Strongly geodesic preinvexity and Strongly Invariant {\eta}-Monotonicity on Riemannian Manifolds and its Application

In this paper, we present strongly geodesic preinvexity on Riemannian manifolds (RM) and strongly {\eta}-invexity of order m on RM. Furthermore, we define strongly invariant {\eta}-monotonicity of order m on RM. Under Condition C, an important characterization of these functions are studied. We construct several non-trivial examples in support of these definitions. Afterwords, an important and significant characterization of a strict {\eta}-minimizers ({\eta}-minimizers)of order m for MOP and a solution to the variational like-inequality problem (VVLIP) has been derived.


Introduction
The variational inequality problem and monotonicity play a vital role in the existence of the solution to the practical problems in many fields of mathematical science and physics such as optimization theory, science, engineering, etc.The generalized monotonicity is effective equipment for the existence and analysis of a solution to the variational inclusion and complementarity problems.The convexity is closely related to monotonicity.It is observed that the monotonicity of the corresponding gradient function is equivalent to the convexity of the realvalued function.Recently, the extension of convexity and monotonicity have been developed by authors, see [6,7,8].Generally, a manifold is a different space from a linear space.Rapcsak [3] and Udriste [4] extended convexity with techniques from linear space to RM, which is called geodesic convexity.The different kinds of invariant monotone vector fields on RM presented by Barani [2].It has been shown that many the results and properties established from the Euclidean space to Riemannian manifolds preserves for invariant monotone vector fields.Also, optimization has been more developed on RM, see [22,23].The concept of invex function on RM defined by Pini [5] and several properties have been discussed.The generalized invexity has developed the convex analysis, and the generalized invexity is closely related to the generalized invariant monotonicity, which has been studied in [18].Yang et al. [22] presented invariant monotonicity, which is the generalization of monotonicity.The existence of a solution to V LIP was derived under generalized monotonicity [18].Nemeth [9], presented a monotone vector field on RM, which is an important generalization monotone operator.Noor [14] discussed the notion of α-invexity and α-monotonicity.Iqbal et al. [21] extended it on RM, which is called strong α-invexity and invariant αmonotonicity.
Motivated by research works, see in [2,11,12,14,16,21,24], we introduce the strongly geodesic preinvexity of order m, strongly η-invexity of order m, strongly quasi η-invexity of order m and strongly pseudo η-invexity of order m on RM, which are generalization of strongly geodesic convex function of order m defined by [10], strongly α-invexity and invariant α-monotonicity defined by Iqbal et al. [21].The article is as follows: Section 2 contains few definitions and facts are undersigned, mainly our concentrate is to know the basic concept about Riemannian geometry which play the main role in this article.Nontrivial suitable example are constructed in support of these definitions and several interesting properties and results are proved.An important characterization of strongly geodesic preinvexity of order m and strongly η-invexity of order m has been introduced in Section 3.
The generalized invariant η-monotonicity of order m such as strongly invariant η-monotonicity and strongly invariant pseudo η-monotonicity are defined on the RM.A relationship between the strongly η-invexity (strongly pseudo η-invexity) and strongly invariant η-monotonicity (strongly invariant pseudo η-monotonicity) has been established, which show the strongly η-invexity (strongly pseudo ηinvexity) is closely related to strongly invariant η-monotonicity (strongly invariant pseudo η-monotonicity) in Section 4. In Section 5, we introduce η-minimizers of order m on the RM.The MOP for strongly η-invex function of order m has been presented an application.A relationship between the strict η-minimizers of order m for MOP and a solution to V V LIP has been introduced.

Preliminaries
In this section, we recall few definitions and basic results regarding RM, which will be used everywhere in this article.For the standard material on differential geometry, consult, [18].Here, M is considered as C ∞ smooth manifold modelled on a Hilbert space H, either finite dimensional or infinite dimensional, endowed with Riemannian metric •, • p on T p M at point p ∈ M, T p M ∼ = M. Thus, we get a smooth assignment of •, • p to every T p M. Usually, we can write Therefore, M is a RM.The length of tangent vector corresponding norm of inner product ., .p is denoted by • p .The length of piece wise we define the distance between x 1 and x 2 .
Then, the original topology on M is induced by the distance d.The set of all vector fields over smooth manifold M is denoted by χ(M).The metric induces a map h → grad h ∈ χ(M) which associates with every h its gradient.i.e., dh, X p = dh(X), ∀X ∈ χ(M).
It is known that on every RM, ∃ exactly one covariant derivation called Levi-Civita connection denoted by ∇ X Y for any X, Y ∈ χ(M).Again recall that a geodesic is a C ∞ smooth path r joining is called a length of curve C, it is called minimal geodesic which satisfies the equation ∇ dr(t) dt dr(t) dt = 0.The existence theorem for ordinary differential equations states that ∀v ∈ T M, ∃ an open interval J(v) containing 0 and exactly one geodesic r v : J(v) → M with dr(0) dt = v.This implies that ∃ an open neighborhood T M of the manifold M s.t.∀ v ∈ T M, the geodesic r v (t) is defined for |t| < 2. By using parallel translation operator of vectors along smooth curve, for a smooth curve r : I → M, a vector v 0 ∈ T r(t 0 ) M, ∀ t 0 ∈ I, ∃ exactly one parallel vector field V (t) along r(t) s.t.V (t 0 ) = v 0 .The Hilbert space (T p M, • p ) is a linear isometric identification between its dual space (T * p M, • p ).The linear isometric map between the tangent spaces T r(t 0 ) M and T r(t) M, ∀ t ∈ I is defined by v 0 → V (t) and denoted by P t t 0 ,r which call the parallel translation from T r(t 0 ) M to T r(t) M along r(t).There exists a linear map dh p : T p M → T h(p) N if h is differentiable map from the manifold M into the manifold N, where dh p represents the differential of h at p.A RM of finite-dimensional is called complete if its geodesic is defined for all values of t.Hopf-Rinow's theorem assures that the RM M is complete if all pairs of points in M can be joined by a (not necessarily unique) minimal geodesic segment.M considered RM, η : M × M → T M be map s.t.∀ u, v ∈ M, η(u, v) ∈ T M and some basic definitions are as follows: Pini [5] defined the definition is given as follows: Definition 1. [5] Let r u,v : [0, 1] → M be a curve on RM M s.t.r u,v (0) = v and r u,v (1) = u.Then, curve r u,v is called possess the property (P ) w.r.
In the case, if r u,v (t) is a geodesic, then satisfy Condition C Together C 1 and C 2 is called Condition C defined by Barani et al. [1].
The strongly geodesic convex of order m defined by Akhlad et al. [10].

Definition 3. [10] A function
The concept of geodesic invex sets was defined by Barani et al. [1], which is given as follow:

Strongly geodesic preinvexity on Riemannian manifolds
In this section, we introduce strongly geodesic preinvexity on RM M and strongly η-invexity of functions of order m on RM M. Definition 7. Let m be a positive integer.Let S ⊆ M be a geodesic invex subset of RM M. A function h : then it reduces to strongly geodesic convexity of order m, see [10].
.., h k , are strongly geodesic preinvex functions of order m on geodesic invex set S, then h = k j=1 a j h j and H = max 1≤j≤k h j are also strongly geodesic preinvex functions of order m, where a j > 0, 1 ≤ j ≤ k.
Proof.Since h j , 1 ≤ j ≤ k, are strongly geodesic preinvex functions of order m, for every u, v ∈ S, s ∈ [0, 1], we have a j on both sides, we get a j δ > 0. Hence, the function h is strongly geodesic preinvex of order m.
For other part, since h j , 1 ≤ j ≤ k, are strongly geodesic preinvex functions of order m, we have Theorem 2. Let M be a complete RM, S ⊆ M be a geodesic invex set w.r.t.η and F : S × S → R be a continuous strongly geodesic preinvex function of order m w.r.t.(η, η), i.e., F is strongly geodesic preinvex function of order m to each variable.Then, the function Ψ : S → R defined by is strongly geodesic preinvex function of order m w.r.t.η.
By geodesic invexity of S w.r.t.η, ∃ a geodesic Clearly, the curve where the map η 0 : (M × M) × (M × M) → T M × T M. By Definition of infimum and the strongly geodesic preinvexity of order m of F , we have For m=2, then Definition 8 reduce to strongly η-invex of order 2 w.r.t.η defined by [2].
In the following example, we show the existence of strongly η-invex function of order m.
Since M is RM, then for every u, v ∈ M, we get For any u = (u 1 , u 2 ) ∈ M and any x = (x 1 , x 2 ) ∈ T u (M), the map exp u : T u (M) → M is given by Then, h is strongly η-invex of any order w.r.t η (see Definition 8), where η : 2 ).However, h is not strongly geodesic preinvex function of any order.For this, let , 1 9 and s = 1 10 , then for any δ > 0 and any m ≥ 1, it is easy to see the following holds: , where the geodesic segment joining u = (u 1 , u 2 ) and v = (v 1 , v 2 ) is given by In the following theorem, we show differentiable strongly geodesic preinvex function is strongly η-invex.Theorem 3. Let S ⊆ M be an open geodesic invex set and h : S → R be continuously differentiable function.If h is strongly geodesic preinvex of order m w.r.t.η, then h is strongly η-invex of order m w.r.t.η.
Proof.Assume h is strongly geodesic preinvex of order m w.r.t.η, for every u, v ∈ S, ∃ exactly one geodesic r u,v : [0, 1] → M s.t. and Since h is differentiable, dividing by s on both side and taking s → 0, we get However, the converse of Theorem 3 holds when η satisfies Condition C as follows: Theorem 4. Let S ⊆ M be an open geodesic invex set w.r.t.η.Assume h : S → R is continuously differentiable function.The function h is strongly η-invex of order m w.r.t.η and η satisfies Condition C if and only if h is strongly geodesic preinvex of order m w.r.t.η.
Proof.By Theorem 3, then the function h is strongly η-invex of order m w.r.t.η.Conversely, suppose h is strongly η-invex of order m w.r.t.η on open geodesic invex set S w.r.t.η i.e., for every u, v ∈ S ∃ a exactly one curve r u,v : (0, 1) → M s.t.
Fixed s ∈ (0, 1) and setting v = r u,v (s).Then, we have By multiplying s in (3.1) and (1 − s) in (3.2) respectively, adding and applying Case (ii) m > 2, then the real valued function φ(s) = s m−1 is convex on (0, 1), thus we have 1 2 It follows that (3.4), ∃ δ ′ > 0, which is independent from u, v, s such that Hence, the function h is strongly geodesic preinvex function of order m.Now, we generalize strongly η-invex function of order m as follows : Definition 9. Let m be a positive integer.Let h : M → R be a differentiable function on RM M.Then, the function h is called : In the following example, we show the existence of strongly pseudo η-invex function of order m.
Then, h is strongly pseudo η-invex type 1 of any order w.r.t η, where η : However, h is not strongly η-invex function of any order.For this, let u = 1 2 , 1 2 , v = (1, e 2 ), then it is easy to see the following Let m be a positive integer.Let h : M → R be a differentiable function on RM M.Then, the function h is called : In the following example, we show the existence of strongly quasi η-invex function of order m.
Then, h is strongly quasi η-invex type 1 of any order w.r.t η, where η : and However, h is not strongly η-invex function of any order.For this, let u = 1, 1 e 5 , v = (1, 1), then it is easy to see the following

Strongly invariant η-monotone on Riemannian manifolds
The monotonicity of vector field on RM defined by Nemeth [9] as follows : where r u,v is a geodesic joining u and v.
Barani et al. [2] generalized it and defined invariant monotonicity on RM M. Later Iqbal et al. [10] extended the notion of invariant monotonicity to strongly invariant α-monotonicity on RM M. Motivated by Iqbal et al. [10], we extend it as follows: Definition 12. Let m be a positive integer.Let M be a RM.A vector field X on M is called : it reduces to strongly invariant monotone defined by Barani et al. [2].
For m = 2, it reduces to strongly invariant pseudo monotone defined by Barani et al. [2].
In the support of our Definition 12, we give the following example.
Given M is RM, then for every u, v ∈ M, we have 2) By adding (4.1) and (4.2), we get In the next Theorems, we discuss a relationship between strongly η-invex of order m and strongly invariant η-monotone of order m.Theorem 5. Let h : M → R be a differentiable function on M. Suppose h is strongly η-invex of order m.Then, dh is strongly invariant η-monotone of order m.
Proof.Let h be a strongly η-invex of order m on M.Then, (4.4) Adding (4.3) and (4.4), we get Thus, dh is strongly invariant η-monotone of order m.Theorem 6.Let h : M → R be a differentiable function on geodesically complete RM M. If a map η : M × M → T M is integrable and dh is strongly invariant η-monotone of order m.Then, h is strongly η-invex of order m.

Vector variational-like inequality problem on Riemannian manifolds
In this section, we consider the multi-objective optimization problem (MOP ) as an application for strongly η-invex functions of order m, known as strongly η-invex multi-objective optimization problem, which generalizes the results obtained by Iqbal et al. [10].Suppose H = (h 1 , h 2 , ..., h k ), where h i : M → 2 T M are set valued vector fields on M. The vector variational-like inequality problem (V V LIP ) is to find a solution u * ∈ M, and X ∈ H(u * ) s.t.
The MOP is to find a strict η-minimizers of order m for : Motivated by Iqbal et al. [10] and Bhatia et al. [17], we define a local strict η-minimizers of order m with respect to a nonlinear function on RM for MOP .In next Theorem, we show an important characterization of a solution of V V LIP and a strict η-minimizers of order m for MOP .Theorem 8. Let h i , 1 ≤ i ≤ k, be a strongly η-invex functions of order m on M.Then, u * ∈ M is a solution of V V LIP ⇐⇒ u * is a strict η-minimizers of order m for MOP .
Proof.Assume u * is a solution of V V LIP but u * is not a strict η-minimizers of order m for MOP .Then, for all δ > 0, ∃ some ū ∈ M, s.t.

Conclusion
The strong concept of strongly geodesic preinvexity of order m w.r.t.η on geodesic invex sets, strongly η-invex functions of order m and strongly invariant ηmonotonicity of order m on RM have been introduced.The definitions presented in this paper are supported by non-trivial examples.Several interesting properties have also been discussed.An interesting application to MOP for strongly η-invex functions of order m has been presented and the characterization of a strict ηminimizers of order m for MOP and a solution to V V LIP has been derived.Our results generalize the previously known results proven by different authors.
strongly geodesic preinvex of order m w.r.t.η.Now, we define strongly invex function of order m w.r.t.η.Definition 8. Let m be a positive integer.Let M be a RM.A differentiable function