Open access peer-reviewed chapter

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

Written By

Svetlin G. Georgiev

Submitted: 09 September 2020 Reviewed: 22 December 2020 Published: 08 September 2021

DOI: 10.5772/intechopen.95621

From the Edited Volume

Recent Developments in the Solution of Nonlinear Differential Equations

Edited by Bruno Carpentieri

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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.


  • integro-dynamic equations
  • time scale
  • BVP
  • existence
  • positive solution
  • fixed point
  • cone
  • sum of operators

1. 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 integro-differential 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 T be a time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let also, a,bT, a<b. In this chapter we will investigate the following second-order integro-dynamic equation


subject to the boundary conditions



(H1)kCrdaσ2b×aσ2b, α,βR, α0.

(H2)fCaσ2b×R2 and


ajCrdaσ2b, j1,2,3, are non-negative functions, p1,p20.

We will investigate the BVP (1), (2) for existence of non-negative solutions. Our main result in this chapter is as follows.

Theorem 1.1.SupposeH1-H2. Then the BVP(1), (2)has at least two non-negative solutions.

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.


2. 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].

Definition 2.1.A time scaleTis an arbitrary nonempty closed subset of the real numbers.

Definition 2.2.

1. The operatorσ:TTgiven by


will be called the forward jump operator.

2. The operator ρ:TT defined by


will be called the backward jump operator.

3. The function μ:T0 defined by


will be called the graininess function.

We set


Observe that σtt for any tT and ρtt for any tT. Below, suppose that T 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 pointtTis said to be

  1. right-scattered if σt>t.

  2. right-dense if t<supT and σt=t.

  3. left-scattered if ρt<t.

  4. left-dense if t>infT and ρt=t.

  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.Letf:TRbe a given function andtTκ. The delta or Hilger derivative offattwill be called the numberfΔt, provided that it exists, if for anyε>0there is a neighborhoodUoft,U=tδt+δTfor someδ>0, such that


IffΔtexists for anytTκ, then we say thatfis delta or Hilger differentiable inTκ. The functionfΔ:TRwill be called the delta derivative or Hilger derivative, shortly derivative, offinTκ.

Remark 2.6.The delta derivative coincides with the classical derivative in the case whenT=R.

Note that the delta derivative is well defined.

Theorem 2.7.Letf:TRbe a given function andtTκ.

  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.

  1. Let t is right-dense. Then the function f is differentiable at t if and only if the limit


exists as a finite number. In this case, we have


  1. We have


if f is differentiable at t.

Definition 2.8.Letf:TtoRis a given function.

  1. We say thatfis regulated if its right-sided limits exist (finite) at all right-dense points inTand its left-sided limits exist (finite) at all left-dense points inT.

  2. We say thatfis pre-differentiable with region of differentiationDif

    1. it is continuous,

    2. DTκ,

    3. Tκ\Dis countable and contains no right-scattered elements ofT,

    4. fis differentiable at eachtD.

To define indefinite integral and Cauchy integral on time scale we have a need of the following basic result.

Theorem 2.9.Lett0T,x0R,f:TκRbe a given regulated function. Then there exists unique functionFthat is pre-differentiable and


Definition 2.10.

1. Letf:TRis a regulated function. Then any functionFin Theorem 2.9. is said to be a pre-antiderivative of the functionfand the indefinite integral of the regulated functionfis defined by


Herecis an arbitrary constant. Define the Cauchy integral as follows


2. A function F:TR is said to be an antiderivative of the function f:TR if


Definition 2.11.Letf:TRbe a given function. If it is continuous at right-dense points inTand its left-sided limits exist (finite) at left-dense points inT, then we say thatfis rd-continuous. WithCrdTwe will denote the set of all rd-continuous functionsf:TRand withCrd1Twe will denote the set of all functionsf:TRthat 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.Leta,b,cT,αRandf,gCrdT. Then we have the following.

  1. abft+gtΔt=abftΔt+abgtΔt,

  2. abαftΔt=αabftΔt,

  3. abftΔt=baftΔt,

  4. abftΔt=acftΔt+cbftΔt,

  5. abfσtgΔtΔt=fgbfgaabfΔtgtΔt,

  6. abftgΔtΔt=fgbfgaabfΔtgσtΔt,

  7. aaftΔt=0,

  8. If ftgt on ab, then


  9. If ft0 for all at<b, then abftΔt0.



We have


In [12], it is proved that G is the Green function for the BVP


3. Auxiliary results

Let X be a real Banach space.

Definition 3.1.A mappingK:XXthat 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 thatΩXis the class of all bounded sets ofX. The functionα:ΩX0that is defined in the following manner


where diamYj=supxyX:xyYj is the diameter of Yj, j1m, 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 mappingK:XXis continuous and bounded and there exists a nonnegative constantksuch that


for any bounded set YX, then we say that it is a k-set contraction.

Note that any completely continuous mapping K:XX is a 0-set contraction (see [15]).

Definition 3.4.Suppose thatXandYare real Banach spaces. Then the mapK:XYis called expansive if there exists a constanth>1for which


for any x,yX.

Definition 3.5.A closed, convex setPinXis said to be cone if.

  1. αxPfor anyα0and for anyxP,

  2. x,xPimpliesx=0.

Denote P=P\0,


for positive constants r1,r2 such that 0<r1r2. The following result will be used to prove our main result. We refer the reader to [16, 17] for more details.

Theorem 3.6.LetPbe a cone in a Banach spaceE. LetΩbe a subset ofP,0Ωand0<r<L<Rare real constants. Let also,T:ΩEis an expansive operator with a constanth>1,S:PR¯Eis ak-set contraction with0k<h1andSPR¯ITΩ. Assume thatPr,LΩØ,PL,RΩØand there exist anu0Psuch thatTxλu0Pfor allλ0andxPrΩ+λu0and the following conditions hold:

  1. Sxxλu0, xPr, λ0,

  2. Sx+T0h1x and Tx+Fxx, xPLΩ,

  3. Sxxλu0, xPR, λ0.

ThenT+Shas at least two fixed pointsx1Pr,LΩ,x2PL,RΩ, i.e.,








Suppose that E=Crd1aσ2b is endowed with the norm


provided it exists. Next two lemmas give integral representations of the solutions of the BVP (1), (2).

Lemma 3.7.IfxEis a solution to the integral equation


then x is a solution to the BVP (1), (2).

Proof. Since G is the Green function of the BVP (3) and ϕΔ2t=0, taσ2b, we get




Thus, x is a solution to the BVP (1), (2). This completes the proof.□

For xE, define the operator


Lemma 3.8.IfxEis a solution to the integral equation


then x is a solution to the BVP (1), (2).

Proof. We have




taσ2b. 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 F1.

Lemma 3.9.IfxEandxcfor some positive constantc, then


Proof. We have


taσ2b, and


taσ2b. Thus,


This completes the proof.□

Below, suppose

(H3) Suppose that the positive constants A, m, ε, r1, L1, R1 and R satisfy the following conditions


In the next section, we will give an example for constants A, m, ε, r1, L1, R1 and R that satisfy H3. For xE, define the operator


By Lemma 3.9, we get the following result.

Lemma 3.10.IfxEandxcfor some positive constantc, then


Lemma 3.11.IfxEis a solution to the integral equation


then it is a solution to the BVP (1), (2).

Proof. We have




taσ2b, and


taσ2b. Now, the assertion follows from Lemma 3.7. This completes the proof.□


4. Proof of the Main result



With P we will denote the set of all equi-continuous families in P˜. For vE, define the operators


tt0. Note that any fixed point vE of the operator T+S is a solution to the IVP (1). Define


  1. For v1,v2Ω, we have


whereupon T:ΩE is an expansive operator with a constant 1+>1.

  1. For vP¯R1, we get


Therefore SP¯R1 is uniformly bounded. Since S:P¯R1E is continuous, we have that SP¯R1 is equi-continuous. Consequently S:P¯R1E is a 0-set contraction.

  1. Let v1P¯R1. Set


Note that by the second inequality of H3 and by Lemma 3.10, it follows that εFv1+εL150 on t0. We have v20 on t0 and


Therefore v2Ω and




Consequently SP¯R1ITΩ.

  1. Suppose that there exists an v0P such that Tvλv0P for all λ0, vPr1Ω+λv0 and Sv=vλv0 for some λ0 and for some vPr1. Then


because by the second inequality of H3 and by Lemma 3.10, it follows that εFv+εL1200 on t0.

  1. Let vPL1. Then


Note that in the last inequality we have used the third inequality of H3.

  1. Now, assume that vPL1Ω is such that




Since vPL, we have that v0 on t0 and by the second inequality of H3 and by Lemma 3.10, it follows that Fv+L15>Fv+L1200 on t0.This is a contradiction.

  1. Suppose that there exists an v0P such that Tvλv0P for all λ0, vPR1, vPR1Ω+λv0 and Sv=vλv0 for some λ0 and for some vPR1. Then


which is a contradiction.

Therefore all conditions of Theorem 3.6 hold. Hence, the IVP (1) has at least two solutions u1 and u2 so that


5. An example

In this section we will illustrate our main result with an example. Firstly, we will give an example for the constants A, m, ε, r1, L1 and R1 that satisfy the hypothesis H3. Let T=Z, a=0, b=10, α=β=1,


s012, s1011, and






i.e., H1-H3 hold. Consequently the BVP


has at least two non-negative solutions.


6. Conclusion

In this chapter we introduce a class of BVPs for a class second-order integro-dynamic 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


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Written By

Svetlin G. Georgiev

Submitted: 09 September 2020 Reviewed: 22 December 2020 Published: 08 September 2021