Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative

1 Introduction

The fractional differential equations have become a significant field of investigation due to their frequent use in biophysics, economics, chemistry, mechanics, control theory, image processing and signal, etc. [1–3]. Recently, existence theory of solutions to initial and boundary value problems for fractional system has received considerable attention from many researchers [4–11].

As is known to all, Riesz-Caputo derivative refers to a fractional derivative that reflects both the future and the past memory effects; consequently, the Riesz fractional operator plays an important role in characterizing anomalous diffusion, owing to successful applications to subdiffusive, superdiffusive, and evolution problems. Space fractional quantum mechanics is a natural generalization of standard quantum mechanics, which arises when the Brownian trajectories in Feynman path integrals are replaced by Levy flights. The classical Levy flight is a stochastic processes, which in one dimension, is described by a jump length probability density function. The position space representation of the α th power of the momentum operator is given by:

where D ∣ x ∣ α is the Riesz fractional derivative operator of order α . The Riesz fractional derivative is regarded as an effective tool for studying nonlocal and memory effects in physics, engineering, and applied sciences. Therefore, many scholars are engaged in the study of the solutions of differential equations with the Riesz fractional derivative. Many works carried out so far discuss the numerical solutions of diffusion equations, which contain the Riesz derivative, and fractional variational problems, which contain the Riesz-Caputo derivative, and only a few works report on the existence results of fractional boundary value problems, which contain the Riesz-Caputo derivative.

Gu et al. [12] discussed a new class of differential equations that contains the Riesz-Caputo fractional derivative:

On the other hand, hybrid differential equation is a class of dynamical systems with quadratic perturbation. Nowadays, due to the wide range of application of hybrid differential equations in several areas of real-life problems, more researchers began to study the existence of solutions for hybrid differential equations with different perturbations (readers can refer to [13–20]).

Houas [21] studied the results for the coupled hybrid system that contains integral boundary conditions as follows:

where D α i ( i = 1 , 2 ) stands for the Caputo’s derivative.

Derbazi et al. [22] considered the solutions for the fractional boundary value problem as follows:

where C D 0 + α and D 0 + β C stand for the Caputo fractional derivative.

Recently, Baleanu et al. [23] discussed the hybrid fractional coupled system as follows:

where D ω , D ε , D μ , and D ν stand for the Caputo’s fractional derivative.

By virtue of the aforementioned documents, in this work, the authors present the solutions for the hybrid fractional coupled system that contains the Riesz-Caputo derivative as follows:

where 0 < γ ≤ 1 , 0 < ν ≤ 1 , 0 ≤ t ≤ 1 , D 1 γ 0 R C , and D 1 ν 0 R C are Riesz-Caputo’s fractional derivative of orders γ , ν , f 1 , f 2 , h 1 , h 2 ∈ C ( [ 0 , 1 ] × R 2 , R ) , g 1 , g 2 ∈ C ( [ 0 , 1 ] × R 2 , R \ { 0 } ) and a 1 , a 2 , b 1 , and b 2 are the real constants. We will use “Dhage’s fixed point theorem” to find the sufficient conditions for the existence of solutions. This is the first time for us to present the existence of solutions of Systems (1)–(4).

2 Some lemmas

Suppose n − 1 < α ≤ n , n ∈ N for α > 0 , and n = [ ω ] and [ ⋅ ] stands for the ceiling of a number.

From the aforementioned equations, thus we have

Particularly, for x ( t ) ∈ C ( 0 , T ) , then

if 0 < α ≤ 1 .

3 Main results

The Banach space C [ 0 , 1 ] is recorded as X . The norm is defined as follows: ‖ x ‖ = max t ∈ [ 0 , 1 ] ∣ x ( t ) ∣ . Under the following norm definition ‖ ( x , y ) ‖ = ‖ x ‖ + ‖ y ‖ , we have E = X × X , which is a Banach space.

Define the operator T mapping E to E

where

We give the hypotheses of this study.

( H 1 )   f 1 , f 2 , h 1 , h 2 : [ 0 , 1 ] × R 2 → R and g 1 , g 2 : [ 0 , 1 ] × R 2 → R \ { 0 } are continuous functions;

( H 2 )   ϕ 0 , ϕ 1 , ϕ 0 ¯ , ϕ 1 ¯ are four positive functions such that

for all t ∈ [ 0 , 1 ] , x , x ¯ , y , and y ¯ are the elements in R ;

( H 3 ) Functions p 1 , p 2 ∈ L ∞ ( J , R + ) and four continuous nondecreasing functions ψ 1 , ψ 2 , ξ 1 , ξ 2 : [ 0 , ∞ ) → ( 0 , ∞ ) satisfy

for all t ∈ [ 0 , 1 ] , x and y are the elements in R ;

( H 4 ) The positive number r satisfies

where

and ‖ ϕ 1 ‖ Λ + ‖ ϕ 0 ‖ < 1 2 , ‖ ϕ 1 ¯ ‖ Λ + ‖ ϕ 0 ¯ ‖ < 1 2 , f 1 = sup t ∈ [ 0 , 1 ] ∣ f 1 ( t , 0 , 0 ) ∣ , f 2 = sup t ∈ [ 0 , 1 ] ∣ f 2 ( t , 0 , 0 ) ∣ , g 1 = sup t ∈ [ 0 , 1 ] ∣ g 1 ( t , 0 , 0 ) ∣ , g 2 = sup t ∈ [ 0 , 1 ] ∣ g 2 ( t , 0 , 0 ) ∣ .