Fixed point theorems in b-metric spaces and their applications to non-linear fractional differential and integral equations

Abstract

In this paper, we modify L-cyclic $(\alpha ,\beta )_s$-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,

$$\begin{aligned} ^{c}D^{\sigma }(x(t))=f(t,x(t)),\ \ \text {for all}\ \ t\in (0,1), \end{aligned}$$

(1)

with the integral boundary conditions,

$$\begin{aligned} x(0)=0,\ \ x(1)=\int _{0}^{\rho }x(r)\mathrm{d}r,\ \ \text {for all}\ \rho \in (0,1), \end{aligned}$$

where $x\in C(\left[ 0,1\right] ,\mathbb {R})$, $^{c}D^{\alpha }$ denotes the Caputo fractional derivative of order $\sigma \in (1,2]$, $f : [0,1] \times \mathbb {R}\rightarrow \mathbb {R}$ is a continuous function. Furthermore, we established existence result of a unique common solution to the system of non-linear quadratic integral equations,

$$\begin{aligned}{\left\{ \begin{array}{ll} x(t)&{}= \int _{0}^{1}H(t,\tau )f_{1}(\tau ,x(\tau )) \mathrm{d}\tau ,\ \text {for all}~ t\in [0,1]; \\ x(t)&{}= \int _{0}^{1}H(t,\tau )f_{2}(\tau ,x(\tau )) \mathrm{d}\tau ,\ \text {for all}~ t\in [0,1], \end{array}\right. } \end{aligned}$$

where $H : \left[ 0,1\right] \times \left[ 0,1\right] \rightarrow [0,\infty )$ is continuous at $t\in \left[ 0,1\right]$ for every $\tau \in \left[ 0,1\right]$ and measurable at $\tau \in \left[ 0,1\right]$ for every $t\in \left[ 0,1\right]$ and $f_{1}, f_{2}: \left[ 0,1\right] \times \mathbb {R}\rightarrow [0,\infty )$ are continuous functions.