Our purpose in this paper is to establish some new common fixed point theorems for four self-mappings of a dislocated metric space.

In this paper, we investigate the existence of positive solutions for second-order nonlinear three-point integral boundary value problems. By using the Leray-Schauder fixed point theorem, some sufficient conditions for the existence of positive solutions are obtained, which improve the results of literature Tariboon and Sitthiwirattham [J. Tariboon, T. Sitthiwirattham, Boundary Value Problems, 2010 (2010), 1-11].

In this work we study integral boundary value problem involving Caputo differentiation \[ \begin{cases} ^c D^q_t u(t)= f(t,u(t)),\,\, 0<t<1,\\ \alpha u(0)-\beta u(1)=\int^1_0 h(t)u(t)dt, \gamma u'(0)-\delta u'(1)\int^1_0 g(t)u(t)dt, \end{cases} \] where \(\alpha,\beta,\gamma,\delta\) are constants with \(\alpha>\beta>0,\gamma>\delta>0, f\in C([0,1]\times \mathbb{R}^+,\mathbb{R}), g,h\in C([0,1],\mathbb{R}^+)\) and \( ^c D^q_t\) is the standard Caputo fractional derivative of fractional order \(q(1 < q < 2)\). By using some fixed point theorems we prove the existence of positive solutions.

In this paper, we study the existence of positive solutions for a class of coupled integral boundary value problems of nonlinear semipositone Hadamard fractional differential equations \[D^\alpha u(t) + \lambda f(t, u(t), v(t)) = 0,\quad D^\alpha v(t) + \lambda g(t, u(t), v(t)) = 0,\quad t \in (1, e),\quad \lambda > 0\] \[u^{(j)}(1) = v^{(j)}(1) = 0, 0 \leq j \leq n - 2; u(e) = \mu\int^e_1 v(s) \frac{ds}{ s} , v(e) = \nu\int^e_1 u(s) \frac{ds}{ s},\] where \(\lambda,\mu,\nu\) are three parameters with \(0<\mu<\beta\) and \(0<\nu<\alpha,\quad \alpha,\beta\in (n - 1; n]\) are two real numbers and \(n\geq 3, D^\alpha, D^\beta\) are the Hadamard fractional derivative of fractional order, and \(f; g\) are sign-changing continuous functions and may be singular at \(t = 1\) or/and \(t = e\). First of all, we obtain the corresponding Green's function for the boundary value problem and some of its properties. Furthermore, by means of the nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorems, we derive an interval of \(\lambda\) such that the semipositone boundary value problem has one or multiple positive solutions for any \(\lambda\) lying in this interval. At last, several illustrative examples were given to illustrate the main results.

In this paper, we prove coupled coincidence and coupled common fixed point theorems for compatible mappings in partially ordered G-metric spaces. The results on fixed point theorems are generalizations of some existing results. We also give an example to support our results.

This paper deals with almost periodic Hematopoiesis dynamic equation on time scales. By applying a novel method based on the fixed point theorem of decreasing operator, we establish sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence which converges to the almost periodic positive solution. Moreover, we investigate global exponential stability of the almost periodic positive solution by means of Gronwall inequality.

In this paper we present a new extension of coupled fixed point theorems in metric spaces endowed with a reflexive binary relation that is not necessarily neither transitive nor antisymmetric. The key feature in this coupled fixed point theorems is that the contractivity condition on the nonlinear map is only assumed to hold on elements that are comparable in the binary relation. Next on the basis of the coupled fixed point theorems, we prove the existence and uniqueness of positive definite solutions of a nonlinear matrix equation.

Boundary value problems for the elliptic sinh-Gordon equation formulated in the half plane are studied by applying the so-called Fokas method. The method is a significant extension of the inverse scattering transform, based on the analysis of the Lax pair formulation and the global relation that involves all known and unknown boundary values. In this paper, we derive the formal representation of the solution in terms of the solution of the matrix Riemann-Hilbert problem uniquely defined by the spectral functions. We also present the global relation associated with the elliptic sinh-Gordon equation in the half plane. We in turn show that given appropriate initial and boundary conditions, the unique solution exists provided that the boundary values satisfy the global relation. Furthermore, we verify that the linear limit of the solution coincides with that of the linearized equation known as the modified Helmhotz equation.