This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order $\beta \in (1,2)$, ${}_0^C{D}_t^{\beta }u(t,\zeta )=Au(t,\zeta )+f(t,u(t,\zeta ))d{C}_t+Bx\left(t\right)Cx\left(t\right)dt$ with initial conditions $u\left(0\right)=u_0,u^{\prime }\left(0\right)=u_1,$ where $u(t,\zeta )$ takes values from $U(\subset E_N),V(\subset E_N)$ is the other bounded space, and $E_N$ represents the set of all upper semi-continuously convex fuzzy numbers on $\mathbb{R}$. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations.

In this paper, we define $(p,q)$-integrals for continuous functions of two variables. Then, we prove the Hermite-Hadamard type inequalities for coordinated convex functions by using $(p,q)$-integrals. Many results obtained in this paper provide significant extensions of other related results given in the literature. Finally, we give some examples of our results.

In this paper, we propose the solutions of nonhomogeneous fractional integral equations of the form $I_{0^{+}}^{3\sigma }y\left(t\right)+a·I_{0^{+}}^{2\sigma }y\left(t\right)+b·I_{0^{+}}^{\sigma }y\left(t\right)+c·y\left(t\right)=f\left(t\right),$ where $I_{0^{+}}^{\sigma }$ is the Riemann–Liouville fractional integral of order $\sigma =1/3,1,f\left(t\right)=t^n,t^n{e}^t,n\in \mathbb{N}\cup \left\{0\right\},t\in {\mathbb{R}}^{+},$ and $a,b,c$ are constants, by using the Laplace transform technique. We obtain solutions in the form of Mellin–Ross function and of exponential function. To illustrate our findings, some examples are exhibited.

In this paper, we establish some new versions of Hermite–Hadamard type inequalities for co-ordinated convex functions via $q_1,q_2$-integrals. Since the inequalities are newly proved, we therefore consider some examples of co-ordinated convex functions and show their validity for particular choices of $q_1,q_2\in (0,1)$. We hope that the readers show their interest in these results.

Symmetry plays an essential role in determining the correct methods for the oscillatory properties of solutions to differential equations. This paper examines some new oscillation criteria for unbounded solutions of third-order neutral differential equations of the form ${(r_2(\varsigma ){({(r_1(\varsigma ){(z^{\prime }(\varsigma ))}^{{\beta }_1})}^{\prime })}^{{\beta }_2})}^{\prime }$ + ${\displaystyle \sum _{i=1}^n}{q}_i(\varsigma )x^{{\beta }_3}({\varphi }_i(\varsigma ))=0.$ New oscillation results are established by using generalized Riccati substitution, an integral average technique in the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems.

In this paper, we derive an identity of the q-definite integral of a continuous function f on a finite interval. We then use such identity to prove some new quantum integral inequalities for $(p,h)$-convex function. The results obtained in this paper generalize previous work in the literature.