| Summary: | Abstract In this manuscript, we obtain sufficient conditions required for the existence of solution to the following coupled system of nonlinear fractional order differential equations: D γ ω ( ℓ ) = F ( ℓ , ω ( λ ℓ ) , υ ( λ ℓ ) ) , D δ υ ( ℓ ) = F ‾ ( ℓ , ω ( λ ℓ ) , υ ( λ ℓ ) ) , $$ \begin{gathered} D^{\gamma}\omega(\ell)= \mathcal{F} \bigl( \ell,\omega(\lambda\ell), \upsilon(\lambda\ell) \bigr), \\ D^{\delta}\upsilon(\ell)=\mathcal{\overline{F}} \bigl(\ell,\omega ( \lambda\ell), \upsilon(\lambda\ell) \bigr), \end{gathered} $$ with fractional integral boundary conditions a 1 ω ( 0 ) − b 1 ω ( η ) − c 1 ω ( 1 ) = 1 Γ ( γ ) ∫ 0 1 ( 1 − ρ ) γ − 1 ϕ ( ρ , ω ( ρ ) ) d ρ and a 2 υ ( 0 ) − b 2 υ ( ξ ) − c 2 υ ( 1 ) = 1 Γ ( δ ) ∫ 0 1 ( 1 − ρ ) δ − 1 ψ ( ρ , υ ( ρ ) ) d ρ , $$ \begin{gathered} \mathfrak{a}_{1}\omega(0)- \mathfrak{b}_{1}\omega(\eta)-\mathfrak {c}_{1}\omega(1)= \frac{1}{\varGamma(\gamma)} \int_{0}^{1}(1-\rho )^{\gamma-1} \phi \bigl( \rho, \omega(\rho) \bigr)\, d\rho\quad\text{and} \\ \mathfrak{a}_{2}\upsilon(0)-\mathfrak{b}_{2} \upsilon (\xi)-\mathfrak{c}_{2}\upsilon(1)=\frac{1}{\varGamma(\delta)} \int _{0}^{1}(1-\rho)^{\delta-1} \psi \bigl( \rho, \upsilon(\rho) \bigr) \,d\rho, \end{gathered} $$ where ℓ ∈ Z = [ 0 , 1 ] $\ell\in\mathfrak{Z}=[0,1]$ , γ , δ ∈ ( 0 , 1 ] $\gamma, \delta\in(0,1]$ , 0 < λ < 1 $0<\lambda<1$ , D denotes the Caputo fractional derivative (in short CFD), F , F ‾ : Z × R × R → R $\mathcal{F}, \mathcal{\overline{F}}: \mathfrak{Z}\times \mathfrak{R}\times\mathfrak{R} \rightarrow\mathfrak{R}$ and ϕ , ψ : Z × R → R $\phi , \psi:\mathfrak{Z}\times\mathfrak{R}\rightarrow\mathfrak{R}$ are continuous functions. The parameters η, ξ are such that 0 < η , ξ < 1 $0<\eta, \xi<1$ , and a i , b i , c i $\mathfrak{a}_{i}, \mathfrak{b}_{i}, \mathfrak {c}_{i}$ ( i = 1 , 2 $i=1, 2$ ) are real numbers with a i ≠ b i + c i $\mathfrak{a}_{i}\neq\mathfrak {b}_{i}+\mathfrak{c}_{i}$ ( i = 1 , 2 $i=1, 2$ ). 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.
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