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In this paper, we study a higher-order fractional differential equation with integral boundary conditions and a parameter. Under different conditions of nonlinearity, existence and nonexistence results for positive solutions are derived in terms of different intervals of parameter. Our approach relies on the Guo–Krasnoselskii fixed point theorem on cones.

In this paper, using a mixed monotone operator method, we study the existence and uniqueness of positive solutions to a nonlinear arbitrary order fractional differential equation. An example is provided to illustrate the main result.

We study the existence of nonnegative solutions for a system of impulsive differential equations subject to nonlinear, nonlocal boundary conditions. The system presents a coupling in the differential equation and in the boundary conditions. The main tool that we use is the theory of fixed point index for compact maps.

Numerical solutions of, unsteady laminar free convection from an incompressible viscous fluid past a vertical cone with uniform surface heat flux is presented in this paper. The dimensionless governing equations of the flow that are unsteady, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicolson type. The velocity and temperature fields have been studied for various parameters Prandtl number and semi vertical angle. The local as well as average skin-friction and Nusselt number are also presented and analyzed graphically. The present results are compared with available results in literature and are found to be in good agreement.

Natural convection effects of the numerical solution for unsteady, laminar, free convection flow over an incompressible viscous fluid past a non-isothermal vertical cone with surface temperature T′ w(x) = T ′∞ + axn varying as power function of distance from the apex (x = 0) is presented here. The dimensionless governing equations of the flow that are unsteady, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicolson type. The velocity and temperature fields have been studied for various parameters Prandtl number, semi vertical angle 0 ◦ < φ < 90◦ and n. The local as well as average skin-friction and Nusselt number are also presented and analyzed graphically. The present results are compared with available results in literature and are found to be in good agreement.

Abstract: In this paper we consider a second-order boundary value problem of the form , , . We demonstrate that if the nonlinear functional satisfies certain asymptotic behavior, then the problem will possess at least one positive solution. These results generalize and improve on some recent results in the literature on boundary value problems with nonlocal, nonlinear boundary conditions, and we provide some examples, which illustrate these generalizations and improvements. [Copyright &y& Elsevier]

Abstract: While the nonlinearities are bounded below, we establish the existence and multiplicity results of doubly periodic solutions for a coupled nonlinear telegraph system with a parameter. The proof is based on a well known fixed theorem in a cone. [Copyright &y& Elsevier]

Abstract: We obtain critical point variants of the compression fixed point theorem in cones of Krasnoselskii. Critical points are localized in a set defined by means of two norms. In applications to semilinear elliptic boundary value problems this makes possible the use of local Moser–Harnack inequalities for the estimations from below. Multiple solutions are found for problems with oscillating nonlinearity. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the boundary value problem subject to the boundary conditions and . In this setting, is a continuous functional, which represents a nonlinear nonlocal boundary condition. By imposing sufficient structure on and the nonlinearity , we deduce the existence of at least one positive solution to this problem. The novelty in our setting lies in the fact that may be strictly nonpositive for some . Our results are achieved by appealing to the Krasnosel’skiĭ fixed point theorem. We conclude with several examples illustrating our results and the generalizations that they afford. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a first-order delta dynamic boundary value problem of the form , for , subject to either the boundary condition , the boundary condition , or the boundary condition for , where is a time scale, and are continuous functions, is a given linear functional with the restriction that depends upon , and is a given continuous function. In this case, the function is the so-called one-dimensional -Laplacian. Our results here generalize recent results in the literature on this type of problem. We conclude with several numerical examples illustrating the results and the improvements and generalizations that they provide. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the existence of periodic solutions for the th-order ordinary differential equation where the nonlinear term is a continuous odd function and is -periodic in . Some existence results for odd -periodic solutions are obtained under the condition that satisfies some linear, superlinear or sublinear growth conditions on . [Copyright &y& Elsevier]

Abstract: This paper concerns the existence of positive solution for a class of second-order -point boundary value problems under different resonant conditions. By using the Leggett–Williams norm-type theorem due to O’Regan and Zima, we obtain the existence of positive solution. An example is given to demonstrate the main results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the multiplicity of positive solutions for a class of singular higher-order perturbed differential systems with different orders. By employing a well-known fixed point theorem, some new existence results are given under the case where nonlinearity can be sign changing. [Copyright &y& Elsevier]

Abstract: This paper investigates the existence of positive solutions of singular multi-point boundary value problems of fourth order ordinary differential equation with -Laplacian. A necessary and sufficient condition for the existence of positive solution as well as pseudo- positive solution is given by means of the fixed point theorems on cones. [Copyright &y& Elsevier]

Abstract: In this paper, we study a class of one-dimensional -Laplacian boundary value problem with generalized Sturm–Liouville boundary conditions. Some results are obtained for existence of at least three positive solutions by using the related fixed point theory. The interesting point is the nonlinear term which involves the first order derivative explicitly. [Copyright &y& Elsevier]

Abstract: In this paper, by using fixed-point index theory and a new fixed-point theorem in cones, some sufficient conditions for the existence of countably many positive solutions for some nonlinear singular impulsive boundary value problems are established. In the case of the -Laplace operator, our results for some generalize previous known results. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point theorem in cones, the existence of positive solutions for systems of nonlinear second-order semipositone boundary value system which nonlinearity and are both semipositone, is obtained under the condition that and are suplinear or sublinear. The results obtained in this paper essentially improve and generalize many well-known results. [Copyright &y& Elsevier]

Abstract: This paper deals with the eigenvalue problems for a class of positive nonlinear operators. The projective metric technique is used to prove the existence of a unique fixed point of the operator. We then show how to apply our results to the study of nonlinear elliptic problems. [Copyright &y& Elsevier]

Abstract: We show the existence of positive solutions to some Dirichlet problem of semilinear elliptic equations on annulus with singular nonlinearity on the boundary, and emphasize the sharpness of the results in the sublinear case. [Copyright &y& Elsevier]

Abstract: We establish the existence of periodic solution of a class of non-autonomous second-order systems, , where , if , uniformly in and is bounded below or above for appropriate ranges of . [Copyright &y& Elsevier]

Abstract: We study the existence of positive solutions for a system of perturbed Hammerstein integral equations. Our setting is quite general and covers a wide class of systems of boundary value problems. The main ingredient in our theory is the classical fixed point index theory for compact maps. [Copyright &y& Elsevier]

Abstract: In this paper, by using the Avery–Peterson fixed point theorem on a cone, we establish the existence of three positive solutions for -point boundary value problems on infinite intervals where is a continuous function, are given. [Copyright &y& Elsevier]

Abstract: We consider existence of nonzero solutions to the following boundary value problem where and are positive parameters, . We prove that solutions lose positivity as the parameter or increases. In particular, we study problems where the associated integral equation has a kernel that changes sign. The proof is based on the fixed point theorem in cones. [Copyright &y& Elsevier]

Abstract: This paper presents some new methods of computation of the fixed point index for cone mapping. As applications of our main results, the existence of positive solutions for the fourth-order boundary value problem both in scale spaces and in abstract spaces is considered. [Copyright &y& Elsevier]

Abstract: Using the cone theory and the lattice structure, we establish some methods of computation of the fixed point index and the topological degree for the unilaterally asymptotically linear operators which are not assumed to be cone mappings. As applications, the existence results of boundary value problems for nonlinear elliptic partial differential equations are given. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Finally we apply the main theorem to a nonlinear parabolic partial differential equation. Some results are new even for increasing or decreasing operators. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of countably many positive solutions for some nonlinear singular three-point boundary problems on the half-line where is an increasing homeomorphism and positive homomorphism and and , and has countably many singularities on . By using the fixed-point index theorem and a new fixed-point theorem in cones, the existence of countably many solutions for singular three-point boundary value problem are obtained under conditions weaker than those used by Liu and Zhang [B.F. Liu, J.H. Zhang, The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions, J. Math. Anal. Appl. 309 (2005) 505–516]. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of positive solutions for a class of nonlinear second-order four-point boundary value problems with alternating coefficient. Our approach relies on the Krasnosel’skii fixed point theorem. The result of this paper is new and extent the previously known result. [Copyright &y& Elsevier]

Abstract: Let be a time scale with . We investigate the existence of solutions to the nonlinear even-order three-point boundary value problem where , , and are given constants. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of positive solutions is investigated for the following singular fourth-order Sturm–Liouville boundary value problem with changing sign nonlinearity: where is continuous, is continuous, is Lebesgue integrable. Without any nonnegative assumption on nonlinearity, by choosing a proper cone and creating an available integral operator, the existence of the positive solutions for the above singular Sturm–Liouville boundary value problem is established. Some examples are worked out to indicate the application of our main result. [Copyright &y& Elsevier]

Abstract: We introduce a first-order delta dynamic equation on time scales involving the one-dimensional -Laplacian, and prove the existence of at least one positive solution. An example applying our result is also given. [Copyright &y& Elsevier]

Abstract: Using the lattice structure, we give some methods of computation of the topological degree for the operator which is quasi-additive on lattice. The operator is not assumed to be a cone mapping. As applications, the existence of nontrivial solutions for the nonlinear Sturm–Liouville problem is discussed. [Copyright &y& Elsevier]

Abstract: In this paper, a new monotonicity, -monotonicity, is introduced, and the resolvent operator of an -monotone operator is proved to be single valued and Lipschitz continuous. With the help of the resolvent operator, an equivalence between the variational inequality VI and the fixed point problem of a nonexpansive mapping is established. A proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method, which is based on the assumption that the projection mapping is semismooth, is given for calculating -solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable. [Copyright &y& Elsevier]

Abstract: In this paper, we study the nonlinear four-point boundary value problem with the -Laplacian where . By applying a fixed point theorem in cones, sufficient conditions are given for the existence of a positive solution and multiple positive solutions. The interesting point is the Sturm–Liouville-like boundary condition, which was rarely treated until now. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence of positive solutions for a class of singular fourth-order -Laplacian equation with multi-point boundary conditions. By means of a monotone iterative technique, a necessary and sufficient condition for the existence of a pseudo- positive solution is established. Also, the existence and uniqueness of the pseudo- positive solution, the iterative sequence of the solution and an error estimation for the pseudo- positive solution are obtained. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the following nonlinear third-order three-point boundary value problem: where and . First, the Green’s function for the associated linear boundary value problem is constructed, and then, some useful properties of the Green’s function are obtained by a new method. Finally, existence results for at least one positive solution for the above problem are established when is superlinear or sublinear. [Copyright &y& Elsevier]

Abstract: We study the existence of positive solutions to the operator equation , where is a linear Fredholm mapping of index zero and is a nonlinear multivalued operator. [Copyright &y& Elsevier]

Abstract: In this paper we provide some properties of a class of concave operators and apply these results in discussing three-point boundary value problems for differential equations and nonlinear Volterra integral equations. [Copyright &y& Elsevier]

Abstract: We study the nonlinear boundary value problem with multi-point boundary condition Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions of this problem using fixed point theorems on cones. Applications of our results to the special case where is a power function of and its derivatives are also discussed. Moreover, similar conclusions for a more general higher order boundary value problem are established. Our results extend some recent work in the literature on boundary value problems for ordinary differential equations. [Copyright &y& Elsevier]

Abstract: We consider the existence of positive solutions for the following fourth-order singular Sturm–Liouville boundary value problem: where may be singular at and/or 1. Moreover may also have singularity at . The existence and multiplicity theorems of positive solutions for the fourth-order singular Sturm–Liouville boundary value problem are obtained by using the first eigenvalue of the corresponding linear problems. Our results significantly extend and improve many known results including singular and nonsingular cases. [Copyright &y& Elsevier]

Abstract: The singular fourth-order nonlocal boundary value problem is considered under some suitable conditions concerning the first eigenvalue of the corresponding linear operator, where is continuous, symmetric on (0, 1) and may be singular at and is continuous and is symmetric on [0,1] for all . The existence of at least one symmetric positive solution is obtained by the application of the fixed point index in cones. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of sign-changing solutions for the fourth-order nonlinear boundary value problem where is a continuously odd function. Some conditions on guaranteeing the existence and multiplicity of sign-changing solutions are presented. The existence and multiplicity conditions concern the eigenline of the associated two-parameter linear eigenvalue problem. The discussion is based on the fixed point index theory in cones and an anti-symmetrical extension method of solution. [Copyright &y& Elsevier]

Abstract: The paper is concerned with a two-delay singular differential system with a twin parameter. Applying fixed-point index theory, we show the relationship between the asymptotic behaviors of nonlinearities (at zero and infinity) and the open regions (eigenvalue regions) of parameters, which are correlated with delays, such that the system has zero, one and two positive solution(s). [Copyright &y& Elsevier]

Abstract: For some fourth-order boundary value problems, several new existence theorems on multiple positive, negative and sign-changing solutions are obtained. The critical point theory and the supersolution and subsolution method are employed to discuss this problem. [Copyright &y& Elsevier]

Abstract: In this paper, some new multiple solutions theorems are obtained by introducing some new concepts such as the O-bounded cone. The abstract results obtained here are applied to ordinary differential equations and nonlinear systems of differential equations. [Copyright &y& Elsevier]

Abstract: We employ a partial order method and cone theory to prove an existence theorem for a fixed point for a semicontinuous operator; also the decreasing and increasing semicontinuous operators are studied. [Copyright &y& Elsevier]

Abstract: In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions for the initial value problem of the impulsive integro-differential equation of Volterra type in a Banach space where , , , and , . Under wide monotone conditions and the noncompactness measure condition of nonlinearity , we obtain the existence of extremal solutions and a unique solution between lower and upper solutions. Our result improves and extends some relevant results in abstract differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point index method, we establish the existence of at least one or at least two positive solutions to the three-point boundary value problem for a second order differential equation with an advanced argument where and . As an application, we also give an example to demonstrate our result. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a vector version of generalized Minty''s lemma and obtain existence theorems of solutions for two kinds of vector variational-like inequalities. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of positive solutions in case of the nonlinear fractional differential equationwhere , is the standard Riemann–Liouville fractional derivative, , , may change sign, and is a parameter. Our analysis relies on a nonlinear alternative of Leray–Schauder type. [Copyright &y& Elsevier]