Existence and uniqueness of solutions for coupled system of fractional differential equations involving proportional delay by means of topological degree theory

In this manuscript, we obtain sufficient conditions required for the existence of solution to the following coupled system of nonlinear fractional order differential equations: Dγω(ℓ)=F(ℓ,ω(λℓ),υ(λℓ)),Dδυ(ℓ)=F‾(ℓ,ω(λℓ),υ(λℓ)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{gathered} D^{\gamma}\omega(\ell)= \mathcal{F} \bigl( \ell,\omega(\lambda\ell), \upsilon(\lambda\ell) \bigr), \\ D^{\delta}\upsilon(\ell)=\mathcal{\overline{F}} \bigl(\ell,\omega ( \lambda\ell), \upsilon(\lambda\ell) \bigr), \end{gathered} $$\end{document} with fractional integral boundary conditions a1ω(0)−b1ω(η)−c1ω(1)=1Γ(γ)∫01(1−ρ)γ−1ϕ(ρ,ω(ρ))dρanda2υ(0)−b2υ(ξ)−c2υ(1)=1Γ(δ)∫01(1−ρ)δ−1ψ(ρ,υ(ρ))dρ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{gathered} \mathfrak{a}_{1}\omega(0)- \mathfrak{b}_{1}\omega(\eta)-\mathfrak {c}_{1}\omega(1)= \frac{1}{\varGamma(\gamma)} \int_{0}^{1}(1-\rho )^{\gamma-1} \phi \bigl( \rho, \omega(\rho) \bigr)\, d\rho\quad\text{and} \\ \mathfrak{a}_{2}\upsilon(0)-\mathfrak{b}_{2} \upsilon (\xi)-\mathfrak{c}_{2}\upsilon(1)=\frac{1}{\varGamma(\delta)} \int _{0}^{1}(1-\rho)^{\delta-1} \psi \bigl( \rho, \upsilon(\rho) \bigr) \,d\rho, \end{gathered} $$\end{document} where ℓ∈Z=[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell\in\mathfrak{Z}=[0,1]$\end{document}, γ,δ∈(0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma, \delta\in(0,1]$\end{document}, 0<λ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\lambda<1$\end{document}, D denotes the Caputo fractional derivative (in short CFD), F,F‾:Z×R×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}, \mathcal{\overline{F}}: \mathfrak{Z}\times \mathfrak{R}\times\mathfrak{R} \rightarrow\mathfrak{R}$\end{document} and ϕ,ψ:Z×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi , \psi:\mathfrak{Z}\times\mathfrak{R}\rightarrow\mathfrak{R}$\end{document} are continuous functions. The parameters η, ξ are such that 0<η,ξ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\eta, \xi<1$\end{document}, and ai,bi,ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak{a}_{i}, \mathfrak{b}_{i}, \mathfrak {c}_{i}$\end{document} (i=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i=1, 2$\end{document}) are real numbers with ai≠bi+ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak{a}_{i}\neq\mathfrak {b}_{i}+\mathfrak{c}_{i}$\end{document} (i=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i=1, 2$\end{document}). Using topological degree theory, sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, an appropriate example is presented in the end.

where ∈ Z = [0, 1], γ , δ ∈ (0, 1], 0 < λ < 1, D denotes the Caputo fractional derivative (in short CFD), F, F : Z × R × R → R and φ, ψ : Z × R → R are continuous functions. The parameters η, ξ are such that 0 < η, ξ < 1, and a i , b i , c i (i = 1, 2) are real numbers with a i = b i + c i (i = 1, 2). Using topological degree theory, sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, an appropriate example is presented in the end.

Introduction
It has been proved that fractional differential equations (in short FDEs) are a powerful tool for modeling various phenomena of physical and chemical as well as biological sciences.
Many mathematicians give much attention to the existence theory of FDEs with multipoint boundary conditions, and there is rapidly growing area of research due to its wide range of applications in real world problems [6][7][8][9][10]. For the existence and uniqueness of solutions of FDEs, different methods are used like topological degree theory and fixed point theory. Here we use topological degree theory. After studying the present literature, we noticed that FDEs having fractional integral type boundary conditions are not well examined through topological degree theory. Very few articles used topological degree theory for simple initial and boundary value problems (BVPs) having CFD [11][12][13][14][15]. If viewed carefully, the existence of solutions to FDEs having integral boundary conditions has a wide range of applications in optimization theory, viscoelasticity, fluid mechanics, and quantitative theory which have been studied by many researchers [16][17][18][19][20][21]. Keeping in mind the applications of topological degree theory, Ali et al. [22] studied the existence of solutions to the following FDE: where F 1 , F 2 : C(Z, R) → R and F : Z × R → R are continuous functions and a i , b i are real numbers with a i + b i = 0, i = 1, 2. Using fixed point theory, Cabada et al. [23] discussed the following problem: , Motivated by [22] and [23], we examine the results for the existence of solution to the following nonlinear coupled system of FDEs through topological degree theory: where ∈ Z, γ , δ ∈ (0, 1], 0 < λ < 1, D denotes the CFD. Further F, F : Z × R × R → R and φ, ψ : Z × R → R are continuous functions. The parameters η, ξ are such that 0 < η, ξ < 1 and a i , b i , c i (i = 1, 2) are real numbers with a i = b i + c i .

Preliminaries
In this section we recollect some facts, definitions, and results. Throughout this work U = C(Z, R), V = C(Z, R) represent the Banach spaces for all continuous function defined on Z into R with norm ω = sup{|ω( )| : 0 ≤ ≤ 1}. The product space U × V is a Banach space with norm (ω, υ) = ω + υ . Moreover, H : V → U is Lipschitz whenever there is > 0 such that Further H will be a strict contraction if < 1.
If Λ is bounded in U , so there exists r > 0 such that Λ ⊂ S r (0), then the degree Consequently, H has at least one fixed point which lies in S r (0).

Definition 2.2 ([26])
The fractional order integral of a function F : R + → R is defined by

Main results
In this section, we discuss the existence and uniqueness criteria for BVP (1.1). Before we start our main work, we need the following hypotheses.
with integral type boundary conditions has a solution Proof Applying fractional integrable operator I γ to D γ ω( ) = h( ) and using Lemma 2.1, we get On applying boundary conditions to (3.1), we have By rearranging, we get By Lemma 3.1, the solution of system (1.1) is a solution of the following system of integral equations: Further, we define T = J + G. Then the system of integral equations (3.2) can be written as an operator form which is the solution of system (1.1) in the operator form.

Lemma 3.2 The operator J satisfies the Lipschitz condition
which implies that Similarly, From (3.4) and (3.5), we have Thus J is Lipschitz with constant k, and therefore by Proposition 2.2, J is σ -Lipschitz with constant k.
For every ∈ Z and by using (C 3 ), we get Similarly other terms approach 0 as n → +∞. It follows that That is, G 1 is continuous. Proceeding the same way as above, we can show that That is, G 2 is continuous and hence G is continuous.

Lemma 3.4
The operators J and G satisfy the following growth conditions: and Proof For the growth condition on J , consider , which is the growth condition for J . Now, for the growth condition on G, we have Similarly, Now, from (3.8) and (3.9), we have Hence G satisfies the growth condition.

Lemma 3.5 The operator
Proof Let B be a bounded subset of B r ⊆ U × V and {(ω n , υ n )} n∈N be a sequence in B, then by using the growth condition of G, it is clear that G(B) is bounded in U × V. Now, we need to show that G is equicontinuous. Let 0 ≤ ≤ τ ≤ 1, then we have Taking limit as → τ , we get That is, there exists > 0 such that Similarly, From (3.10) and (3.11), it follows that G(ω n , υ n )( ) -G(ω n , υ n )(τ ) < . (3.12) Hence G is equicontinuous. Therefore G(B) is compact in U × V and hence by Proposition 2.1, G is σ -Lipschitz with constant zero.
We have to show that B is bounded in U × V. Choose (ω, υ) ∈ B, then by using (3.6) and (3.7) we have Thus B is bounded in U × V. Therefore Theorem 2.1 guarantees that T has at least one fixed point; consequently, BVP (1.1) has at least one solution.

Theorem 3.2
Under assumptions (C 1 )-(C 4 ), assume that G * < 1, then BVP (1.1) has a unique solution, where Proof To find the unique solution of system (1.1), we use the Banach contraction theorem, that is, we have to show that T is a contraction. For this, let (ω, υ), (ω, υ) ∈ U × V, then from (3.3) in Lemma 3.2, we showed that which implies that Similarly, From (3.14) and (3.15), it follows that which implies that Now, from (3.13) and (3.16), it follows that which implies that Thus T is a contraction and hence problem (1.1) has a unique solution.
To illustrate our results, we provide the following example.