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Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces

Dumitru Baleanu123*, Ravi P Agarwal4, Hakimeh Mohammadi5 and Shahram Rezapour5

Author Affiliations

1 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

2 Department of Mathematics, Cankaya University, Ogretmenler Cad. 14 06530, Balgat, Ankara, Turkey

3 Institute of Space Sciences, Magurele, Bucharest, Romania

4 Department of Mathematics, Texas A&M University, University Blvd., Kingsville, 78363-8202, USA

5 Department of Mathematics, Azarbaidjan Shahid Madani University, Azarshahr, Tabriz, Iran

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Boundary Value Problems 2013, 2013:112  doi:10.1186/1687-2770-2013-112

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/112


Received:9 August 2012
Accepted:16 April 2013
Published:3 May 2013

© 2013 Baleanu et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By using fixed point results on cones, we study the existence and uniqueness of positive solutions for some nonlinear fractional differential equations via given boundary value problems. Examples are presented in order to illustrate the obtained results.

1 Introduction

The field of fractional differential equations has been subjected to an intensive development of the theory and the applications (see, for example, [1-6] and the references therein). It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions. There are some papers dealing with the existence of solutions of nonlinear initial value problems of fractional differential equations by using the techniques of nonlinear analysis such as fixed point results, the Leray-Schauder theorem, stability, etc. (see, for example, [7-19] and the references therein). In fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena and aerodynamics (see, for example, [20-23] and the references therein). The main advantage of using the fractional nonlinear differential equations is related to the fact that we can describe the dynamics of complex non-local systems with memory. In this line of taught, the equations involving various fractional orders are important from both theoretical and applied view points. We need the following notions.

Definition 1.1 ([1,4])

For a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M1">View MathML</a>, the Caputo derivative of fractional order α is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M2">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M5">View MathML</a> denotes the integer part of α.

Definition 1.2 ([1,4])

The Riemann-Liouville fractional derivative of order α for a continuous function f is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M6">View MathML</a>

where the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M7">View MathML</a>.

Definition 1.3 ([1,4])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M8">View MathML</a> be an interval in ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M9">View MathML</a>. The Riemann-Liouville fractional order integral of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M10">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M11">View MathML</a>

whenever the integral exists.

Suppose that E is a Banach space which is partially ordered by a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M12">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M13">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M14">View MathML</a>. We denote the zero element of E by θ. A cone P is called normal if there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M15">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M16">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M17">View MathML</a> (see [24]). Also, we define the order interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M19">View MathML</a>[24]. We say that an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M20">View MathML</a> is increasing whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M13">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M22">View MathML</a>. Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M23">View MathML</a> means that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M25">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M26">View MathML</a> (see [24]). Finally, put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M28">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M29">View MathML</a> is convex and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M30">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M31">View MathML</a>. We recall the following in our results. Let E be a real Banach space and let P be a cone in E. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M32">View MathML</a> be an interval and let τ and φ be two positive-valued functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M33">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M35">View MathML</a> is a surjection. We say that an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M36">View MathML</a> is τ-φ-concave whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M37">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M39">View MathML</a>[13]. We say that A is φ-concave whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M40">View MathML</a> for all t[13]. We recall the following result.

Theorem 1.1 ([13])

LetEbe a Banach space, letPbe a normal cone inE, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M36">View MathML</a>be an increasing andτ-φ-concave operator. Suppose that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M42">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M43">View MathML</a>. Then there are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M44">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M45">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M47">View MathML</a>, the operator A has a unique fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M48">View MathML</a>, and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M49">View MathML</a>and the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M50">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M51">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M52">View MathML</a>.

2 Main results

We study the existence and uniqueness of a solution for the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M53">View MathML</a>

on partially ordered Banach spaces with two types of boundary conditions and two types of fractional derivatives, Riemann-Liouville and Caputo.

2.1 Existence results for the fractional differential equation with the Riemann-Liouville fractional derivative

First, we study the existence and uniqueness of a positive solution for the fractional differential equation

(2.1)

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M56">View MathML</a> is the Riemann-Liouville fractional derivative of order α. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M57">View MathML</a>. Consider the Banach space of continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M58">View MathML</a> with the sup norm and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M59">View MathML</a>. Then P is a normal cone.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M63">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M64">View MathML</a>. Then the problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M65">View MathML</a>with the boundary value condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M66">View MathML</a>has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M67">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M67">View MathML</a>is a solution of the fractional integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M69">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M70">View MathML</a>

Proof From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M65">View MathML</a> and the boundary condition, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M72">View MathML</a>. By the definition of a fractional integral, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M73">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M74">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M75">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M76">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M77">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M78">View MathML</a>

This completes the proof. □

Now, we are ready to state and prove our first main result.

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M60">View MathML</a>be given and letτandφbe two functions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M80">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M81">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M82">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M83">View MathML</a>is a surjection and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M84">View MathML</a>is increasing inufor each fixedt, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M86">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M87">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M88">View MathML</a>. Assume that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M90">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M91">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M92">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M62">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M94">View MathML</a>is the green function defined in Lemma 2.1. Then the problem (2.1) with the boundary value condition (2.2) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M95">View MathML</a>. Moreover, for the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M96">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M97">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M98">View MathML</a>.

Proof By using Lemma 2.1, the problem is equivalent to the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M99">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M100">View MathML</a>

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M101">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M102">View MathML</a>. Then u is a solution for the problem if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M103">View MathML</a>. It is easy to check that the operator A is increasing on P. On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M104">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M88">View MathML</a>. Thus, the operator A is τ-φ-concave. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M107">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M62">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M109">View MathML</a>. Now, by using Theorem 1.1, the operator A has a unique positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M95">View MathML</a>. This completes the proof. □

Here, we give the following example to illustrate Theorem 2.2.

Example 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M111">View MathML</a> be given. Consider the periodic boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M112">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M113">View MathML</a>, g is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M115">View MathML</a>. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M116">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M117">View MathML</a>. Now, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M121">View MathML</a> and also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M122">View MathML</a> for all t. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M123">View MathML</a> is a surjection and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M124">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M125">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M126">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M127">View MathML</a>

Now, put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M130">View MathML</a>. Then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M131">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M132">View MathML</a>

Thus, by using Theorem 2.2, the problem has a unique solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M133">View MathML</a>.

2.2 Existence results for the fractional differential equation with the Caputo fractional derivative

Here, we study the existence and uniqueness of a positive solution for the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M134">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M135">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M136">View MathML</a> is the Caputo fractional derivative of order α. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M137">View MathML</a> be the Banach space of continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M138">View MathML</a> with the sup norm and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M139">View MathML</a>

It is known that P is a normal cone. Similar to the proof of Lemma 2.1, we can prove the following result.

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M143">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M144">View MathML</a>. Then the problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M145">View MathML</a>with the boundary value conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M146">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M147">View MathML</a>has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M67">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M67">View MathML</a>is a solution of the fractional integral equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M150">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M151">View MathML</a>

Theorem 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M61">View MathML</a>be given and letτandφbe two positive-valued functions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M153">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M154">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M155">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M156">View MathML</a>is a surjection and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M84">View MathML</a>is increasing inufor each fixedt, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M158">View MathML</a>whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M159">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M160">View MathML</a>otherwise, and also<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M161">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M88">View MathML</a>. Assume that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M90">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M91">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M167">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M142">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M94">View MathML</a>is the green function defined in Lemma 2.3. Then the problem (2.3) with the boundary value conditions (2.4) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M95">View MathML</a>. Moreover, for the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M96">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M97">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M98">View MathML</a>.

Proof It is sufficient to define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M101">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M175">View MathML</a>

Now, by using a similar proof of Theorem 2.2, one can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M176">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M142">View MathML</a>, and also the operator A is τ-φ-concave. By using Theorem 1.1, the operator A has a unique positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M95">View MathML</a>. This completes the proof by using Lemma 2.3. □

Below we present an example to illustrate Theorem 2.4.

Example 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M180">View MathML</a>. Consider the periodic boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M181">View MathML</a>

where g is a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M182">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M183">View MathML</a>. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M185">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M186">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M187">View MathML</a>. Now, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M192">View MathML</a>. Then it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M193">View MathML</a> is a surjection map and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M194">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M195">View MathML</a>. Also, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M196">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M126">View MathML</a>. Now, put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M199">View MathML</a> and also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M200">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M201">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M202">View MathML</a>

Thus, by using Theorem 2.4, the problem has a unique solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/112/mathml/M133">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Authors contributed equally in writing this article. Authors read and approved the final version of the manuscript.

Acknowledgements

This work is partially supported by the Scientific and Technical Research Council of Turkey. Research of the third and forth authors was supported by Azarbaidjan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions which improved final version of this paper.

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