Multiple Solutions for Some Classes Integro-Dynamic Equations on Time Scales

In this chapter we study a class of second-order integro-dynamic equations on time scales. A new topological approach is applied to prove the existence of at least two non-negative solutions. The arguments are based upon a recent theoretical result.


Introduction
Many problems arising in applied mathematics and mathematical physics can be modeled as differential equations, integral equations and integro-differential equations.
Integral and integro-differential equations can be solved using the Adomian decomposition method (ADM) [1,2], Galerkin method [3], rationalized Haar functions method [4], homotopy perturbation method (HPM) [5,6] and variational iteration method (VIM) [7]. ADM can be applied for linear and nonlinear problems and it is a method that represents the solution of the considered problems in the form of Adomian polynomials. Rationalized Haar functions and Galerkin methods are numerical methods that can be applied in different ways for the solutions of integral and integro-differential equations. VIM is an analytical method and can be used for different classes linear and nonlinear problems. HPM is a semi-analytical method for solving of linear and nonlinear differential, integral and integrodifferential equations.
In recent years, time scales and time scale analogous of some well-known differential equations, integral equations and integro-differential equations have taken prominent attention. The new derivative, proposed by Stefan Hilger in [8], gives the ordinary derivative if the time scale is the set of the real numbers and the forward difference operator if the time scale is the set of the integers. Thus, the need for obtaining separate results for discrete and continuous cases is avoided by using the time scales calculus.
This chapter outlines an application of a new approach for investigations of integro-differential equations and integro-dynamic equations on time scales. The approach is based on a new theoretical result. Let  be a time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let also, a, b ∈ , a < b. In this chapter we will investigate the following second-order integro-dynamic equation subject to the boundary conditions where a j ∈ C rd a, σ 2 b ðÞ ½ ðÞ , j ∈ 1, 2, 3 fg , are non-negative functions, p 1 , p 2 ≥ 0. We will investigate the BVP (1), (2) for existence of non-negative solutions. Our main result in this chapter is as follows. Linear integro-dynamic equations of arbitrary order on time scales are investigated in [9] using ADM. Nonlinear integro-dynamic equations of second order on time scales are studied in [10] using the series solution method. Asymptotic behavior of non-oscillatory solutions of a class of nonlinear second order integro-dynamic equations on time scales is considered in [11].
The chapter is organized as follows. In the next section, we will give some basic definitions and facts by time scale calculus. In Section 3, we give some auxiliary results which will be used for the proof of our main result. In Section 4, we will prove our main result. In Section 5, we will give an example. Conclusion is given in Section 6.

Time scales revisited
Time scales calculus originates from the pioneering work of Hilger [8] in which the author aimed to unify discrete and continuous analysis. Time scales have gained much attention recently. This section is devoted to a brief introduction of some basic notions and concepts on time scales. For detailed introduction to time scale calculus we refer the reader to the books [12,13]. 1. The operator σ :  !  given by will be called the forward jump operator.
2. The operator ρ :  !  defined by will be called the backward jump operator.
3. The function μ : will be called the graininess function. We set Observe that σ t ðÞ≥ t for any t ∈  and ρ t ðÞ≤ t for any t ∈ . Below, suppose that  is a time scale with forward jump operator and backward jump operator σ and ρ, respectively. Definition 2.3. We define the set Using the forward and backward jump operators, one can classify the elements of a time scale.
Definition 2.4. The point t ∈  is said to be 2. right-dense if t < sup and σ t ðÞ¼t.
5. isolated if it is left-scattered and right-scattered at the same time.
6. dense if it is left-dense and right-dense at the same time.
Definition 2.5. Let f :  !  be a given function and t ∈  κ . The delta or Hilger derivative of f at t will be called the number f Δ t ðÞ , provided that it exists, if for any ε > 0 there is a neighborhood U of t, U ¼ t À δ, t þ δ ðÞ ∩  for some δ > 0, such that |f σ t ðÞ ðÞ À fs ðÞÀf Δ t ðÞσ t ðÞÀs ðÞ |≤ε |σ t ðÞÀs| for all s ∈ U: If f Δ t ðÞexists for any t ∈  κ , then we say that f is delta or Hilger differentiable in  κ . The function f Δ :  !  will be called the delta derivative or Hilger derivative, shortly derivative, of f in  κ .
Remark 2.6. The delta derivative coincides with the classical derivative in the case when  ¼ .
Note that the delta derivative is well defined. Theorem 2.7. Let f :  !  be a given function and t ∈  κ .
1. The function f is continuous at t, if it is differentiable at t.
2. The function f is differentiable at t and if f is continuous at t and t is tight-scattered.
3. Let t is right-dense. Then the function f is differentiable at t if and only if the limit lim s!t ft ðÞÀfs ðÞ t À s (11) exists as a finite number. In this case, we have 4. We have if f is differentiable at t. Definition 2.8. Let f : |to is a given function.

We say that f is pre-differentiable with region of differentiation D if
To define indefinite integral and Cauchy integral on time scale we have a need of the following basic result. Theorem 2.9. Let t 0 ∈ ,x 0 ∈ ,f :  κ !  be a given regulated function. Then there exists unique function F that is pre-differentiable and Definition 2.10.

Let f :  !  is a regulated function. Then any function F in Theorem 2.9. is said to be a pre-antiderivative of the function f and the indefinite integral of the regulated function f is defined by
Here c is an arbitrary constant. Define the Cauchy integral as follows 2. A function F :  !  is said to be an antiderivative of the function f : Definition 2.11. Let f :  !  be a given function. If it is continuous at right-dense points in  and its left-sided limits exist (finite) at left-dense points in , then we say that f is rd-continuous. With C rd  ðÞ we will denote the set of all rd-continuous functions f :  !  and with C 1 rd  ðÞ we will denote the set of all functions f :  !  that are differentiable and whose derivative are rd-continuous.
We will note that if f is rd-continuous, then it is regulated Below, we will list some of the properties of the Cauchy integral.
Theorem 2.12. Let a, b, c ∈ , α ∈  and f , g ∈ C rd  ðÞ . Then we have the following.
We have In [12], it is proved that G is the Green function for the BVP

Auxiliary results
Let X be a real Banach space. Definition 3.1. A mapping K : X ! X that is continuous and maps bounded sets into relatively compact sets will be called completely continuous.
The concept for k-set contraction is related to that of the Kuratowski measure of noncompactness which we recall for completeness. Definition 3.2. Suppose that Ω X is the class of all bounded sets of X. The function α : Ω X ! 0, ∞ ½Þ that is defined in the following manner fg , is said to be Kuratowski measure of noncompactness.
For the main properties of measure of noncompactness we refer the reader to [14]. Definition 3.3. If the mapping K : X ! X is continuous and bounded and there exists a nonnegative constant k such that for any bounded set Y ⊂ X, then we say that it is a k-set contraction. Note that any completely continuous mapping K : X ! X is a 0-set contraction (see [15] for any x, y ∈ X. Definition 3.5. A closed, convex set P in X is said to be cone if.
Denote P * ¼Pn 0 fg , P r 1 ,r 2 ¼ u ∈ P : r 1 < ∥u∥ < r 2 fg for positive constants r 1 , r 2 such that 0 < r 1 ≤ r 2 . The following result will be used to prove our main result. We refer the reader to [16,17] for more details.
Theorem 3.6. Let P be a cone in a Banach space E, ∥ Á ∥ ðÞ .LetΩ be a subset of P, 0 ∈ Ω and 0 < r < L < R are real constants. Let also, T : Ω ! E is an expansive operator with a constant h > 1,S : P R ! E is a k-set contraction with 0 ≤ k < h À 1 and S P R ÀÁ ⊂ I À T ðÞ Ω ðÞ . Assume that P r,L ∩ Ω 6 ¼ Ø, P L,R ∩ Ω 6 ¼ Ø and there exist an u 0 ∈ P * such that TxÀ λu 0 ðÞ ∈ P for all λ ≥ 0 and x ∈ ∂P r ∩ Ω þ λu 0 ðÞ and the following conditions hold: Then T þ S has at least two fixed points x 1 ∈ P r,L ∩ Ω,x 2 ∈ P L,R ∩ Ω, i.e., Let and ϕ t ðÞ¼ Suppose that E ¼C 1 rd a, σ 2 b ðÞ ½ ðÞ is endowed with the norm provided it exists. Next two lemmas give integral representations of the solutions of the BVP (1), (2). Proof. Since G is the Green function of the BVP (3) and ϕ Δ 2 t ðÞ¼0, t ∈ a, σ 2 b ðÞ ½ , we get and xa ðÞ¼ϕ a ðÞ¼ ασ 2 b ðÞ Àβa þ β À α ðÞ a σ 2 b ðÞÀa ¼ α, Thus, x is a solution to the BVP (1), (2). This completes the proof.
then x is a solution to the BVP (1), (2). Proof. We have t ∈ a, σ 2 b ðÞ ½ . Hence and Lemma 3.7, we conclude that x is a solution to the BVP (1), (2). This completes the proof. □ Now, we will give an estimate of the norm of the operator F 1 . Lemma 3.9. If x ∈ E and ∥x∥≤c for some positive constant c, then t ∈ a, σ 2 b ðÞ ½ , and t ∈ a, σ 2 b ðÞ ½ . Thus, This completes the proof. □ Below, suppose (H3) Suppose that the positive constants A, m, ε, r 1 , L 1 , R 1 and R satisfy the following conditions In the next section, we will give an example for constants A, m, ε, r 1 , L 1 , R 1 and R that satisfy H3 ðÞ . For x ∈ E, define the operator By Lemma 3.9, we get the following result.
Lemma 3.11. If x ∈ E is a solution to the integral equation then it is a solution to the BVP (1), (2).

Proof of the Main result
LetP ¼fu ∈ E : u ≥ 0on t 0 , ∞ ½Þ g : With P we will denote the set of all equi-continuous families inP. For v ∈ E, define the operators t ∈ t 0 , ∞ ½Þ . Note that any fixed point v ∈ E of the operator T þ S is a solution to the IVP (1). Define 1. For v 1 , v 2 ∈ Ω, we have whereupon T : Ω ! E is an expansive operator with a constant 1 þ mε > 1.

5.
Let v ∈ ∂P L 1 . Then Note that in the last inequality we have used the third inequality of H3 ðÞ .

Suppose that there exists an
ðÞ and Sv ¼ v À λv 0 for some λ ≥ 0 and for some v ∈ P R 1 . Then which is a contradiction. Therefore all conditions of Theorem 3.6 hold. Hence, the IVP (1) has at least two solutions u 1 and u 2 so that

Conclusion
In this chapter we introduce a class of BVPs for a class second-order integrodynamic equations on time scales. We give some integral representations of the solutions of the considered BVP. We apply a new multiple fixed point theorem and we prove that the considered BVP has at least two nontrivial solutions. The approach in this chapter can be applied for investigations of IVPs and BVPs for dynamic equations and integro-dynamic equations of arbitrary order on time scales.

Additional classifications AMS Subject Classification: 39 A 10, 39 A 99
Author details Svetlin G. Georgiev Sorbonne University, Paris, France *Address all correspondence to: svetlingeorgiev1@gmail.com © 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.