Existence Theorems Of Positive Solutions Research Articles

Solvability for a Hadamard-type fractional integral boundary value problem

In this paper, we study an integral boundary value problem involving a Hadamard-type fractional differential equation. Using fixed point theory and upper-lower solutions, we present some sufficient conditions to obtain existence theorems of positive solutions for the problem. Examples are provided to illustrate our results.

A class of iterative functional fractional differential equation on infinite interval

In this paper, a class of iterative functional fractional differential equations on the infinite interval with integral boundary conditions are presented. The existence theorem of positive solutions are established by using monotone iterative technique. The results obtained in this paper reflect the influence of feedback iteration term on the existence of solutions. Finally, an example is given out to illustrate the adaptability and universality of our main results.

Existence of positive solutions for a system of nonlinear Caputo type fractional differential equations with two parameters

The main purpose of this paper is to prove the existence of positive solutions for a system of nonlinear Caputo-type fractional differential equations with two parameters. By using the Guo–Krasnosel’skii fixed point theorem, some existence theorems of positive solutions are obtained in terms of different values of parameters. Two examples are given to illustrate the main results.

Existence of positive solutions for boundary value problems of nonlinear fractional functional integro-differential equations

In this paper, we study the existence of positive solutions for a class of fractional functional integro-differential equations with two fractional derivative terms. First, we transform the boundary value problem into an equivalent integral equation, establish the operator T and prove its full continuity, then the existence theorems of positive solutions of boundary value problems is established by using the fixed point theorems of cone extension and cone compression.

Positive solutions to nonlinear inclusion problems in Orlicz–Sobolev spaces

In this paper, we study the existence of positive solutions to a nonlinear elliptic inclusion problem driven by an elliptic differential operator with Dirichlet boundary and a multivalued term depending on the solution and its gradient in reflexive Orlicz–Sobolev spaces by using a linear functional analysis method and a subsolution–supersolution method. We provide some sufficient conditions to ensure that a subsolution and a supersolution exist. An existence theorem for positive solutions of the problem is established.

Some existence results of positive solutions for p-Laplacian systems

In this paper, we establish several new existence theorems for positive solutions of systems of (2n,2m)-order of two p-Laplacian equations. The results are based on the Krasnosel’skii fixed point theorem and mainly complement those of Djebali, Moussaoui, and Precup.

Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis.

Existence results and a priori estimates for solutions of quasilinear problems with gradient terms

In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in \(\mathbb{R}^N\) with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of \(u\) and by removing the growth condition for the nonlinearity \(f\) at \(u=0\).

Existence criteria of solutions for a fractional nonlocal boundary value problem and degeneration to corresponding integer-order case

In this paper, we mainly discuss the existence and uniqueness results of solutions to fractional differential equations with multi-strip boundary conditions. When the fractional order α becomes integer, the existence theorem of positive solutions can be established by a monotone iterative technique. Also, some examples are presented to illustrate the main results.

Existence of solutions for quasilinear Dirichlet problems with gradient terms

In this paper we prove an existence theorem for positive solutions of a nonlinear Dirichlet problem involving the p-Laplacian operator on a smooth bounded domain when a nonlinearity depending on the gradient is considered. Our main theorem extends a previous result by Ruiz in [ 19 ], in which a slight modification of the celebrated blowup technique due to Gidas and Spruck, [ 11 ] and [ 12 ] is introduced.

Positive Solutions for a System of Semipositone Fractional Difference Boundary Value Problems

Using the fixed point index, we establish two existence theorems for positive solutions to a system of semipositone fractional difference boundary value problems. We adopt nonnegative concave functions and nonnegative matrices to characterize the coupling behavior of our nonlinear terms.

Positive solutions to nonlinear nonhomogeneous inclusion problems with dependence on the gradient

The goal of the paper is to study a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with the Dirichlet boundary condition and a convection multivalued term. An existence theorem for positive solutions of the problem is established by applying the method of subsolution–supersolution, together with truncation and comparison techniques.

Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary (M,∂M,〈,〉) and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the so called Conformal Method.

Positive solubility of some classes of non-linear integral equations of Hammerstein type on the semi-axis and on the whole line

We study certain classes of non-linear Hammerstein integral equations on the semi-axis and the whole line. These classes of equations arise in the theory of radiative transfer in nuclear reactors, in the kinetic theory of gases, and for travelling waves in non- linear Richer competition systems. By combining special iteration methods with the methods of construction of invariant cone segments for the appropriate non- linear operator, we are able to prove constructive existence theorems for positive solutions in various function spaces. We give illustrative examples of equations satisfying all the hypotheses of our theorems.

Positive solutions for a coupled system of nonlinear fractional differential equations

Using the Krasnoselskii–Zabreiko fixed point theorem, we establish two existence theorems for positive solutions of a coupled system of nonlinear fractional differential equations. Power functions and nonnegative matrices are used to characterize coupling behavior of our nonlinearities, so nonlinearities may grow differently; in fact, one may grow superlinearly, and the other may grow sublinearly. Copyright © 2014 John Wiley & Sons, Ltd.

Removable singularities and singular solutions of semilinear elliptic equations

This note proves that a closed set with appropriate properties is removable for solutions of semilinear elliptic equations satisfying a certain growth condition near that set. Also, we give the existence theorem of positive solutions with singularities on a prescribed compact set, which shows that a growth condition in our removability theorem is optimal.

Existence of positive solutions for second order multi-point boundary value problems

We study the second order nonlinear differential equation: (E) y′′ +h(t) f (y) = 0, 0 < t < 1, h(t) ∈ L(0,1), f (y) ∈C(R,R+) subject to multi-point boundary conditions: (i) y(0) = m ∑ i=1 αiy(ξi), y(1) = m ∑ i=1 βiy(ξi), (ii) y′(0) = m ∑ i=1 αiy′(ξi), y(1) = m ∑ i=1 βiy(ξi), (iii) y(0) = m ∑ i=1 αiy(ξi), y′(1) = m ∑ i=1 βiy′(ξi), where 0 < ξ1 < · · · < ξm < 1,αi 0,βi 0, i = 1,2, · · · ,m . We also assume that h(t) is nonnegative and can be singular at t = 0 or t = 1 or both and α1 + · · ·+αm < 1 , β1 + · · ·+βm < 1 . We prove existence theorems for positive solutions of (E) (i), (E) (ii) and (E) (iii) when the limit f (y)/y as y → 0 and y→ ∞ do not necessarily exist. Our results extend recent results of Zhang and Sun [18] for the boundary value problem (E)(i) when αi ≡ 0 for all i = 1, . . . ,m . Mathematics subject classification (2010): 34B10, 34B15.

Nonlinear oscillations of fourth order quasilinear ordinary differential equations

We consider fourth order quasilinear ordinary differential equations. Firstly, we classify positive solutions into four types according to their asymptotic properties. Then we derive existence theorems of positive solutions belonging to each type. Using these results, we can obtain an oscillation criterion, which is our main objective. Moreover, applying such criteria for ordinary differential equations to binary elliptic systems, we establish nonexistence theorems for positive solutions.

Existence of positive solutions of singular elliptic boundary value problems in a ball

This paper deals with existence theorems of positive solutions for singular elliptic boundary value problems in a ball. The results of this paper are generalizations of those proved by authors cited in the references.

Local existence of multiple positive solutions to a singular cantilever beam equation

By constructing suitable cone and control functions, we prove some local existence theorems of positive solutions for a singular fourth-order two-point boundary value problem. In mechanics, the problem is called cantilever beam equation. Furthermore, we improve a famous method appeared in the studies of singular boundary value problems. The approximation theorem of completely continuous operators and the Guo–Krasnosel'skii fixed point theorem of cone expansion–compression type play important parts in this work.