Abstract
By means of the method of upper and lower solutions together with the Schauder fixed point theorem, the conditions for the existence of at least one positive solution are established for some higher-order singular infinite-point fractional differential equation with p-Laplacian. The nonlinear term may be singular with respect to both the time and the space variables.
Highlights
1 Introduction We investigate the existence of positive solutions for the following fractional differential equations containing a p-Laplacian operator (PFDE, for short) and infinite-point boundary value conditions: Dβ +(φp(Dα +u(t))) + f (t, u(t)) =, < t
In [ – ], by means of a fixed point theorem and the theory of the fixed point index together with the eigenvalue with respect to the relevant linear operator, the existence and multiplicity of positive solutions, pseudo-solutions are obtained for the m-point boundary value problem of the fractional differential equations (A) Dα +u(t) + q(t)f t, u(t) =, < t
Motivated by [ ], by introducing height functions of the nonlinear term on some bounded sets, we considered the local existence and multiplicity of positive solutions for BVP (A) with infinitepoint boundary value conditions in [ ]
Summary
We investigate the existence of positive solutions for the following fractional differential equations containing a p-Laplacian operator (PFDE, for short) and infinite-point boundary value conditions: Dβ +(φp(Dα +u(t))) + f (t, u(t)) = , < t < , u( ) = u ( ) = · · · = u(n– )( ) = , Dα +u( ) = , u(i)( ) =. Motivated by [ ], by introducing height functions of the nonlinear term on some bounded sets, we considered the local existence and multiplicity of positive solutions for BVP (A) with infinitepoint boundary value conditions in [ ]. Definition A continuous function (t) is called a lower solution of the PFDE ( ) if it satisfies. Lemma (Leray-Schauder fixed point theorem) Let T be a continuous and compact mapping of a Banach space E into itself, such that the set.
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