We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as $(p,h)$ fractional type inequality and many others, as the $(p,h)$-convexity is the generalization of the other convex inequalities. As a consequence of the $(h,m)$-convexity, the fractional inequality of the $(s,m)$-type is obtained. Many consequences of such fractional inequalities and generalizations are obtained.

Fixed-point results on covariant maps and contravariant maps in a $C^{★}$-algebra-valued bipolar metric space are proved. Our results generalize and extend some recently obtained results in the existing literature. Our theoretical results in this paper are supported with suitable examples. We have also provided an application to find an analytical solution to the integral equation and the electrical circuit differential equation.

Fixed point results and metric fixed point theory play a vital role to find the unique solution to differential and integral equations. Likewise, fractal calculus has vast physical applications. In this article, we introduce the concept of bipolar-controlled metric space and prove fixed point theorems. The derived results expand and extend certain well-known results from the research literature and are supported with a non-trivial example. We have applied the fixed point result to find the analytical solution to the integral equation and fractional differential equation. The analytical solution has been supplemented with numerical simulation.

In this paper, a new type of convexity is defined, namely, the left–right-(k,h-m)-p IVM (set-valued function) convexity. Utilizing the definition of this new convexity, we prove the Hadamard inequalities for noninteger Katugampola integrals. These inequalities generalize the noninteger Hadamard inequalities for a convex IVM, (p,h)-convex IVM, p-convex IVM, h-convex, s-convex in the second sense and many other related well-known classes of functions implicitly. An apt number of numerical examples are provided as supplements to the derived results.

In this work, various fractional convex inequalities of the Hermite–Hadamard type in the interval analysis setting have been established, and new inequalities have been derived thereon. Recently defined p interval-valued convexity is utilized to obtain many new fractional Hermite–Hadamard type convex inequalities. The derived results have been supplemented with suitable numerical examples. Our results generalize some recently reported results in the literature.

In the present work, simulation function is applied to establish fixed point results of $ \mathcal{F} $-contraction in the setting of Bi-polar metric space. Our results are extensions or generalizations of results proved in the literature. The derived results are substantiated with suitable examples and an application to find the solution to the integral equation.