In this paper, we investigate a coupled system of Hilfer-type nonlinear proportional fractional differential equations supplemented with mixed multi-point and integro-multi-point boundary conditions. We used standard methods from functional analysis and especially fixed point theory. Two existence results are established using the Leray-Schauder's alternative and the Krasnosel'skii's fixed point theorem, while the existence of a unique solution is achieved via the Banach's contraction mapping principle. Finally, numerical examples are constructed to illustrate the main theoretical results. Our results are novel, wider in scope, produce a variety of new results as special cases and contribute to the existing literature on nonlocal systems of nonlinear $\psi$-Hilfer generalized fractional proportional differential equations.

In this research, a new class of fractional boundary value problems is introduced and studied, which combine Hilfer fractional derivatives with iterated Riemann-Liouville and Hadamard fractional integrals boundary conditions. Existence and uniqueness results are obtained by using standard tools from fixed point theory. The obtained results are well illustrated by numerical examples.

In this paper, we studied the existence of solutions for a coupled system of nonlinear sequential proportional $\psi$-Hilfer fractional differential equations with multi-point boundary conditions. By using a Burton's version of the Krasnosel'ski $\breve{\mathrm{i}}$'s fixed-point theorem we established sufficient conditions for the existence result. An example illustrating our main result was also provided.

In this paper, we investigate the existence and uniqueness of solutions to a nonlinear coupled systems of $(k,\phi )$-Hilfer fractional differential equations supplemented with nonlocal integro-multi-point boundary conditions. We make use of the Banach contraction mapping principle to obtain the uniqueness result, while the existence results are proved with the aid of Krasnosel'ski $\text{{rm{mathord{buildrel{lower3pthbox{}$\scriptscriptstyle\smile $\text{}} over i} }}}$'s fixed point theorem and Leray-Schauder alternative for the given problem. Examples demonstrating the application of the abstract results are also presented. Our results are of quite general nature and specialize in several new results for appropriate values of the parameters ${\beta }_1,$ ${\beta }_2,$ and the function $\phi$ involved in the problem at hand.

In this article, we aim to introduce and explore a new class of preinvex functions called $\mathfrak{n}$-polynomial $m$-preinvex functions, while also presenting algebraic properties to enhance their numerical significance. We investigate novel variations of Pachpatte and Hermite-Hadamard integral inequalities pertaining to the concept of preinvex functions within the framework of the Caputo-Fabrizio fractional integral operator. By utilizing this direction, we establish a novel fractional integral identity that relates to preinvex functions for differentiable mappings of first-order. Furthermore, we derive some novel refinements for Hermite-Hadamard type inequalities for functions whose first-order derivatives are polynomial preinvex in the Caputo-Fabrizio fractional sense. To demonstrate the practical utility of our findings, we present several inequalities using specific real number means. Overall, our investigation sheds light on convex analysis within the context of fractional calculus.

In the present paper, we establish the existence criteria for solutions of single valued and multivalued boundary value problems involving a ${\overline{\psi }}_{\ast }$-Hilfer fractional proportional derivative operator, subject to nonlocal integro-multistrip-multipoint boundary conditions. We apply the fixed-point approach to obtain the desired results for the given problems. The obtained results are well-illustrated by numerical examples. It is important to mention that several new results appear as special cases of the results derived in this paper (for details, see the last section).

This paper describes a new class of boundary value fractional-order differential equations of the $q$-Hilfer and $q$-Caputo types, with separated boundary conditions. The presented problem is converted to an equivalent integral form, and fixed-point theorems are used to prove the existence and uniqueness of solutions. Moreover, several special cases demonstrate how the proposed problems advance beyond the existing literature. Examples are provided to support the analysis presented.

In this paper, we studied the generalizations of quantum calculus on finite intervals. We presented the new definitions of the quantum derivative and quantum integral of a function with respect to another function and studied their basic properties. We gave an application of these newly defined quantum calculi by obtaining a new Hermite-Hadamard inequality for a convex function. Moreover, an impulsive boundary value problem involving quantum derivative, with respect to another function, was studied via the Banach contraction mapping principle.

In this study, the subject of the multivalued ${\rho }_{\ast }$-interpolative Ćirić-Reich-Rus-type fuzzy contractions is introduced and investigated in which the $\vartheta$-comparison functions and the property of the ${\rho }_{\ast }$-admissibility play an important role. In the first step, the existence of fixed point theorems is proven for such a type of contractions in the context of the complete fuzzy metric spaces. Then, some of the results are extended in the framework of the fuzzy metric spaces equipped with a partial order. In this direction, we give some examples to clarify the obtained results and definitions. Additionally, we demonstrate an application about the solutions of non-linear matrix equations on the basis of fixed points of these new contractions.

Based on the generalized fractional integrals, we develop some Hermite-Hadamard-type inequalities for interval-valued s-convex functions. Further, we verify our results with graphical representation and concrete examples. Our research not only generalizes and extends the existing literature but also provides valuable insights for further research on interval-valued integral inequalities.