]) is the space of continuous functions defined on [0, T] with norm

||u||C = supt∈[0,T] |u(t)|.

Corollary 3.1

Let the assumptions of Theorem 2.1 satisfied. Then the coupled system of weighted Cauchy-type problems (2) has a solution (u, v) ∈ C1−α([0, T]) × C1−α([0, T]).

Proof

Define

Y={(u(t),v(t))|(u(t),v(t))∈C1−α×C1−α:||(u,v)||C1−α×C1−α=max(||u||C1−α,||v||C1−α)≤r′},

and define the subset Qr by

Qr={(u(t),v(t))∈Y:||(u(t),v(t))||Y≤r′},

where r′=max{r1′,r2′} , ( r1′ and r2′ will be indicated in the proof). The set Qr is nonempty, closed and convex.

Let T : Qr → Qr. For (u, v) ∈ Qr, T is a continuous operator: indeed, if {(un(t), vn(t)} is a sequence in Qr which converges to (u(t), v(t)) for every t ∈ [0, T]. Then

limn→∞T1vn(t)=k1 tα−1+limn→∞∫0t (t − s)α−1Γ(α) f1(s,vn(s),∫0sg1(s,θ,vn(θ))dθ) ds,

from the assumptions and Lebesgue dominated convergence theorem, we get that

limn→∞T1vn(t) = T1v(t).

Similarly

limn→∞T2un(t) = T2u(t).

Then

limn→∞T(un,vn)(t) = T(u,v)(t).

For (u, v) ∈ Qr, we have

|t1−α T1v(t)|≤k1 + t1−α ∫0t (t − s)α−1Γ(α) |f1(s,v(s),∫0sg1(s,θ,v(θ)) dθ)| ds≤k1 + t1−α ∫0t (t − s)α−1Γ(α) [a(s) + b1 |v(s)| + b2 ∫0s|g1(s,θ,v(θ))| dθ] ds≤k1 + t1−α Iα|a(t)| + b1 t1−α ∫0t (t − s)α−1Γ(α) |v(s)| ds+b2 t1−α∫0t (t − s)α−1Γ(α) ∫0s[k(s,θ) + b3|v(θ)|] dθ ds=k1 + t1−α Iα−γIγ|a(t)| + b1 t1−α ∫0t (t − s)α−1Γ(α) sα−1 s1−α |v(s)| ds+b2 t1−α∫0t (t − s)α−1Γ(α) ∫0sk(s,θ) dθ ds+b2 b3 t1−α ∫0t (t − s)α−1Γ(α) ∫0sθα−1θ1−α |v(θ)| dθ ds≤k1 + T1−α Iα−γ a∗ + b1 T1−α ||v||C1−α ∫0t (t − s)α−1Γ(α) sα−1 ds+b2 T1−αk∗Γ(α) + b2 b3 T1−α ||v||C1−α ∫0t (t − s)α−1Γ(α) ∫0sθα−1 dθ ds≤k1 + T1−α a∗Γ(α−γ+1) Tα−γ + b1 T1−α ||v||C1−α Γ(α)Γ(2α)T2α−1+b2 T1−αk∗Γ(α) + b2 b3 T1−α ||v||C1−α ∫0t (t − s)α−1Γ(α) sαα ds=k1 + a∗Γ(α−γ+1) T1−γ + b1 ||v||C1−α Γ(α)Γ(2α)Tα+b2 T1−αk∗Γ(α) + b2 b3 T1−α ||v||C1−α Γ(α)Γ(2α+1)T2α=k1 + a∗Γ(α−γ+1) T1−γ + b1 ||v||C1−α Γ(α)Γ(2α)Tα+b2 T1−αk∗Γ(α) + b2 b3 ||v||C1−α Γ(α)Γ(2α+1)Tα+1≤k1 + a∗Γ(α−γ+1) T1−γ + b1 r1′ Γ(α)Γ(2α)Tα+b2 T1−αk∗Γ(α) + b2 b3 r1′ Γ(α)Γ(2α+1)Tα+1,

where a* = supt∈[0,T] Iγ |a(t)|. Then

||T1 v||C1−α ≤ r1′,

where

r1′ = k1 + a∗Γ(α−γ+1) T1−γ + b2 k∗Γ(α)T1−α1 − (b1 Γ(α) Γ(2α)Tα + b2 b3 Γ(α)Γ(2α+1)Tα+1),   0<γ<α.

Similarly

|t1−β T2u(t)|≤k2 + a∗Γ(β−γ+1) T1−γ + b1 r2′ Γ(β)Γ(2β)Tβ+b2 T1−βk∗Γ(β) + b2 b3 r2′ Γ(β)Γ(2β+1)Tβ+1,||T2 u||C1−α≤r2′,

where

r2′ = k2 + a∗Γ(β−γ+1) T1−γ + b2 k∗Γ(β)T1−β1 − (b1 Γ(β)Γ(2β)Tβ + b2 b3 Γ(β)Γ(2β+1)Tβ+1),   0<γ<β.

Then T1v(t) is uniformly bounded in Qr, similarly T2u(t) is uniformly bounded in Qr.

Since

||T(u,v)(t)||=||T1v(t),T2u(t)|| = max(||T1v||C1−α,||T2u||C1−α)≤max(r1′,r2′) ≤ r′.

Therefore, T is uniformly bounded in Qr.

Now, we show that T is a completely continuous operator.

Indeed, let τ1, τ2 ∈ [0, T], τ1 < τ2 such that |τ2 − τ1| < δ, we have

τ21−αT1v(τ2)−τ11−αT1v(τ1)=τ21 − α∫0τ2(τ2−s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ))dθ) ds− τ11 − α∫0τ1(τ1−s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ))dθ) ds=τ21 − α∫0τ1(τ2−s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ))dθ) ds+τ21 − α∫τ1τ2(τ2−s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ))dθ) ds− τ11 − α∫0τ1(τ1−s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ))dθ) ds≤(τ21−α−τ11−α)∫0τ1(τ1−s)α−1Γ(α)f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)ds+τ21 − α∫τ1τ2(τ2−s)α−1Γ(α) f1(s,v(s),∫0sg1(s,θ,v(θ))dθ) ds,
|τ21−αT1v(τ2)−τ11−αT1v(τ1)|≤(τ21−α−τ11−α)∫0τ1(τ1−s)α−1Γ(α)|f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)|ds+τ21 − α∫τ1τ2(τ2−s)α−1Γ(α) |f1(s,v(s),∫0sg1(s,θ,v(θ))dθ)| ds≤(τ21−α−τ11−α)∫0τ1(τ1−s)α−1Γ(α)[a(s)+b1|v(s)|+b2∫0s|g1(s,θ,v(θ))|dθ]ds+τ21−α∫τ1τ2(τ2−s)α−1Γ(α)[a(s)+b1|v(s)|+b2∫0s|g1(s,θ,v(θ))|dθ]ds≤(τ21−α−τ11−α) Iα−γIγ a(τ1)+b1(τ21−α−τ11−α)∫0τ1(τ1−s)α−1Γ(α)sα−1s1−α|v(s)|ds+b2(τ21−α−τ11−α)∫0τ1(τ1−s)α−1Γ(α)∫0s[k(s,θ) + b3|v(θ)|]dθ ds+τ21−α Iτ1α−γ Iτ1γa(τ2)+b1τ21−α∫τ1τ2(τ2−s)α−1Γ(α)sα−1s1−α|v(s)|ds+b2τ21−α∫τ1τ2(τ2−s)α−1Γ(α)∫0s[k(s,θ) + b3|v(θ)|]dθ ds≤a∗Γ(α−γ+1)(τ21−α−τ11−α)τ1α−γ+b1(τ21−α−τ11−α)||v||C1−αΓ(α)τ12α−1Γ(2α)+b2k∗1Γ(α)(τ21−α−τ11−α)+b2b3(τ21−α−τ11−α)||v||C1−α∫0τ1(τ1−s)α−1Γ(α)∫0sθα−1dθ ds+a∗(τ2 − τ1)α−γΓ(α−γ+1)τ21−α+b1τ21−α||v||C1−αΓ(α)(τ2−τ1)2α−1Γ(2α)+b2 k∗ 1Γ(α) τ21−α+b2b3τ21−α||v||C1−α∫τ1τ2(τ2−s)α−1Γ(α)∫0sθα−1dθ ds≤a∗Γ(α−γ+1)(τ21−α−τ11−α)τ1α−γ+b1(τ21−α−τ11−α)||v||C1−αΓ(α)τ12α−1Γ(2α)+b2 k∗ 1Γ(α) (τ21−α−τ11−α)+b2b3(τ21−α−τ11−α)||v||C1−α Γ(α) τ12αΓ(2α+1)+a∗(τ2 − τ1)α−γΓ(α−γ+1)τ21−α+b1τ21−α||v||C1−αΓ(α)(τ2−τ1)2α−1Γ(2α)+b2k∗1Γ(α)τ21−α+b2b3τ21−α||v||C1−α Γ(α) (τ2−τ1)2αΓ(2α+1).

Therefore {T(u, v)(t)} is equi-continuous. By Arzela-Ascoli Theorem then {T(u, v)(t)} is relatively compact. Therefore, the conditions of the Schauder fixed point Theorem hold, which implies that T has a fixed point in Qr. Then (2) has a solution (u, v) ∈ C1−α([0, T]) × C1−α([0, T]).

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Received: 2019-08-15
Accepted: 2019-10-01
Published Online: 2019-12-31

© 2019 Ahmed M. A. El-Sayed and Sheren A. Abd El-Salam, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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https://doi.org/10.1515/math-2019-0120
Keywords for this article
Riemann-Liouville differential operator; coupled system; weighted Cauchy type problem; continuous dependence
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Articles in the same Issue

  1. Regular Articles
  2. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one
  3. Centralizers of automorphisms permuting free generators
  4. Extreme points and support points of conformal mappings
  5. Arithmetical properties of double Möbius-Bernoulli numbers
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  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
  87. Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
  88. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
  89. Differential polynomials of L-functions with truncated shared values
  90. Exclusion sets in the S-type eigenvalue localization sets for tensors
  91. Continuous linear operators on Orlicz-Bochner spaces
  92. Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
  93. Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
  94. A quantitative obstruction to collapsing surfaces
  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
  97. Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
  98. On the two-term exponential sums and character sums of polynomials
  99. F-biharmonic maps into general Riemannian manifolds
  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
  101. Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
  102. Power graphs and exchange property for resolving sets
  103. On nearly Hurewicz spaces
  104. Least eigenvalue of the connected graphs whose complements are cacti
  105. Determinants of two kinds of matrices whose elements involve sine functions
  106. A characterization of translational hulls of a strongly right type B semigroup
  107. Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
  108. Lp estimates for maximal functions along surfaces of revolution on product spaces
  109. Path-induced closure operators on graphs for defining digital Jordan surfaces
  110. Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
  112. Injective hulls of many-sorted ordered algebras
  113. Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
  114. Global properties of virus dynamics with B-cell impairment
  115. The monotonicity of ratios involving arc tangent function with applications
  116. A family of Cantorvals
  117. An asymptotic property of branching-type overloaded polling networks
  118. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
  119. Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
  120. L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
  137. The general position problem and strong resolving graphs
  138. Connected domination game played on Cartesian products
  139. On minimum algebraic connectivity of graphs whose complements are bicyclic
  140. A novel method to construct NSSD molecular graphs
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