In this paper, we modify L-cyclic (α,β)s\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha ,\beta )_s$$\end{document}-contractions and using this contraction, we prove fixed point theorems in the setting of b-metric spaces. As an application, we discuss the existence of a unique solution to non-linear fractional differential equation, 1cDσ(x(t))=f(t,x(t)),for allt∈(0,1),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} ^{c}D^{\sigma }(x(t))=f(t,x(t)),\ \ \text {for all}\ \ t\in (0,1), \end{aligned}$$\end{document}with the integral boundary conditions, x(0)=0,x(1)=∫0ρx(r)dr,for allρ∈(0,1),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} x(0)=0,\ \ x(1)=\int _{0}^{\rho }x(r)\mathrm{d}r,\ \ \text {for all}\ \rho \in (0,1), \end{aligned}$$\end{document}where x∈C(0,1,R)\documentclass[12pt]{minimal}
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\begin{document}$$x\in C(\left[ 0,1\right] ,\mathbb {R})$$\end{document}, cDα\documentclass[12pt]{minimal}
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\begin{document}$$^{c}D^{\alpha }$$\end{document} denotes the Caputo fractional derivative of order σ∈(1,2]\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \in (1,2]$$\end{document}, f:[0,1]×R→R\documentclass[12pt]{minimal}
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\begin{document}$$f : [0,1] \times \mathbb {R}\rightarrow \mathbb {R}$$\end{document} is a continuous function. Furthermore, we established existence result of a unique common solution to the system of non-linear quadratic integral equations, x(t)=∫01H(t,τ)f1(τ,x(τ))dτ,for allt∈[0,1];x(t)=∫01H(t,τ)f2(τ,x(τ))dτ,for allt∈[0,1],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}{\left\{ \begin{array}{ll} x(t)&{}= \int _{0}^{1}H(t,\tau )f_{1}(\tau ,x(\tau )) \mathrm{d}\tau ,\ \text {for all}~ t\in [0,1]; \\ x(t)&{}= \int _{0}^{1}H(t,\tau )f_{2}(\tau ,x(\tau )) \mathrm{d}\tau ,\ \text {for all}~ t\in [0,1], \end{array}\right. } \end{aligned}$$\end{document}where H:0,1×0,1→[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$H : \left[ 0,1\right] \times \left[ 0,1\right] \rightarrow [0,\infty )$$\end{document} is continuous at t∈0,1\documentclass[12pt]{minimal}
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\begin{document}$$t\in \left[ 0,1\right] $$\end{document} for every τ∈0,1\documentclass[12pt]{minimal}
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\begin{document}$$\tau \in \left[ 0,1\right] $$\end{document} and measurable at τ∈0,1\documentclass[12pt]{minimal}
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\begin{document}$$\tau \in \left[ 0,1\right] $$\end{document} for every t∈0,1\documentclass[12pt]{minimal}
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\begin{document}$$t\in \left[ 0,1\right] $$\end{document} and f1,f2:0,1×R→[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$f_{1}, f_{2}: \left[ 0,1\right] \times \mathbb {R}\rightarrow [0,\infty )$$\end{document} are continuous functions.
