In this paper, we are concerned with the study of the existence and uniqueness of fixed points for the class of functions $f:C\to C$ satisfying the inequality

$$\ell \left(\alpha f(t)+(1-\alpha )f(s)\right)\le \sigma \ell (\alpha t+(1-\alpha )s)$$

for every $t,s\in C$ with $f(t)\ne f(s)$, where $C$ is a closed subset of $[0,\mathrm{\infty })$, $\alpha ,\sigma \in (0,1)$ are constants, and $\ell :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a function satisfying the condition $\underset{t>0}{inf}\frac{\ell (t)}{t^{\rho }}>0$ for some constant $\rho >0$. Namely, under a weak continuity condition imposed on $f$, we show that $f$ possesses a unique fixed point, and for every $t_0\in C$, the Picard sequence defined by $t_{n+1}=f(t_n)$, $n\ge 0$, converges to this fixed point. Next, we study the special cases when $C$ is a closed interval and $\ell$ is a convex or concave function. Namely, making use of the Hermite-Hadamard inequalities, we obtain several new fixed point theorems. To the best of our knowledge, the considered class of functions was never previously investigated in the literature.

We consider the class of functions $u\in C^2((0,\mathrm{\infty }))$ satisfying second-order differential inequalities in the form $u^{″}(x)+\frac{k}{x^2}u(x)\ge 0$ for all $x>0$. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases ( $k=\frac{1}{4}$ and $0<k<\frac{1}{4}$). We next extend our obtained results to the two-dimensional case. In the limit case $k\to 0^{+}$ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations.

This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erdélyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying periodic contraction rate has been explored by using the Banach contraction principle. Finally, two numerical examples demonstrate the practical application of these findings in the realm of fractional integral equation theory.

This work primarily delves into three key areas: the presence of mild solutions, exploration of the topological and geometrical makeup of solution sets, and the continuous dependency of solutions on a second-order semilinear integro-differential inclusion. The Bohnenblust-Karlin fixed-point method has been integrated with Grimmer's theory of resolvent operators. Ultimately, the study delves into a mild solution for a partial integro-differential inclusion to showcase the achieved outcomes.

In this manuscript, we use the notion of orthogonality in a super metric space and prove some fixed point theorems that focus on orthogonal contractions and orthogonal $\mathcal{F}$-contractions, which are types of mappings exhibiting particular contraction properties while satisfying orthogonality. To illustrate and support our theoretical results, we provide concrete examples that demonstrate the application of these findings. Furthermore, we show how our results can be utilized in practical scenarios by presenting a solution to a deformable implicit differential equation, highlighting the relevance of our work in both theoretical and applied contexts.