Study of an implicit type coupled system of fractional differential equations by means of topological degree theory

In this work, a sufficient condition required for the presence of positive solutions to a coupled system of fractional nonlinear differential equations of implicit type is studied. To study sufficient conditions essential for the existence of unique solution degree theory is used. Two examples are given to illustrate the established results.


Introduction
The concept of fractional differential equations (abbreviated as FDEs) has been examined and considered seeing its usefulness and plentiful presentations in different disciplines of applied science, engineering, and technology such as computer networking, fluid dynamics, control theory, mathematical biology, economics, viscoelasticity, optimization theory, and control theory [1][2][3][4][5][6][7][8]. Nonlinear fractal oscillator is recognized in a fractal space by fractal derivative, and its variational principle is gained for a thin film equation [9]. In a fractal space He's fractional derivative [10] is assumed to originate evolution equations involving fractional order [11]. In a fractal process, the Fornberg-Whitham fractional equation through He's fractional derivative is considered [12], and future challenges of fractal calculus have been illustrated from two-scale thermodynamics to fractal variational principle by Ji-Huan He [13]. Substantial consideration has been given to the presence of solutions of initial and boundary value problems (BVPs) having CFD.
Diverse sort of problems dedicated to FDEs, like local and nonlocal BVPs, Dirichlet and Neumann BVPs, integral BVPs, and impulsive BVPs, have been explored so far. An indispensable class of FDEs named implicit fractional differential equations (shortly IFDEs) has been considered by numerous writers. This is because of the point that many problems of finances and decision-making can be modeled by using IFDEs. Recently more courtesy has been given to scrutinizing sufficient conditions essential for the existence of solutions to IFDEs. It was observed sensibly that the existence of solutions to IFDEs had a lot of solicitations in optimization theory, quantitative theory, viscoelasticity, and fluid mechanics [14][15][16][17][18][19]. Nonlocal Cauchy problem via a fractional operator including power kernel in Banach spaces was considered in [20]. The fractional hampered generalized regularized long wave equation in the sagacity of Caputo, Atangana-Baleanu, and Caputo-Fabrizio fractional derivatives was investigated in [21]. In [22] authors presented a method for nonlinear fractional regularized long-wave (RLW) models. Mehmet Yavuz [23] inspected innovative solutions of fractional order best valuing models and their fundamental mathematical studies.
Fixed point concept has been used to probe the existence and uniqueness for some problems. Operating these notions, one needs strong compact settings due to which the area is limited to some BVPs. To spread the methods to additional classes of BVPs, mathematicians have been attracted to finding a tool of nonlinear analysis. One of the strong tools is the degree method. After studying the present literature, we pointed out that IFDEs having integral boundary conditions have not been properly studied by the degree method.
There are very few results in the literature which utilized the degree method for the existence of solutions to initial and some BVPs having CFD [1,[24][25][26][27]. Therefore, inspired by the applications of IFDEs, Samina et al. [28] investigated the presence of solutions to the following coupled system "of IFDEs through fixed point theory where κ, δ ∈ (1, 2], ξ ∈ (0, ∞), ∈ [0, ξ ] and F, F : J × R × R → R are nonlinear continuous functions. " Using fixed point theory, Cabada et al. [29] discussed the following problem: Motivated by [28] and [29], we use degree theory and investigate some suitable conditions for uniqueness and existence of solutions to the following IFDEs: where κ, δ ∈ (2, 3], D denotes the CFD, F, F : J × R × R → R, g 1 , g 2 : J × R → R, and r, h : J → R are continuous functions.

Preliminaries
To prove the main results, we need some definitions and results in the sequel from the existing literature. Throughout the work the notations M = C(J, R) and N = C(J, R) are used for Banach spaces having the norm u = sup{|u( )| : ∈ J}. The product M × N is a Banach space with the norm (u, w) = u + w .
Further, if K < 1, then W is a strict contraction. " It means that W has at least one fixed point.
The arbitrary order (κ > 0) integral of a function F : R + → R is given by

Main results
Before studying the existence results for BVP (1.1), we list the following assumptions.
has a solution Proof Applying the operator I κ to D κ u( ) = h( ), and by Lemma 2.1, we have Utilizing the boundary conditions to (3.1), we get By Lemma 3.1, the solutions of coupled system (1.1) are solutions of the following system of integral equations: (1 -s) κ-1 g 1 s, u(s) ds Then the solution of (1.1) in the operator form becomes

Lemma 3.2
The following Lipschitz condition is satisfied for the operator A: Proof For any (u, w), (u, w) ∈ M × N , we have where k θ = max{k r , k h }. Thus A is Lipschitz having constant k θ , and in view of Proposition 2.2, A is σ -Lipschitz having constant k θ . For this, we have (1 -s) κ-1 g 1 s, u(s) ds which implies that (1 -s) κ-1 g 1 s, u n (s) -g 1 s, u(s) ds From the continuity of F it follows that F s, w n (s), ω n (s) → F s, w(s), ω(s) as n → ∞.

Lemma 3.4 The following growth conditions are valid for the operators A and B:
and respectively, where c θ = max{c r , c h }, θ = max{z g 1 + 2d 1 1-d 2 , z g 2 + 2c 1 1-c 2 }, and Proof For the growth condition, consider where M = M r + M h + |u o | + |w o |, hence the operator A satisfies the growth condition. Now which implies that B(u, w) ≤ θ (u, w) + , (3.8) which is the required growth condition on B. (1 -s) κ-1 F s, w n (s), D κ u n (s) ds (1 -s) κ-1 g 1 s, u n (s) ds which implies that (1 -s) κ-1 g 1 s, u n (s) ds (1 -s) κ-1 F s, w n (s), D κ u n (s) ds Taking limit as τ → , we get So there exists > 0 such that (1 -s) κ-1 g 1 s, u(s) -g 1 s, u(s) ds g 1 s, u(s) -g 1 s, u(s) which implies that where C = max{a 1 , it means that T is a contraction. Therefore system (1.1) has a unique solution.
Example 3.1 Consider the given problem as follows: