We study the three-point boundary value problem of higher-order fractional differential equations of the form ${Dc}_{0+}^{\rho }u\left(t\right)+f\left(t, u\left(t\right)\right)=0$, , , $u^{\prime }(0)=u^{\prime \prime }(0)=\cdots =u^{\left(n-1\right)}(0)=0$, $u(1)+p{u}^{\prime }(1)=q{u}^{\prime }(\mathrm{\xi })$, where ${\mathrm{ }}^c{D}_{0+}^{\rho }$ is the Caputo fractional derivative of order $\rho$, and the function $f:[\mathrm{0,1}]\times [0,\mathrm{\infty })\mapsto [0,+\mathrm{\infty })$ is continuously differentiable. Here, $0⩽q⩽p$, , . By virtue of some fixed point theorems, some sufficient criteria for the existence and multiplicity results of positive solutions are established and the obtained results also guarantee that the positive solutions discussed are monotone and concave.

We consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form: $x(0)=\beta x(\theta ),\hspace{0.17em}x(\xi )=\alpha {\int }_{\eta }^1\mathrm{‍}x(s)ds$, and . According to these conditions, the value of the unknown function at the left end point $t=0$ is proportional to its value at a nonlocal point $\theta$ while the value at an arbitrary (local) point $\xi$ is proportional to the contribution due to a substrip of arbitrary length $(1-\eta )$. These conditions appear in the mathematical modelling of physical problems when different parts (nonlocal points and substrips of arbitrary length) of the domain are involved in the input data for the process under consideration. We discuss the existence of solutions for the given problem by means of the Sadovski fixed point theorem for condensing maps and a fixed point theorem due to O’Regan. Some illustrative examples are also presented.

We give a Ćirić-Berinde type contractive condition for multivalued mappings and analyze the existence of fixed point for these mappings.

In 2012, Samet et al. introduced the notion of α-ψ-contractive mapping and gave sufficient conditions for the existence of fixed points for this class of mappings. The purpose of our paper is to study the existence of fixed points for multivalued mappings, under an α-ψ-contractive condition of Ćirić type, in the setting of complete b-metric spaces. An application to integral equation is given.

We introduce a generalized proximal weak contraction of rational type for the non-self-map and proved results to ensure the existence and uniqueness of best proximity point for such mappings in the setting of partially ordered metric spaces. Further, our results provides an extension of a result due to Luong and Thuan (2011) and also it provides an extension of Harjani (2010) to the case of self-mappings.

In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature.