Our purpose in this paper is to use \(\psi-\)Riemann-Liouville fractional integral operator which is the fractional integral of any function with respect to another increasing function to establish some new fractional integral inequalities of Hermite-Hadamard, involving concave functions. Using the concave functions, we establish some new fractional integral
inequalities related to the Hermite-Hadamard type inequalities via \(\psi-\)Riemann-Liouville fractional integral operator.

Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \(D^\alpha u(t)=\lambda\sqrt{I^\beta[\sigma^2(t,u(t))]}\dot{w}(t),\,\,0<t<1\) with boundary conditions \(u(0)=0,\,\,u'(0)=u'(1)=0,\) where \(\lambda>0\) is a level of the noise term, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(\dot{w}(t)\) is a generalized derivative of Wiener process (Gaussian white noise), \(D^\alpha\) is the Riemann-Liouville fractional differential operator of order \(\alpha\in (3,4)\) and \(I^\beta,\,\,\beta>0\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \(\alpha=2\) and \(\beta=0\) with \(u(0)=u(1)=0\) is also studied.

In this paper, the mathematical formulation of \(2^{nd}\) law of thermodynamics has been explained, and the mathematical formulation of \(1^{st}\) law has been revisited from this noble perspective. It is not claimed that the \(2^{nd}\) law of thermodynamics is a redundant of the \(1^{st}\) law, rather I shown here how we can extract the mathematical formulation of the \(2^{nd}\) law from the mathematical formulation of the \(1^{st}\) law of thermodynamics. The Clausius statement of the \(2^{nd}\) law of thermodynamics is, it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. An alternative statement of the law is, “All spontaneous processes are irreversible” or, “the entropy of an isolated system always increases”. Having strong experimental evidences, this empirical law is obvious, which tells us the arrow of time and the direction of spontaneous changes.

This paper investigates the existence results and uniqueness of solutions for a class of boundary value problems for fractional differential equations with the Hilfer fractional derivative. The reasoning is mainly based upon Mönch’s fixed point theorem associated with the technique of measure of noncompactness. We illustrate our main findings, with a particular case example, included to show the applicability of our outcomes. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.

The purpose of this article is to construct fixed point theorems and prove the existence and uniqueness of common fixed point results of \(s-\alpha\) contraction for a pair of maps in the setting of \(b\) – dislocated metric spaces. Our results extend and generalize several well-known comparable results in the literature. The study procedure we used was that of Zoto and Kumari [1]. Furthermore, we provided an example in support of our main result.

In this paper, we find some Hermite-Hadamard type inequalities for co-ordinated harmonically convex functions via fractional integrals.

In this paper, we study the existence of positive solutions for an initial value problem of a state-dependent neutral functional differential equation with two state-delay functions. The continuous dependence of the unique solution will be proved. Some especial cases and examples will be given.

This paper furthers the study on a new graph parameter called the degree affinity number. The degree affinity number of a graph \(G\) is obtained by iteratively constructing graphs, \(G_1,G_2,\dots,G_k\) of increased size by adding a maximal number of edges between distinct pairs of distinct vertices of equal degree. Preliminary results for certain \(2\)-regular graphs are presented.

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

The concept of weak implicative UP-filters in UP-algebras is introduced and analyzed. Some characterizations of weak implicative UP-filters are derived with the using of some other filter types in such algebras.