Positive solutions for a fourth order boundary value problem, Electronic Journal of Qualitative Theory of differential Equations

This paper deals with the existence and multiplicity of positive solutions for the fourth-order boundary value problem . Here . We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some integral identities and integral inequalities.

(3) By using the monotone iterative technique, the authors proved that problem (3) has at least one symmetric positive solution under certain conditions.In [17], Pang et al. studied the existence and multiplicity of nontrivial solutions for the fourth-order boundary value problem  (4) =   − ′′    () =  (1) =  ′′ () =  ′′ (1) =  (4) where   ( 2  ).Making use of the theory of Leray-Schauder degree, under appropriate conditions on the nonlinearity , the authors proved that problem (4) has at least six different nontrivial solutions, including two positive ones and two negative ones.
Motivated by [18], in this paper, we will discuss the existence and multiplicity of positive solutions for problem (1).As in [18], we �rst use the method of order reduction to transform (1) into a boundary value problem for a secondorder integro-differential equation and then seek the existence and multiplicity of positive solutions for the resultant problem.To overcome the difficulties stemming from the presence of derivatives of all orders and the difference between (1) and ( 5) in their boundary value conditions, we have to prove that the maximum of every nonnegative concave function  can be dominated by the integrals (see Lemmas 5 and 6 below for more details).Based on a priori estimates achieved by utilizing some integral identities and integral inequalities, we use �xed point index theory to prove the existence and multiplicity of positive solutions for (1).is paper is organized as follows.In Section 2, we transform (1) into a boundary value problem for a secondorder integro-differential equation and then establish some basic integral identities and integral inequalities that are useful in deriving the priori estimates in the next section.Our main results, namely eorems 10-12, are stated and proved in Section 3.

Preliminaries
Let Now       4 +   + ) implies that  is a completely continuous operator.In our setting, the existence of positive solutions for (1) is equivalent to that of positive �xed points of     .
To establish the priori estimates of positive solutions for some problems associated with (11), we need several integral identities and integral inequalities below.
Proof.Integrating by parts and using  ′ )  )  , we have from which (13) follows.is completes the proof.