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We present some new common fixed point theorems for a pair of nonlinear mappings defined on an ordered Banach space. Our results extend several earlier works. An application is given to show the usefulness and the applicability of the obtained results.

The problem of existence of common fixed points to a pair of nonlinear mappings is now a classical theme. The applications to differential and integral equations made it more interesting. A considerable importance has been attached to common fixed point theorems in ordered sets [

Let

Let

In our considerations the following definition will play an important role. Let

A function

The family

If

The family

A measure of weak noncompactness

The first important example of a measure of weak noncompactness has been defined by De Blasi [

Notice that

By a measure of noncompactness on a Banach space

Let

Clearly, every

A map

The concept of ws-compact mappings arises naturally in the study of both integral and partial differential equations (see [

A map

Let

It is easy to prove that a nonlinear contraction mapping is a nonlinear set-contraction with respect to the Kuratowskii measure of noncompactness. We will prove that the same property holds for the De Blasi measure of weak noncompactness provided that the nonlinear contraction mapping is ww-compact.

Let

Let

The following theorem is a sharpening of [

Let

Let

In [

As an application of Theorem

As easy consequences of Theorem

Let

Let

Let

Note that in Corollary

Let

From Corollary

Let

Let

Let

As easy consequences of Theorem

Let

Let

Let

This follows from Theorem

Let

Let

It is worth noting that, in some applications, the weak sequential continuity is not easy to be verified. The ws-compactness seems to be a good alternative (see [

Let

Let

As easy consequences of Theorem

Let

This follows from Theorem

Let

This follows From Theorem

Let

Let

Note that if

Let

Let

The purpose of this section is to study the existence of integrable nonnegative solutions of the integral equation given by

Integral equations like (

for any fixed

for almost any

Let

Let

If

Although the Nemytskii operator

The problem of existence of nonnegative integrable solutions to (

the function

for all

If

From assumption (b) it follows that

Assume that the conditions (a–d) are satisfied. Then the implicit integral equation (

The problem (

Note that

Here

Let

Note that, for any

Our strategy consists in applying Corollary

The use of the dominated convergence theorem allows us to conclude that the sequence

Consequently,

Note that the

The first and second authors gratefully acknowledge the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.